Improvement of Political Campaigning by a Computer Simulation Model Based on Game Theory Darko DUKIĆ Josip Juraj Strossmayer University of Osijek, Department of Physics Trg Ljudevita Gaja 6, 31000 Osijek, Croatia Bojan BODRAŽIĆ Promotor Ltd. Osijek Virovitička 10, 31000 Osijek, Croatia ABSTRACT
INTRODUCTION
There are numerous areas where the decision-making process can be significantly improved by using computer simulation. This paper presents a simulation model intended for resolving the issue of optimal advertising strategy in political confrontations. It is a well-known fact that voters' opinions are greatly influenced by the media. With this in mind, in any election campaign it is of great importance to properly assess the effects of particular types of promotion, taking into account the opponents' activities and possible reactions. There is important support for solving such problems, namely models developed in the framework of game theory. The model presented here, theoretically and on a hypothetical example, is based on an assumption that two opposing sides are taking part in an election race. Another assumption is that the percentage of votes gained by one contestant is equivalent to the percentage of votes lost by the other contestant. The values of percentage changes are defined as random variables that follow a normal distribution. In order to assess its parameters as correctly as possible, an adequate database needs to be modeled and constructed. Simulated values are used as a basis for formulating the model. Solving it by means of one of the available program applications leads one to the optimal advertising strategy, which is the result of the first simulation set. By repeating the procedure a number of solutions are generated, which allows strategy distributions to be formed. The total optimal strategy is reached by calculating the means of each of the determined distributions. Since it incorporates a quantitative model and the application of a computer system required for its efficient solution and analysis, the construction described in this paper represents a decision support system.
The media play a very important role in the political life. Their power is primarily manifested during election campaigns. They allow candidates to present their political programs, attitudes, and ideas to the public, and thus attract the voters. On the other hand, the media can damage a politician's image and reputation, influencing directly the elections in this way. Since election races are frequently uncertain to the very end, media promotion is definitely imperative for any political campaign.
Keywords: political campaigning, computer simulation model, game theory, database, optimal advertising strategy, decision support system
From the early stages of modern democracies, which have election processes at its core, politicians have accepted the importance of the media and used them to achieve their goals. One of the basic tasks of their teams is to harmonize the effects of advertising with the available funds, which are as a rule insufficient to carry out all the desired promotional activities. Rivals' strategies need to be taken into account as well, but they cannot be anticipated with certainty. There is, however, support for solving such complex problems, namely models developed within game theory. Although there had been earlier precursors, the true development of game theory started around mid-20th century. Game theory can be defined as a mathematical theory that applies to certain situations in which there are conflicts of interests between two or more individuals or groups (M.M. Helms (eds.) [6]). According to F. Carmichael [3], strategic game is a scenario or situation where for two or more individuals their choice of action or behaviour has an impact on the other (or others). In such circumstances are individuals' decisions mutually conditioned, i.e. interdependent. Participants in strategic games are called players, and the plan of actions they can undertake are called strategies. The goal of each player is to choose a strategy that in a given situation ensures the best result. The basic task of game theory is to develop the criteria for choosing the optimal strategy.
RESEARCH METHODOLOGY The model we have developed here has its origins in game theory. Its solution is based on quantitative optimization methods. Since the model is founded on the assumption that players' strategies cannot be predicted with certainty, it belongs to the group of stochastic models. In the proposed model the percentage changes regarding voters who have changed their affiliation under the influence of a particular form of advertising are simulated. It is assumed here that random variables representing them follow a normal distribution. In conducting the simulation process and solving the model, after its transformation into the linear programming problem, the computer and adequate software play the key role. In this paper, we used the Mathematica software package. Its usage allows simple and fast generation of solutions. To make the solution procedure more accessible to a wider circle of managers, who frequently lack adequate knowledge of computer simulation and quantitative optimization, the hypothetical example quotes the Mathematica notebook with the input and output of the analyzed problem. In this way, we sought to increase the usage value of the model.
MODEL FORMULATION In this paper, we have focused on two-person, zero-sum games (A. Kelly [8], C. Schmidt (ed.) [12], J.N. Webb [15]). As the name implies, there are two players in such a game, whereby one player's gain is equivalent to another's loss. Thus, the percentage of voters attracted by one of the two opponents in an election process corresponds to the percentage of voters lost by the other side, which means that the sum of their net winnings is zero. Let us assume that the first player has m strategies at his disposal, and the second one has n strategies. Let gij be defined as the first player's gain in case he chooses the strategy i, and his opponent the strategy j. In such a case, the second player's gain is - gij. The first player's payoff table has the form shown in Table 1.
PLAYER 1
PLAYER 2 strategy
1
2
...
1 2 . ..
g 11
g 12
g 21 . ..
g 22 . ..
... ...
m
g m1
g m2
...
n g1n g 2n . .. g mn
Table 1. Payoff table for the first player In our model, the payoff table is formed on the basis of the estimated effects of combinations of all advertising strategies on the percentage change in voters that the first player (a political candidate or a party) has gained or lost. Given that in the proposed decision support system the percentage change values g*ij are defined as random variables following a normal distribution, whose parameters are the mean (μ) and standard deviation (σ), the payoff table needs to be set up in the way shown in Table 2. PLAYER 2 strategy PLAYER 1
An optimal strategy can be pure or mixed. In a pure strategy, players always choose the same activity in every match of the game, since this allows them to maximize their own minimal gain, simultaneously minimizing the maximum possible loss. In case of a mixed strategy this choice is not the same; rather, it changes during the game. The reason for this is the players' inability to predict with certainty the strategies their opponents will choose in given circumstances. In such circumstances, a player's optimal strategy is therefore defined as a portion of different activities at his/her disposal.
1 2 .. . m
1
2
μ11 ±σ 11 μ12 ±σ 12 μ 21 ±σ 21 μ 22 ±σ 22
.. .. . . μ m1 ±σ m1 μ m 2 ±σ m 2
...
n
...
μ1n ±σ 1n μ 2n ±σ 2 n
... ...
.. . μ mn ±σ mn
Table 2. Payoff table for the first player in the case when the percentage change values are defined as random variables that follow a normal distribution The parameters of normal distribution have to be estimated as correctly as possible, because this will directly affect the efficiency of the proposed model. In doing the estimates, we need to consider all the relevant data, with special emphasis on past indicators, i.e. those obtained by the analysis of previous confrontations between political opponents. For this reason, the election management team should model and construct an appropriate database. In this way, the possibility of choosing a non-optimal advertising strategy is reduced to a minimum degree. The solution of two-person, zero-sum games that have a saddle point can be determined quite simply by means of minimax criterion, which is based on the assumption that players strive to maximize their minimum gain, and minimize the maximum possible loss at the same time. In such a case, the solution to a problem is a pure strategy. If there is no saddle point, this implies that the solution is given in the form of a mixed strategy. One of the ways to resolve such a game is to transform it into a linear
programming model. Numerous publications provide the procedure for formulating a game theory problem as a linear programming model (R. Bronson, G. Naadimuthu [2], F.S. Hillier, G.J. Lieberman [7], H.A. Taha [13]). If xi (i =1, 2, …, m) is defined as the probability of the first player to choose the strategy i, it is then necessary to solve the following maximin problem in order to determine his optimal strategy: ⎧⎪ ⎛ max ⎨min⎜⎜ xi ⎪⎩ ⎝
m
m
∑ g x ,∑ g * i1 i
i =1
m
* i2
i =1
∑g
xi ,K,
* in
i =1
⎞⎫⎪ xi ⎟⎟⎬ ⎠⎪⎭
m
∑ x =1 i
i =1
A linear programming model is formulated based on computer-simulated values of percentage exchanges, which represent its parameters. H.J. Harrington, K. Tumay [5], M. Pidd [11] and J.R. Thompson [14] have dealt with different aspects of the simulation process in detail, whereas P. Weirich [16] has explored the possibilities of applying computer simulations in game theory. By solving the model set up in this way, we arrive at the optimal strategy, which is the result of the first simulation set. Repeating the simulation process and solving the linear programming model will probably result in different optimal strategies being generated, which in turn allow us to form their distributions. The total optimal strategy can ultimately be determined by calculating the mean of each distribution determined in this way.
xi ≥ 0 , i =1, 2, K,m
Thus, to find the first player's optimal strategy means simply to determine the probability of choosing the strategies that will yield the biggest of all minimum gains. Another element that can be introduced into the described model is the variable v, which represents the value of the game: m
∑
m
g *i1 x i ,
i =1
∑ i =1
m
g *i2 x i ,K ,
∑g i =1
* in
⎞ x i ⎟⎟ ⎠
As the first player's objective is to maximize his minimum gain, for any of the actions it cannot be lower than v: m
∑g i =1
* ij
max z = v m
∑g i =1
The first politician's campaign management team has estimated the effects of advertising if the identical amount is invested into each of the four media, taking into account possible activities of their opponent. The following table is actually the payoff table referring to the first politician. It states the estimated values of the means and standard deviations for all the strategy combinations that might be undertaken by the politicians.
xi ≥ v , j =1, 2, K,n
From the above stated we derive the following linear programming model:
v−
Let it be assumed that two politicians are running in the election race, with four advertising media at their disposal – the press, radio, television and the Internet. It will also be assumed that the percentage of voters attracted by one of them is equivalent to the percentage lost by the other candidate.
* ij
x i ≤ 0 , j =1, 2, K,n m
∑ x =1 i
i =1
xi ≥ 0 , i =1, 2, K,m v - unrestricted
Since there is a possibility of loss, the value of the game v is unrestricted in sign.
POLITICIAN 2
POLITICIAN 1
⎛ v = min⎜⎜ ⎝
AN EXAMPLE OF MODEL CONSTRUCTION AND SOLUTION
strategy
press
radio
press
5±2
0±1
-3±2
1±2
radio
-2±1
2±2
-3±1
4±2
television
6±3
3±2
5±1
-2±1
Internet
-1±2
-3±1
-4±3
-8±3
television Internet
Table 3. First politician's payoff table with the estimated values of the means and standard deviations of percentage changes of voters who changed allegiance
If, for example, both politicians opt for press advertising, the first one will attract 5% of voters who were previously leaning towards the other politician, which
represents the expected value of the percentage change. In that case the estimated standard deviation is 2%. After the first set of simulations, the first politician's payoff table can be formed, which is then used as a basis for setting up the following linear programming model: max z = v v − 4.79129 x1 + 1.91047 x 2 − 6.86226 x 3 − 1.67245 x 4 ≤ 0 v + 1.41107 x1 − 2.38003x 2 − 2.50499 x 3 + 1.93302 x 4 ≤ 0 v + 1.12121x1 + 2.91994 x 2 − 3.45183x 3 + 6.92283x 4 ≤ 0 v − 4.42841x1 − 5.10747 x 2 + 1.44145 x 3 + 6.77104 x 4 ≤ 0 x1 + x 2 + x 3 + x 4 = 1 x1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , x4 ≥ 0 v - unrestricted
The values of percentage changes are generated by means of the Mathematica software package, using the function RandomReal. The values obtained in this way represent the parameters used in the linear programming model to express the decision variables effects within a set of constraints. The model formulated here has also been solved using the above-mentioned application. Within this application, the problem of linear optimization can be solved by using the function Maximize. Figure 1 shows the Mathematica notebook with the input and output of the analyzed problem. In addition to the model solution, it states the percentage change values that are generated from normally distributed random variables. The first set of simulations yielded the x2 = 0.117594, following results: x1 = 0.323079, x3 = 0.559327, x4 = 0 and v = 1.2251. Thus, the first politician's mixed optimal strategy consists of investing 32.3079% of the disposable funds into press advertising, 11.7594% into radio, 55.9327% into television, whereas Internet advertising proved to be cost-ineffective.
Figure 1. Mathematica notebook with the input and output of the analyzed problem
The value of the game for the first politician, determined on the basis of the first simulation set, is 1.2251, which means that in the given circumstances he can expect to take over 1.2251% of his opponent's voters. The procedure was repeated 150 times, which made it possible to form strategy distributions that constituted the optimal strategy. After that, the means of strategy distributions were calculated. In this way, we obtained the total optimal mixed strategy: x1 = 0.097491, x2 = 0.411833, x 3 = 0.490676 and x4 = 0. The results indicate that the first politician should by no means invest into Internet advertising. The average value of the game for the first politician is v = 0.730766. Since the second politician's value of the game has the opposite sign, it can be expected that he will lose 0.730766% of his voters, if the stated model assumptions are satisfied.
CONCLUSIONS By using the decision support system presented in this paper it is possible to improve the management of political campaigns in a case where only two candidates are confronted in an election process. This does not mean that the proposed model is an infallible construction that the decision makers in the realm of political marketing can completely rely on. Its efficiency will largely depend on the data used to form the payoff table. A wrong estimate of the parameters will have wrong decisions as its consequence. These problems can be largely overcome by constructing an adequate database and by simulating the values of percentage changes of voters who changed their allegiance under the influence of the media. Computer simulation of these values, i.e. the definition of percentage changes as random variables is one of the model's specific qualities. It should be noted that some model assumptions reduce the usage value of the decision support system developed here. However, if the political campaign teams take into consideration these limitations, the advantages of the proposed system will come into their own. Because of their importance in constructing, solving and implementing the model, information technologies have received particular attention in our analysis. To make the model more accessible to a wider circle of users, the paper quoted the Mathematica notebook with the input and output of the problem defined on the basis of a hypothetical example. By using this program application, it is quite simple to determine the optimal advertising strategy even in much more complex situations.
REFERENCES [1] Bogle, I.D.L., Žilinskas, J. (eds.): Computer Aided Methods in Optimal Design and Operations (Series on Computers and Operations Research), World Scientific Publishing Company, Singapore, 2006. [2] Bronson, R., Naadimuthu, G.: Theory and Problems of Operations Research (Schaum's Outline Series), Second Edition, McGraw-Hill, New York, 1997. [3] Carmichael, F.: A Guide to Game Theory, Pearson Education Limited, Harlow, 2005. [4] Chavas, J.-P.: Risk Analysis in Theory and Practice, Elsevier Academic Press, San Diego, 2004. [5] Harrington H.J., Tumay K.: Simulation Modeling Methods - To Reduce Risks and Increase Performance, McGraw-Hill, New York, 2000. [6] Helms, M.M.: Encyclopedia of Management, Fifth Edition, Thomson Gale, Detroit, 2006. [7] Hillier, F.S., Lieberman, G.J.: Introduction to Operations Research, Seventh Edition, McGrawHill, Boston, 2001. [8] Kelly, A. (ed.): Decision Making Using Game Theory - An Introduction for Managers, Cambridge University Press, Cambridge, 2003. [9] McCarty, N., Meirowitz, A.: Political Game Theory - An Introduction, Cambridge University Press, Cambridge, 2007. [10] Mora, M., Forgionne, G.A., Gupta, J.N.D. (eds.): Decision Making Support Systems Achievements, Trends and Challenges for the New Decade, Idea Group Publishing, Hershey, 2003. [11] Pidd, M.: Computer Simulation in Management Science, Fourth Edition, John Wiley & Sons Ltd, Chichester, 1998. [12] Schmidt, C. (ed.): Game Theory and Economic Analysis, Routledge, London, 2002. [13] Taha, H.A.: Operations Research - An Introduction, Eighth Edition, Pearson Prentice Hall, Upper Saddle River, 2007. [14] Thompson, J.R.: Simulation - A Modeler's Approach (Wiley Series in Probability and Statistics), John Wiley & Sons, Inc., New York, 2000. [15] Webb, J.N.: Game Theory - Decisions, Interaction and Evolution, Springer, London, 2007. [16] Weirich, P.: “Computer Simulations in Game Theory”, Conference “Models and Simulations”, Paris, June 12-13, 2006., Institut d'Histoire des Sciences et des Techniques/Centre for Philosophy of Natural and Social Science, London School of Economics, http://philsci-archive.pitt.edu/archive/ 00002754/01/Weirich,_Simulations.pdf