Aenorm 58 by VSAE

Preface

Behold...a new year is upon us A

s I’m sitting here in a room at the University of Amsterdam on a very calm night, I just realized that another year has passed. A year which was quite turbulent for us econometricians, I might say. The “war on terrorism” policy of president Bush, with a gigantic amount of dollars on expenses involving, eventually didn’t turn out to be the worst thing for the American economy during 2007. Even the numerous scandals around American companies assisting the American government in Iraq, like Blackwater, couldn’t contribute to this. No, the most dreadful event for us econometricians in 2007 has to be the collapse of the American housing market. Up till now the rate of the Dollar is dropping down and exchanges all over the world are affected by this crisis in America. Do we have to fear another stock market crash or are we just overreacting? Also Aenorm went through a turbulent year. One of the main concerns of the redactional staff of Aenorm during last year, was to broaden the group of readers and to make this group a bit more international. This resulted in an edition especially for the participants of the Econometric Game and a rise of the circulation of our magazine upto 1900 copies. The content of Aenorm always was morealess in English, but last year we made the decision to no longer publish any, not even one, Dutch words in Aenorm, which makes it a lot more readable for our international subscribers all over the world. A new board will bring the VSAE to new hights during the upcoming year and it's time for me to pass the ambt of chief editor of Aenorm through to one of these boardmembers with a lot of fresh and new ideas. I can only wish him all the best and sure hope he continues to improve this magazine on certain levels. A chief editor always has to realize that he would be nothing without his full redactional staff. David Hollanders states in his article in this edition that fortunately we observe cooperation in the real world, although economic theory doesn’t always take it into account. Hereby, I would like to thank my staff acting not too economical with all the cooperation they gave me during last year. Erik Beckers

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January 2008

Limit Cycles and Multiple Attractors in Logit Dynamics

There is a growing body of experimental research documenting non-Nash equilibrium play, non-convergence and cyclical patterns in various learning algorithms. This article shows that, even for 'simple' three-strategy games, periodic attractors do occur under an alternative, rationalistic way of modelling evolution in games, namely the Logit Dynamics. Secondly, it presents numerical evidence for multiple interior equilibria in a Coordination game created via a sequence of saddle-node bifurcations. Marius Ionut-Ochea

The secondary insurance market

In the case of surrender, a policyholder can only sell the right of ownership of his policy to the insurer which has written the policy; insurers have a so-called buyerâ&#x20AC;&#x2122;s monopoly. In the past decade, secondary insurance markets are rising in foreign countries. This article will first of all give a historical overview of the origin of the market and afterwards will go into the subject of the current situation. Subsequently it will set apart the advantages and disadvantages of the secondary insurance market. Finally the article accounts for the possibilities of the arising of such a market in the Netherlands. Martijn Visser

A Probabilistic Analysis of a Fortune-wheel Game

In this article a probabilistic analysis is given for a simple fortune-wheel game. Each player tries to get a higher score than any of his opponents. To achieve this goal each player must decide to turn the wheel a second time or not. Turning the wheel a second time has its price, because it can result in a zero score. In the article the optimal strategies are calculated for the case of two and three players. Simulations are added for the case of more than three players. Rein Nobel and Suzanne van der Ster

Quantifying Operational Risk in Insurance Companies 24 Most empirical evidence suggests that the Fisher effect, stating that inflation and nominal interest rates should cointegrate with a unit slope on inflation, does not hold, a finding at odds with many theoretical models. This paper argues that these results can be attributed in part to the low power of univariate tests, and that the use of panel data can generate more powerful tests. For this purpose, we propose two new powerful panel cointegration tests that can be applied under very general conditions. Youbaraj Paudel

Effects of invalid and possibly weak instruments

Cover design: Carmen CebriĂĄn Aenorm has a circulation of 2000 copies for all students Actuarial Sciences and Econometrics & Operations Research at the University of Amsterdam and for all students in Econometrics at the Free University of Amsterdam. Aenorm is also distributed among all alumni of the VSAE. Aenorm is a joint publication of VSAE and Kraket. A free subsciption can be obtained at www.aenorm.nl. Insertion of an article does not mean that the opinion of the board of the VSAE, the board of Kraket or the redactional staff is verbalized. Nothing from this magazine can be duplicated without permission of VSAE or Kraket. No rights can be taken from the content of this magazine. ÂŠ 2008 VSAE/Kraket

This article examines IV (instrumental variables) estimation when instruments may be invalid. The limiting normal distribution of inconsistent IV is derived and provides a firstorder asymptotic approximation to the density in finite sample. In a specific simple model this approximatioen will be scanned and compared with the simulated empirical distribution with regard to measures for model fit, simultaneity, instrument invalidity and instrument weakness. Jan Kiviet and Jerzy Niemczyk

Dice Games and Stochastic Dynamic Programming

This article considers stochastic optimization problems that are fun and instructive for teaching purposes on the one hand and involve challenging research questions on the other hand. These optimization problems arise from the dice game Pig and the related dice game Hog. The article concentrates on the practical question of how to compute an optimal control rule for various situations. This will done by using the technique of stochastic dynamic programming. Henk Tijms

Aenorm 58

Contents List

A Bayesian Approach to Medical Reasoning

Medical reasoning is a complex form of human problem-solving that requires the physician to take appropriate action in a world that is characterized by uncertainty. Dynamic Bayesian networks are put forward as a framework that allows the representation and solution of medical decision problems, and their use is exemplified by a case study in oncology. This paper makes the case for dynamic Bayesian networks as a formalism for the representation of medical knowledge and the execution of medical tasks such as diagnosis and prognosis. Marcel van Gerven

Micro-foundations are useful, up to a point

Economic analysis rests on the assumption of methodological individualism. So, no matter what the macro-phenomena under study might be, economic analysis tries to explain it as the (equilibrium)outcome of the games rational individuals play. It is totally true that cooperation is the direct result of individual choices, but whether cooperation can be understood as such, is exactly the topic under discussion. David Hollanders

Introduction of the no-claim protector: Research of the consequences for the Dutch car-insurance market 52 During the start of 2007, a new product was introduced in the insurance market: the no-claim protector. The no-claim protector would lead to significant cost savings for the policyholders according to the insurers.. This article summarises a research of the efficiency of a bonus-malus system including the ncno-claim -protector. The goal of this research is to describe for which drivers the no-claim protector is a profitable investment. Hein Harlaar

Conflicting interests in supply chain optimization

An elementary aspect of production planning is the relation between setup costs for starting a production run and the costs for holding inventory. One can link two of these problems in a supply chain where one producer supplies an intermediate product for the next producer. In this case we can still handle this multi-level production problem if we are allowed to see it as one optimization problem. But what if the supplier and customer have different goals in mind? Reinder Lok

Coalitional and Strategic Bargaining: the Relevance of Structure 60 This article aims to be a first initiative and obviously must leave many questions unanswered. It focuses on the relevance of the degree of bargaining structure. In particular, the goal is to know whether a lower degree of structure increases the predictive power of coalitional solution concepts and whether it has an influence on the relative bargaining power of the players. Concretely the article looks at two bargaining situations with the same bargaining problem. Adriaan de Groot Ruiz

Puzzle 67 Facultative 68

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January 2008

Econometrics

Limit Cycles and Multiple Attractors in Logit Dynamics There is a growing body of experimental research documenting non-Nash equilibrium play, non-convergence and cyclical patterns in various learning algorithms (see Camerer (2003) for an overview). On the theory side, within the evolutionary game dynamics literature an important result is Zeeman (1980) conjecture that there are no generic Hopf bifurcations in the case of three strategies games: “When n = 3 all Hopf bifurcations are degenerate under Replicator Dynamics”. This means that no limit cycles are possible under Replicator Dynamics for 3 x 3 games. Here we show that, even for such 'simple' three-strategy games, periodic attractors do occur under an alternative, rationalistic way of modelling evolution in games, namely the Logit Dynamics. Second, we present numerical evidence for multiple interior equilibria in a Coordination game created via a sequence of saddle-node bifurcations.

Volume 15 Edition 57 Marius-Ionut Ochea October 2007 1568-2188 isISSN a PhD student at the Center for Nonlinear Dynamics in Economics and Finance(CeNDEF), University of Amsterdam. Chief editor: about this article the author can be contacted For comments Erik Beckers vai e-mail: m.i.ochea@uva.nl Editorial Board: From Daniëlla BralsReplicator to Logit Dynamics Siemen van der Werff

Evolutionary game theory deals with games

Design: played within a large population over long time Carmen Cebrián

fi(x) =(Ax)i

(1)

Sandholm (2006) rigorously defines an evolutionary dynamics as a map assigning to each population game a differential equation x = V ( x ) on the simplex Δ n −1: In order to derive such an ‘aggregate’ level vector field from individual choices he introduces a revision protocol ( (x), x) indicating, for each pair ( , ), the rate of switching( ) from the currently played strategy to strategy : The mean vector field is obtained as:

horizon(evolution scale). Its main ingredients Lay-out: are the underlying normal form game - with = x i V= inflow into strategy i i (x) payoff matrix A[n x n] - and the evolutionary Jeroen Buitendijk outflow from strategy i dynamic class which defines a dynamical sysn n Editorial tem staff:on the state of the population. In a symRaymon Badloe metric framework, the strategic interaction = x j ρ ji ( f ( x ) , x ) − xi ρij ( f ( x ) , x ) (2) Erik Beckers takes the form of random matching with each j =1 j=1 Daniëlla Brals of the two players choosing from a finite set of Jeroen Buitendijk available strategies E = {E1,E2,...,En} For every Based on the computational requirements/qualNynke de Groot time t; x(t) denotes the n-dimensional vector ity of the revision protocol the set of evoluMarieke Klein of frequencies for each strategy/type Ei and betionary dynamics splits into two large classes: Hylke Spoelstra longs to the n - 1 dimensional simplex: Siemen van der Werff imitative dynamics and pairwise comparison . The first class is represented n Advertisers: ⎧⎪ ⎫⎪ by the Replicator Dynamic (Taylor and Jonker − 1 n n Aegon Δ = ⎨x ∈ \ : xi = 1⎬ . (1978) ) which can be easily derived by substiAON ⎪⎩ ⎪⎭ i =1 tuting into (2) the pairwise proportional reviDeloitte sion protocol: Delta Lloyd = (player switches to De Nederlandsche Bank The assumption of random interactions proves strategy at a rate proportional with the probIbis crucial for the linearity of strategy Ei payoff: IMC ability of meeting an -strategist( ) and with this would be simply determined by averaging Nationale the Nederlanden the excess payoff of opponent if payoffs from each strategic interaction with PricewaterhousCoopers positive): weights given by the state of the population x. Mercer Denoting with f(x) the payoff vector, its compoORTEC nents - individual payoff or fitness of strategy i x i= xi ⎣⎡fi ( x ) − f ( x ) ⎦⎤= xi ⎡⎣( Ax )i − xAx ⎤⎦ (3) PGGM SNS Reaalin biological terms - are: TNO where f ( x ) = xAx is the average population

∑

Towers Perrin Watson Wyatt Worldwide

Econometrics

payoff. Replicator Dynamics found applications in biological, genetic or chemical systems, those domains where organisms, genes or molecules evolve over time via . From the perspective of strategic interaction the main criticism of the ‘biological’ game-theoretic models is targeted at the intensive use of preprogrammed, simple imitative play with no role for optimization and innovation. Specifically, in the transition from animal contests and biology to human interactions and economics the Replicator Dynamics seems no longer adequate to model the rationalistic and ‘competent’ forms of behaviour. Best Response Dynamics would be more applicable to human interaction as it assumes that agents are able to optimally compute and play a (myopic) ‘best response’ to the current state of the population. Apart from the highly unrealistic assumptions regarding agents capacity to compute a perfect best reply to a given population state there is also the drawback that it defines a differential inclusion, i.e. a set-valued function. The best responses may not be unique and multiple trajectory paths can emerge from the same initial conditions. A ‘smoothed’ approximation of the Best Reply dynamics - the Logit dynamics - was introduced by Fudenberg and Levine (1998); it was obtained by stochastically perturbing the payoff vector and deriving the Logit revision protocol: exp ⎡η-1f j ( x ) ⎤ ⎣ ⎦ -1 ⎡ exp η fk ( x ) ⎤ ⎣ ⎦ k

ρij ( f ( x ) , x ) =

∑

exp ⎡η-1 ( Ax )i ⎤ ⎣ ⎦ -1 ⎡ exp η ( Ax )k ⎤ ⎣ ⎦ k

(4)

∑

where > 0 is the noise level. Here represents the probability of player switching to strategy when provided with a revision opportunity. For high levels of noise the choice is fully random (no optimization) while for close to zero the switching probability is almost one. Plugging the Logit revision protocol (4) back into the general form of the mean field dynamic (2) and making the substitution we obtain a well-behaved system of ode’s, the Logit dynamics as a function of the intensity of choice (Brock and Hommes (1997) ) parameter :

x i =

exp ⎡⎣ β ( Ax )i ⎤⎦

∑

exp ⎡⎣ β ( Ax )k ⎤⎦

− xi

(5)

→ ∞ the probability of switching to the discrete ‘best response’ is close to one while for a very low intensity of choice ( → 0) the switching rate is independent of the actual performance of the alternative strategies (equal prob-

AENORM

January 2008

ability mass is put on each of them). Mathematically it is a ‘smoothed’, well-behaved dynamics while from the strategic interaction point of view it models a player. Rock-Scissors-Paper The first 3-strategy example we look at is a generalization of the classical game of cyclical competition, Rock-Scissor-Paper as discussed in Hofbauer and Sigmund(2003) with > 0. In general, any game can be transformed into a ‘simplest’ form by substracting a constant from each column such that all the elements of the main diagonal are zeros (Zeeman (1980)): −ε ⎞ ⎛0 δ ⎜ ⎟ A = ⎜ −ε 0 δ ⎟ ⎜δ −ε 0 ⎟⎠ ⎝

(6)

The replicator equation (3) with the game matrix (6) induce on the 2-simplex the following vector field: ⎡ x = x[yδ − zε − ( x ( yδ − zε ) + ⎤ ⎢ ⎥ y ( − xε + zδ ) + z ( xδ − yε ))]⎥ ⎢ ⎢ ⎥ ⎢y = y[− xε + zδ − ( x ( yδ − zε ) + ⎥ ⎢ ⎥ y ( − xε + zδ ) + z ( xδ − yε ))]⎥ ⎢ ⎢ z = z[xδ + yε − (x ( yδ − zε ) + ⎥ ⎢ ⎥ ⎢⎢ ⎥⎥ )] y xε zδ z xδ yε − + + − ( ) ( ) ⎣ ⎦

(7)

Proposition 1

Proof It hinges on the computation of the first Lyapunov coefficient as = 0 which means that there is a first degeneracy in the third order terms from the Taylor expansion of the normal form. The detected bifurcation is a or bifurcation (assuming away other higher order degeneracies: technically, the second Lyapunov coefficient should not vanish). Although, in general, the orbital structure at a degenerate Hopf bifurcation may be extremely complicated for our particular vector field induced by the Replicator a of cycles is born at the critical parameter value. The logit evolutionary dynamics (5) applied to our normal form game (6) leads to the following vector field: ⎡ x = x[yδ − zε − ( x(yδ − zε ) + ⎤ ⎢ ⎥ y(− xε + zδ) + z(xδ − yε ))]⎥ ⎢ ⎢y = y[− xε + zδ − ( x(yδ − zε ) + ⎥ ⎢ ⎥ y(− xε + zδ) + z(xδ − yε ))]⎥ ⎢ ⎢ z = z[xδ − yε − (x(yδ − zε ) + ⎥ ⎢ ⎥ y(− xε + zδ) + z(xδ − yε ))]⎦⎥ ⎣⎢

(8)

Econometrics

Figure 1. Replicator Dynamics and RSP game for fixed Hopf bifurcation for = 1, (d) stable focus

Figure 2. Logit Dynamics and RSP. fixed (c)-stable limit cycle

= 1,

Proposition 2

= 1 and different . (a) unstable focus, (b) degenerate

= 0,8, free . (a) stable focus, (b) generic Hopf bifurcation,

ble limit cycle emerges around the now unstable steady state. Unlike Replicator Dynamics, stable cyclic behavior does occur under the Logit dynamics even for three-strategy games. 3 x 3 Coordination Game

Proof. In order to show that Hopf bifurcation is non-degenerate we have to compute again the first Lyapunov coefficient and check whether it is non-zero. The analytical form of this coefficient takes a complicated expression of exponential terms which, after some tedious computations, boils down to:

(

l1 β

Hopf

)

(

864 δε + δ2 + ε 2 , ε, δ = − < 0, 2 3 (3ε − 3δ )

)

(∀) ε

> δ > 0.

Typical trajectories of this route to cycling are shown in Figure 2 below. We notice that as moves up from 10 to 35 (i.e. the noise level is decreasing) the interior stable steady state looses stability via a supercritical, noncatastrop-hic Hopf bifurcation and a small, sta-

Using topological arguments, Zeeman (1980) shows that stable games have at most one interior isolated fixed point under Replicator Dynamics(Theorem 3). In particular, fold catastrophe (two isolated fixed points which collide and disappear when some parameter is varied) cannot occur within the simplex. In this section we provide - by means of the classical coordination game - numerical evidence for the occurrence of multiple, isolated interior steadystates under the Logit Dynamics and conjecture that fold catastrophe is possible when we alter the intensity of choice . The coordination game we consider is given by the following payoff matrix:

Wat als zijn overboeking naar Hong Kong halverwege de weg kwijtraakt? Een paar miljoen overmaken is zo gebeurd. Binnen enkele seconden is het aan de andere kant van de wereld. Hij twijfelt er niet aan dat zijn geld de juiste bestemming bereikt. Het gaat immers altijd goed. Maar wat als het toch van de weg af zou raken? Door hackers, fraude of een computerstoring? Daarom levert de Nederlandsche Bank (DNB) een bijdrage aan een zo soepel en veilig mogelijk betalingsverkeer. We onderhouden de betaalsystemen, grijpen in als problemen ontstaan en onderzoeken nieuwe betaalmogelijkheden. Het betalingsverkeer in goede banen leiden, is niet de enige taak van DNB. We houden ook toezicht op de financiële instellingen en dragen – als onderdeel van het Europese Stelsel van Centrale Banken – bij aan een solide monetair beleid. Zo maken we ons sterk voor de financiële stabiliteit van Nederland. Want vertrouwen in ons financiële stelsel is de voorwaarde voor welvaart en een gezonde economie. Wil jij daaraan meewerken? Kijk dan op www.werkenbijdnb.nl.

Werken aan vertrouwen. 8

AENORM

January 2008

Econometrics

⎛1 − ε 0 0 ⎞ ⎜ ⎟ 0 1 0 ⎜ ⎟ , ε ∈ ( 0,1) ⎜0 ⎟ 0 1 ε + ⎝ ⎠

Proposition 3 Unlike Replicator, the Logit Dynamics may display multiple, interior isolated steady states created via a fold bifurcation. In the particular case of a 3-strategy Coordination game, three interior stable steady states emerge through a sequence of two saddle-node bifurcations. Figure 3 depicts the fold bifurcations scenario by which the multiple, interior fixed points appear when the intensity of choice(Panel (a)) or the payoff parameter(Panel (b)) changes. For small values of the unique, interior stable steady state is the barycentrum (1/3, 1/3, 1/3). A fold bifurcation occurs at = 2.7 and two new fixed points appear, one stable and one unstable. If we increase even further ( ≈ 3.5) a second fold bifurcation takes place and two additional equilibria emerge. A third saddle-node bifurcation occurs at ≈ 3.5 completing the steady states unfolding process. A similar pattern is visible in the payoff parameter space (Panel (b)) where a family of fold bifurcations is obtained for different values of the switching intensity.

The numerical computation of the basins of attraction for different equilibria reveals desirable properties of the Logit dynamics from an social welfare perspective. While for extreme values of the switching parameters the basins of attraction are similar in size with the Replicator Dynamics for moderate levels of rationality the population manages to coordinate close to the Pareto optimal Nash equilibria. Conclusions This research was motivated by the idea of identifying periodic under a ‘rationalistic’ evolutionary dynamics, namely the smoothed best-reply or Logit dynamics. Consequently, in an evolutionary Rock-Scissors-Paper game, we showed, by means of normal form computations, that, unlike Replicator, the Logit dynamics is generic even for three strategy games and stable cycles emerge via a supercritical Hopf bifurcation. These periodic attractors can be generated by varying either the payoff or behavioral parameters. Moreover, via numerical computations on a 3x3 Coordination game, we showed that the Logit may display multiple, isolated, interior steady states (created via a sequence of fold bifurcations), a phenomenon known not to occur under the Replicator Dynamics. Interestingly, bounded rationality (i.e. small ) may help coordination close to the Pareto optimal equilibrium irrespective of the initial mixture of the population. References Brock, W. and Hommes, C.H. (1997). A Rational Route to Randomness, Econometrica, 65, 1059-1095. Camerer, C.F. (2003). Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press. Hofbauer, J., and Sigmund, K. (2003). Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UK. Sandholm, W. (2006). Population Games and Evolutionary Dynamics. Unpublished manuscript, University of Wisconsin. Schuster, P., Schnabl, W., Stadler, P.F. and Forst, C. (1991). Full characterization of a strange attractor: chaotic dynamics in low-dimensional replicator systems, Physica D, 48, 65-90. Taylor, P.D. and Jonker, L. (1978). Evolutionarily stable strategies and game dynamics. Mathematical Biosciences, 40, 145-156.

Figure 3. Equilibria curves as function of model parameters

Zeeman, E.C. (1980). Population dynamics from game theory. In: Global Theory of Dynamical Systems. Lecture Notes in Mathematics 819. New York: Springer.

Actuarial Sciences

The secondary insurance market In the case of surrender, a policyholder can only sell the right of ownership of his policy to the insurer which has written the policy; insurers have a so-called buyer’s monopoly. In the past decade, secondary insurance markets are rising in foreign countries. On this market, policyholders can also sell their right of ownership to other parties as well as the original insurer. This development may influence the way in which insurers determine the surrender values of insurance products. Up to this moment there hasn’t been published much in the Netherlands on so-called secondary insurance markets and their possible consequences. In this article first of all I will give a historical overview of the origin of the market and then I will discuss the current situation. Subsequently I will set apart the advantages and disadvantages of the secondary insurance market. Finally I amplify the possibilities of the arising of such a market in the Netherlands.

Several countries – mostly the United States, but for example also Great Britain – know an alternative for the surrender of insurance policies: sale on the secondary insurance market. This article discuss this market, where the right of ownership of a life insurance policy can be sold to a third party, who does not have any interest in the life of the insured. There hasn’t been published much on the secondary market for life insurances. The first actuarial article on this kind of markets appeared in 1996 (McGurk, 1996). In 2003, the first study on the economical effects of this market appeared (Bhattacharya et al., 2003). Since 2005 there has been published a lot more on this subject. The origin market

the

secondary

insurance

Formerly, a secondary insurance market, on which the right of ownership of policies is sold to third parties, only existed as an “underground” market. It was a market on which a lot of speculation took place. This speculation evolved out of the origin of the insurance, where gambling played a crucial role. To prohibit the substantial gambling market on human lives, in 1774 the Life Assurance Act was adopted in Great Britain. This law prohibited policies issued on lives in which the policyholder had no interest. Through jurisprudence the legal interpretation of this “insured interest” was developed. Eventually, in the case Dalby versus India & London Life in 1854, it was determined that this interest only had to exist when the policy was issued. Consequently, policies in Great Britain are transferable after they are issued. The first public auction of policies in Great Britain already took place in 1843 (McGurk, 1996).

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January 2008

Martijn Visser has gained his Master in actuarial sciences at the University of Amsterdam (UvA) in February 2007. He has written his Master thesis “Afkoop: een ondergewaardeerd onderwerp” under supervision of Willem Jan Willemse. This thesis was partly based on an internal research for a Dutch insurer. This article is based on a chapter of this thesis.

Elizur Wright, the founder of the American actuarial school, visited such an auction of existing life insurance policies on the London “Royal Exchange” in 1844. Wright worried about the lack of regulations involving these transactions and even compared them to slavery. At his return in the United States he introduced the right of surrender in his homestate Massachusetts (the possibility of surrender already existed at several insurers). This right compelled the insurer to always buy back their issued policies. Furthermore, Wright enacted the insurers to calculate the surrender value through the so-called “cash value formula”. Only policies with a substantially higher market value than the value attached by the insurer were allowed to be traded on the secondary market. The main purpose of Wright’s reformations was to create a fair situation for policyholders who wished to end their policies. However, these reformations also created a buyer’s monopoly: the policyholder is only able to sell his policy to the insurer. To breach this monopoly, buyers who were willing to pay more than the surrender value should be found. Therefore, the right to sell no longer desired insurance policies to third parties was acknowledged in the United States around 1900 (Belth, 2002; Coventry, 2006).

Wij bieden je

Dat wil niet zeggen dat je van Mars moet komen Als afgestudeerde wil je graag direct aan de slag. Bij ORTEC

Vanwege onze constante groei is ORTEC altijd op zoek

hoef je hier niet lang op te wachten. Je wordt direct op

naar enthousiaste studenten en afgestudeerden die de

projecten ingezet en krijgt veel eigen verantwoordelijkheid.

ruimte zoeken om zich te ontwikkelen en willen bijdragen

Bij ORTEC werken veel studenten. Sommigen schrijven bij

aan de volgende generatie optimalisatietechnologie.

ons een afstudeerscriptie, anderen werken enkele dagen per week als studentassistent.

Hiervoor denken we aan bèta’s in de studierichtingen: • Econometrie

Maar je staat er nooit alleen voor. Je kunt rekenen op

• Operationele Research

de expertise van je collega’s: stuk voor stuk experts

• Informatica

op het gebied van complexe optimalisatievraagstukken in

• Wiskunde

diverse logistieke en financiële sectoren. Hoogopgeleide,

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veelal jonge mensen die weten wat ze doen en jou naar een

Kijk voor vacatures en afstudeerplaatsen eens op

hoger niveau zullen brengen. Samen met je collega’s help je

www.ortec.com/atwork. Zit jouw ideale functie of afstu-

klanten gefundeerde beslissingen te nemen. Dit doe je met

deerplek er niet bij, stuur dan een open sollicitatie of

gebruik van wiskundige modellen en het toepassen van

scriptievoorstel naar recruitment@ortec.com.

simulatie- en optimalisatietechnieken.

EPROFESSIONALS IN PLANNING

Actuarial Sciences

Strong marketgrowth through aids-epidimic For a long time the secondary insurance market remained limited in both Great Britain as the United States. The strong growth of the market in both countries is a relatively recent phenomenon and arose in the late eighties as a response on the aids-epidemic. Many young HIV-infected people were in a rapid need of money to pay for their medication and to attain their standard of living. For these people, long term assets had lost their value and in a search for cash they attempted to sell their life insurance policies. Because of their strongly decreased life expectancy, the discounted value of the payment at death greatly increased and broadly exceeded the surrender value. Thus an underground, speculative market arosed, which still contributes to the negative image of the modern secondary insurance market (Doherty & Singer, 2002). The market grew rapidly when official companies started to play an active role in the valuation of policies and acted as an intermediary between buyers and sellers during the end of the eighties,. “Policy Network” became the first company which played this active role in Great Britain in 1988 (McGurk, 1996) and “Living Benefits” was the first one to act on the secondary market in the United States in 1989 (Belth, 2002). In addition, in Germany at least one organisation is active dealing in policies on the secondary insurance market. The German trade association “Bundesverband Vermögens-anlagen im Zweitmarkt Lebensversicherungen” (BVZL) acts in the same way as players on the American and the British secondary markets. Search for new growth opportunities At the turn of the century, the medication for aids-patients improved substantially, so the life expectancy of aids-patients increased as well and the secondary market became less profitable. This led to a search for new growth opportunities. Through further development of their straightforward models and methods, investors are nowadays able to buy policies of non-terminal patients with a declined life expectancy on the secondary market as well (Doherty & Singer, 2002). The secondary market for terminal patients is called the “viatical” market. Improved techniques and a search for new markets, because of diminishing profit margins, led to the creation of the so-called “life settlements” market. Policies of individuals aged older than 65 years could now also be purchased, given that they have experienced a decline in their health and

a remaining life expectancy of between six and twelve years. With a longer life expectancy, the longevity risk is too much substantial, which leads to a spreading of the yield over additional years, so that the investment return will be to low (Doherty & Singer, 2002). Table 1 provides a clear summary of the characterizations of both markets in the United States. Characteristics

Viatical market

“Life settlement” market

Insured amount

< $100.000, mostly between $25.000 and $50.000

> $100.000, mostly more than $250.000

Policyholder

Terminal patients between 25 and 40 years old

Individuals older then 65 years.

Remaining life expectancy

< 2 years, mostly less than 12 months.

> 2 years, mostly up to 12 or 15 years old

Table 1: Viatical market versus “Life settlement” market in the United States Source: Deloitte (2005)

Negative attitude of insurers Insurers responded mainly in a negative way to the presence of the secondary market. A substantial part of this negativism can probably be explained by the loss of the buyer’s monopoly for the insurers, which put their surrender results under pressure. Insurers have lobbied a lot in the United States to prevent further growth of the secondary market. Moreover, insurers have taken arrangements themselves to diminish the attractiveness of the market. Since the end of the nineties, there exist so-called ADB’s (Accelarated Death Benefits) in the primary market: options to transfer the payment at death at a rate of 25 to 100% in advance, when certain conditions have been fulfilled (Doherty & Singer, 2002). Functioning of the secondary insurance market: the relation with the surrender value As described before, on the secondary market the right of ownership of life insurance policies is sold to a third party, who hasn’t got any direct interest in the insured life. The policy liabilities, in contradiction to the case of surrender, still apply for the insurer. As a consequence of the transaction, the buyer receives a financial interest in an early death of the insured. For which reason do secondary insurance markets really exist? Why do profits arise for buyers on this market? Aren’t insurers able to fill

Viatical is an old term for travelling money. For catholic people, the viaticum is the last Communion and part of the last sacraments. 1

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Actuarial Sciences

this gap in the market? To answer these questions it is necessary to look closer to the surrender values assigned by insurers to surrendering policyholders. When a policyholder judges he no longer needs his policy, he wish to receive the value which is build up until that moment. When this option on surrender would not exist, the request for life insurance policies would diminish substantially through uncertainty upon future needs. This explains why insurers make surrender possible. The surrender value for an individual policyholder is calculated by means of the passed duration of the policy, the policy characteristics etc. However, an important aspect in explaining the existence of a secondary market is that the surrender value for an individual policyholder is calculated by the insurer based on the expected mortality trend and development of expenses and returns. When the expectations for an individual insured differ from this average expectation, the economical value of the individual policy will differ from the surrender value. Given that the mortality expectations for an individual insured are worse than the average mortality expectations, a payment at death will take place earlier than average. Therefore, the economical value of this policy is higher than the average economical value, which is illustrated in figure 1.

Value of policy Face Value

Policy Value for 65 yr old in very poor health

Economic Value based on Normal Health

40 years

Surrender Value based on Normal Health

65 years

Age

Figure 1: Economic value and surrender value (Doherty & Singer, 2002)

The volume of the secondary market What is the size of the secondary insurance market? During 2006, policies worth of 200 million pound were traded in Great Britain. This was estimated to be approximately 25% of the potential at that time (McGurk, 1996). During 2002, a total policy value of two billion dollar was traded on the American life settlements market. An estimated 20% of the policyholders aged older than 65 years owns a policy for which the economical value exceeds the sur-

"During 2002, a total policy value of two billion dollar was traded on the American life settlements market. " The difference between the economical value of the average policy and the surrender value is the margin for the insurer, including a cost charge. Other opportunities for a secondary market This suggests that it is possible to sell policies with other deviations from the average expected parameters through the secondary market. When, for example, the current expected interest rates are lower than the interest rates on which the surrender value is based, the economical value of policies is higher than the surrender value as well. In Germany, the trade association BVZL experienced an enormous increase in purchases on the secondary market, when the interest rates and â&#x20AC;&#x201C;expectations were low (AM, 2006).

render value. The estimated value of all policies of elderly people with a declined health was around $100 billion during 2002 (Conning, 2003). According to the â&#x20AC;&#x153;Viatical Association of Americaâ&#x20AC;?, $50 million was sold on policies on the secondary market in 1990, during 1999 it was $1 billion and in 2001 it was between $2 and $4 billion (Doherty & Singer, 2002). Conforming Belth, many articles state that the secondary market is growing, while available data show the opposite. In table 2 are the transactions of one of the key players in the viatical market. Supervision on the secondary market Because of the strong growth of the secondary market, this market is under supervision since a while. The British supervisor is auditing intermediaries on the British secondary market since 1992. 23 states in the United States have adopted laws for the life settlements market in 2005. Moreover, the national organizations of supervisors and intermediaries are developing

Actuarial Sciences

Year

Transactions

1995

$ 65.5 mln

1996

$ 46.3 mln

1997

$ 19.1 mln

1998

$ 27.4 mln

1999

$ 21.3 mln

2000

$ 10.2 mln

Table 2: Transactions at the American viatical company Life Partners (1995-2000 in millions of U.S. Dollars) Source: Belth (2002)

regulations for this market, such as for market entry, basic training, obligations for all insurers to indicate the existence of the market to insured people and new information brochures (Deloitte, 2005). Advantages and disadvantages of a secondary market For insured people with a declined health who don’t want to be insured any more, the secondary market provides a substantial higher value for a policy than the surrender value. At this moment the market is legal and organised. Generally, the liquidity on the insurance market is therefore increasing and the value of insurance products will increase through this option for the consumer. Finally, insurers maintain their contracts and advisors their commission for these policies, since those policyholders don’t lapse anymore. However, the secondary market encounters several ethical objections. The market would stimulate assassinating an insured, the policy can be resold several times without knowledge of the insured and confidential medical dossiers are accessible for buyers. Moreover, the market, such as it has arisen in the United States, is very sensitive to fraud. The insured is able to show a more declined health than is the case in reality. In addition, the market still should get better organised, since for the time being it lacks effective supervision and solid mathematical methods. Finally, insurers protest against the market, because of a decrease in their lapse results. Opportunities for a Dutch market Finally, I want to discuss the opportunity of the arising of a secondary insurance market in the Netherlands. In principle, the Wet financieel toezicht and other laws don’t hinder the existence of a Dutch secondary insurance market. The right to transfer the policy to a third party after enclosure is captured in the policy conditions. Insurers will not be eager to narrow this opportunity, since the value of an insurance as a financial instrument for consumers will then

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January 2008

be lowered. In 2004 the Authoriteit Financiële Markten (AFM) made clear that in their opinion a viatical market for insurances would be in breach with “decent morals” in the Netherlands. The reason for this decision is that for viatical settlements the investor obtains an interest in an early death of the insured. Likewise, the Minister of Finance has commented negatively on the desirability of a Dutch secondary market in the past (Ministerie van Financiën, kamervragen 29-3-2004). Dutch entrepreneurs are already acting on the American market and in april 2007 the Dutch national news agency ANP reported that more and more Dutchmen invest in secondary insurance policies in the United States. Summary On the secondary insurance market, the right of ownership of a life insurance policy can be sold to a third party, who does not have any interest in the life of the insured. Countries like the United States and Great Britain have such a market as an alternative for the surrender of insurance policies. For a long time this market remained limited and “underground”, but it grew rapidly in the eighties as a response to the aids-epidemic. Currently, the market is also accessible for the elderly with a strongly declined health. The market exists because of the difference of the individual economic value of a policy (which is increased because of the expected early death) and the surrender value, which is calculated based on average (mortality) assumptions. The market offers advantages as well as disadvantages for insurers and policyholders. In the Netherlands, AFM and the Minister of Finance strongly reject the origin of a secondary market.

ORM

A Probabilistic Analysis of a Fortune-wheel Game A so-called fortune wheel is given. This is a disc divided into twenty wedges, numbered from 1 to 20, and provided with a pointer which can be turned. When the pointer stops, it always points to one of the twenty wedges and thus points to an integer between 1 and 20. Now let n persons be given who play the following game: every player turns the wheel either one or two times; after the first turn he can decide to stop; in this case, his score is the result of the first turn. He can also decide to turn the wheel a second time; then his score is the sum of the first and the second turn, provided that this sum is at most 20; if this sum is larger than 20, his score is 0. The player with the highest score wins, on the understanding that when multiple players get the same score, the player who got this score first wins. We are interested in the optimal strategies for all players, that is strategies that maximize their probability of winning.

Rein Nobel is an assistant professor in operations research at the Vrije Universiteit Amsterdam. He graduated in pure mathematics and in computer science. His main research interests are probability, Markov decision theory, queueing theory and simulation.’ Suzanne van der ster is a bachelor student in Operations Research. She is currently finishing her bachelor thesis under supervision of Rein Nobel. The article was written during the period in which Suzanne was a student assistent for Rein Nobel.’

The two-player game First we consider the case of two players [n = 2], A and B. Player A begins, so when both players get the same score A wins. Now we define a so-called m-strategy. A player follows the m-strategy if he only turns the wheel a second time in case the result of the first turn is at most m. Let Xm be the score of a player who follows the m-strategy. We determine the probability distribution of Xm under the obvious conditions [all results of a turn are equally probable and separate turns are independent]. Then the collection of possible values for Xm, say WXm, is equal to the collection {0, 1, ..., 20}. Now we have to determine the probability IP(Xm = i) for every i ∈ WXm. This gives

IP( X m= i= ) m 1 k 1 ⎧ = ⋅ m(m + 1)= i 0, ⎪ k =1 20 20 800 ⎪ i −1 1 1 (i − 1) ⎪ i 1,..., m, ⋅ = ⎨ = k =1 20 20 400 ⎪ m ⎪1 1 1 1 m i =m + 1,...,20. + ⋅ = + ⎪ k = 1 20 20 20 20 400 ⎩

∑ ∑

∑

Next we introduce the random variable Y, being the score of player B. Naturally, the probability distribution of Y depends on m and on the strategy of player B. Since player B knows the score of player A, he will adjust his strategy to this. Reasoning tells us that the only rational strategy for player B is to play an a-strategy when player A gets score a. Therefore the conditional distribution of Y given Xm = a is equal to the distribution of Xa. Now we can calculate the probability that player B wins by splitting according to the partitioning events {Xm = i}, i = 0, 1, ..., 20 and subsequently conditioning on these events, 1 1 1−( [( m(m + 1))2 IP(player B wins) = 160000 2 m(m + 1))2m + 1) m 1 − + ]+( ) 6 20 400 m(m + 1)(2m + 1) 1 1 [2870 − ]), 400 6

where we use the following equalities m

∑i i =1

m(m + 1)(2m + 1) 6

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AENORM

January 2008

ORM

and

60240/160000

38567/80000

31517/80000

38237/80000

6571/16000

15029/32000

6823/16000

1827/4000

14109/32000

7021/16000

18151/40000

2077/5000

37177/80000

61769/160000

3787/8000

1121/3200

15341/32000

4923/16000

7719/16000

4123/16000

1 = i 3 ( m(m + 1))2. 2

∑ i =1

The result above immediately gives the probability that player A wins. Introduce = IP(player A wins). Then we obtain = α(m)

1 1 [( m(m + 1))2 − 160000 2 m(m + 1)(2m + 1) m 1 + ]+( )* 6 20 400 m(m + 1)(2m + 1) 1 [2870 − ]. 400 6

Table 1: Winning probabilities player A in the two-player game.

287 5733 1 α(m) =+ m− m2 800 320000 15360 7 1 m3 − m4. 160000 1920000

j-1

1 i IP(Y = j) = [ ]IP( X m = i) + + 20 400

∑

i=0 20

We can easily determine m* := arg max{ m = 1, ..., 20} by calculating the value of for the different values of m. See Table 1.

i= ) (**)

i=j

To be able to continue, we need to distinguish two cases, m < j - 1 and m ≥ j - 1. First we consider the case m < j - 1. Splitting the sum-

∑

=i 0 20

IP( X i > i )IP( X= i) m =

0 i=

∑

=i 0=i 0 20 20 20

IP( X= i) m

∑

IP(= Xi = k)

0 i= k= i +1 20 2

∑ IP(X

1 i i (20 − i )[ + i )[1 ] =IP( X m =− ]= 1− 20 400 400 =i 0

1 1−( 160000

∑ (i − 1)i

=i 1

=i 0 m

∑

IP( X= i) m

j= )

=i 1

∑ IP(X=

i) * m 0 i= 20 2

=i m +1

=i 1=i 1

−

(

1 m i + ) )= 20 400 400

m(m + 1))2m + 1) ]+ 6

1 m 1 m(m + 1)(2m + 1) + ) [2870 − ]), 20 400 400 6 j-1

∑

IP(Y=

∑

1 i j −1 [ + ]IP( X m = (**) IP( X i == j)IP( X m i ) = i) + IP( X m = i) = 20 400 400

j ) | X m= i )IP( X m= i= )

∑

−1 i + ∑ ∑ i400 400

i = 1 −( 400

i) m =

1 1 1−( [( m(m + 1)) ∑ i − ∑ i ]) = 160000 2

1 1 m ) [ + 20 400 400

IP(Y=

∑

1 i + = [ ] 20 400

0 i= k= i +1 m 2 2

(

IP( X= i

i 0=i 0 j-1 20 i=j

IP(player B wins) = IP(Y > X m ) = IP(Y > X m; X m == i) IP(Y > i; X m == i) IP(Y > i | X m = i )IP( X m == i) 20

j −1

∑ 400 IP(X =

Next, we calculate the probability distribution of

∑

for

Y, the score of player B as a function of the mstrategy of player A. We start with calculating the probability that player B obtains a score j ≠ 0, by splitting and conditioning on the events {Xm = i},

We see that the probability that player A wins is a fourth-degree expression in m. Some simple algebra [MAPLE] gives

1 1 IP(Y = j) = ( m(m + 1) + 400 40

∑

i −1 (20 + i ) + 400

∑ IP(X= i

∑

j −1

∑

i=j

1 m (20 + i )[ + ]+ 20 400

i= 1 i= m +1

1 (10m(m + 1) + 160000

m ]= ∑ ( j − i)[ 201 + 400 i= j

j −1

∑

j)IP( X m= i ) +

i=0

∑ i=0

j)IP( X m= i= )

(20 + i )(i − 1) + (20 + m)[

i= 1

∑

(20 + i ) +

i= m +1

∑ ( j − 1)]) = i= j

1 1 21 2 1 3 m − m ). (10(1 − j)( j − 82) − (3 j2 − 249 j + 2708)m − 160000 6 2 6

ORM

mations (**) gives 1 (10(1 − j)( j − 82) − IP(Y = j) = 160000 1 21 2 1 3 (3 j 2 − 249 j + 2708)m − m − m ). 6 2 6

To check these answers it is very useful to simulate the game and compare the answers to the simulated values. In Table 2 we compare IP(Xm = i), i = 0, 1, ..., 20, and IP(Y = j), j = 0, 1, ..., 20 to simulated values for these probabilities where we simulated 1 million times the two-player game. Player A always follows his optimal m-strategy, so m = 10. In these 1 million simulations player A has won 482113 times, and we see that this is in agreement with the calculated probability 7719/16000 = 0.482.

Now we consider the case m ≥ j - 1. The other necessary split of (**) gives 1 2209 ( IP(Y = j) = j + 11 j 2 − 160000 6 1 3 1 4 j − 379 + (21 − j)m(m + 1)). 6 2

The three-player game Next we consider the same game for three players, A, B and C [n = 3]. Again the problem is to determine the optimal m-strategy for player A. But first we consider the optimal strategy for player B. Player B knows the score of player A, say a. Some thinking tells us that B should follow a q-strategy, where q = max {a,m*} and m* = 10 the value we found above for the optimal strategy for player A in the two-player case. Finally we consider the strategy for player C. He knows the scores of the players A and B, say a and b. Then player C will follow a r-strategy, where r = max{a,b}. We will assume that player A follows a m-strategy and we look for the value of m that will maximize his probability of winning. The random variable Xm is the individual score of a player following a m-strategy. Let Y be the score of player B and Z the score of player C [naturally, the probability distributions of Y and Z also depend on m, but we suppress this in the notation]. Now, for player B the conditional probability distribution of Y given Xm = a is equal to the probability distribution of Xmax{a,m*}. Furthermore, the conditional probability distribution of Z given Y = b and Xm = a is equal to the probability distribution of Xmax{a,b}. Again, let be the probability that player A wins when he plays the m-strategy. Then we get, again by splitting according to the partitioning events {Xm = i},

Finally, we need to calculate the probability that player B obtains a score 0. 1 ( 320000

∑ i(i

IP(Y = 0) =

− 1) +

i =1

(20 + m)

∑

i(i + 1)).

=i m +1

Some algebra leads to 1 18397 (61600 + m 320000 6 251 2 43 3 1 4 − m − m − m ). 12 6 12

IP(Y = 0) =

We resume the results.

= IP(Y

) j=

1 13897 ⎧ m− ⎪ 320000 (61600 + 6 ⎪ j = 0, 251 2 43 3 1 4 ⎪ − − m m m ) ⎪ 12 6 12 ⎪ 1 2209 1 2 ⎪ j + 11 j − j 3 − ( ⎪⎪160000 j = 1,..., 6 6 ⎨ m + 1, 1 ⎪ 379 + (21 − j )m(m + 1)) 20 i i ⎪ 2 ⎪ = α ( m ) = IP( Xmax{i , j} k ) 1 1 ⎪ (10(1 − j )( j − 82) − ( j 3 − = = = i 0 j 0 k 0 = j m + 2, ⎪160000 6 6 ⎪ IP( Xmax{i , m* } = j)IP( X m = i ) 21 1 ...,20. 2 3 ⎪ 249 j + 2708)m − m − m ) ⎪⎩ 2 6 For clarity we first calculate the inner summa-

∑∑ ∑

IP(Y = j) =

1 (10m(m + 1) + 160000

=i 1

=i j

=i m +1

∑ (20 + i)(i − 1) + ∑ ( j − 1)(i − 1) + ∑

( j − 1)(20 + m)) =

1 2209 1 1 ( j + 11 j2 − j3 − 379 + (21 − j)m(m + 1)). 160000 6 6 2 5

∑

=i 1=i 1 20

∑

=i 1 m 1 2

AENORM

∑

1 m i(i + 1)[ = + ]) ( 20 400 320000

i= m +1

January 2008

−1 + ∑ i(i + 1) i400

1 1 IP(Y = 0) =IP( X i = 0)IP( X m = i) = i(i + 1)IP(X m = i) = ( 800 800

i(i − 1) + (20 + m)

∑

i(i + 1)).

i= i= m +1 1

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4389_1

R i s i c o m a n a g e m e n t

•

E m p l o y e e

B e n e f i t s

•

V e r z e k e r i n g e n

ORM

IP(Xm=i)

simulation

IP(Y = j)

simulation

0.1375

0.1376

0.2568

0.2574

0.0069

0.0068

0.0025

0.0090

0.0050

0.0049

0.0113

0.0075

0.0137

0.0136

0.0100

0.0099

0.0162

0.0161

0.0125

0.0188

0.0189

0.0150

0.0151

0.0216

0.0175

0.0173

0.0244

0.0200

0.0199

0.0273

0.0225

0.0222

0.0303

0.0301

0.0750

0.0333

0.0335

0.0750

0.0752

0.0389

0.0391

0.0750

0.0748

0.0444

0.0442

0.0750

0.0749

0.0496

0.0750

0.0751

0.0547

0.0545

0.0750

0.0753

0.0596

0.0592

0.0750

0.0749

0.0643

0.0645

0.0750

0.0753

0.0688

0.0685

0.0750

0.0751

0.0731

0.0750

0.0749

0.0772

0.0775

Table 2: Calculated and simulated probabilities for the scores of player A and player B.

tion, using the probability distribution of Xm [see first page]. Introduce Mij := max{i, j}. i

∑

IP( X M = k ) = ij

k =0

To be able to continue, we split the summations over i in ∑i≤m* and ∑i>m*. We get m*

1 Mij (Mij + 1) + 800

= α(m)

1 ( = IP( X m i )2i 2[m*(m* + 1) + 640000

∑ i =0

1 k −1 [Mij (Mij + 1) + i(i − 1)]. = 400 800

∑

k =1

1 800

∑ IP(X= m

i )[(i(i − 1) + i(i + 1)] *

i =0

∑

α(m) =IP( X m ≥ Y ; X m ≥ Z ) =

∑ IP(Z ≤ i;Y ≤ i; X

IP(X m ≥ Y ; X m ≥ Z; X m =i ) =

m =i ) =

=i 0=i 0 20 i 20 i

∑∑ IP(Z ≤ i;Y =j; X

j 0 =i 0=

α(m) =

1 800

∑

∑∑

IP(Z ≤ i | Y =j; X m =i )IP(Y =j | X m =i )IP( X m =i ) = IP(Xmax{i , j} ≤ i ) * m =i ) = j 0 =i 0= =i 0=j 0

∑

= k )IP( X ∑∑ ∑ IP(X 1 j) = IP( X = i )[(i(i − 1) + i(i + 1)] * 800 ∑

IP( Xmax{i , m* } = j)IP( X m= i= ) 20

max{i , j}

=i 0=j 020 = k 0

IP( X m = i ) = [i(i + 1) + i(i − 1)]IP( X M im* =i 0= =i 0 j 0

max{i, m* }

j)IP( X m= i )

1 [ [M * (Mim* + 1) + i(i − 1)]]. 800 im

AENORM

To be able to do the further calculations we need to know whether here [i.e. in the threeplayer case] the optimal value for m for player A will be greater or less than m* [the optimal value for the two-player case]. Intuitively we expect that the optimal m we are looking for will be greater than m*, because player A has

7 1 [ [M * (Mim* + 1) + i(i − 1)]] 800 im 6

IP( X m = i)4i 4 ).

=i m* +1

Now we get ) α(m =

∑

i(i − 1)] +

January 2008

ORM

more opponents in this case. He will thus try to reach a higher score than in the two-player case. Therefore we will now continue the calculations assuming that the m we are looking for is greater than the m* = 10 we found earlier [of course the formula for can also be calculated for m < m* but this is omitted]. From now on we will insert the value m* = 10 in calculations and assume m > 10. We continue as following (m) α=

IP(Y=

∑ IP(X i =0

IP( X ∑= i

j −1 1 m(m + 1) + [ 400 800 1 m(m − 1) + (20 − m) * 800 1 m j −1 ( + )] = . 20 400 400

IP(Y = j) =

1 1 1 *3 [ m*m(m + 1) + m + 8000 40 30 1 *2 1 * 21 2 21 m + m + m + m+ 20 12 40 40 1 1 3 11 2 mj(m − 3) − j + j + 40 120 20 2209 379 12 ]. − 120 20

Now, for j > m

∑

IP( X ∑=

i )2i 2[

=i m* +1

i =0

1 [i(i + 1) += i(i − 1)]] 800

∑

1 ( = IP( Xm i)2i2[m*(m* + 1) + i(i − 1)] + 640000 i =0

∑

IP( X m = i )4i 4 +

i =0

∑ 2i

=i 1 10

11 1 ( 640000 20

∑

=i 1

2 i −1

400

[110 + i(i − 1)] +

−1 4i ∑ i400

=i 11

i 2(i − 1) +

1 200

∑

i 3(i − 1) +

=i 1

i)4i 4 ).

1 100

∑

IP( X m = i )4i 4= )

=i m +1

1 ( 640000

IP( X ∑=

=i m +1

IP( X m = i )2i 2[m*(m* + 1) + i(i − 1)] + 10

Now the case m* < j ≤ m*

At this point we need to distinguish three cases; j ≤ m*, m* < j ≤ m* and j > m. We start with j ≤ m*

1 724481 7226667 ( m− + 640000 5 1000 1 7 5 m2 − m3 − m4 − 2000 100 48 21 5 1 9 m − m6 ). 500 3000

1 α(m= ) ( 640000

j= )IP(X m i ).

=i m +1

1 1 ( = [m*(m* + 1) + i(i − 1)]] + = α(m) IP( X m i )2i 2[ 800 800

j)IP(X m= i ) +

We see that , the probability that player A wins when he follows a m-strategy, is a sixthdegree polynomial in m. Now we can find m** := arg max{ : m = 1, 2, ..., 20} by calculating the expression for the different values of m > 10. The result is m** = 13 with corresponding probability = 4370213/12800000 ≈ 0.341. In Table 3 we show all the winning probabilities for player A for m ≥ 10. Just as in the two-player case we can determine the probability distribution of Y, the score of player B. We start with calculating the probability that player B obtains a score j ≠ 0. As before we do this by splitting according to and conditioning on the events {Xm = i}.

j= )

∑

=i m +1

1 m [ ]4i 4 ) = + 20 400

i 4(i − 1) +

=i 11

20 + m 100

∑

i 4) =

=i m +1

1 724481 7226667 1 7 5 21 5 1 ( + m− m2 − m3 − m4 − m − m6 ). 640000 5 1000 2000 100 48 500 3000 20

∑

IP(Y = j ) =IP(Y = j | Xm = i ) =IP(Y = j | Xm = i )IP(X m = i) +

∑

IP(Y = j | Xm = i) *

=i m* +1

=i 0=i 0

∑

IP( X m = i) = IP( X m* = j)IP(X m = i) + IP( X i = j)IP(Xm = i). i =0

=i m* +1

ORM

1 1 [ m*m(m + 1) + 8000 40 3 2 1 1 1 * m* − m* + m − 120 40 60 1 21 2 677 m3 − m − m− 120 40 30 1 1 83 mj( j − 83) − j 2 + j − 41]. 40 2 2

IP(Y = j) =

Finally, for

10575891/ 32000000

5094419/ 16000000

10770827/ 32000000

19102553/ 16000000

2178889/ 6400000

8646677/ 32000000

4370213/ 12800000

7423467/ 32000000

5419431/ 16000000

5823467/ 32000000

10604789/ 32000000

= 0 we obtain

IP(Y = 0) =

1 [m*(m* + 1)m(m + 1) + 640000 3 2 1 *4 1 m − m* − m* + m* − 2 2 1 4 43 3 251 2 m − m − m + 6 3 6 14 18397 m + 123200]. 3

We resume the results. 11

IP(Y = j ) =

j −1 1 m(m + 1) + 400 800 1 400

j −1 i −1 + ∑ ∑ 400 400

=i m* +1

=i 1 20

∑ (i − 1) + ∑

=i 1

=i m +1

m 1 + (= )] 20 400

1 m* 1 ) IP(Y = j ) = ( + m(m + 1) + 20 400 800

∑

(

∑

j −1 i −1 + 400 400

=i m +1

j −1 1 m j −1 1 ( + )= [ m(m + 1) + 400 20 400 400 800

j −1 1 m j −1 1 1 m(m + 1) + m(m − 1) + (20 − m)( + )] = . [ 400 400 800 800 20 400

1 m* i − 1 ) + + 20 400 400

j −1

∑

(

1 i i −1 ) + + 20 400 400

j −1 i −1 + ∑ 400 400

=i 1= i j =i m* +1 20

∑

=i m +1

1 1 1 1 j −1 1 m m* m* ( )= ( ) ( ) + + m(m + 1) + + 400 20 400 20 400 800 400 20 400

∑

j −1 160000

j −1 1 m = ( + ) ∑ (i − 1) + (20 − m) 400 20 400

(i − 1) +

j −1

∑

1 160000

i(i − 1) +

=i m* +1

i =1

1 1 * 1 *3 1 *2 [ m m(m + 1) + m + m + 8000 40 30 20

i= j

1 * 21 2 21 1 1 3 11 2 2209 379 m + m + m+ mj(m − 3) − j + j + − ]. 12 40 40 40 120 20 120 20 13

IP(Y = j ) = ( 20

∑ i= j

1 m* 1 ) + m(m + 1) + 20 400 800

∑

(

1 m* i − 1 ) + + 20 400 400

j −1

∑

(

=i m* +1

=i 1

* * m

1 1 1 1 j −1 1 m m* m ( )= ( ) ( ) + + m(m + 1) + + 400 20 400 20 400 800 400 20 400

∑

( j − m − 1)

1 1 1 1 m m ( )+ ( ) + + 20 20 400 400 20 400

j −1

∑

=i m +1

1 i i −1 ) + + 20 400 400

(i − 1) +

1 8000

j −1

∑

(

=i m +1

1 i 1 m )( )+ + + 20 400 20 400

∑

i(i − 1) +

=i m* +1

i =1

1 160000

∑

i(i − 1) +

=i m* +1

3 1 1 * 1 j −1 1 m ( ) [ i + (20 − j + 1) * = + m m(m + 1) + m* − 400 20 400 8000 40 120

1 1 83 1 *2 1 * 1 21 2 677 m − m− mj( j − 83) − j 2 + j − 41]. m + m − m3 − 30 40 2 2 40 60 120 40 14

1 1 IP(Y = 0) = m*(m* + 1) m(m + 1) + 800 800

∑

=i 1

1 i −1 m*(m* + 1) + 800 400

∑

=i m* +1

1 i −1 i(i + 1) + 800 400

∑

= i(i + 1) =i m +1

AENORM

i =1

∑

=i m +1

1 i(i + 1) * 800

1 ∑ (i − 1) + 320000 ∑

1 1 1 m = ( + ) m*(m* + 1)m(m + 1) + m*(m* + 1) 20 400 640000 320000

=i m* +1

i(i + 1)(i − 1) +

1 1 m ( + )* 800 20 400

4 3 2 1 1 1 1 43 3 251 2 18397 m − m + m + 123200]. [m*(m* + 1)m(m + 1) + m* − m* − m* + m* − m4 − 640000 2 2 6 3 6 3

January 2008

ORM

= IP(Y

j= )

1 ⎧ * * ⎪ 640000 [m (m + 1) * ⎪ 1 *4 ⎪ ⎪m(m + 1) + 2 m − ⎪ ⎪m*3 − 1 m*2 + m* − 0, j = ⎪ 2 ⎪ ⎪ 1 m4 − 43 m3 − 251 m2 + ⎪6 3 6 ⎪ 18397 ⎪ m + 123200] ⎪ 3 ⎪ j = 1,..., ⎪ j −1 ⎪ 400 m*, ⎪ 1 ⎪ 1 * ⎪ 8000 [ 40 m m(m + 1) + ⎪ ⎪ 1 *3 1 *2 1 * ⎨ m + m + m + 20 12 ⎪ 30 ⎪ 21 21 1 = j m* + 1, ⎪ m2 + m+ mj * 40 40 ⎪ 40 ..., m, ⎪ 1 3 11 2 j + j + ⎪(m − 3) − 120 20 ⎪ ⎪ 2209 379 ]. − ⎪ 20 ⎪ 120 ⎪ 1 1 [ m*m(m + 1) + ⎪ 8000 40 ⎪ 3 2 ⎪ 1 1 1 * m* − m* + m − ⎪ = j m + 1, 40 60 ⎪120 21 2 677 ...,20. ⎪ 1 3 ⎪120 m − 40 m − 30 m − ⎪ 1 2 83 ⎪1 ⎪⎩ 40 mj( j − 83) − 2 j + 2 j − 41].

Table 4: Probability distribution of the score of player B in the three-player game.

More than three players

Again, we compare these results with simulation results. We simulate the game 1 million times, where all players play their optimal strategy. This means m* = 10 and m** = 13. In 1 million simulations player A has won 341327 times, which is again in good agreement with the theoretical probability of 4370213 / 12800000 = 0.341. Now the final issue is to calculate the ty

IP(= Z k= )

20 20

∑∑ IP(X= m

i;= Y

j;= Z k)

=i 1=j 1

ORM

m** = 13 and a is the score of player A. Player C knows the scores of players A and B, say a and b, but also has to keep in mind that player D will play the game last. This means that player C will follow a r-strategy, where r = max{a, b,m*} and m* = 10. Finally player D will follow a s-strategy, where s = max{a,b,c}. We try different values for m and discover [by simulation!] that the probability of winning for player A is optimal when m*** = 14. This gives a winning probability for player A of â&#x2030;&#x2C6; 0.2715. See Table 6 for the winning probabilities for the other values of m. We see from Table 6 that the winning probabilities for player A do not differ very much for m = 13, 14, 15. So, around the optimal value for m the function is rather flat.

0.2281

0.2008

0.2291

0.1704

0.2267

0.1938

0.2193

Table 7: Simulations for the winning probability of player A for different values of m.

wheel game for n players in which each player can increase his score by turning the wheel a second time. Turning a second time can result in a zero score. This asks for finding an optimal threshold strategy for each player which maximizes his probability to win. For the case of two and three players we have presented a detailed

"The optimal threshold for the first player is increasing in the number of players, whereas the maximum winning probability of the first player is decreasing in the total number of players." m

0.2692

0.2499

0.2715

0.2278

0.2698

0.1938

0.2637

0.1441

Table 6: Simulations for the winning probability of player A for different values of m.

For the five-player case we do the same thing. Here every player is again playing his optimal strategy and we determine the optimal m-strategy for player A. Now player B is using the optimal value of m from the four-player case in his optimal strategy. Different values of m give the probabilities of winning for player A that appear in table 7. The highest probability appears where m = 15. This would be the optimal m for player A. This gives a probability of winning â&#x2030;&#x2C6; 0.2291. From the previous simulations for the four- and five-player cases it becomes clear that for higher numbers of players, the optimal value for m for player A increases too. Conclusion In this paper we have analyzed a simple fortune-

AENORM

January 2008

probabilistic analysis. The optimal strategy for the first player has been calculated, and also the probability distributions of the score of the second player have been derived. The mathematical results have been confirmed by simulations. For the case of four and five players we have only presented simulations. All the results confirm the conjecture that the optimal threshold for the first player is increasing in the number of players, whereas the maximum winning probability of the first player is decreasing in the total number of players. The paper also illustrates the fruitful cooperation between probabilistic calculations and stochastic simulations: mathematical results can be checked by simulations, and simulations can give a better insight in the characteristics of a problem when mathematical results are (still) lacking. So, simulation results can be a guide for further understanding.

Actuarial Sciences

Quantifying Operational Risk in Insurance Companies The main objective of this paper is to illustrate how financial institutions including insurance companies can quantify Operational Risk (OR) that meets the regulatory conditions. The lack of loss data is one of the main obstacles for quantifying OR. We briefly discuss the Loss Distribution Approach (LDA) and Bayesian Approach (BA) to estimate the OR losses. In conclusion we will provide some OR estimation results provided by our model and discuss them briefly.1

Youbaraj Paudel was born in Nepal and came to the Netherlands in 2000. He joined VASVU in 2001 and in 2002 he started his study Business Mathematics and Informatics (BMI) at the Vrije Universiteit Amsterdam. He obtained his master’s degree in July 2007. At this moment he works for Watson Wyatt Worldwide in the Insurance Practice. His interests are among others financial risk management and life insurance.

Introduction The financial risk within financial institutions consists of various components like Credit Risk, Market Risk, Business Risk and other types of risks such as Operational Risk (OR). There is a growing pressure from prudential authorities on the management and measurement of OR by both insurance and banking institutions. Both Basel II and Solvency II encourage the banking and insurance sector respectively to measure their OR losses based on an internal model. The OR is mainly based on the organisational management of internal processes. As internal processes vary within companies a companyspecific OR model for estimation and management is required. Therefore, the statistical models used by different companies are required to reflect company-specific factors. Contrary to other risks, OR is managed by means of changing existing processes such as technology, organisation and people. To be able to manage these, it is necessary to have a lot of company specific information available. Quantifying the informative and qualitative operational data is complex and, even if it is possible, the uncertainty factor is very high. This makes it difficult to quantify and express OR in numbers. We observed that the OR quantifying models

developed within the banking sector are mostly based on either scenario-based or curve fitting techniques, and sometimes on the combination of both. The main reason for using this method is that the companies do not want to rely completely on historical data, since that might fail to represent the new emerging risk events in a rapidly changing market. Moreover, many companies do not yet have an internal database of historical loss records. The dynamic underlying causes of OR need a progressive and interactive approach to risk management where all different kinds of information like expert opinion, internal and external historical data, key indicators and other informative data as well as quantitative data might be involved. As this method is thought to be conceptually sound and, if implemented with integrity, it would be recognised under Solvency/Basel II for Advanced Measurement Approach (AMA) status and under Solvency II for an internal model. The OR estimation model we developed is based on two quantification, the Loss Distribution and Bayesian approach, methods and is discussed in the following chapter. For more detailed information, the reader is referred to my internship paper ‘Quantifying operational Risk within Insurance industry’ July 2007. OR Estimation Before we start modelling the OR losses using some statistical methods we should realise that the quality of the outputs of such models depend on the quality of the input. Therefore, the first part – the identification/classification of risk variables is very crucial work to be done to provide optimal results. Sound Operational Risk Management consists of the following steps2:

The views expressed in this paper are my own, and not necessarily reflects the opinions of Watson Wyatt Worldwide or its members. 2 This topic is discussed in more detail in my internship paper. 1

Actuarial Sciences

-Objective setting; -Risk identification and classification; -Qualitative/quantitative data collection and data cleaning; -Risk measurement; -Risk control monitoring and mitigation process; -System infrastructure. A sound Operational Risk management system can clearly analyse the cause-effect relationship in detail. The causes should be defined in various levels so that even the smallest details can be incorporated in the model. Figure 1 defines the different steps to connect the specific causes with their respective consequences.

types4 are defined j, where j =1…m. The probability function for number of losses (loss frequency N(i,j)) is given by pi,j and the overall loss frequency distribution is then Pi,j(n), which gives: Pi , j (n) =

s=n

∑p

i , j (s)

(1)

s =0

As there is not enough loss data, it is necessary to use the Monte Carlo simulation to compensate for this. Therefore we use different frequency fitting models like: Poisson, Gamma and Negative Binomial5. Then the total loss, for each business line i and event type j in the given time interval [t, t + ] is given by: N(i , j )

θ(i, j) =

∑ ψ (i, j) n

(2)

n =0

Where denotes the loss amount for event j and business line i.

Figure 1: Classification: consequences for OR

cause,

events

and

The entire OR management framework is not discussed in this article. My internship paper can be consult for more details on this topic. Loss Distribution Approach (LDA) LDA refers to a statistical actuarial bottom-up approach for modelling OR events losses and their impact. In this chapter we provide a brief description of how this method should be implemented in practice and how quantitative and qualitative should be combined to provide inputs for the model. Here follows a brief description. An insurance company offers various types of products, and they belong to a specific line of business3 i, where i = 1,..,n. Also, different risk

Figure 2: Loss Frequency/Severity for event j and line of business

For instance, the different lines of business are life/non-life/health insurance within the insurance industry. The types of risks I include in my model are the same as introduced by Basel II. These are: 5 In my internship paper you can find more information about how the parameters for the fitting models concerned are estimated and which inverse functions are used. For detail about how the various parameters for different fitting models are estimated and which inverse functions are used to draw random numbers, you are advised to consult the abovementioned paper. 3 4

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January 2008

Actuarial Sciences

Data available

Statistical Approach

Prior Inference

Empirical Inference

Internal Data only

General LDA

None

Expert Data only

General LDA

None

External Data only

General LDA

None

Internal + External Data

Bayesian inference

External Data

Internal Data

External Data + Expert Opinion

Bayesian inference

External Data

Expert Data

Expert Opinion Internal + External Data

Bayesian inference

External Data

Internal Data + Expert Opinion

Table 1: Different alternatives for data manipulation applying different approaches

Then the compound aggregated loss capital is: ⎧ ∞ ⎪⎪ pi , j (n) *Ψn(i, j) if x > 0 Gi , j (x ) = ⎨ n =1 ⎪ p (0) if x 0 = ⎪⎩ i , j

∑

(3)

F is the severity function and Fin, j (x) = θ (x; i, j)n 6 denotes the severity amount x for line of business i and event j for nth occurrence (frequency).

Equation (4) denotes the expected losses that is caused by event type j with frequency n emerged in business line i. And the unexpected losses that may arise from OR loss events are given by: UL= (i, j; x ) ⎡Gi−,1j ( x ) − θ(i, j ) ⎤ ⎣ ⎦

(5)

Where x denotes the amount that lies at some specific quintile (i.e. 99.5%) level, also referred to as . Writing the equation in open form we get: = UL(i , j; α) [inf{x | Gi , j ( x ) ≥ α} − EL(i , j )]

(6)

Now, it is possible to calculate the required capital as defined by Solvency II by using all these ingredients mentioned above. RC (= i , j; α) EL(i, j ) + UL(i, j; α)

(7)8

This is equal to: RC = Gi−,1j (x )

Figure 3: Aggregated capital

The last step in LDA is the calculation of the capital charge, which is a VaR measure of risk. We can calculate this amount in two ways: at overall level or at business level. Calculation under the overall level may include the correlation information matrix that could arise through the risk event’s dependency. For that reason, a correlation matrix should be integrated in the model to include that dependency in overall capital [1]7. Capital calculation at business level is done by estimating the losses at each business level and then summing them up. Theoretically, the overall capital is the sum of expected (EL) and unexpected (UL) OR losses. Based on the previous calculation we can define the EL and UL as following: ⎡ N (i , j ) ⎤ EL(i, j) = Ε ⎢ ψn(i, j) ⎥ ⎢ ⎥ ⎣ n=0 ⎦

∑

(4)

(8)

Where x is equal to some predefined threshold (i.e. 99.5%). As shown in the previous section, the LDA approach needs appropriate data fitting models. To determine which model to use for modelling frequency and severity the characteristics of each risk event type should first be precisely understood. Bayesian Approach Most traditional and classical approaches to estimate occurrence rates and their impact are mainly based on historical data. If sufficient historical data is available, the methods discussed in the previous chapter can be used to provide optimal results, as they possess the asymptotic properties. With sufficient data, representative parameters can be estimated with a large degree of certainty. However, this is not possible if not much data is available. Since OR losses

This severity function is a function of losses . For technical background about the dependency/independency among the loss events we refer the interested reader to the paper “Coherent Measures of Risks An Exposition for the Lay Actuary” by Glen Meyers. 8 Actually, we can calculate the RC as stated in Basel II, which is equal to: = + EL(i,j). 6 7

T o e t s h e t a a nwe z i g e v e r mo g e n v a n e e n p e n s i o e n f o n d s om d e i n d e x a t i e voo r gepen s i onee rden te bepa len .

Hoeveel is er nodig om onze pensioenen in de toe-

investeringsstrategieĂŤn. We werken voor toonaan-

komst te kunnen betalen? Rekening houdend met

gevende bedrijven, waarmee we een hechte relatie

de vergrijzing en de economische ontwikkelingen?

opbouwen om tot de beste oplossingen te komen.

Kunnen we straks nog steeds zorgeloos een potje

Onze manier van werken is open, gedreven en infor-

biljarten? Bij Watson Wyatt kijken we verder dan de

meel. We zijn op zoek naar startende en ervaren

cijfers. Want cijfers hebben betrekking op mensen.

medewerkers, bij voorkeur met een opleiding Actu-

En op maatschappelijke ontwikkelingen. Dat maakt

ariaat, Econometrie of (toegepaste) Wiskunde. Kijk

ons werk zo interessant en afwisselend. Watson

voor meer informatie op werkenbijwatsonwyatt.nl.

Wyatt adviseert ondernemingen en organisaties wereldwijd op het gebied van â&#x20AC;&#x2DC;mens en kapitaalâ&#x20AC;&#x2122;: pensioenen, beloningsstructuren, verzekeringen en

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January 2008

Watson Wyatt. Zet je aan het denken.

Actuarial Sciences

Internal Data

Frequency + Severity Data Horizon (yrs)

5,000

Loss reported: how many times

9,000

Standard Deviation Frequency

0,500

0,200

1,000

3,000

2,100

2,000

Mean Loss Severity

12,000

SD Loss Severity

2,500

Table 2: Input Internal loss data for the OR estimation model

are rare and asymmetric, it is quite difficult to estimate these based on only a few historical data [2]. For this approach we have three kinds of loss data: Internal, External and Expert or subjective data. We use the external loss data as posterior information, while the internal and expert data are modelled as prior information. The Bayesian method is based on a data-combination approach and integrates SBA and SMA in it to estimate OR losses for different events [3]. The Bayesian approach can mainly be used in the following situations: there is no or not much loss data, it is necessary to include subjective information in results, and the prior and posterior information is consistent. With these three types of data we have the different possibilities either for LDA or for Bayesian inference. By nature, the Bayesian method automatically considers the uncertainty associated with various estimation parameters of a probability model that originated from different information sources. Hence, the Bayesian method is currently mainly recommended as an appropriate manner to mix qualitative and quantitative data and information, such as expert opinion, internal and external data. At the same time we have to consider some shortcomings of this approach, i.e. generally it produced extremely overconfident results and has technical and computational limitations and it is generally known that this method overuses the averaging and aggregating information. The application of the Bayesian approach is based on the so-called Bayesian theorem. This connects the conditional9 and marginal10 probability distribution of random variables. The Bayesian theorem tells us how to update or revise viewpoints in light of new evidence. For two stochastic events, A and B, respectively having the probability P(A) ≠ 0 and P(B) ≠ 0, the definition of the conditional probability A giving B is given by [4]:

P( A | B) =

P( A ∩ B) P(B)

(9)

Equation (9) can also be written in a different way, and this provides us with:

P( A | B) =

P(B | A)P( A) P(B)

(10)

Equation (10) literally states11: Posterior information = (likelihood * prior) / normalizing constant. We will now translate the above mentioned Bayesian concept into a Bayesian computational approach for OR loss estimation [4]. Using the same Bayesian theorem we can write a conditional probability distribution of B given the observed data A as following: P(B | A) =

P( A | B)P(B) P( A)

(11)

The P(B) (i.e. internal loss data or other subjective information) represents information that is known without knowledge of data (i.e. external data). Therefore, P(B) is called prior information and the distribution of B a priori. The P(A) represents the distribution function for prior information based on available data and is also known as a priori. The core point of the Bayesian theorem, subsequent distribution based on these two types of data is then given by equation (9) and (10). As equation (11) now depends on the given data B, this is no longer the function of A, but of B. From that point onwards we call this function the ‘Likelihood’ function of B for giving historical data A (internal or external data). Now, we can write equation (11) as [4]: p(B | A) = l ( B | A ) p ( B )

(12)12

This implies:

P(A | B) is a conditional probability, and also to mentioned as likelihood (L(A | B)) of A given event B. P(A) and P(B) are the marginal probabilities. 11 Most Bayesian based calculations have been done in a numerical way as it is too complex for an analytical calculation. Based on the type of data it is possible to use either a continuous or a discrete form of this approach. The following expression can be used for these two forms: p(A) = E[p(A | B)] which equals if B is continuos and ∑ p(A | B)p(B) if B is discrete. In this case E[p(A | B)] denotes the mathematical expression for expectation of p(A | B) with respect to function p(B). 12 The Bayesian theorem tells us that the probability distribution for B posterior to the data A is proportional to the product of the distribution for B prior to the data and the likelihood for B given A. 9

Actuarial Sciences

Posterior Distribution ∝ likelihood*prior distribution

However, estimating the parameters for both prior and posterior distribution is not an easy task. In this article we do not discuss the parameters estimation techniques. Results In this chapter we provide some results generated by above mentioned model. The OR loss estimation model concerned was written in Excel using VBA-language. As there are several combinations in this model, it is not possible to discuss all of them in this paper. Therefore, we discuss here only one alternative in which only the internal loss data serves as input and LDA for modelling purpose. We have used the inputs from table 2 to provide the following results: Severity

Frequency

E(X)

Lognormal

Poisson

± 34

Lognormal

Neg-Binomial

± 48

Pareto

Poisson

± 38

Pareto

Neg-Binomial

± 75

Normal

Poisson

± 25

Normal

Neg-Binomial

± 30

Table 3: Expected loss Severity with different fitting models.

The results in table 3 describe the differences among the different alternatives for OR estimation. As might be expected, the loss data generated by the combination of normal and Poisson distribution for loss severity and frequency respectively is the lowest one. As both distributions have a symmetric and smooth character and Poisson is one parametric, the difference in result is quite easy to describe. We can also observe that the loss severity generated through the combination of Pareto and Negative-binomial provides the largest loss amount. Since both fitting models are defined by two parameters and have a fat tail, the results should apparently be higher in this case than in other cases. Bibliography [1] Franchot, A., Roncalli, T. and Salomon, E. (2003). “The Correlations Problem in Operational Risk”. [2] Quigley , J., Bedford, T. and Walls L (2005). “Estimating Rate of Occurrence of Rare Events withy Empirical Bayes”.

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January 2008

[3] Ferson, S. (2003/2004). Methods in Risk Assessment”.

“Bayesian

[4] Yasuda, Y. (2003). “Application of Bayesian Inference to Operational Risk Management”. [5] Trip, M.H. and others (2004). “Quantifying Operational Risk in general Insurance Companies”. [6] Bank of Mauritius (2005). “Guideline on Operational Risk Management and Capital Adequacy determination”. [7] Tunkey, J. (2003). “Operational Risk and Professional Risk Management”. [8] Frachot, A., Georges, P. and Roncalli, T. (2001). “Loss Distribution Approach for Operational Risk”. [9] Quigley, J., Bedford, T. and Walls, L. (2005). “Estimating Rate of Occurrence of Rare Events withy Empirical Bayes”.

of weet jij* een beter moment voor de beste beslissing van je leven? www.werkenbijpwc.nl

Assurance • Tax • Advisory

Econometrics

Effects of invalid and possibly weak instruments When in a regression model some of the explanatory variables are contemporaneously correlated with the disturbance term, and this correlation is unknown, then one needs further variables in order to find consistent estimators by the method of moments. These instrumental variables should have a known (usually zero) correlation with the disturbances. In practice, however, it is usually difficult to assess whether an instrumental variable satisfies the moment condition. Firstly, instrument validity or orthogonality tests are only viable under just identification or overidentification by truly valid instruments. Moreover, orthogonality tests will have reasonable power only when the instruments employed and those under test are not too weak (are sufficiently correlated with the regressors) and the sample size is substantial. Therefore, it seems very likely that IV estimation will often be employed when some of the instruments are in fact invalid.

In this paper we consider the asymptotic distribution of an IV estimator in a linear regression model when some of its exploited orthogonality conditions actually do not hold. We focus on a single structural equation that otherwise has been specified correctly in the sense that its implied series of error terms is IID (independent and identically distributed) with unconditional expectation equal to zero. We cover the general case where the number of moment conditions exploited (l) is at least as large as the number of unknown coefficients, k ≤ l. In terms of parameters and data moments, we derive an expression for the inconsistency of the IV estimator and the asymptotic variance of the limiting normal distribution around this inconsistency. These results yield a first-order asymptotic approximation to the actual distribution of inconsistent IV estimators in finite sample. We verify by simulation whether these analytic findings are accurate regarding the actual estimator distribution in finite sample. In the experiments, we focus on a simple model with one explanatory variable and one instrument. From our simulations we establish that an invalid but reasonably strong instrument yield IV estimator which has a distribution in small samples that is rather close to the asymptotic approximation derived here. Hence, the distribution of this estimator is often close to normal, but instead of the true value, has its probability mass centered around the pseudo-true-value. However, when the instrument is very weak, we establish that the accuracy of standard large-sample asymptotics is very poor, as had already been established for the valid instrument case, see for example Bound, Jaeger, and Baker (1995).

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January 2008

Jan Kiviet is Professor of Econometrics in the Department of Quantitative Economics of the University of Amsterdam. He is a Fellow of the Tinbergen Institute and of the Journal of Econometrics, and Director of the research group UvA-Econometrics. Jerzy Niemczyk is a PhD student in the Department of Quantitative Economics of the University of Amsterdam. Before starting his PhD in 2004, he obtained an M.Sc in Mathematics at the Wroclaw University of Technology, and an M.Sc in Economics and Statistics at the Free University of Brussels.

Model, assumptions and theorems We consider data generating processes for variables for which n observations have been collected in the rows of y = (y1, ..., yn)’, X = (x1, ..., xn)’ and Z = (z1, ..., zn)’. The matrices X and Z have k and l columns respectively, with l ≥ k. Column vector xi contains the explanatory variables for yi in a linear structural model with structural disturbance , i = 1, ..., n. The l variables collected in Z will be used as instrumental variables for estimating the k structural parameters of interest . The basic framework is characterized by the following parametrization and regularity conditions, which involve linearity and stationarity. Framework 2.1 We have: (i) the structural equation y = ; (ii) with disturbances having (for i ≠ h ≠ 1, ..., n) the (finite) unconditional moments E( ) = 0, E( ) = 0, E( ) = , E( )= and E( ) = ; (iii) while E(xi | ) = and E(zi | ) = , with and fixed

Econometrics

parameter vectors of k and l elements respec≡ plim 1/n X’X, tively. Moreover, (iv) ≡ plim 1/n Z’Z and ≡ plim 1/n Z’X have all full column rank, and (v) so have X’X, Z’Z and Z’X with probability one. Finally, (vi) we have E(1/n Z’Z | Z ) = op(1 / n ) and E(1/n X’X | X , Z) = op(1 / n ), where X = ( x 1, ... , x n)’ and Z = (z 1, ..., z n)’ with x i ≡ E( | ) and z i ≡ zi E(zi | ). Note that the latter definitions imply the decompositions X ≡ X + and Z ≡ Z + . From (iii) we find E( X ) = 0 and E( Z ) = 0, whereas E(X ) =

and E(Z’ ) =

(1)

Hence, if = 0 for some j ∈ {1, ..., k} then the j-th regressor in X is predetermined and will establish a valid instrument; otherwise, when ≠ 0, the j-th regressor is endogenous. Likewise, if = 0 for some g ∈ {1, ..., l} then the g-th column of Z establishes a valid instrument, and an invalid instrument otherwise. It can be shown that (vi) boils down to the mild regularity assumptions 1/n Z ’ Z plim 1/n Z ’ Z = op(1 / n ) and 1/n X ’ Z plim 1/n X ’ Z = op(1 / n ). Since l ≥ k the generalized instrumental variable (GIV) or 2SLS estimator of exists and is given by

ˆ = [X’Z(Z’Z)-1Z’X]-1X’Z(Z’Z)-1Z’y β GIV ˆ y, ˆ )-1 X ˆ’ X = (X

(2)

ˆ ≡ ZΠ ˆ , where Π ˆ = with X contains the (reduced form) coefficient estimates of the first-stage regressions. We define the pseudoˆ as true-value of β * ˆ βGIV ≡ plim β GIV

= β + σ ε2[Σ X ' Z Σ−Z1' Z Σ Z ' X ]−1 *

(3)

Σ X ' Z Σ−Z1' Z ς,

ˆ by and we denote the inconsistency of β GIV * β − β σ ε2Σ−Xˆ1' Xˆ Π ' ς, GIV ≡ βGIV = -1 where we used Xˆ ' Xˆ ≡ and ∏ ≡ -1 . Note that the GIV estimator is consistent if and only if = 0. For the special case l = k (just identification), the above GIV results specialize to simple IV, -1 ˆ = i.e. β and β = . When IV OLS in fact Z = X (all regressors are used as instruments), i.e. , then IV specializes to OLS, -1 ˆ = (X’X)−1X’y and β i.e. β = . OLS OLS Here, for the sake of simplicity, we only present

the result for models with disturbances that have 3th and 4th moments corresponding to those of the normal distribution, and then obtain: Theorem 2.1 If μ3 = 0 and μ4 = 3, we have n1/2( ˆ ˆ → N(0,V N ) with β β GIV GIV GIV N VGIV = σ ε2c5(1 − c3 + c4 )Σ−Xˆ1' Xˆ + σ ε2c42 *

Σ−Xˆ1' XˆΣ X ' X Σ−Xˆ1' Xˆ − c4[Σ−Xˆ1' Xˆ * ΣX ' X β β' GIV β'GIV + βGIV GIV * -1

Σ X ' X Σ−Xˆ1' Xˆ ] + σ ε4c4(1 − 2c4 ) *

Σ−Xˆ1' Xˆ ξξ ' Σ−Xˆ1' Xˆ + σ ε2c4(1 − 2c5 ) * −1 [Σ−Xˆ1' Xˆ ξβ' GIV + β'GIV ξ ' Σ X ˆ'X ˆ] +

-1

[c5(1 − 2c5 ) − c3 + σ ε-2 ( β' GIV Σ X ' X * βGIV )]βGIV β'GIV , -1 where c1 ≡ c c and c ≡ 1 ≡ 1 2 5

≡

c4.

c ≡ β GIV, 3

, c β GIV 4

ˆ We find that the limiting distribution of β is GIV still genuinely normal when instruments are invalid, although no longer centered at but at the pseudotrue-value * . When all instruments are valid, i.e. = 0, then Theorem 2.1 ˆ specializes to the standard result n1/2( β ) GIV -1 → N(0, ˆ'X ˆ ). For the special case l = k we X -1 β have = , and as a result the GIV N above asymptotic variance, VIV , simplifies to σ ε2 (1 − c3 )2 Σ−Z1' X Σ Z ' Z Σ−X1' Z − [2c32 − 2c3 + Σ 1 − σ ε-2 (β' IV X ' X βIV )]βIV β'IV ,

(4)

-1 = where c3 ≡ β . Since for arbitrary IV ≠ 0 and ≠ 0 the scalar c3 can either be positive or negative, no general conclusions can be drawn on the behavior of (4) in comparison -1 -1 to the reference case . Depending on the particular parametrization and data moment matrices the asymptotic variance of individual coefficient estimates may either increase or decrease. ˆ = β ˆ , When Z = X, which gives and β IV OLS N the resulting VIVN = VOLS is the same as the formula found for an inconsistent OLS estimator when the disturbances are (almost) normal, as derived in Kiviet and Niemczyk (2007a). For the case where the disturbances may have general 3rd and 4th moment see Kiviet and Niemczyk (2007b). The result in Theorems 2.1 in fact address special case of Theorem 2 in (Hall and Inoue, 2003, p.369). The latter theorem is more general, but more implicit at the same time. It concerns GMM estimation of a possibly nonlinear misspecified model and expresses its asymptotic

Econometrics

variance matrix in a few model characteristics and a matrix Ω. This matrix, which is not further specified, is the variance of the limiting distribution of a particular (2k +l) × 1 vector v* . For the linear model, our approach allows to evaluate explicitly the covariance matrix Ω, and consequently the asymptotic variance of GIV. Experiments

�� 1 V N ) ˆ ∼ N(β+β, β IV IV n

is completely determined by n and the four model characteristics z, , and SN. In order to generate ( , , zi)’ ∼ IID(0, Ω), with appropriate 3 × 3 covariance matrix Ω, we can first generate vi = (vi,1, vi,2, vi,3)’ ∼ IIN(0, I3) and then parameterize as follows:

To illustrate the asymptotic finding presented in the foregoing section, we examine a simple model with one regressor and one instrument,

= xi = zi =

= yi βxi + ε i ⎫ ⎪ x= xi + ξ ε i ⎬ i z= zi + ζ ε i ⎪⎭ i

The above implies

(5)

where and are scalars1. In the simulation, we employ disturbances that are normally distributed, and without loss of generality we fix = 1. For this model, writing for , or for , etc, and using = / , = / , we find that the expression for the inconsistency and the asymptotic variance can be expressed as

σ zε = = β σ ε2Σ−X1= IV ' Zζ σ xz

ρzε σ ε ρxz σ x

(6)

N VIV

σ ε2 (1 − ρ2zε )(ρxz − ρzε ρxε )2 4 ρxz

σ x2

4 ρzε (1 − ρ2xε ) 4 ρxz

+ (7)

shows The expression for the inconsistency β GIV that its sign is determined by the sign of / , whereas its magnitude is inversely related to the strength of the instrument, cf. Bound, Jaeger, and Baker (1995). VIVN is unaffected by the signs of , and as long as the sign of the product remains the same, or when either or is zero. Self-evidently, VIVN diverges for approaching zero. An important characteristic of the model is the signal-to-noise ratio (SN), which is here equal to SN =

β 2σ x2 = σ ε2

σ x2 σ ε2

, , +

⎛ εi ⎞ ⎜ ⎟ Ω1 / 2vi = ⎜ xi ⎟ = ⎜z ⎟ ⎝ i⎠ ⎛ σε ⎜ ⎜ σεξ ⎜σ ζ ⎝ ε

0 0 ⎞ ⎛ vi,1 ⎞ ⎟ ⎟⎜ α1 0 ⎟ ⎜ vi,2 ⎟ . ⎜ ⎟ α2 α3 ⎟⎠ ⎜ vi,3 ⎟ ⎝ ⎠

(8)

are proFrom (6) and (7) we see that VIVN and β portional to (the square root of) the inverse of SN and that the approximation to the distribution of the IV estimator

(10)

Further, in this simple model we have

ˆ = β IV

zy ∑ = ∑z x i i

i i

and

(9)

β+

∑ z ε , ∑z x i i

(11)

i i

which clarifies that, irrespective of the sample ˆ is invariant to the size, the distribution of β IV scale of zi. We may also change the sign of all ˆ . Therefore, we may the zi without affecting β IV restrict ourselves in the illustrations to cases with > 0 and = + + = 1, from which = |(1 )1/2|. Also, without loss of generality, see the full version of the paper, we can restrict ourselves to 0 and normalize = 1, which leads to SN = . The rest of the parameters of the DGP will be obtained through the relationships ζ = ρzε , ξ = ρxε σ x , = α1 σ x | (1 − ρ2xε )1 / 2 |,

(12)

α2 = (ρxz − ρxε ρzε ) / | (1 − ρ2xε )1 / 2 | . This reparametrization is useful, because the parameters , and and SN have a direct econometric interpretation, viz. the degree of simultaneity, instrument (in)validity, and instrument strength, whereas SN is directly related to the model fit, which can be expressed as SN/(SN + 1). From the above it follows that by varying the four parameters we can examine the limiting and finite sample distributions ˆ over the entire parameter space of this of β IV model. Note, however, that not all admissible

This paper contains a few series of graphs only; more complete and animated pictures are available via the web, see: www.feb.uva.nl/ke/jfk.htm 1

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Econometrics

values of these parameters will be compatible. For example, when is large and is small, this cannot be compatible with being very large. Moreover, has just an effect on the scale ofN , V and β ˆ , so we fix / = 10, β GIV IV IV yielding a population fit of the model of 10/11 = 0.909. ˆ The empirical distribution of β and the asIV ymptotic approximation (9) can now be calculated for any set of compatible values of n, , ˆ to see , and can be compared in orderβ how close they are. Since, when not have finite first or second moments in finite sample, instead of analyzing mean squared error of the estimator, which does not exist then, we analyze its median absolute error (MAE). For ˆ of , MAE( β ˆ) is defined as a scalar estimator β

ˆ − β| ≤ MAE(β)}= ˆ Pr{| β 0.5.

(13)

ˆ of independent Monte Carlo reFrom a series β ˆ) ˆ MAE( β aliz we estimate β by taking the median of the sorted | valˆ) of MAE( β ˆ ues. The asymptotic version AMAE( β ) we asses in the following way. Let the CDF of the normal approximation to the distribution of ˆ be indicated by Φ β,σ ( x). Then, for ≡ β ˆ β ˆ), we have AMAE( β α

ˆ − β| ≤ m}= 0.5 = Pr{| β Φ β,σ (m) − Φ β,σ (−m) ˆ β

so that we solve

ˆ β

from

Φ β,σ (m = ) 0.5 + Φ β,σ (−m). ˆ β

ˆ β

(14)

For several different parameter values we will ˆ examine the empirical density functions firstβ ˆ ˆ and the asymptotic approximation (9). β β Then, we will compare MAE with AMAE ˆ parameter space. ˆ over almost the entire We will β β also compare MAE with . The latter estimator always uses extremely strong instruments that at the same time are invalid in case of simultaneity. The Figures 1 through 4 contain 4 panels each. In every panel, four densities are presented, black lines for = 50 and red lines for = 200, whereas are for the empirical distribution and for its asymptotic approximation. Four panels at each figure have different combinations of and values. The left-hand panels have = 0 (the instrument is valid and the standard asymptotic result applies), and the right-hand panels have = 0.2 (the instrument is invalid and the IV estimator is inconsistent). The two rows of panels cover the cases = 0.3 and = 0.6 (although we do not have symmetry with respect to , for a sake of brevity and because we noticed that

only the magnitude of can matter, we present our results for positive only. In Figure 1 = 0.8, so the instrument is certainly not weak. In Figure 2, 3 and 4, = 0.3, 0.1 and 0.01 respectively. Hence, in the Figure 4 the instrument is certainly weak. In the simulations we used 1,000,000 replications. From Figure 1 we see in the left-hand column that the standard asymptotic approximation of IV when using a valid and strong instrument is quite accurate when the simultaneity is not very serious, but deteriorates when increases, especially when is small. We note some skewness and one fat tail (which can not be captured by the first-order normal asymptotic approximation), but the asymptotic distribu tion is never extremely bad for the cases examined. In the right-hand column we see that the new result of Theorem 2.1 is almost of the same quality but slightly less accurate. In Figure 2, where the instrument is weaker, we find that when the instrument is valid the distribution is more skewed, and more so for serious simultaneity. There is a substantial but not a dramatic difference between the actual distribution and its approximation. The discrepancies are more pronounced in Figure 3, and affect both the standard ( = 0) and the new ( ≠ 0) asymptotic approximations. From Figure 4 it is clear that the asymptotic approximations are useless (at the sample sizes examined) when the instrument is really weak. When the instrument is valid the actual distributions show some median bias, but theyNare much less dispersed than suggested by . From Figures 1 through 4 we conclude that, irrespective of whether the instruments are valid or not, one should avoid to use standard large sample asymptotics when instruments are really weak. If one replaces the weak instrument with a strong one that is invalid (which is always possible by reverting to OLS), one obtains an inconsistent estimator, such as depicted in the right-hand column of Figure 1, which has a distribution that is actually much more concentrated around the true value than that of the consistent estimator depicted in the lefthand column of Figure 4. Figure 5 provides an overview of the (in)accuracy in finite sample of the asymptotic approximation (9) to the actuˆ ˆfor = 100, expressed as al distribution of IVβ β log[MAE /AMAE ]. These figures (based on 10,000 replications) cover all compatible positive values of and , for = 0, 0.1, 0.3 and 0.6. Hence, positive values (yellow, amber) indicate larger absolute errors in finite sample than indicated by the asymptotic approximation and negative values (blue) indicate that standard asymptotics is too pessimistic ˆ in finite samabout the absolute errors of β IV ple. Note that this log-ratio is invariant regarding the value of SN = / . We find that the degree of simultaneity has little effect, and

Econometrics

ˆ β

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ˆ β

January 2008

ˆ β

Econometrics

neither has the (in)validity of the instrument . Just instrument weakness (roughly, when | | < 1 / n ) seriously deteriorates the accuracy of the large-n asymptotic approximation. ˆ )/MAE( β ˆ )], Figure 6 examines log[MAE( β IV IV which is also invariant with respect to SN. It shows that in finite sample the absolute estimation errors committed by OLS are larger than those of IV only when both and are large. The area where IV beats OLS gets smaller for larger . We also note that OLS may beat IV by a much larger margin (when the instrument is weak and the simultaneity not so serious) than IV will ever beat OLS (which happens when the instrument is strong, the simultaneity serious, and the instrument not severely invalid). Conclusions In this study we present an explicit formula for the asymptotic variance of the generalized instrumental variable estimator when some of the employed instruments are invalid. We showed that the limiting distribution of such an inconsistent estimator is normal, and is centered at the pseudo-true-value, whereas its asymptotic variance includes a number of terms and factors additional to the standard result. To obtain our results we assumed covariance stationarity of all variables. In the simple illustrative models which we used, the data observations are in fact IID, as is often assumed in cross-section applications. Note, however, that our theorems also hold for time-series applications. Our study shows that it is possible to obtain an explicit large sample asymptotic approximation to the distribution of IV estimators when some of the exploited moment conditions are invalid. Not surprisingly, however, that approximation is found to be vulnerable when instruments are weak. One option now would be to replace it by an approximation that aims to cope with weakness of instruments. However, our illustrations suggest an alternative approach in which the employment of weak instruments, which invariably yields estimators with flat distributions, is abandoned altogether. We saw that exclusively exploiting strong instruments, even if these constitute invalid instruments, we can produce a reasonably accurate approximation to the finite sample distribution. But, to render this approximation feasible one requires information on the simultaneity parameter and the instrument invalidity parameter . That seems hard to obtain, and if such information was available other estimators than those obtained by minimizing an (in)appropriate GMM criterion function might be better for producing accurate inference on the coefficient .

References Bound, J., Jaeger, D.A. and Baker R.M. (1995). Problems with Instrumental Variables Estimation When the Correlation Between the Instruments and the Endogeneous Explanatory Variable is Weak, Journal of the American Statistical Association, 90(430), 443–450. Hall, A.and Inoue, A. (2003). The large sample behaviour of the generalized method of moments estimator in misspecified models, Journal of Econometrics, 114(2), 361–394. Kiviet, J. and Niemczyk, J. (2007a). The asymptotic and finite sample distributions of OLS and simple IV in simultaneous equations, Journal of Computational Statistics and Data Analysis, 51(7), 3296–3318. Kiviet, J. and Niemczyk, J. (2007b). On the limiting and empirical distribution of IV estimators when some of the instruments are invalid, UvA-Econometrics discussion paper 2006/02.

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Dice Games and Stochastic Dynamic Programming This article considers stochastic optimization problems that are fun and instructive for teaching purposes on the one hand and involve challenging research questions on the other hand. These optimization problems arise from the dice game Pig and the related dice game Hog. Both games are very popular board games in the USA; see reference 1.

The game of Pig The game of Pig involves two players who in turn roll a die. The object of the game is to be the first player to reach 100 points. In each turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds. If the player rolls a 1, the player gets a score zero for that turn and it becomes the opponent’s turn. If the player holds after having rolled a number other than 1, the total number of points rolled in that turn is added to the player’s total score and it becomes the opponent’s turn. At any time during a player’s turn, the player must choose between the two decisions “roll” or “hold”. The game of Hog The game of Hog (fast Pig) is a variation of the game of Pig in which players have only one roll per turn but may roll as many dice as desired. The number of dice a player chooses to roll can vary from turn to turn. The player’s score for a turn is zero if one or more of the dice come up with the face value 1. Otherwise, the sum of the face values showing on the dice is added to the player’s score. In this paper we will concentrate on the practical question of how to compute an optimal control rule for various situations. This will done by using the technique of stochastic dynamic programming. The computations reveal that the optimal control rule has certain structural properties, but the interesting research problem of giving a mathematical proof of the structural properties is left open. The optimality equations of dynamic programming provide not only the tool for numerical computations, but are also the key to the theoretical optimality proofs. An even more exciting open research problem is the game-theoretic variant of the problems above. In this variant the two players have to take simultaneously decisions in each round the game rather than sequentially, where the players cannot observe each other’s actions in any given round of the game but know each other’s

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January 2008

Henk Tijms is professor in Operations Research at the Vrije University in Amsterdam. He studied mathematics at the University of Amsterdam and got his PhD degree in 1972 at the same university. His research interests are in the fields of applied probability and stochastic optimization. He published several textbooks in these fields, including his recent book Understanding Probability. Also, he has a strong interest in the popularization of probability and operations research and developed for this purpose the software package ORSTAT2000 that can be freely downloaded from his homepage.

score after each round. How is the paper organized? First we will analyze for the game of Pig the single-player version for various optimality criteria. Then the twoplayers version of the game of Pig is analyzed. Next, we discuss the game of Hog. The analysis is very similar to that of the game of Pig. Finally, we briefly discuss the game-theoretic variant of the game of Hog. The game of Pig For the single-player version the following two optimality criteria can be considered: • minimal expected number of turns to reach 100 points • maximal probability of reaching 100 points in a given number of turns. The optimal control rules can be calculated from the optimality equations from stochastic dynamic programming, but these optimal rules are rather complex and difficult to use in practice. Therefore we also consider the simple “hold at 20” heuristic and compare the performance of this heuristic with the performance of the optimal rule. The “hold at 20” rule is as follows: after rolling a number other than 1 in the current turn, the player holds that turn when the accumulated number of points during the turn

ORM

is 20 or more. The rationale of this simple heuristic is easily explained. Suppose that points have been accumulated so far in the current turn. If you roll again, the expected number of points you gamble away is 1/6 × , while the expected number of additional points you gain is equal to 5/6 × 4, using the fact the expected value of the outcome of a roll of a die is 4 given that the outcome is not 1. The first value of for which 1/6 × 5/6 × 4 is = 20. It turns out that the “hold at 20” heuristic performs very well when the criterion is to minimize the expected number of turns to reach 100 points. As will be shown below, the expected value of the number of turns to reach 100 point is 12.545 when the “hold at 20” heuristic is used and this lies only 0.7% above the minimal expected value 12.367 that results when an optimal control rule is used. The situation is different for the criterion of maximizing the probability of reaching 100 points when no more than turns are allowed with a given integer. Under the “hold at 20” heuristic the probability of reaching 100 points within turns has the respective values 0.0102, 0.0949, 0.3597, 0.7714, and 0.9429 for = 5, 7, 10, 15, and 20, whereas this probability has the maximum values 0.1038, 0.2198, 0.4654, 0.8322, and 0.9728 when an optimal rule is used.

= the minimal expected value of the number of turns including the current turn to reach 100 points starting from state . We wish to compute together with the optimal decision rule. This can be done from Bellman’s optimality equations. For any and < 100, min[V (i + k , 0), V (i, k ) =

= the player’s score at the start of the current turn = the number of point obtained so far in the current turn. We first consider the criterion of minimizing the expected number of turns to reach 100 points. For this criterion, the value function is defined by

∑ 6 V (i, k + r )],

r =2

where = 0 for those with 100. The first term in the right side of the last equation corresponds to the decision “hold” and the second term corresponds to the decision “roll” (of course in state , the action “roll” is always taken). The optimality equation can be solved by the method of successive substitutions. Starting with = 0 for all , the functions , , . . . are recursively computed from min[Vn −1(i + k , 0), Vn(i, k ) = 6

1 Vn −1(i, 0) + 6 1

∑6V

Dynamic programming for the single-player version In the dynamic programming analysis of the single-player version, a state variable should be defined together with a value function. The state s of the system is defined by a pair = , where

1 V (i, 0) + 6

n −1(i , k

+ r )].

r =2

By a basic result from stochastic dynamic programming, lim Vn(s) = V (s) for all .

n →∞

In the literature bounds are known for the difference , providing a stopping criterion for the method of successive substitutions. It appears from the numerical computations that the structure of the optimal policy is of the control-limit type. An open research question is to give a mathematical proof that the optimal policy is always of this specific structure. Another interesting problem is to prove a kind of turnpike-result, that is, the optimal rule uses the decision from the “hold at 20” heuristic as long as the player is sufficiently far away from

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the desired number of points. Let us next consider the optimality criterion of maximizing the probability of reaching 100 points in no more than N turns with N a given integer. Then, we define for m = 0, 1, . . . , N the value functions Pm(s) by Pm(s) = the maximal probability of reaching 100 points from state s when no more than m turns can be used including the current turn, where Pm(s) = 1 for all s = (i,k) with i + k ≥ 100. The desired probability PN(0, 0) and the optimal decision rule can be calculated from Bellman’s optimality equation. For i = 99, 97, . . . , 0 and k = 100 - i, . . . , 0, min[Pm −1(i + k , 0), Pm(i, k ) = 6

1 Pm −1(i, 0) + 6

∑6P

m (i , k

+ r )].

re i and has k points accumulated so far in the current turn and j indicates that the opponent’s score is j. Define the value function P(s) by P(s) = the probability of the player winning whose turn it is given that the present state is state s, where P(s) is taken to be equal to 1 for those s = ((i,k), j) with i + k ≥ 100 and j < 100. To write down the optimality equations, we use the simple observation that the probability of a player winning after rolling a 1 or holding is one minus the probability that the other player will win beginning with the next turn. For state s = ((i, k), j) with k ≥ 0 and i + k, j < 100, P((i, k ), j ) = min[1 − P(( j, 0), i + k ), 1 [1 − P(( j, 0), i )] + 6 6

∑ P((i, k + r ), j)],

r =2

The value functions P1(s), P2(s), . . . , PN(s) can be recursively calculated, using the fact that Pm(i, k) = 1 if i + k ≥ 100 and starting with ⎧1 if i + k ≥ 100 P0(i, k ) = ⎨ ⎩0 if i + k < 100

The analysis for the “hold at 20” heuristic proceeds along similar lines as the analysis for an optimal rule. In the case of the heuristic there is only one decision possible for each state. Thus, defining the function H(s) as the minimal expected number of turns including the current turn needed to reach 100 points from state s when using the “hold at 20” heuristic, the equations for V(s) are easily modified to get the equations for H(s). The same observation applies to the probability of reaching 100 points within n turns. Another approach for the “hold at 20” heuristic is to use an absorbing Markov chain; see reference 2. Dynamic programming for the two-players case To conclude this section, we consider for the game of Pig the original case of two players. The players alternate in taking turns rolling the die. The first player to reach 100 points is the winner. Since there is an advantage in going first in Pig, it is assumed that a toss of a fair coin decides which player begins in the game of Pig. Then, under optimal play of both players, each player has a probability of 50% of being the ultimate winner. But how to calculate the optimal decision rule? The dynamic programming solution proceeds as follows. The state s is now defined by s = ((i,k), j), where (i,k) indicates that the player whose turn it is has a sco-

r =2

where the first expression in the right side of the last equation corresponds to the decision “hold” and the second expression corresponds to the decision “roll”. Using the method of successive substitution, these optimality equations can be numerically solved, yielding the optimal decision to take in any state s = ((i,k), j). The game of Hog In the game of Hog (Fast Pig) the player has to decide in each turn how many dice to roll simultaneously. A similar heuristic as the “hold at 20” rule manifests itself in the game of Hog (Fast Pig). This heuristic is the “five dice’ that prescribe to roll five dice in each turn. The rationale of this rule is as follows: five dice are the optimal number of dice to roll when the goal is to maximize the expected value of the score in a single turn. The expected value of the total score in a single turn with d dice is (1-(5/6)d)×0+(5/6)d × 4d and this expression is maximal for d = 5. In the single-player version of the game, the number of turns needed to reach 100 points has the expected value 13.623 when the “five dice” rule is used, while the expected value of the number of turns needed under 13.039 when an optimal decision rule is used. Again, a very good performance of the heuristic rule when the criterion is to minimize the expected number of turns. However, the story is different when the criterion is to maximize the probability of reaching 100 points in no more than N turns with N given. The analysis of the game of Hog is very similar to that of the game of Pig both for the single-player version and the the two-players version. Remark: A challenging variant of the game of

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Hog arises when the two players have to take simultaneously a decision in each round of the game. Think of the following television game show. Two contestants each sit behind a panel with a battery of buttons numbered as 1, 2, . . . , 10. In each stage of the game, the two contestants must simultaneously press one of the buttons, where they cannot observe each otherâ&#x20AC;&#x2122;s decision. The number pressed is the number of dice the contestant must throw. The score of the contestantâ&#x20AC;&#x2122;s throw is added to his/her total provided that no 1 was thrown; otherwise no points are added to the current total of the candidate. The candidate who first reaches a total of 100 points is the winner. In case both candidates reach the goal of 100 points in the same move, the winner is the candidate who has the largest total. In the event of a tie, the winner is determined by a toss of a fair coin. At each stage of the game both candidates have full information about his/her own current total and the current total of the opponent. What does the optimal strategy look like? The computation and the structure of an optimal strategy is far more complicated than in the problems discussed before. The optimal rules for the decision problems considered before were deterministic, but for the problem of the television game show the optimal strategy will involve randomized actions. This problem has still many open questions; see reference 3. Literature [1] Neller, T.W. and Presser, C.G.M. (2004). Optimal play of the dice game Pig, The UMAP Journal, 25,25-47 (see also the material on the website http://cs.gettysburg.edu/projects/pig/). [2] Tijms, H.C. (2007). Understanding Probability, Chance Rules in Everyday Life, 2nd edition, Cambridge University Press, Cambridge. [3] Tijms, H.C. and Van der Wal, J. (2006). A realworld stochastic twoperson game, Probability in the Engineering and Informational Sciences, 25, 1-12.

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Geloof het of niet, maar bij ons begint je dag met het lezen van de krant. De kleinste gebeurtenis aan de andere kant van de wereld kan de grootste gevolgen hebben op jouw werk. Je moet op alles voorbereid zijn. Daarom begint je dag met het lezen van de krant. IMC is gespecialiseerd in aandelen- en derivatenhandel voor eigen rekening. Naast Trading zijn we toonaangevend in Brokerage, Consultancy in derivaten en Asset Management. De ruim 400 medewerkers zijn verdeeld over het hoofdkantoor in het Financiële centrum van Amsterdam - Zuid WTC - en vestigingen in Chicago, Sydney, Zug en Hong Kong. We bestaan sinds 1989 en zijn een niet-hiërarchische, dynamische en jonge organisatie, waar innovatie en ondernemerschap voorop staan. Onze cultuur kenmerkt zich door professionaliteit, gedrevenheid en teamspirit. Vanwege de groei zijn we voortdurend op zoek naar:

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Trading globally IMC (International Marketmakers Combination), Strawinskylaan 377, 1077 XX Amsterdam, www.imc.nl

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A Bayesian Approach to Medical Reasoning Medical reasoning is a complex form of human problem-solving that requires the physician to take appropriate action in a world that is characterized by uncertainty. Dynamic Bayesian networks are put forward as a framework that allows the representation and solution of medical decision problems, and their use is exemplified by a case study in oncology.

“You will have to undergo chemotherapy”. If your physician delivers you this message then this will come to you as a shock, but having faith in the physician’s expertise you will comply and undergo this dangerous intervention. However, there are many factors that influence the desirability of the intervention such as patient history, presence of symptoms, availability of alternatives, and the uncertainty about all these factors. How then does our physician come up with such a decision? In this paper, I will make the case for dynamic Bayesian networks as a formalism for the representation of medical knowledge and the execution of medical tasks such as diagnosis, prognosis, monitoring, and treatment of patients. Note that this formalism is not meant to be descriptive (i.e., describing how the physician thinks) but rather normative in the sense that it formalizes how the physician should act in the light of available evidence. In the following, I will describe (dynamic) Bayesian networks in a medical context and demonstrate their use by means of a case study in oncology. I will end this paper with some thoughts about medical reasoning. (Dynamic) Bayesian networks A Bayesian network B = (G,P) is a pair where G is an acyclic directed graph, with nodes corresponding to a set of random variables X, and P is a joint probability distribution (JPD) of variables in X (Pearl, 1988). For the sake of simplicity, we will assume that random variables are discrete, such that P(· | ·) can be specified by a finite look-up table. The JPD factorizes as: P (X) =

∏ P (X | π

( X ))

X ∈X

where πG(X) denotes the parents of X in G. For the sake of simplicity, we will assume that random variables are discrete, such that the con-

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Marcel van Gerven studied cognitive science at the Radboud University Nijmegen and has recently obtained his PhD degree at the computer science department of the same university. He conducted research at the Artificial Intelligence Department at the UNED, Madrid and worked in collaboration with researchers at the Netherlands Cancer Institute on the use of dynamic Bayesian networks in the domain of clinical oncology. Currently, he is working as a postdoctoral researcher on the topic of braincomputer interfacing in collaboration with researchers at the F. C. Donders Institute for Cognitive Neuroimaging.

ditional probability distributions can be specified by a finite look-up table. The factorized JPD embodies our domain knowledge. For example, the medical knowledge that some disorder D leads to a symptom F can be represented by a simple structure D → F and associated probability distributions P(F|D) and P(D). Often, the arcs can be given a causal interpretation, as in this example, but this is not required. For instance, the structure F → D has no such causal interpretation, but its associated distributions P(D|F) and P(F) can model exactly the same JPD. The graph and associated distributions can be learnt from data or constructed by hand. This means that we may use patient data to learn a Bayesian network or interview expert physicians in order to obtain the required knowledge. In our research, we have taken the latter approach, which amounts to the assignment of subjective (Bayesian) priors to the distributions of interest. The factorized representation of a JPD generally reduces the number of parameters that need to be estimated and allows for efficient probabilistic inference. For example, we may compute the probability of a disorder D given an observed finding F, which is a simple example of diagnostic reasoning. Note that this form of diagnostic reasoning goes against the direction of the arcs and is directly interpretable as the

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application of Bayes’ rule. An example of a system that is based on these principles is Promedas (Kappen and Neijt, 2002), which covers a large diagnostic repertoire of internal medicine by relating risk factors Ri, disorders Dj and findings Fk (Fig. 1). In similar ways, we could define network structures for monitoring, prognosis and treatment.

Figure 1: Promedas models associations between risk factors Ri, disorders Dj and findings Fk.

Although systems such as Promedas perform well in clinical tasks such as differential diagnosis, they often make unrealistic assumptions such as the independence between findings given the disease. This affects both the accuracy of computed posterior probability distributions and the ability to understand how domain variables interact. In practice, one often needs detailed information about the causal mechanisms that are responsible for observed findings. The use of causality as a guiding principle when building a Bayesian network for clinical decision support is advantageous, since knowledge concerning pathophysiology and the effect of treatment is normally described in the medical literature in terms of causes and effects. Causal models also facilitate the explanation of drawn conclusions, which may increase the acceptance of automated assistance in medicine, both by the physician and by the patient (Teach, 1984). It is important to realize that the representation of pathophysiology as such is insufficient for guiding treatment, since clinical decision support often requires the suggestion of appropriate action. In other words, automatically obtaining a differential diagnosis is beneficial in the sense that the physician is less likely to misdiagnose, but does not always give insight into the optimal treatment given the differential diagnosis. Hence, it is often necessary to represent the decision-theoretical notions of actions and utility as well. Fortunately, this is easily achieved by representing treatment options as well as the desirability of treatment outcomes as random variables themselves. One last and important factor that is not addressed by standard Bayesian networks is the temporal nature of medical problems. During diagnosis, to know the temporal order and du-

ration of symptoms can influence the diagnostic conclusions, the selection of treatments or tests may depend on the time at which the selection is made, during prognosis, the disease dynamics is described as the unfolding of events over time, and during monitoring, we need to track the patient’s pathophysiological status over time. In case we are dealing with problems of a temporal nature, we explicitly include time within a Bayesian network, by reasoning over random processes = { X(t) : t ∈ T } instead of random variables. We will focus here on discrete-time and discrete-space random processes. The resulting model is known as a dynamic Bayesian network. If it is assumed that the Markov property holds, which states that the future is independent of the past given the present, we obtain the following factorization of the JPD: P (X) =

(

∏ ∏

(

P X ( t ) | πG X ( t )

t ∈T X (t )∈X ( t )

))

with X(t) = { X(t) : X ∈ X }. We will focus on discrete-time and discretespace random processes, which implies that T ⊆ N. If the structure of the dynamic Bayesian network is invariant for all times t ∈ {0,1,2,...} then it can be specified in terms of: - a prior model Bt = (G0,P0) such that

(

)

P0 X ( 0 ) =

∏

X (0 )∈X ( 0 )

(

P X ( 0 ) | πG

( X (0) ) )

specifies the initial distribution of the joint process at time 0, and - a transition model B0 = (Gt,Pt) such that

(

)

Pt X ( t ) | π ( X ( t ) ) =

∏

X ( t )∈X ( t )

(

Pt X ( t ) | πG

( X (t ) ) )

specifies the evolution of the process as it moves from time t - 1 to time t for t ∈{1,2,...}. For dynamic Bayesian networks, posterior probabilities can be computed by means of specialized inference algorithms such as the exact interface algorithm (Murphy, 2002) or approximate particle filtering (Koller and Lerner, 2001). For medical tasks we are often interested in filtering (computing posterior probabilities of unobserved random variables at the current time given evidence) and prediction (computing posterior probabilities of unobserved random variables at some future time given evidence).

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Case study: High-grade carcinoid tumours Here we demonstrate how the desiderata of causality, decision-making, and temporal evolution can be combined in a dynamic Bayesian network that is able to solve problems in clinical oncology. A carcinoid tumour is a type of neuroendocrine tumour that is predominantly found in the midgut and is normally characterized by the production of excessive amounts of biochemically active substances, such as serotonin. These neuroendocrine tumours are often differentiated according to the histological findings and in a small minority of cases tumours are of high-grade histology. For these tumours, aggressive chemotherapy is the only remaining treatment option (Moertel et al., 1991). We have constructed a high-grade carcinoid model in collaboration with an expert physician at the Netherlands Cancer Institute and Fig. 2 depicts the structure of this model. Since patients return to the clinic for follow-up every three months, we assume that each time-slice represents the patient status at three-month intervals, at which time treatment can be adjusted. For a detailed description of all variables, see van Gerven (2007).

Figure 2: A dynamic Bayesian network for highgrade carcinoid tumour pathophysiology with prior model B0 and transition model Bt, where shaded variables are observable and unshaded variables are hidden.

Figure 2 represents causal knowledge such as the fact that the response (resp) of the patient to chemotherapy (chemo) influences the tumour mass (mass). This in turn influences the general health status (ghs), which determines the patientâ&#x20AC;&#x2122;s quality of life (qol). This model can be used to make a prognosis about the patientâ&#x20AC;&#x2122;s future situation, but also to automatically suggest whether or not to administer chemotherapy, as this is captured by the posterior over chemo after patient evidence has been entered. Whether or not to administer chemotherapy is thought to depend on patient health, treatment history (treathist), and past findings with regard to bone-marrow depression (bmdhist). An important task related to treatment is to find an

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optimal policy for chemo that determines which treatment decision to make under all possible conditions. This policy is expressed as the conditional distribution P(chemo | ghs, treathist, bmdhist). In van Gerven (2007), it is shown that the optimal policy can be approximated using simulated annealing, where the impact of local changes in P(chemo | ghs, treathist, bmdhist) on the desired outcome (in terms of quality of life and treatment cost) is computed. Using this algorithm, we have found the same treatment policy that is used by physicians in clinical practice. Some thoughts about medical reasoning In this paper, we have examined the use of dynamic Bayesian networks as a framework for reasoning under uncertainty in medicine. This begs the question of how physicians reason in practice. Do they act according to probability theory when making an inference and do they act according to decision theory when making a decision? In other words, are probability and decision theory just normative, or are they descriptive as well? The literature about the cognitive biases and heuristics displayed in humans in general (Kahneman et al., 1982) and physicians in particular (Borstein and Emier, 2001) suggests no. However, recent research has also demonstrated that some of the biases disappear when questions are posed in a less artificial way (Gigerenzer, 2000). The emerging framework of naturalistic decision-making (Klein et al., 1993) recognizes the importance of these observations, and dictates that we should consider decision-making in a natural setting, where we need to deal with stress, time pressure, fatigue, and communication patterns, as well as with the bounded rationality of humans due to information-processing constraints (Simon, 1955). Under that interpretation, heuristics are not viewed as erroneous, but rather as effective strategies for real-world decisionmaking (Patel et al., 2002). One particularly influential view of problemsolving in medicine is the hypothetical-deductive approach (Elstein et al., 1978), which is an iterative process where hypotheses are generated according to the available data, and hypotheses in turn guide the selection of new data. It is found that expert physicians generate the correct hypothesis early on and use the remaining time to confirm and refine the hypothesis, whereas less experienced physicians take longer to decide upon the final hypothesis due to an inability to eliminate incorrect alternatives (Joseph and Patel, 1990). Another observation is that although expert physicians have more extensive knowledge about pathophysiological processes, they tend to make less use of it than non-experts, and base themselves more

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on clinical experience. One explanation of this effect is the notion of knowledge encapsulation, which suggests that explicit pathophysiological knowledge is represented by the expert in compiled form, while still being retrievable if necessary (Boshuizen and Schmidt, 1992). The picture which emerges is one where expert physicians rapidly recognize the correct hypothesis, while still being able to give a causal explanation of how they arrive at a hypothesis. Our own experiences suggest that expert physicians may indeed operate in this way. During the initial phase of knowledge elicitation the physician often jumped to conclusions, associating findings with expected outcomes, whereas after requiring a causal explanation, it became possible to explain associations in terms of cause-effect relations. From the point of view of knowledge engineering, we emphasize that the translation of a physician’s knowledge into a dynamic Bayesian network is difficult and time-consuming, which implies a trade-off between the amount of time one is willing to spend and the quality of the resulting system. When one believes intervention to be the ultimate goal of clinical reasoning, associative models such as Promedas can perform just as well as causal models provided that they lead to the same actions. However, often, not only intervention but also the explanation of intervention is required. Furthermore, associative models are difficult to extend as new knowledge becomes available. Therefore, if the aim is to create a flexible system that represents domain knowledge to a high degree of detail, then one should consider the incorporation of causality, decision-making, and time into the framework of dynamic Bayesian networks, as is advocated in this paper. References

Gigerenzer, G. (2000). Adaptive Thinking: Rationality in the real world. Oxford University Press, New York, NY. Joseph, G.-M. and Patel, V.L. (1990). Domain knowledge and hypothesis generation in diagnostic reasoning. Med Decis Making, 10, 31-46. Kahneman, D., Slovic, P., and Tversky, A., editors (1982). Judgment under uncertainty: Heurstics and biases. Cambridge University Press, Cambridge, UK. Kappen, H.J. and Neijt, J.P. (2002). Promedas, a probabilistic decision support system for medical diagnosis. Technical report, Stichting Neurale Netwerken, Nijmegen, The Netherlands. Klein, A., Orasanu, J., Calderwood, R. and Zsambok, C.E., editors (1993). Decision Making in Action: Models and Methods. Ablex Publishing Corporation, Norwood, NJ. Koller, D. and Lerner, U. (2001). Sampling in factored dynamic systems. In Doucet, A., de Freitas, N., and Gordon, N., editors, Sequential Monte Carlo Methods in Practice}, chapter 21, pages 445—464. Springer-Verlag, San Francisco, CA. Moertel, C.G., Kvols, L.K., O’Connell, M.J. and Rubin, J. (1991). Treatment of neuroendocrine carcinomas with combined etoposide and cisplatin. Evidence of major therapeutic activity in the anaplastic variants of these neoplasms. Cancer, 68(2), 227-232. Murphy, K.P. (2002). Dynamic Bayesian Networks. PhD thesis, UC Berkeley, Berkeley, CA.

Borstein, B.H. and Emier, A.C. (2001). Rationality in medical decision making: A review of the literature on doctors’ decision-making biases. J Eval Clin Pract, 7, 97-107.

Patel, V.L., Kaufman, D.R. and Arocha, J.F. (2002). Emerging paradigms of cognition in medical decision-making. J Biomed Informat, 35, 52-75.

Boshuizen, H.P.A. and Schmidt, H.G. (1992). On the role of biomedical knowledge in clinical reasoning by experts, intermediates and novices. Cognit Sci, 16, 153-184.

Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, CA, 2nd edition.

Elstein, A.S., Shulman, L.S., and Sprafka, S.A. (1978). Medical Problem Solving: An Analysis of Clinical Reasoning. Harvard University Press, Cambridge, MA.

Simon, H.A. (1955). A behavioral model of rational choice. Q J Econ, 69(1), 99-118.

Gerven, M. A. J. van (2007). Bayesian Networks for Clinical Decision Support. PhD thesis, Radboud University Nijmegen, Nijmegen, the Netherlands.

Teach, R.L. and Shortliffe, E.H. (1984). An analysis of physicians’ attitudes. In Buchanan, B. G. and Shortliffe, E. H., editors, Rule-Based Expert Systems: The MYCIN Experiments of the Stanford Heuristic Programming Project. Addison-Wesley, Reading, Mass.

Economics

Micro-foundations are useful, up to a point Economics consists of the trinity rationality, equilibrium and individualism. When looking for a characterization of the latter, the movie Monty Pyton’s Life of Brian is not a bad place to start. The main character Brian tries to get rid of a mass of people following him, believing he is the chosen one. When they have followed him up to his house, he shouts in utter despair: ‘You don’t need to follow me. You don’t need to follow anybody! You’ve got to think for yourselves. You’re all individuals!’ The followers echo in chore: ‘Yes, we’re all individuals!’. Brian: ‘You’re all different!’. ‘Yes, we are all different!’ reply the followers again.

Economic analysis rests on the assumption of methodological individualism, which is the scientific name for Brian’s exclamation that we are all individuals, who not only should think for ourselves but indeed do. A more formal definition is given by Blaug: ‘

’ [as cited in Earl, 2005]. So, no matter what the macrophenomena under study might be -be it unemployment, inflation, or non-market interactions as state-formation or war- economic analysis tries to explain it as the (equilibrium)outcome of the games rational individuals play. This is a diligent task, as lots of assumptions can be relaxed. Agents may be bounded rational, information is never perfect and out-of-equilibrium dynamics matters for the sustainability of the equilibrium, indeed there may be multiple equilibria. Understanding democratization, implementation of welfare programs or election outcomes is equivalent to providing microfoundations. Many economists feel their model is incomplete without micro-foundations or as Rizvi comments on the period 1970-1985 ‘ ’. These approaches have been very fruitful according to some, imperialistic to others. Be that as it may, methodological individualism as such is seldom seen as a problematic aspect. What is more, it is not really seen as something that has to be argued for, it is simply taken as a given that can go without comment. I believe it can’t. To be sure, it is entirely obvious, a verum factum est, that every macro-phenomenon in the end consists of individual behavior. Elections are indeed the outcome of individuals voting, unemployment is the macro-result of people of-

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David Hollanders gained his degree in History and Econometrics in 2003. Momentarily he is a promovendus at the UvA and is soon finishing the Masterprogram of the Tinbergen-Institute.

fering or hiring labour, and war has to start with some head of state refusing to take it anymore. So what I will definitely not argue is that the macro-phenomena as mentioned do not ultimately and solely consist of individual actions. But I will argue that it cannot be understood as such and that micro-foundations are sometimes even counterproductive. Sometimes individual behavior cannot be understood by contemplating and modeling preferences, rationality or information, unless the social context is taken into account. The strength of micro-economic foundations The prisoner’s dilemma is the textbook example of rational, self-centered persons arriving at an inefficient equilibrium. In this 2x2 game two players have two strategies, to cooperate or not to cooperate. Cooperating is the strictly dominated strategy, hence the (unique) equilibrium is that both players do not cooperate. The resulting equilibrium of both not cooperating is however Pareto inefficient (both cooperating is better for both of them). This can be used to model such varying matters as arms race, advertisement and price competition of firms. Both countries are both better off with no arms expenditure, but no matter what the other does you’d better arm yourself. This might be the end of the matter, if not we do observe cooperation, both in experiments and the real world. The question then is: how come? How to understand the macro-phenomena of social cooperation? Some words about words.

Economics

I call this a macro-phenomenon as cooperation is strictu sensu supra-individual; one cannot cooperate alone. It is totally true that cooperation is the direct result of individual choices, but whether cooperation can be understood as such, is exactly the topic under discussion. One might want to argue that cooperation results from altruism, social capital or simply a proper education. This might all be true, but the point of economic analysis is that social cooperation can perfectly be rationalized if the game is infinitely repeated. The folk theorem states that many equilibria are sustainable. This procedure, simple it may be in the present context, in essence constitutes the core of economics. Some outcome -social cooperation- is not understood as resulting from socialization or something the like, but is endogenously derived from a model, where all individuals are rational (and in this case also self-centered). I hold this to be a good example, as the relation between the assumptions and the outcome is intuitive, the longer the interaction horizon, the more people cooperate. That makes sense. Of course, the model is not claiming to be true or realistic in the immediate sense of the word. It cannot explain why people cooperate in a one-shot game, as people tend to do in experiments (Camerer, 2003). So it is not descriptive in every single case, and is without predictive power deserving the name. But it formalizes a plausible mechanism, which is the name of the game in economics. It does not tell us when cooperation occurs, but provides a way to understand it, when it does. Cooperation in the prisoner’s dilemma might not be the best example of truly a macro- phenomenon. Now look at one that is: segregation in cities. This is as macro as it gets. There are a lot of individuals involved, together determining the outcome, without the ability to influence it. Discrimination, group identity and the reproduction of social classes are some among many factors that might have their role to play. However, this hardly counts as an explanation in economics. For a micro-underpinning one can turn to the work of Schelling (Batten, 2000). The model consists of a chess board, representing a city. There are two classes of agents, in the original model the dividing line is black and white, but any two classes might be thought of. Preferences are a dislike for being separated from the own class, rather than a dislike for the other class. People have the following rule: ‘

’ These preferences are reconcilable with fully integrated cities (for example, free corners and each even diagonal occupied by one class, and each odd

Economics

by the other one). But a few permutations can give way to a pattern of unraveling and highly segregated city (a segregated chess-board that is). Key here is that the overarching occurrence of segregation can be derived from preferences and behavior at the micro-level, and that these are plausible in at least the following sense: people are not totally xenophobic, but do not want to be fully isolated either. This arguably is a satisfactory explanation since large segregation is observed, while many people seem to dislike such segregation. Segregation is explainable as the collective outcome of maximizing individual behavior, where this segregation at the aggregate level is the unintended and perhaps unforeseen consequence of intended action. So far, no appeal has to be made to any social psychological mechanism. So far and, for economics, so good. Still, the model does not explain where the dividing line comes from, and why some dividing lines matter and others don’t. Individuals are said to have ‘ ’. Economists in turn seem to have some difficulty in explaining those difficulties. Here might be a role to play for social psychology and sociology. But let’s first turn to another complication of micro-economic foundations. The difficulty of micro-foundations Often, agents are groups of individuals, not individuals. For example firms or states are treated as if being, acting and thinking as an individual. Not so of course, as is explicitly addressed by agency-literature. CEO’s do not fully internalize the interests of share holders, and representatives are chosen by the voters, which is not the same as acting as voters choose. Two well known results - the Concordet paradox and Arrow’s impossibility theorem- mean further trouble. The Condorcet paradox may arise in pair-wise voting by three (rational) individuals over three alternatives. The three alternatives are and and all three individuals have both complete and transitive preferences. These preferences are described, in the case of the first agent, by: . This means that he prefers , and in turn over (hence, ). The preferences of the second agent are described by and and finally and for the third individual. The voting mechanism consists of pair-wise voting. The result is that the aggregated preferences are not transitive, as is chosen over over and over . So, though the individuals are rational, social preferences are not. Rationality of social preferences is not assured, even if individual preferences are. More general result is Arrow’s impossibility theorem. It first formulates five reasonable properties.

Economics

The theorem states that there is no voting mechanism that is transitive and satisfies the five conditions. One rather pessimistic consequence is that â&#x20AC;&#x2DC; â&#x20AC;&#x2122; (Schotter). Another consequence is, again, that rationality of all individuals involved is no guarantee that social preferences are rational.

"If agents do not think as economists think they should, all the more necessary that economists think as the agents do."

The limitation of micro-foundations

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January 2008

Economics

Conclusion

Economics

is not only difficult, but sometimes stretching methodological individualism to it’s limits, simply does not make sense, as the voter paradox shows. It runs the risk of concentrating on the first two criteria, internal consistency and congruence with the stylized facts (some at least), at the expense of propagating a mechanism that is not consistent with even casual observation of –in this case- voting behavior. Yes, we are all individuals, and yes, at least to my own extra-scientific opinion, we should think for ourselves. But sometimes we simply do not. And if agents do not think as economists think they should, all the more necessary that economists think as the agents do. Literature

since modelers are very well aware of those facts when modeling. It says more, to paraphrase Friedman, about the ingenuity of the model-builders. The second necessary condition for a good model is it’s internal consistency, in other words, the math should work out. Although the difficulty of these two tasks are not to be underestimated, as anyone who has tried it can testify, these are not the final, not even most important criteria to judge the qualities of a model. Whether the model is satisfying is the subjective answer to the question whether the mechanism is convincing, plausible and intuitive. For example, the textbook monopoly model is convincing as the formalization of the notion that a monopolist raises prices above marginal costs. That the mechanism makes sense, does not mean the same can be said of the assumptions. In this case the assumptions are debatable: linear demand, constant marginal costs, common knowledge of demand, profit-maximization as the only goal. So, the third criterion means that not the assumptions as such but the mechanism should be plausible. This criterion is subjective in nature, and science is then the art of persuasion. In economics to understand is to model. It means endogenizing something in a model of (boundedly) rational agents. This is less restrictive as it seems, as lots of assumptions can and indeed are relaxed. Economists do not take every individual to be rational, all information to be perfect or every equilibrium to be unique. This has resulted in many elegant models, with the (repeated) prisoner’s dilemma, Schellings segregation model and the principal-agent literature as examples. As the paradox of Condorcet and Arrow’s impossibility theorem show, countries and firms cannot without problem be treated as individuals. This is to say that providing true micro-foundations is difficult and that models have to be understood under that proviso. It

Batten, D.F. (2000). Discovering Artificial Economics, Westview Press. Bendor, J., Diermeier, D. and Ting, M. (2003). A Behavioral Model of turnout, American Political Science Review, 97, 261-280. Camerer, C.F. (2003). Behavioural studies of strategic thinking in games, TRENDS in cognitive Sciences, 7, 225-231. Camerer, C.F., Loewenstein, G. and Prelec, D. (2004). Neuroeconomics: Why Economics Needs Brains, Scandinavian Journal of Economics, 106, 555-579. Earl, P.E. (2005), Economics and psychology in the twenty-first century, Cambridge Journal of Economics, 29, 909-926. Hendry, D. (1980). Econometrics-Alchemy or Science?, Economica, 47, 387-406. Rizvi, S.A.T., Microeconomics.

Postwar

neoclassical

Schotter, A.F. (1994). Microeconomics, a modern approach.

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Actuarial Sciences

Introduction of the no-claim protector: Research of the consequences for the Dutch carinsurance market During the start of 2007, a new product was introduced in the insurance market: the noclaim protector. The no-claim protector is mainly used in the car-insurance market and would lead to significant cost savings for the policyholders according to the insurers. In most car insurances, the premium depends on the number of claims and, in some cases, the total amount that has been claimed in the preceding years. The premium system is split up in several classes, each having an own reduction percentage on the basic-premium. This system is called a bonus-malus system.

Hein Harlaar is a Master student in Actuarial Sciences at the University of Amsterdam. He started his study in 2004. In 2006 Hein joined Towers Perrin where he works at the Employee Benefits Services department. This article is a summary of his bachelor thesis.

People who do not submit any claims for several years are rewarded with an increasing reduction in their premiums. However, when a policyholder submits a claim, these reductions strongly decrease for the upcoming years. The no-claim protector allows the insured to submit one claim every year without losing their ‘damage-free status’. Therefore it could be seen as an additional insurance to prevent for premium increases when a claim is submitted. This article summarises a research of the efficiency of a bonus-malus system including the no-claim protector. The efficiency is a criterion which describes in which extent the paid premiums match the expected claims. The goal of this research is to describe for which drivers the no-claim protector is a profitable investment. A comparison between a system with and without the no-claim protector is made. Furthermore, populations of policyholders with a (uniform) fixed claim frequency and populations with a stochastic claim-frequency will be taken into account. All scenarios are implemented in a model which simulates the development of the distribution of the policyholders over the different premium classes. The considered bonusmalus system originates from the ANWB which

is one of the leading car insurance providers in the Netherlands. Methodology To analyse the efficiency and expected premiums in a bonus-malus system, the system is defined as a Markov chain. A Markov chain is a stochastic process of which the next state only depends on the present state and is independent from previous states. The range of n premium classes is described →

by the vector C = (C1,....Cn ). According to the property of memorylessness the transition probabilities from state Ci to Cj are independent of the way the insured ended up in state Ci. The transitions within a bonus-malus system are described by the transition matrix Tk = tij(k). The entries of Tk are equal to one when the insured transfers from Ci to Cj by submitting k claims. The probability of submitting k claims is , with as the annual claim frequency. A distribution which is commonly used to model these probabilities is the Poisson distribution. (Lemaire, 1989)

pk ( λ) = e− λ

λk k!

(1)

The matrix provides the probability to transfer from class i to j within a period (pij). The elements of this matrix are determined as follows

Actuarial Sciences

∞

= pij ( λ)

∑

= pk ( λ) * tij (k ), i, j 1,2,...., n

(2)

k =0

If the distribution of the insured over the premium classes in the initial state is denoted as →

the vector l (0) with fj the fraction of the population which are in premium class j. →

l (0) = (f1, f2 ,...., fn ) met f j ≥ 0,

∑f

(3) = 1

j =1

Then the distribution of the insured on a random time point is determined by the equation below →

→

l (n) = l (n − 1)P(λ ) =

l (n − 2)P(λ )P(λ )

(4)

→

lim l (t + 1) = lim l (t ) * P(λ )

t →∞

(5)

t →∞

The steady state premium is the expected premium income of the insurer in the steady state. →

The vector k is defined as the vector with the reduction on the base premium for each of the n premium classes. The steady state premium b corresponding with claim frequency follows →

by multiplying the basic-premium q with k and →

l (∞ ) .

b( λ)

→

(140%,110%,.....,20%) l (∞) * q

(6)

λ db( λ) d log b( λ) = b( λ) d( λ) d log λ

(7)

Ideally the elasticity is equal to one, which means that an increment of x% leads to a similar increment in the expected premiums. In this case the insured pays an annual premium which corresponds with the expected claim amounts. Assumptions Determining the steady state premium and efficiency requires some assumptions about several parameters. Each claim amount is assumed to be equal. In practice, the car-insurance pre-

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January 2008

(8)

Finally a population with a stochastic claim frequency is considered, by taking a Gamma distributed random variable with parameters and . The Gamma distribution is commonly used to model claim frequencies (Lemaire, 1989). The assumed values of the parameters and are respectively 1.6313 and 16.1384 and are based on a maximum likelihood estimation of the claims in a Belgian insurance portfolio (Kaas et.al, 2001). Mathematically the derivation of the probability of submitting k claims goes as follows.

P(K = k |= Λ λ= ) e− λ

λk k!

(9)

For k = 0 this leads to

P(K = 0) =

The elasticity of the steady state premium with respect to the claim frequency is called the efficiency. = e( λ)

= pk P= (K k | Good )P(Good ) + P(K = k | Bad)P(Bad )

A Markov chain ultimately converges to a steady state which is independent of the distribution of the policyholders in the initial state. (Tijms, 2003) When the system is in the steady state the population remains (approximately) equally distributed over all premium classes. →

miums depend on several factors. The current value of the car, the year of construction and the residence of the insured all have some influence on this amount. Because it’s difficult to model these factors, a uniform basic-premium is assumed. This premium is estimated by the average gross received premium as determined by ‘Centrum voor Verzekeringsstatistiek’ (CVS). As mentioned before, three different models for the claim frequencies are considered. The first is a fixed claim-frequency which is equal for all insured’s. Furthermore, we calculate with a population which distinguishes between good and bad drivers. The claim probability in this case leads to the equation below. The claim frequency will be a weighted average of the and .

∫ ∫

∞

0 ∞ 0

P(K = 0|= Λ λ)f Λ( λ)dΛ = e− λf Λ(0)dΛ = E(e − λ )

(10)

The expression above is equal to the momentgenerating function (mgf) of in t = -1 . For a Gamma distributed random variable this function has a simple expression β

= M Λ(t ) ( β − t )α , = M Λ(−1) ( β +1)α

(10)

It appears to be possible to express the claimprobabilities for all positive, integer values of k in the known parameters using the mgf. For all three scenarios mentioned above the steady state premium and the corresponding efficiency are calculated in bm-systems with and without the no-claim protector.

Amrit Jadnanansing, actuaris bij AEGON

‘Ik wil een bijdrage leveren om AEGON tot dé verzekeraar te maken!’ www.aegon.nl

Werken bij AEGON Ik koos bewust voor AEGON. Vanwege de vrijheid, het ondernemerschap en de eigen ruimte. Je kunt hier echt zelf invulling geven aan je werk. Er wordt veel geïnvesteerd in mogelijkheden om ‘up to date’ te blijven en voorop te lopen in kennisontwikkeling.

AEGON krijgt hiervoor gemotiveerde medewerkers terug, die kansen zien en aangrijpen. Er wordt veel en op een leuke manier samengewerkt. Ik heb het hier prima naar mijn zin.

Actuarial Sciences

Results Since the results of the three models did not differ significantly, the results of only one of the models are presented below.

Figure 1: The steady state premium

Figure 1 shows the development of the steady state premium for expected claim frequencies between 0 and 0.5. It considers the case of a population with good and bad drivers. The pink line displays the premium in the situation without a no-claim protector. The two lines intersect at a of 0.155. This means that for claim frequencies above this value the expected premium income in the steady state is smaller when the no-claim protector is applied.

the insured’s expected claim amounts. In this case there is no fair price, the good drivers subsidize the bad ones. In practice the average annual claim frequency lies between the 0.05 and 0.2 (Kaas et.al, 2001). Within this domain the efficiency shows a strong upward trend when the no-claim protector is not applied. However, when the no-claim protector is applied, the efficiency remains close to zero. This low efficiency is caused by the fact that in the steady state, over 90% of the drivers ends up in the highest premium class so almost all insured’s pays the same premium. This is not a favorable situation, as well for the insurer as for the policyholders. The question is, to what extent the good drivers want to switch to another insurer in this case. Empirical research showed that annual about 12.5% of the insured switches to another insurer (Barone & Bella, 2005). An unfair price and disputes between insured and insurer appear to be the main motives to consider a switch. However for insured’s with a no-claim protector the cost of switching could be relatively high. This is because the new insurer bases its initial premium on the true amount of submitted claims. Summarizing, the introduction of the no-claim protector has great disadvantages for the efficiency in a bonus-malus system because almost all insured’s will end up in the same premium class. Only for drivers who submit significantly more claims than average this is a profitable situation. In spite of this, the increased switching-cost will prevent the insurer for losing all the good risks. References Anton, C., Camarero, C. and Carrero, M. (2005). Analysing firms’ failures as determinants of consumer switching intentions. University of Valladolid, Spain. ANWB (2007). March 2007.

Figure 2: Efficiency of the bonus-malus systems

In figure 2 the efficiency of both systems is compared. The efficiency is an important measure for the fairness of bm-systems. The graph shows that the system without no-claim protector (pink line) is much more efficient for claim frequencies below 0.35 and reaches the ideal efficiency 1 for a of 0.18. Conclusions When the efficiency of a bonus-malussystem is low, the paid premiums do not correspond with

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January 2008

Homepage

(www.anwb.nl),

Barone, G. and Bella M. (2004). Price-elasticity based customersegmentation in the Italian auto insurance market. Journal of Targeting, Measurement and Analysis for Marketing, 131, 21–31. Centrum voor verzekeringsstatistiek (2006). Statistieken schadeverzekeringen. (http://www.verzekeraars.org/smartsite. dws?id=43&mainpage=3145) Kaas, R., Goovaarts, M., Dheane, J. and Denuit, M. (2001). Modern actuarial risk theory. Kluwer adademic publishers. Lemaire J. (1989). Auto-Mobile Insurance: Actuarial Models. Dordrecht: Kluwer.

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Conflicting interests in supply chain optimization An elementary aspect of production planning is the relation between setup costs for starting a production run and the costs for holding inventory. The problem is to find the right balance between these costs. One can link two of these problems in a supply chain where one producer supplies an intermediate product for the next producer. In such a setting you also need a balance between the supplier’s and customer’s costs. We can still handle this multi-level production problem if we are allowed to see it as one optimization problem. But what if the supplier and customer have different goals in mind? In the following we investigate how to handle this situation of conflict.

Reinder Lok studied Econometrics at the University of Groningen. From February 2003 to January 2007 he did research at the university of Maastricht. Last June he successfully defended his PhD thesis ‘Auction Mechanisms in Supply Chain Optimization’. Currently he is employed by Statistics Netherlands in Heerlen. The author can be contacted via e-mail: rbl.um@keldor.nl

The problem Let us focus on the setup costs and inventory holding costs. According to setup costs, one prefers to minimize the number of production runs. However, this might result in high inventory holding costs if the time between producing and supplying the product gets long. The problem is therefore to find the right balance between setup and holding costs. This so-called lot-sizing problem is already computationally hard in discrete-time settings where capacity restrictions hold. The problem gets even more complicated in supply chain settings. In that setting we not only need to find the right balance between setup costs and inventory holding costs, but also between the costs at the different production levels. Many researchers considered the multi-level lot-sizing problem and the complexity of finding the optimal (lowest cost) solution. However, this approach assumes that two crucial conditions hold. - One, there is one authority that is responsible for the production planning. - Two, the authority has all information that is needed for finding the optimal solution. In our competitive and specialized world, these

conditions are easily violated. In the following we discuss the way how we can handle these situations. We start with the problem of conflicting interests and the need to handle this problem. Then we explain the principles from the field of mechanism design that should be applied. However, we will see that a straightforward application of mechanisms is too expensive. Therefore, we present the combinatorial auction as a practical implementation where the planning problem and the procurement problem are integrated. Conflicting interests In operational research we often think in terms of finding the optimal solution. But in general, there might be no agreement on what is optimal. On the contrary, in economic terms we are all competing for the same scarce goods. So instead of working together, we might be fighting to reach our own goals. These conflicting interests play a role in the production setting in supply chains. There we want to implement the cheapest solution for the whole chain, but we also know that the links within the chain are aiming for there own profits. Unfortunately, most companies wrongly assume that behaving in their own interest is also in the interest of the supply chain, as Narayanan and Raman (2004) pointed out: “Every firm behaves in ways that maximize its own interests, but companies assume, wrongly, that when they do so, they also maximize the supply chain’s interests. In this mistaken view, the quest for individual benefit leads to collective good, as Adam Smith argued about markets more than two centuries ago. Supply chains are expected to work efficiently without interferen-

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ce, as if guided by Smith’s invisible hand. But our research over the last ten years shows that executives have assumed too much. We found, in more than 50 supply chains we studied, that companies often didn’t act in ways that maximized the network’s profits; consequently, the supply chains performed poorly.” It is the problem of selfish behaviour of supply chain partners that we discuss in the following. Mechanism design There are ways to overcome the incentive problem in supply chain planning. The idea is to reshape the circumstances in such a way that the selfish behaviour of links is indeed the best for the chain. These ways are studied in the field of mechanism design. Mechanism design

of the pie. (..) Our research, however, shows that a company can increase the size of the pie itself by aligning partners’ incentives. (..) If the companies work together to efficiently deliver goods and services to consumers, they will all win. If they don’t, they will all lose to another supply chain.” Revelation principle A mechanism is a game that is designed in such a way that its equilibrium yields the outcome wanted by the mechanism designer. Usually in mechanism design it is assumed that the only action of the participants is to reveal their preferences to a neutral officer. The idea behind this assumption is that the neutral officer will play the game for all players according to their preferences and the rules of the game. This

"Solutions from the field of mechanism design are not suitable for the situation in which there is only one supplier and one buyer." is more or less a generalization of the field of game theory. Where game theory studies the behaviour of players in a certain situation, mechanism design handles the matter how to shape the situation such that the rational behaviour of players yields a desired outcome. This field of research is an important part of contemporary economic research. In December 2007 even the Nobel price in economics was awarded to researchers that initiated, refined and applied the theory of mechanism design on the allocation of goods. We apply the theories of mechanism design to production planning in supply chains. The goal is to shape the business rules in a supply chain such that individual interests and supply chain’s interest are reconciled. Assume that companies are profit maximizers. Then, companies are willing to change their behaviour if that increases their individual profit. Shortly, good behaviour should be rewarded, inconvenient behaviour should be fined. However, business is voluntarily. There is no government that is willing to act as a police officer in production planning. Businesses should shape the situation themselves. Rewards and fines should be a natural consequence of the way they interact. Narayanan and Raman (2004) showed that this is really possible: “In recent years, many companies have assumed that supply costs are more or less fixed and have fought with suppliers for a bigger share

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January 2008

idea is called the revelation principle. In our setting of a supply chain, this gives the following steps: - First, the two links of the chain reveal their costs and capacities to the neutral officer. - Second, the multi-level lot-sizing problem is solved by the officer. - Third, production takes place according to the optimal planning found by the officer. - Finally, goods and money are transferred according to the optimal plan and the payment rules of the mechanism. (Note that in fact the neutral officer is just the optimization tool used for finding the optimal planning.) The payment rule The first step is a very crucial one. Only when the participants tell the truth, only then, the following steps will lead to the optimal production planning. The question is whether the participants have an incentive to tell the truth. Actually, this depends on the payment rules of the last step. The payment rules of the so-called VCG-mechanism will do this job. The idea of these payments is that you pay an amount equal to your own alleged benefit from participating, and re-

ORM

ceive an amount equal to the alleged benefit to the whole system resulting from your participation. As a result, your net benefit equals your own true benefit from participation plus the alleged benefit for the others. This is exactly what is maximized if you tell the truth. So, the VCG-mechanism gives the incentive to tell the truth. Let us see what happens in the supply chain setting. Suppose that current supply chain planning results in a cost of 1000 euro for the supplier and 500 euro for the buyer. Furthermore, the optimal planning results in costs of respectively 700 and 600 euro. Now they apply the VCG-mechanism. First consider the payment of the supplier. The first part of the payment is equal to the supplier’s benefit, which is 1000 – 700 = 300 euro, to be paid. The second part is the reduction in overall costs (200), to be received. So the supplier pays 100 euro. As he had a cost reduction of 300 euro, the net benefit for him is 200 euro. The buyer faces an increase in costs, so the fist part of the payment is a receipt of 600 – 500 = 100 euro. The second part of the payment is again the reduction in overall costs, 200 euro. So the buyer receives in total 300 euro. Together with her cost increase of 100 euro, she has a net benefit of 200 euro. Impossibility The example shows that both links in the supply chain have a net benefit equal to the total cost reduction of 200 euro. So, either party claims the whole budget that emerged from using the mechanism at all. As a consequence, there is a budget deficit of 200 euro (one player pays 100 euro, but the other receives 300). This is not a coincidence. In fact, the impossibility result of Myerson and Satterthwaite (1983) says that it is impossible to have budget balance in this setting. The problem is that there are only two participants that are both essential for achieving any savings at all. To overcome the impossibility result, we introduce competition in this setting. As before, we assume that there is only one buyer facing a lot-sizing problem. But now we assume that the buyer can get its input from different suppliers competing for the order. Furthermore, the choice between these suppliers will be made by using a procurement auction mechanism using the VCG-payments. This combines the choice of a supplier and the production planning, in order to be able to coordinate the planning at the two levels of the supply chain.

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Combinatorial auction In my thesis (Lok, 2007) we introduced a combinatorial auction for the lot-sizing setting in a supply chain. The combinatorial auction is appropriate for this setting, i.e. it fits the cost structure implied by the planning problems faced by the bidders. The combinatorial auction allows bidders to fully incorporate their cost structure, in particular balancing setup and holding costs. As a consequence, a combinatorial auction using the rules of the VCG mechanism will yield the cost minimizing production plan of the (multi-level) lot-sizing problem. However, even though the combinatorial auction allows bidders to express their costs appropriately in their bids, it might be better for the auctioneer to use less sophisticated auction mechanisms. Fortunately, on average the combinatorial is by far the best solution. So a supply chain that needs to improve its overall planning, should consider the use of the combinatorial auction. This mechanism is pre-eminently the solution for matching the lot-sizing problems at both levels of the chain. Concluding Multi-level lot-sizing becomes a difficult problem when different companies are involved. In these situations the problem exists that each company aims for its own profit, potentially hindering the optimal planning of the supply chain. Solutions from the field of mechanism design are not suitable for the situation in which there is only one supplier and one buyer. The problem is that then both are a kind of monopolist, giving them the power to block improved planning. Introducing multiple suppliers that compete for the order eliminates the problem. Moreover, the combinatorial auction appears to be a format that fits the cost structure of lotsizing. It is on average also the most preferred mechanism for the buyer. Literature Lok, R.B. (2007). Auction Mechanisms in Supply Chain Optimization. Universitaire Pers Maastricht, ISBN 978-90-5278-638-4, digitally available at the University Library of Universiteit Maastricht . Myerson, R.B. and Satterthwaite, M.A. (1983). Efficient mechanisms for bilateral trading. Journal of Economic Theory, 29(2), 265– 281. Narayanan, V.G. and Raman, A. (2004). Aligning incentives in supply chains. Harvard Business Review, November. DOI: 10.1225/R0411F.

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Econometrics

Coalitional and Strategic Bargaining: the Relevance of Structure How relevant is the structure of a bargaining situation to its outcome? Will the order of the moves or the degree to which the actions of the bargaining parties are restricted affect what will happen? If structure turns out to be highly relevant, one may wonder whether our highly stylized game-theoretic bargaining models make sense at all. If structure has little relevance, then coalitional (cooperative) models may be preferred to strategic (non-cooperative) models. The fact of the matter is, however, that at the present moment we do not have an unambiguous answer to this question, theoretically nor empirically. In any case, the answer is likely to depend on the details of the bargaining situation. To illustrate this let us consider two situations that only differ in the properties of the disagreement point.

Adrian de Groot Ruiz is a PhD candidate at the Center for Experimental Economics and Political Decision Making (Amsterdam School of Economics). He holds a Bachelor in Liberal Arts from University College Utrecht and a Master in Econometrics from the University of Amsterdam. This article is based on his master thesis, supervised by Roald Ramer and Arthur Schram.

Consider a legislature that has to revise the budget for an ongoing war and consists of three factions: doves, moderates and hawks. Doves ideally lower the budget of the war by 20 billion, moderates prefer not to change the budget and hawks would like to increase the budget by 40 billion. They can also choose to stop the war and, in fact, if the legislature does not manage to find an agreement they will be forced to stop the war (â&#x20AC;&#x2DC;retreatâ&#x20AC;&#x2122;). None of the factions holds a majority in the legislature and any coalition of two does.

Figure 1. Preferences of three parties facing a revision of a war budget

In the first situation, retreating is very unattractive and all factions prefer any revision to retreating. (See figure 1) In this case, Duncan

Blackâ&#x20AC;&#x2122;s Median Voter Theorem (1948) tells us that the moderates will get their way. The reason is that a revision of 0 is the unique (strong) core element, as it will beat any revision in a direct vote: doves and moderates prefer 0 to all increases in the budget and moderates and hawks prefer 0 to all decreases in the budget. In the second situation, retreating can be attractive. Each faction prefers some revisions to retreating and retreating to other revisions. If the budget is lowered by too large amount parties believe they cannot win the war, and if the budget is increased by a too large amount parties believe it is too expensive. In this case the Median Voter Theorem does not apply. In fact, if hawks and doves prefer retreating to a zero revision and doves and moderates prefer some revisions to retreating, then the core is empty. What about structure? In the first situation, the precise structure of the legislative process does not seem to be very important. Having a unique strong core element is a very strong result and several authors obtained core-equivalence in their non-cooperative model of a Median Voter Setting (Baron 1991, Banks and Duggan 2006). In the second situation, however, one might expect a larger degree of randomness inherent to the situation. McKelvey (1976; 1979) and Schofield (1978) showed that if the core is empty, under general conditions, all outcomes can be supported by some agenda-institution. This suggests that structure completely determines the outcome if the core is empty. Still, as doves and hawks can only coordinate on the status quo, one might still expect the moderates to wield the highest bargaining power. This seems to make some outcomes (say close to

Econometrics

0) more likely than others (say close to 40). In any case, structure can be expected to play an important role and it seems wise to make a non-cooperative model in this case. However, several questions ought to be answered before such an approach can be taken. Can we solve a game that truthfully models all the fine details of bargaining? Well, probably not. Is a stylized model, in the spirit of Baron and Ferejohn (1989), then justified? Do different stylized non-cooperative models yield the same result? Suppose we can justify a stylized model as differences in structure are found not to matter much; would this then not also imply that we can say something after all without considering a non-cooperative model? Furthermore, we need to know whether real human agents respond in the same way to differences in the bargaining procedure as our models predict. In sum, we ought to know how the structure of bargaining affect its outcome, theoretically and empirically. Theoretically, this question relates to the Nash program, which tries to establish the link between cooperative game theory (which does not take structure into account) and non-cooperative game theory (which explicitly models structure). However, to make this link one should define what a coalitional game is exactly a model of. Interpreting a coalitional game as a model for all types of bargaining situations seems problematic, as we know that the bargaining structure does have some impact – in an ultimatum game it does matter which party makes the offer. Some argue instead that cooperative games are a model for ‘unstructured’ bargaining situations. However, truly unstructured bargaining does not exist. Moreover, the noncooperative games to which coalitional solution concepts have been linked to in the Nash program, such as those in Binmore et al. (1986) or Maskin (1999), surely have structure. We argue that a coalitional game should not be interpreted as a model of the bargaining situation as a whole, but rather as a model of one aspect of the bargaining situation, the bargaining problem. This allows us to systematically investigate the impact of structure and reinterpret the Nash program as a theoretical inquiry into the relevance of structure and the common factors different bargaining situations may share. This theoretical Nash program, moreover, should be complemented with an empirical counterpart. We want to know which aspects of structure are key in influencing human behavior and whether they correspond with the aspects that game-theory would single out. Hence, we need a systematic study of the effects of structure on human bargaining behavior. The study here described aimed to be a first initiative and obviously must leave many questions unanswered. We focused on the relevance

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January 2008

of the degree of bargaining structure. In particular, we wanted to know whether a lower degree of structure increases the predictive power of coalitional solution concepts and whether it has an influence on the relative bargaining power of the players. Concretely, after defining a conceptual framework, we looked at two bargaining situations with the same bargaining problem. One situation was highly structured and the other had a low degree of structure. Theoretically, we analyzed the coalitional game corresponding to the bargaining problem and the strategic games corresponding to highly structured bargaining situation. Experimentally, we compared the two bargaining situations in separate treatments. Coalitional and Strategic Bargaining Conceptually, we can distill from a bargaining situation a bargaining problem and a bargaining structure. The bargaining problem is the outcome set, the disagreement point, the set of players, their preferences and the set of winning coalitions. The set of winning coalitions specifies which coalitions of players can decide on the outcome for all players – for a simple majority rule, this set will consist of all majority coalitions. The bargaining structure will consist of the actions each player can do at which time and the consequences of that. A bargaining problem and a bargaining structure are consistent with each other if the decision rules in the bargaining structure are in accordance with the winning coalitions of the bargaining problem. At this point we can identify a coalitional game with the bargaining problem rather than the bargaining situation as a whole. Solution concepts of the coalitional game are then properties of the bargaining problem and a relevant benchmark; whether a particular coalitional solution concept says something about the outcome of the bargaining situation can depend on the structure and is a theoretical and empirical question. We can identify a strategic game with the bargaining situation as a whole. This allows us to game-theoretically compare different bargaining situations that share the same bargaining problem. However, strategic games have a serious limitation: a bargaining situation may have a structure that is too complex to model as a strategic game – either because we cannot define a game tree or because we cannot solve the game. Finally, using the concept of a bargaining problem also allows us to empirically test the impact of structure by comparing different bargaining situations that share the same bargaining problem in laboratory-experiments.

Start je carrière bij Delta Lloyd en geen dag zal hetzelfde zijn. De ontwikkelingen gaan namelijk snel. Na de succesvolle fusie met Nuts/OHRA volgde de joint venture met ABN AMRO Verzekeringen. En Delta Lloyd gaat verder. We zijn een verzekeraar die risico’s niet uit de weg gaat. Daar zoeken we talentvolle nieuwkomers bij die zelf ook veranderingen durven creëren. Dus, wil jij als (aankomend) actuaris/econometrist bij Delta Lloyd aan de slag? Stuur dan een e-mail met CV aan Frederique_Bovy@deltalloyd.nl. 63

Econometrics

Two bargaining situations The two bargaining situations mentioned in the introduction motivate the following bargaining problem. The outcome set X is the union between a line and a disjoint point , which is also the disagreement point. We have three players, a dove, a moderate and a hawk. Each of the three players will have an ideal point on the line, zi, which gives her a pay-off of 1. The ideal point of the dove is – a, the ideal point of the moderate is 0 and the ideal point of the hawk is b. Without loss of generality we can assume b to be at least as large as a. The pay-off of outcomes on the line of each player decreases linearly with the distance to their ideal point. Hence, the pay-off function of player i for outcomes on the line, which we will call ‘agreements’, is ui = 1 - |zi - x|. At the disagreement point all players receive a pay-off of 0. A coalition is winning if and only if it consists of at least two players. This simple problem is interesting, because it has no obvious outcome, but one suspects that the moderate has some advantage above the others. Now we can consider two bargaining situations with this same bargaining outcome. In the first situation, (High structure), the agenda setting procedure is regulated. The bargaining consists of N rounds. In every round one of the players is randomly chosen to make a proposal, which can be any element in X (also ), after which the other two players cast a vote. If at least one player votes ‘YES’, the game ends and the proposal is the outcome. If both players vote ‘NO’ the bargaining continues to the next round. In the final round, if both players vote ‘NO’ the status quo is the outcome. This situation resembles a stylized parliamentary procedure. In the second situation, (Low structure), agenda setting is not regulated and players receive T units of time to reach an agreement. At any time between 0 and T, any player can make a proposal, which remains on the table until the sane player makes a new proposal. (Hence, there can be three proposals on the table, one for each player.) Moreover, at any time can a player accept the proposal of another player, in which case the bargaining ends and the proposal is the outcome. If after time T, no proposal has been accepted, the disagreement point is the outcome. This situation is a more informal kind of decision making, say as it happens in the corridors and back-rooms. Theory The bargaining problem can be modeled as a coalitional game. We look at the core and the uncovered set of the coalitional game. The core is one of the few generally accepted cooperative solution concepts and the uncovered set has recently attracted significant empirical sup-

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port (Bianco et al. 2006). To define these concepts it is useful to define a dominance relation and a covering relation for this game first. We say that an outcome x (majority) dominates an outcome y if two players strictly prefer x to y. We say that an outcome x covers an outcome y if x dominates y and all points that dominate x also dominate y. The core is the set of outcomes that are not dominated by any other point and the uncovered set is the set of outcomes that are not covered by any other point. It turns out that there are three main cases, that depend on the value of a. If a is small (a < 1) and the disagreement point is relatively unattractive, the median preference (0) is the core and the uncovered set. If a is large (a>2) and the disagreement point is relatively very attractive, then the disagreement point is the core and the uncovered set. If a is medium (1<a<2), then the core is empty and the uncovered set consists of the median preference, the disagreement point and the point closest to the median preference where the dove has zero pay-off: {0, , 1-a}. If a = b, then the uncovered set also contains a - 1. These three cases are outlined in figure 2. Hence, without considering the structure, one can would say that the median player has complete bargaining power when a is small and looses bargaining power when a becomes larger than 1.

The High structure situation can be modeled as a strategic game. In particular, this bargaining situation corresponds with a finite-round, closed agenda-setting rule, of the popular BaronFerejohn (1989) model of legislative bargaining. We looked at a refinement of the Subgame Perfect Nash Equilibrium (from now just Nash equilibrium), which can be found by backward induction and which is unique. The results are that when the core exists, the Nash equilibrium outcome will tend to the core element as the number of rounds become large. If the core is empty, then the Nash equilibrium outcome becomes much less consistent, as it will depend on the exact values of and , as well as on the number of rounds and which player is chosen to make the first proposal. In any case, also this model predicts absolute bargaining power

Econometrics

Median f

Median D

Low

High

Difference

Low

High

Difference

a<1

100%

0.14

0.21

-0.07

a>1

83%

0.36

0.46

-0.10

overall

92%

0.24

0.31

-0.07*

for the moderate for small a, and a loss of her bargaining power when a becomes larger than 1. However, the Nash equilibrium outcome will typically not lie in the uncovered set. So, it is striking that when the core is nonempty, both the uncovered set and the Nash Equilibrium outcome are equal to the core, whereas when the core is empty the Nash Equilibrium typically does not lie in the uncovered set. Hence, from a game-theoretic point of view this means that when the core is empty the bargaining structure is important: it can move the outcome away from those outcomes which one could expect without considering structure. Situation B has too little structure to be properly modeled as a strategic game. As agents can make decisions in a continuous time, one cannot define clear decision nodes. Although continuous time games are being explored, the current techniques do not seem to allow the modeling of this bargaining situation. This means that we cannot compare the role of structure between these two situations game-theoretically. Experiment To investigate the role of structure empirically, we ran an experiment with two treatments: a treatment in which subjects play the Low structure bargaining situation and a treatment in which subjects play the High structure bargaining situation with 10 rounds. For the experiment, the outcome line was discre-

tized in units of 0.05. Within each treatments we look at 12 parameter pairs for and â&#x20AC;&#x201C; six with a small (core is the moderateâ&#x20AC;&#x2122;s ideal point) and six with a medium (core empty). For each treatment we ran three sessions, each of which consisted of two matching groups of 6-9 subjects. Hence, we had 12 independent data points, which are few, but sufficient to perform the proper non-parametric tests. Subjects were paid and earned on average (on top of a showup fee) 11.70 euros. Our main empirical result is that structure matters. In the first place, the moderate player does better in Low than in High. In both treatments, the median agreement frequency was 100% (all outcomes were points on the line) for a small and 83% for a medium. However, the median distance between agreements and the moderate ideal point is smaller in Low (0.24) than in High (0.36), a significant difference at a 10% level for a one-tailed test. (See table 1.) In the second place, for medium a the uncovered set predicts better in Low than in High. Whereas the uncovered set attracts 49% of all outcomes in Low, it attracts 32% of all outcomes in High. (By attract we mean how many outcomes fell in the 0.05 neighborhood of points in the uncovered set). This is significant at the 10% in a two-tailed test. (See table 2.) Our explanation for the difference between the Low and High structure treatments is that the Low structure treatment allows for active negotiation, while in the High structure treatment

% of outcomes attracted by the 0.05-neighbourhood of a point/set Point/Set

Low

High

Dif Low-High

36%

23%

13%*

-0.5a

22%

40%

-18%*

a<1

a>1

UCS

17%

10%

5%*

1-a

10%

a-1

17%

11%

Total

49%

32%

17%*

SPNE -0.5a

22% 17%

17%

Econometrics

subjects must wait for their turn (if it comes at all). This can increase the possibilities for the moderate player to exploit her bargaining position in Low. Moreover, it can enhance the predictive power of the uncovered set in Low, as it is based on the idea that people make and consider several proposals.

34.

, 56(1), 23-

Maskin, E. (1999). Nash Equilibrium and Welfare , 66, 23-38.

"Players who have a good bargaining position are better off in a less-structured bargaining situation" Conclusion Does structure matter? Theory could not provide an answer, as a low-structure situation does not allow for a non-cooperative description. Experiments did give us an answer. The degree of structure indeed matters. In particular, our results seem to suggest two conclusions. First, players who have a good bargaining position are better off in a less-structured bargaining situation. Second, solution concepts based on the coalitional game provide more information on the bargaining outcome when the bargaining situation has less structure. References Banks, J. and Duggan J. (2006). A General Bargaining Model of Legislative Policy-making, , (1), 49-85. Baron, D. and Ferejohn, J. (1989). Bargaining in Legislatures, , 83(4), 1181-206. Baron, D. (1991). A Spatial Bargaining Theory of Government Formation in Parliamentary Systems, , 85(1), 137-64. Bianco, W.T., Lynch, M.S., Miller, G.J. and Sened, I. (2006). A Theory Waiting to Be Discovered and Used? A Reanalysis of Canonical Experiments on Majority-Rule Decision Making, , 68 (4), 838-51. Binmore, K., Rubinstein, A., Wolinksy, A. (1986). The Nash Bargaining Solution in Economic Modelling, , 17(2), 176-188. Black, D. (1948). The Median Voter Theorem,

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McKelvey, R.D. (1976). Intransitivities in Multidimensional Voting Models and Some Implications for Agenda Control, , 12(3), 472-82. McKelvey, R.D. (1979). General Conditions for Global Intransitivities in Formal Voting Models, , 47(5), 1085-12. Schofield, N. (1978). Instability of Simple Dynamic Games, , 45(3), 575-94.

ADV

Puzzle

Puzzle As usual, we challenge you to solve these fresh new puzzles for your own mathematical pleasure. But first we present the solutions of the puzzles of the previous Aenorm. A popular sport among econometricians All of the holes are of a length which is dividible by 25, so it is obvious that the length of the strokes should be a multiple of 25. By some simple reasoning it’s best to use a driver of 150 yards and an approach of 125 yards to complete the nine hole course. By doing so, you can run the course in twenty-six shots. Uncle Sam’s Fob Chain To solve this puzzle it’s important to notice that the coins and the eagle can displayed on two sides, which makes two different combinations. This way the first coin can be displayed in 10 ways, the 2nd in 8 ways, the third in 6 ways and the 4th in 4 ways. This gives 10*8*6*4 = 3,840 combinations. By changing the order of the coins, twenty-four (4!) different strings of coins are to be made. So the correct answer to the puzzle is 3,840 times 24 = 92,160.

Yacht Race Below a race is sketched between two Yacht’s on a triangular course from buoy A to B to C, then back to A again. This puzzle is about three people on the winning yacht who tried to record the speed of this boat, but became seasick before reaching the finish. Therefore, the first one was only able to observe that the yacht sailed the first three-quarters of the race in four and a half hour. The second one noted only that it did the final three-quarters in four and a half hour. The third one observed that the middle of the race (B to C) took ten minutes longer than the first leg. Assuming that the buoys mark an equilaterial triangle and that the boat had a constant speed on each leg, can you tell how long it took the yach to finish the race?

Unfortunately, we only recieved a correct submission for the first puzzle. However, since S. Hoving managed to beat Sam Loyd in finding a new correct answer to the first puzzle he still deserves to be the winner of the book token. Congratulations! The new puzzles for this edition are: Domestic Complications This puzzle involves the ordinary affairs of life and seems to be much harder for a mathematician to solve than it is for a housewife. Smith, Jones and Brown were great friends. After Brown’s wife died, his niece kept house for him. Smith was also a widower and lived with his daughter. When Jones got married, he and his wife suggested that they all live together. Each one of them was to contribute $25.00 monthly for household expenses and what remained at the end of the month was to be equally divided. The first month’s expenses were $92.00. When the remainder was distributed each recieved an even number of dollars without fractions. How much money did they receive and why?

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January 2008

Solutions Solutions to the two puzzles above can be submitted upto March 1st. You can hand them in in the VSAE room, room C6.06, mail them to info@vsae.nl or send them to VSAE, for the attention of Aenorm puzzle 58, Roetersstraat 11, 1018 WB Amsterdam, Holland. Among the correct submissions, one book token will be given. Solutions can be submitted both in English as in Dutch.

Facultive

Free University Amsterdam

University of Amsterdam During the period of exams, we are tying together the last ends of our year in the board of the VSAE. A fresh new board has been found to take over in February. We have all the confidence in them, and are sure that they will give a great follow-up to things accomplished in the past. In November we organized the Financial Econometrics (FinEco) project. Several companies in the energy trading branche presented their respective fields of work during several workshops and a presentation. Three days after FinEco, we left with a group of 48 students for Copenhagen. We visited several companies and the University of Copenhagen. Of course we also explored the nightlife of Copenhagen. In December, the Actuarial congress took place. 200 actuarial students and employees came to the NH Barbizon to listen to presentations and a discussion on Pension Risk Management. The agenda for the coming months is relatively empty. The 5th of February the VSAE will organize the Personal Development day, when 24 students will spend a day in training and discovering they strong and weak points. In march the VSAE will celebrate its 45th anniversary. For this occasion we have planned several activities in the week of 10 to 14 March. Especially worth mentioning are the reception on the 10th of March in Cristofori and the gala on Friday the 14th. Agenda 5 February Personal Development Day 7 February General Members Meeting

And thatâ&#x20AC;&#x2122;s the end of 2007. I assume everyone has had a merry Christmas and a pleasant change of year. We certainly have. It is a bit late, but I would like to wish everyone a happy new year. We from the board of Kraket will provide you with a lot of activities to make 2008 a very good year, starting off with a movie night and a karaoke evening. An indoor football tournament will take place in February. Important for both econometrists looking for a job and for students that would like to know more about possibilities in the business is the LED, which will be held in Eindhoven this year. We would like to welcome you on these activities, their exact dates are listed below. Kraket wouldnâ&#x20AC;&#x2122;t be the same if there were no mystery activities, so start guessing where our Kraketweekend and the ActieveLedenDag will take place. Agenda 7 February Karaoke 21 February LED (Eindhoven) 27 February Futsal sponsored by Mercer 2 April ALD 30 May - 1 June Kraketweekend

12 February Monthly free Drink 21 February LED (Eindhoven) 10-14 March 45th anniversary VSAE 7-8 April Econometric Game