Morfismos, Vol 6, No 2, 2002

VOLUMEN 6 NÚMERO 2 JULIO A DICIEMBRE DE 2002 ISSN: 1870-6525

MORFISMOS Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav Editores Responsables • Isidoro Gitler • Jes´ us Gonz´alez

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VOLUMEN 6 NÚMERO 2 JULIO A DICIEMBRE DE 2002 ISSN: 1870-6525

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Morfismos

Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. MORFISMOS appears twice a year.

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Morfismos

Contenido Approximation on arcs and dendrites going to infinity in C n (Extended version) Paul M. Gauthier and E. S. Zeron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Bayesian procedures for pricing contingent claims: Prior information on volatility Francisco Venegas-Mart´ınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Existence of Nash equilibria in discounted nonzero-sum stochastic games with additive structure Heriberto Hern´ andez-Hern´ andez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Existence of Nash equilibria in some Markov games with discounted payoff Carlos Gabriel Pacheco Gonz´ alez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Morfismos, Vol. 6, No. 2, 2002, pp. 1-23 Morfismos, Vol. 6, No. 2, 2002, pp. 1-23

Approximation on arcs and dendrites going to infinity in Cn (Extended version) ∗ Paul M. Gauthier

1

E. S. Zeron

2

Abstract The Stone-Weierstrass approximation theorem is extended to certain unbounded sets in Cn . In particular, on arcs which are of locally finite length and are going to infinity, each continuous function can be approximated by entire functions.

2000 Mathematics Subject Classification: 32E30, 32E25.

Keywords and phrases: Uniform and tangential approximation.

1

Introduction

This work is the original version of the paper: Approximation on arcs and dendrites going to infinity in Cn [14]. This version could not be published in its extended form because of size limitations. However, we wish to publish it because it contains a sketch of the proof of AlexanderStolzenberg’s theorem, which we announced in [14], and several lemmas on tangential approximation by polynomial and meromorphic functions which could not be included on [14]. For example, we include a not-sowell-known result of Arakelian in Proposition 3.1.

A famous theorem of Torsten Carleman [7] asserts that for each continuous function f on the real line R and for each positive continuous function ϵ on R, there exists an entire function g on C such that |f (x) − g(x)| < ϵ(x), for all x ∈ R. ∗ Invited article. In memoriam: Herbert James Alexander 1940-1999. Research supported by NSERC (Canada), FCAR (Qu´ebec) and Cinvestav (M´exico).

1 Centre de rech`erches math´ematiques, Universit´e de Montr´eal, Canada.

2 Departamento de Matem´ aticas del CINVESTAV-IPN, M´exico.

1

2

Paul M. Gauthier and E. S. Zeron

Carleman’s theorem was extended to Cn by Herbert Alexander [3] who replaced the line R by a piecewise smooth arc going to infinity in Cn and by Stephen Scheinberg [20] who replaced the real line R by the real part Rn of Cn = Rn + iRn . In the present work, we approximate on closed subsets of area zero in Cn and extend Alexander’s theorem to closed connected subsets Γ ⊂ Cn which are of locally finite length and contain no closed curves.

It would be a quite difficult task to give a complete description of all the results before the words of Carleman, Alexander and Scheinberg.

Nevertheless, we have added a special section (§4) at the end of this paper, trying to compile the historic results which have drove us to the theorems we are proving in this paper.

Let X be a subset of Cn . X is a continuum if it is a compact connected set. The length and area of X are the Hausdorff 1-measure and 2-measure of X respectively. The set X is said to be of finite length at a point x ∈ X if this point has a neighbourhood in X of finite length, and X is said to be of locally finite length if X is of finite length at each of its points. Notice that if X is a set of locally finite length, then each compact subset of X has finite length (though X itself need not be of ! finite length). We denote the polynomial hull of a compact set X by X. The algebra of continuous functions defined on X is denoted by C(X). ˇ Finally, the definition and some properties of the first Cech cohomology 1 ˇ group with integer coefficients H (X) are presented in [10] and [23].

2

The Alexander-Stolzenberg theorem

John Wermer laid the foundations of approximation on curves in Cn and prepared the way for a fundamental result of Gabriel Stolzenberg [21] concerning hulls and smooth curves (for history see [22]). In [2], Alexander comments that Stolzenberg’s theorem can be improved to consider continua of finite length instead of smooth curves. We shall refer to the following version as the Alexander-Stolzenberg Theorem.

Theorem 2.1 (Alexander-Stolzenberg) Let X and Y be two compact subsets of Cn , with X polynomially convex and Y \ X of zero area.

Then, A Every continuous function on X ∪Y which is uniformly approximable on X by polynomials is uniformly approximable on X ∪ Y by rational functions.

Approximation on arcs going to infinity

3

Suppose, moreover, there exists a continuum Υ ⊂ Cn such that Υ \ X has locally finite length and Y ⊂ (X ∪ Υ). Then:

!

B X ∪ Y \ (X ∪ Y ) is (if non-empty) a pure one-dimensional analytic subset of Cn \ (X ∪ Y ).

ˇ 1 (X) induced by X ⊂ X ∪ Y is injective, ˇ 1 (X ∪ Y ) → H C If the map H then X ∪ Y is polynomially convex.

Notice that in this Alexander-Stolzenberg Theorem, locally finite length is required only for parts B and C. Moreover, the set Υ \ X may be of infinity length; we only need it to have locally finite length. On the other hand, the main ideas in the proof of points A and C are essentially contained in [22, pp. 187-188]). However, we need to introduce several changes due to the new hypotheses.

2.1

Proof of part A of Theorem 2.1

We shall prove part A by considering two cases, depending on whether Y is itself of zero area or not. We only need to cite Stolzenberg’s ideas when Y has area zero (we obviously replace K by Y in the original paper).

“By the theory of antisymmetric sets (see[15]) it suffices to prove that if p ∈ Y \ X then for each q ̸= p in Y ∪ X there is a real-valued f , with f (q) ̸= f (p), which is uniformly approximable by rational functions on Y ∪ X.” “Since X is polynomially convex there is a polynomial g such that g(p) = 1 and ℜ(g) ≤ 0 on X ∪ {q}. Let c be a real-valued continuous function on g(Y ∪ X) which is identically 0 for ℜ(ζ) ≤ 12 and with c(1) = 1. The following argument of Wermer shows that c is a uniform limit of rational functions on g(Y ∪ X).”

“Namely, it suffices to prove that any measure µ on g(Y ∪ X) which annihilates all uniform limits of rational functions also annihilates c.

This will be done if we can show that any such µ is supported on {ℜ(ζ) ≤ 1 2 }. But Y has !area zero and g is a polynomial, so g(Y ) has area zero and, hence, (z − ζ)−1 dµ(z) = 0 for almost all ζ with ℜ(ζ) > 12 .

Therefore, by Fubini’s Theorem, for almost all open disks ∆ ⊂ {ℜ(ζ) > 1 2 }, if ∂=the boundary of ∆ then " " dµ(z) −1 = dζ 0 = 2πi ∂ z−ζ

4

Paul M. Gauthier and E. S. Zeron

=

!

dµ(z) 2πi

!

∂

dζ = ζ −z

!

χ∆ (z)dµ(z),

where χ∆ is the characteristic function of ∆. It follows that µ = 0 on {ℜ(ζ) > 12 }.”

“Hence c is a uniform limit of rational functions on g(Y ∪ X) and, hence, f = c ◦ g is a continuous real-valued function on Y ∪ X, with f (q) ̸= f (p), which is uniform limit of rational functions.”

This settles part A when Y has area zero.

Now suppose we merely know that Y \ X has zero area. Let f be a continuous function on X ∪ Y which is uniformly approximable on X by polynomials and let ϵ > 0. There exists a polynomial p such that |f −p| < ϵ/2 on X. Since X is polynomially convex, it has a fundamental system of neighbourhoods which are polynomial polyhedra [29, Lemma 7.4]. From the continuity of f − p, it follows that |f − p| < ϵ/2 on some " containing X in its interior. Extend p| ! to a polynomial polyhedron X X " so that |f − p˜| < ϵ/2 on X " ∪ Y . It is easy continuous function p˜ on Y \ X " has area zero because K ⊂ Y \ X, to see that the closure K of Y \ X " " with a compact set of and so X ∪ Y can be written as the union of X area zero K; it follows from the first part of this proof that there is a " ∪ Y . By the triangle rational function h such that |˜ p − h| < ϵ/2 on X inequality, |f − h| < ϵ on X ∪ Y which concludes the proof of A.

2.2

Deduction of part C from part B in Theorem 2.1

Here we also need to replace Lemma 1 of [22, p. 188]) by the following proposition.

Proposition 2.2 Let X and Y be two compact subsets of Cn , with X rationally convex and Y \ X of zero area. Then, X ∪ Y is rationally convex. If, moreover, X is polynomially convex, then given a point p in the complement of X ∪ Y , there is a polynomial f such that f (p) = 0, 0 ̸∈ f (X ∪ Y ) and ℜf (z) < −1 for z ∈ X. Proof: The set X has a fundamental system of neighbourhoods which are rational polyhedra [21, p. 283] or [23]. Given a point p in the " which complement of X ∪ Y , choose a compact rational polyhedron X " Along with X, " the closure K contains X in its interior, but p ̸∈ X. " of Y \ X is also rationally convex because it has zero area (notice that K ⊂ Y \ X and see [10, p. 71] recalling that projections preserve the

Approximation on arcs going to infinity

5

zero area condition), so there are two polynomials g and h such that ! and g(p) = h(p) = 0.

0 ̸∈ g(K), 0 ̸∈ h(X) The rational function (h/g) is smooth on K, and so (h/g)(K) has zero area. Thus, we can find a complex number λ ̸∈ (h/g)(K) whose absolute value |λ| is so small that the polynomial f = h − λg has no ! ∪ K. Since X ∪ Y ⊂ X ! ∪ K and f (p) = 0, it follows that zeros on X X ∪ Y is rationally convex. !

If, in addition, X is polynomially convex, one has just to choose X to be a compact polynomial polyhedron (see [21] or [29, Lemma 7.4]) ! and so, for sufficiently and the polynomial h to satisfy ℜ(h) < −1 on X; small λ, ℜ(f ) < −1 on X. Now we can conclude the proof of point C by citing Stolzenberg’s ideas.

“Consider any p ̸∈ Y ∪ X and choose an f as in Proposition 2.2.

Then f is a continuous invertible function on Y ∪ X with a continuous ˇ 1 (T ) is isomorphic to the group of logarithm on X. But, for any T , H all continuous invertible complex-valued functions of T modulo those ˇ 1 (Y ∪ X) → H ˇ 1 (X) is with continuous logarithms. Therefore, since H injective, there is a continuous branch of log(f ) on all of Y ∪X. However, ! by part B, X ∪ Y \ (X ∪ Y ) is (if non-empty) a one-dimensional analytic n subset of C \ (Y ∪ X); so by the argument principle (see, for instance !

[21, p. 271]) f has no zeroes on X ∪ Y \ (X ∪ Y ). Hence, any such p is ! not in Y ∪ X, so Y ∪ X is polynomially convex.”

2.3

Proof of part B of Theorem 2.1

This proof is implicitly contained in Alexander’s paper [2], but we need to make several remarks. " \ Γ. From Set Γ = X ∪ Y and suppose there is a point p ∈ Γ Proposition 2.2, there is a polynomial f such that f (p) = 0, 0 ̸∈ f (Γ) and ℜ(f ) < −1 on X. Fix the compact set L = f (Γ), the closed half-plane H = {ℜ(z) ≥ −1/2} and the open set Ω to be the connected component of C\L which contains the origin. Alexander’s arguments [2] " ∩ f −1 (Ω) is a one-dimensional can be slightly modified to show that Γ analytic subset of f −1 (Ω). Notice that p ∈ f −1 (Ω). Alexander uses the hypothesis that the set L has finite length in the whole plane C. However, his argument works even if we restrict the set L to have finite length just in the half-plane H. Indeed, the intersection L ∩ H is the polynomial image of the compact set Γ ∩ f −1 (H) of finite length; recall

6

Paul M. Gauthier and E. S. Zeron

that Γ ∩ f −1 (H) = (Y \ X) ∩ f −1 (H) has finite length because it is compact and contained in the set Υ \ X of locally finite length. Now we shall rewrite the preparatory lemmas of [2] with their respective modifications.

Lemma 2.3 (Lemma 1 of [2]) Let X be a second-countable topological space, Y a set, f : X → Y a function, σ a non-zero positive measure on Y such that if V is open in X , then f (V ) is σ-measurable. Then for σ-almost all y ∈ Y, the image under f of each neighbourhood in X of each point of f −1 (y) has positive σ-measure.

Lemma 2.4 (Lemma 2 of [2]) Let D be a closed Jordan domain in C with boundary of finite length, K a compact subset of ∂D of positive length, Q a polynomially convex set in Cn , f a polynomial in Cn , s a positive integer. Assume that Q = (f −1 (∂D) ∩ Q)∧ and that f |Q is at most s-to-1 over points of K (i.e., if λ ∈ K then f −1 (λ) ∩ Q has at most s points). Then f −1 (Do ) ∩ Q is a (possibly empty) pure 1-dimensional analytic subset of f −1 (Do ). Here Do stands for the interior of D. The hypotheses of the previous two lemmas need not be changed, so we refer to their original proofs in Alexander’s paper [2, p. 66]. In the following lemmas, the notation #(E) stands for the number (≤ ∞) of elements of the set E. Lemma 2.5 (Lemma 3 of [2]) Let Γ be a compact set in Cn and f has finite length. For x ∈ R, a polynomial in Cn such that Γ ∩ f −1 (H) !∞ set N (x) = #{p ∈ Γ; ℜf (p) = x}. Then −1/2 N (x)dx < ∞.

For the proof that N is a Lebesgue measurable function, see [18, p. 216].

Proof: By replacing Γ by its homeomorphic image in Cn+1 under the mapping z '→ (f (z), z), a Lipschitz mapping preserving the finiteness of length, we may assume that f (z) = z1 , the first coordinate projection.

Let ϵm ↓ 0. Then for each m there exists a finite collection Cm of closed balls in Cn each of diameter less than ϵm such that Cm covers Γ∩f −1 (H) and if αm denotes the sum of the diameters of the members of −1 Cm , then αm ↑ length(Γ∩f ! ∞ (H)). Let Nm (x) = #{B; B ∈ Cm and x ∈ ℜz1 (B)}. Then clearly −∞ Nm (x)dx = αm . Also limNm (x) ≥ N (x) whenever x ≥ −1/2; in fact, if N (x) ≥ k, and p1 , p2 , . . . pk are distinct

Approximation on arcs going to infinity

7

points in Γ∩(ℜz1 )−1 (x), then Nm (x) ≥ k as soon as ϵm < min{∥pi −pj ∥; i ̸= j}. Thus, by Fatou’s lemma !

∞ −1/2

N (x)dx ≤ lim

!

∞ −1/2

Nm (x)dx ≤

≤ limαm = length(Γ ∩ f −1 (H)) < ∞.

Lemma 2.6 (Lemma 4 of [2]) Let I = [0, 1] be the closed unit interval of the real line and F ∈ C(I) be such that ℜF is of bounded variation.

Define for x ∈ R, N (x) = #{t ∈ I; ℜF (t) = x}. Then "∞ N (x)dx < ∞. −1/2

Proof: Let Γ ⊂ C1 be the set {(ℜF (t), t); t ∈ I} and take f (z) = z in Lemma 2.5.

Definition. Let L be a closed subset of C. Let Ω1 and Ω2 be components of C \ L. We shall say that the pair (Ω1 , Ω2 ) is amply adjacent provided the following holds: there exist real numbers b > a > −1/2 and c2 > c1 , and a compact subset K1 ⊂ [a, b] of positive length such that [a, b] × {cj } ⊂ Ωj for j = 1, 2 and K = (K1 × [c1 , c2 ]) ∩ L is a subset of ∂Ω1 ∩ ∂Ω2 such that the projection π1 maps K homeomorphically (and so 1-to-1) onto K1 (we are identifying C and R × R, so π1 (x, y) = x).

Lemma 2.7 (Lemma 5 of [2]) Let L ⊂ C be compact and such that "∞ N (x)dx < ∞ where N (x) = #{q ∈ L; ℜ(q) = x}. Then, for every −1/2 component Ω of C \ L which meets the half-plane H, there exists a finite sequence Ω0 , Ω1 , . . . Ωm of components of C \ L with Ω0 equal to the unbounded component, Ωm = Ω and (Ωj−1 , Ωj ) amply adjacent through rectangles Rj = [a, b] × [cj−1 , cj ] contained in H for j = 1, 2, . . . , m.

The proof of this lemma is exactly the same as the original one presented by Alexander in his paper [2, " a p. 69]; he chooses a line segment [a, b] × c ⊂ Ω and uses the fact that b N (x) < ∞. Thus, we shall have exactly the same result by choosing a horizontal line segment [a, b] × c ⊂ Ω ∩ H and following the original proof word for word.

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Paul M. Gauthier and E. S. Zeron

Lemma 2.8 (Lemma 6 of [2]) Let Γ be a compact !subset of Cn and f ∞ a polynomial in Cn . Set L = f (Γ) ⊂ C. Suppose that −1/2 N (x)dx < ∞ for N (x) = #{p ∈ Γ; ℜf (p) = x}, and that L ∩ H is contained in a continuum L1 whose intersection L1 ∩ H is of finite length. Let (Ω1 , Ω2 ) be a pair of components of C\(L∪L1 ) which are amply adjacent through " ∩ f −1 (Ωi ) a rectangle R = [a, b] × [c1 , c2 ] with b > a > −1/2. Suppose Γ is a (possibly empty) pure 1-dimensional analytic subset of f −1 (Ωi ) for i = 1. Then, the same is true for i = 2.

Again, the proof of this lemma follows word for word the original one presented by Alexander in [2, p. 70], we only need to add the new " ∩ f −1 (Do ) is trivial condition b > a > −1/2. Alexander proves that Γ −1 o a pure 1-dimensional analytic subset of f (D ), where Do is an open " ∩ f −1 (Ω2 ) is also a set contained in R ∩ Ω2 . He deduces then that Γ −1 pure 1-dimensional analytic set in f (Ω2 ) by using Lemma 11 of [22].

This lemma is quite amazing because the component Ω2 may not be completely contained in H. The following lemma needs no changes in its hypotheses, so we refer its proof to the original paper [2, p. 71].

Lemma 2.9 (Lemma 7 of [2]) Let Γ ⊂ S be two compact sets in Cn and suppose that S" \ S is a pure 1-dimensional analytic subset of Cn \ S. " \ S (if non-empty).

Then so is Γ

We conclude the proof of part B of Theorem 2.1 following Alexander’s arguments. If the equality X ∪ Υ = Γ = X ∪ Y holds, we let L = f (Γ) and Ω be the connected component of C \ L which contains the origin 0 = f (p). Apply Lemmas 2.5 and 2.7 to get a sequence Ω0 , Ω1 , . . . Ωm = Ω. Notice that Υ ∩ f −1 (H) = Y ∩ f −1 (H) has finite length because it is compact and contained in the set Υ \ X of locally finite length; so we can take the continuum L1 = f (Υ) in Lemma 2.8 because " −1 (Ω) L1 ∩H = L∩H has finite length. We conclude inductively that Γ∩f is either empty or a pure 1-dimensional analytic subset of f −1 (Ω), for " ∩ f −1 (Ω0 ) = ∅. Hence X ! ∪ Y \ (X ∪ Y ) is analytic L = L1 ∪ L and Γ ! (empty or pure 1-dimensional) at an arbitrary point p ∈ X ∪ Y \(X ∪Y ). " \Γ Now suppose that X ∪ Y is strictly contained in X ∪ Υ. Let p ∈ Γ as above. Modify Υ to obtain Υ0 such that p ̸∈ Υ0 but Υ0 is a continuum with Y ⊂ X ∪ Υ0 and Υ0 \ X of finite length (say by radial projection to the boundary inside a ball containing p in its interior, centered off Υ, and disjoint from Γ). By the previous paragraph, X! ∪ Υ0 \(X ∪Υ0 ) is a pure

Approximation on arcs going to infinity

9

!

1-dimensional analytic subset of Cn \(X ∪Υ0 ), and so is X ∪ Y \(X ∪Υ0 ) ! because of Lemma 2.9. Therefore, the set X ∪ Y \ (X ∪ Y ) is analytic (pure 1-dimensional) at p.

An arc Υ is the homeomorphic image of a closed interval of the real line. A direct consequence of the Alexander-Stolzenberg theorem is that every compact arc Υ which is of locally finite length everywhere except perhaps at finitely many of its points is polynomially convex and the approximation condition C(Υ) = P (Υ) holds; notice that Υ may be of infinity length.

It is natural to ask whether the connectivity can be dropped in these considerations. In fact, Alexander [4] gave an example of a compact disconnected set Y of finite length in C2 for which Y! \Y is not a pure onedimensional analytic subset of C2 \ Y . Thus, the connectivity cannot be dropped in the Alexander-Stolzenberg theorem. Moreover, the following example shows that we cannot finesse Theorem 2.1 by enclosing Y in a continuum of finite length, although it is known that one can always construct a compact arc Υ which meets every component of Y (so Y ∪Υ is connected) and Υ \ Y is of locally finite length.

Example 2.10 There exists a discrete bounded set in C \ {0} such that no continuum containing this sequence has finite length.

2 √ Consider the set E consisting of the complex numbers wj,k = k/j + −1/j, for j = 1, 2, . . . and k = 0, 1, . . . , j. It is easy to see that E is contained in the disjoint union of the closed balls B j,k with respective 1 centers wj,k and radii 2(j+1) 2 . Hence, each continuum which contains E

has to meet the center and boundary of each ball B j,k , so its length " the j+1 has to be greater than j>1 2(j+1) 2 = ∞.

3

Approximation on unbounded sets

Now we shall analyse approximation on closed subsets of Cn rather than on compact sets. Let Γ be a closed subset of Cn and F be a subclass of C(Γ). We say that a function f : Γ → C can be uniformly (resp. tangentially) approximated by functions in F if for each positive constant ϵ > 0 (resp. positive continuous function ϵ : Γ → R) there is g ∈ F such that |f − g| < ϵ on Γ. We are mainly interested in two subclasses F, that which is the restriction to Γ of the class O(Cn ) of

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Paul M. Gauthier and E. S. Zeron

entire functions, and that which is the restriction to Γ of the class of meromorphic functions on Cn whose singularities do not meet Γ.

Recall that any meromorphic function on Cn whose singularities do not meet Γ can be expressed as a quotient p/q of two entire functions p and q with q(z) ̸= 0 for all z ∈ Γ, for the second Cousin problem can be solved in Cn . If Γ is compact, then of course, uniform and tangential approximation are equivalent; and we may even replace the classes of entire and meromorphic functions on Cn by the classes of polynomials and rational functions respectively.

We say that Γ is a set of uniform (resp. tangential) approximation by functions in the class F if each f ∈ C(Γ) can be uniformly (resp. tangentially) approximated by functions in F. Of course, as we have defined them, such sets Γ cannot have any interior. In the literature, one also finds a more generous notion of sets of uniform or tangential approximation, which allows some sets having interior.

Before going any further, we should point out that, sets of uniform approximation and sets of tangential approximation by holomorphic functions are in fact the same. This was proved by Norair Arakelian in his doctoral dissertation [5] in C. His proof works verbatim in Cn .

Since this fact is not well known and the proof is short we include it.

Proposition 3.1 (Arakelian) Let Γ be a closed subset of Cn and let F be either the class of functions holomorphic on Γ or the class of entire functions. Then, Γ is a set of uniform approximation by functions in the class F if and only if it is a set of tangential approximation by functions in the same class.

Proof: Suppose Γ is a set of uniform approximation, f ∈ C(Γ) and ϵ : Γ → R is a positive continuous function. Set ψ = ln ϵ. There exists a function g1 ∈ F such that |ψ − g1 | < 1 on Γ. Setting h = exp(g1 − 1), consider the functions f /h ∈ C(Γ). There exists a function g2 ∈ F such that |f /h − g2 | < 1 on Γ. Then, |f − hg2 | < |h| = exp(ℜ(g1 ) − 1) < exp ψ = ϵ. This completes the proof. ✷ The following is a non-compact version of the Stone-Weierstrass Theorem.

Proposition 3.2 A closed set Γ ⊂ Cn is a set of tangential approximation by entire functions if and only if one can approximate (in the tangential sense) the real part projections ℜ(zm ) for m = 1, . . . , n.

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11

Proof: The necessity is trivial. Moreover, if one can approximate the real part ℜ(zm ), one can approximate the imaginary part ℑ(zm ) as well, since ℑ(zm ) = i(ℜ(zm ) − zm ). Let I be the natural diffeomorphism of Cn onto the real part R2n of C2n . That is: I1 (z) = ℜ(z1 ), I2 (z) = ℑ(z1 ), I3 (z) = ℜ(z2 ), I4 (z) = ℑ(z2 ), etc., for z ∈ Cn . Given two continuous function f, ϵ ∈ C(Γ) with ϵ real positive, we may extend both of them continuously to all of Cn while keeping ϵ positive. By the theorem of Scheinberg (see introduction), there is an entire function F ∈ O(C2n ) such that |f (z) − F ◦ I(z)| < ϵ(z)/2 for z ∈ Cn .

Since F is uniformly continuous on compact subsets of C2n , and the diffeomorphism I is proper, there is a positive continuous function δ on Cn such that |F ◦ I(z) − F (w)| < ϵ(z)/2, for each z ∈ Cn and each w ∈ C2n for which |I(z) − w| < δ(z).

By hypotheses, we can approximate each component of I on Γ by entire functions and so there exists an entire mapping h : Cn → C2n with |I − h| < δ on Γ. Thus, |F ◦ I − F ◦ h| < ϵ/2 on Γ. By the triangle inequality, |f − F ◦ h| < ϵ on Γ. The function F ◦ h is entire because h and F are holomorphic. ✷ An interesting consequence of this result is that neither projection ℜ(z) nor ℑ(z), in the complex plane z ∈ C, can be tangentially approximated on the classical examples where the tangential approximation fails to hold, although uniform approximation may sometimes be possible.

Example 3.3 Let Γ=

∞ !

Γj ,

j=0

where Γ0 = [0, +∞) × {0}, and for j = 1, 2, · · ·, # " # " 1 1 1 1 Γj = [0, j] × { , } ∪ {j} × [ , ] .

2j 2j + 1 2j 2j + 1 Then, both functions ℜ(z) and ℑ(z) can be approximated uniformly, but not tangentially, by entire functions on Γ.

Proof: In his doctoral thesis, Arakelian [5] gave a complete characterization for sets of uniform approximation, from which it follows that Γ is not a set of uniform approximation and a fortiori not a set of tangential approximation. Thus, by Proposition 3.2, the functions ℜ(z) and

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ℑ(z) cannot be approximated tangentially. We show that they can be approximated uniformly.

Fix ϵ > 0 and set Zϵ = {z ∈ C : |ℑ(z)| ≤ ϵ} and Wϵ = Γ \ Zϵ . We may assume Zϵ and Wϵ disjoint (by choosing an appropriate smaller ϵ if necessary). Now, define the function ! ϵ for x ∈ Zϵ , f (x) = ℑ(x) for x ∈ Wϵ .

Invoking again Arakelian’s work (see [5], [13, p.245] or [9]), we deduce the existence of an entire function g such that |f − g| < ϵ on Zϵ ∪ Wϵ . Hence, |ℑ − g| < 2ϵ on Γ. So ℑ(z) and ℜ(z) = z − iℑ(z) can both be approximated uniformly on Γ by entire functions. ✷ It is interesting to compare Propositions 3.1 and 3.2 in the light of the previous example.

We should also notice that Proposition 3.1 also holds if we consider approximation by functions holomorphic in a neighbourhood of Γ instead of approximation by entire functions. That is, we have that each continuous function f ∈ C(Γ) can be approximated (in the tangential sense) by functions holomorphic in a neighbourhood of Γ if and only if every projection ℜ(zm ) can. This result suggests the following: Proposition 3.4 Every closed set Γ ⊂ Cn of area zero is a set of tangential approximation by meromorphic functions in Cn . That is, every continuous function F defined on Γ can be tangentially approximated by meromorphic functions whose singularities do not meet Γ.

Proof: Let f, ϵ ∈ C(Γ) be two continuous functions with ϵ real and positive. We must construct a meromorphic function F such that |F (z) − f (z)| < ϵ(z) on Γ. Let B0 be the empty set and B k closed balls of radius k and center in the origin.

Lemma 3.5 Each continuous function h ∈ C(B k ∪ Γ) which can be uniformly approximated by polynomials in B k can be uniformly approximated on D = B k ∪ (Γ ∩ B k+1 ) by rational functions whose singularities do not meet Γ.

Proof: From Theorem 2.1.A, and for each δ > 0, there exists a rational function (a/b)(z) such that |(a/b)(z) − h(z)| < δ for z ∈ D and 0 ̸∈ b(D). Notice that b(Γ) has zero area, so we may choose a complex

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13

number λ ̸∈ b(Γ) ! with absolute value so small such that λ ̸∈ b(D) and ! ! ! a(z) ! b(z)−λ − h(z)! < δ for z ∈ D. ✷

The proof of Proposition 3.4 now follows a classical inductive process. There exists a rational function F1 whose singularities do not meet Γ and such that |F1 (z) − f (z)| < ( 32 − 2−1 )ϵ(z) for z ∈ Γ ∩ B 1 by the previous Lemma 3.5. Proceeding by induction, we shall construct a sequence of rational functions Fk which converges uniformly on compact sets to a meromorphic function with the desired properties.

Given a rational function Fk whose singularities do not meet Γ and such that |Fk (z)−f (z)| < ( 23 −2−k )ϵ(z) in Γ∩B k , let hk be a continuous function identically equal to zero on B k and such that |hk (z) + Fk (z) − f (z)| < ( 23 − 2−k )ϵ(z) for z ∈ Γ ∩ B k+1 as well. Fix a real number 0 < λk < 1 strictly less than ϵ(z) for every z ∈ Γ ∩ B k+1 .

Applying Lemma 3.5, there exists a rational function Rk whose singularities do not meet B k ∪ Γ and such that |Rk (z) − hk (z)| < 2−1−k λk for z ∈ B k ∪ (Γ ∩ B k+1 ). Thus, the singularities of the rational function Fk+1 (z) = Fk (z) + Rk (z) do not meet Γ and |Fk+1 (z) − f (z)| < ( 23 − 2−1−k )ϵ(z) for z ∈ Γ ∩ B k+1 by the triangle inequality.

Notice that Fk+1 (z) − Fk (z) is holomorphic and its absolute value is less than 2−1−k inside B k , so the sequence Fk converges to a meromorphic function with the desired properties. ✷ Similar inductive processes were originally employed to prove Carleman’s theorem, stated in the introduction, which asserts that the real line R in C is a set of tangential approximation by entire functions.

Alexander [3] extended Carleman’s theorem to piecewise smooth arcs Γ going to infinity in Cn . That is, Γ is the the image of the real axis under a proper continuous embedding (a curve without self-intersections, going to infinity in both directions). We should mention that this problem had been considered independently by Bernard Aupetit and Lee Stout (see Aupetit’s book [1]). As a consequence of the Alexander-Stolzenberg Theorem, we also have the following further extension of Carleman’s theorem, which was conjectured by Aupetit in [1] and announced by Alexander in [3].

Proposition 3.6 Let Γ be an arc which is of finite length at each one of its points, except perhaps in a discrete subset, and going to infinity in Cn . Besides, let ϵ be a strictly positive continuous function on Γ.

Then, for each f ∈ C(Γ), there exists an entire function g on Cn such

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Paul M. Gauthier and E. S. Zeron

that |f (z) − g(z)| < ϵ(z) for all z ∈ Γ. That is, Γ is a set of tangential approximation by entire functions.

Alexander’s proof (see also [1]), for the case that Γ is smooth, relies ingeniously on the topology of arcs and the original Stolzenberg Theorem for smooth curves. It also works when the arc Γ is of locally finite length everywhere except perhaps in a finite subset. One only needs to rewrite Lemma 1 of [3], using the following corollary of Theorem 2.1.

Corollary 3.7 Let X and Y be two compact subsets of Cn such that X is polynomially convex, Y is connected and Y \ X is of finite length at each one of its points, except perhaps at finitely many of them. If the ˇ 1 (X ∪Y ) → H ˇ 1 (X) induced by X ⊂ X ∪Y is injective, then X ∪Y map H is polynomially convex and every continuous function f ∈ C(X ∪ Y ) which can be approximated by polynomials in X can be approximated by polynomials on the union X ∪ Y .

Proof: Let Y0 be the points where Y \X is not of locally of finite length.

Notice that the inclusion mapping X → X ∪ Y can be decomposed as the composition of the two mappings X → X ∪ Y0 and X ∪ Y0 → X ∪ Y . ˇ 1 (X ∪ Y ) → H ˇ 1 (X) can also Hence, the induced injective function H ˇ 1 (X ∪ Y0 ) and ˇ 1 (X ∪ Y ) → H be decomposed as the composition of H 1 1 ˇ ˇ H (X ∪ Y0 ) → H (X). It is easy to see that the last two functions are injective as well. Now, suppose f ∈ C(X ∪Y ) and f can be approximated by polynomials on X. We have that X ∪ Y0 is polynomially convex because of the Oka-Weil theorem or Theorem 2.1. Moreover, we also have that X ∪ Y is polynomially convex and f can be approximated by polynomials on X ∪ Y by Theorem 2.1 again. ✷ We can also approximate by entire functions on unbounded sets which are more general than arcs, but first, we need to introduce the polynomially convex hull of non-compact sets: Definition. Given an arbitrary subset Y of Cn , its$ polynomially convex # " ! : K ⊂ Y is compact . hull is defined by Y! = K

Proposition 3.8 Let Γ be a closed set in Cn of zero area such that !

D ∪ Γ \ Γ is bounded for every compact set D ⊂ Cn . Let B1 be an open ! \ Γ. That ball with center in the origin which contains the closure of Γ ! is, the set B1 ∪ Γ contains the hull K of every compact set K ⊂ Γ.

Approximation on arcs going to infinity

15

Then, given two continuous functions f, ϵ ∈ C(Γ) such that ϵ is real positive and f can be uniformly approximated by polynomials on Γ ∩ B 1 , there exists an entire function F such that |F (z) − f (z)| < ϵ(z) for z ∈ Γ.

Proof: Let B0 be the empty set, B1 as in the hypotheses and Bk open balls with center in the origin such that each Bk contains the closure ! ! of every of Γ ∪B \ Γ. That is, the set B ∪ Γ contains the hull K k−1

k

compact set K ⊂ (Γ ∪ B k−1 ). Define Xk to be the polynomially convex hull of B k+1 ∩ (Γ ∪ B k−1 ), so Xk ⊂ (Bk ∪ Γ). The compact sets Xk and Xk ∩ B k are both polynomially convex.

The given hypotheses automatically imply that there exists a polynomial F1 such that |F1 (z) − f (z)| < ( 32 − 2−1 )ϵ(z) on Γ ∩ B 1 . Proceeding by induction, we shall construct a sequence of polynomials Fk which converges uniformly on compact sets to an entire function with the desired properties.

Given a polynomial Fk such that |Fk (z) − f (z)| < ( 32 − 2−k )ϵ(z) on Γ ∩ B k , let hk be a continuous function equal to Fk on B k and such that |hk (z) − f (z)| < ( 23 − 2−k )ϵ(z) for z ∈ Γ ∩ B k+1 as well. Fix a real number 0 < λk < 1 strictly less than ϵ(z) for every z ∈ Γ ∩ B k+1 .

Notice that Xk = (Xk ∩ B k ) ∪ (Γ ∩ B k+1 ). Hence, by Theorem 2.1.A, the function hk can be approximated by rational functions on Xk because Xk ∩B k is polynomially convex and Γ has zero area. Moreover, the functions hk can be approximated by polynomials by the Oka-Weil theorem. Thus, there exists a polynomial Fk+1 such that |Fk+1 (z)−hk (z)| < 2−1−k λk for z ∈ Xk , and so |Fk+1 (z) − f (z)| < ( 32 − 2−1−k )ϵ(z) on Γ ∩ B k+1 .

Finally, the inequality |Fk+1 (z)−Fk (z)| < 2−1−k holds for z ∈ B k−1 , so the sequence Fk converges to an entire function with the desired properties. ✷ ! = Γ holds as well in the last On the other hand, if the equality Γ proposition, we can choose the empty set instead of the open ball B1 (because the proof is an inductive process); and so Γ becomes a set of tangential approximation by entire functions. There are many closed sets Γ which satisfy the hypotheses of the last proposition. For example, we have the following.

Theorem 3.9 Let Γ be closed connected set of locally finite length in ˇ 1 (Γ) vanishes (Γ contains no simple Cn whose first cohomology group H

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Paul M. Gauthier and E. S. Zeron

closed curves). Then, Γ is a set of tangential approximation by entire functions.

Proof: The proof strongly uses the topology of Γ. We show that each point of Γ has finite order; that is, has a basis of neighbourhoods in Γ having finite boundaries. Given a point z ∈ Γ, let Br be the open ball in Cn of radius r and center z. Since Γ is locally of finite length, the intersection of Γ with the closed ball Br has finite length, so the intersection of Γ with the boundary of Bs must be a finite set for almost all radii 0 < s < r. Whence, each sub-continuum of Γ is locally connected [17, p. 283]. On the other hand, there are no simple closed ˇ 1 (Γ) = 0; so each sub-continuum of Γ is curves contained in Γ because H a dendrite, that is, a locally connected continuum containing no simple closed curves. In particular, if Γ is compact, then it is a dendrite.

Notice the following lemma.

Lemma 3.10 Each compact subset K ⊂ Γ is contained in a sub-continuum (dendrite) of Γ.

Proof: Since Γ is locally connected, the set K is contained in a finite union of sub-continua of Γ. The lemma now follows since Γ is arcwise connected (see Theorem 3.17 of [16]). ✷ Let D be a compact set in Cn . Notice that D ∪Γ may contain simple closed curves Υ with D ∩ Υ ̸= ∅ but Υ ̸⊂ D. We shall call such a simple closed curve Υ ⊂ (D ∪ Γ) a loop. We show there exists a ball which contains all of these loops. Henceforth, let Br be open balls of radii r and center in the origin, and choose a radius s > 0 such that D ⊂ Bs .

Recall that Γ ∩ B s+1 has finite length, so there exists a ball Bt with s < t < s + 1 such that Γ meets the boundary of Bt only in a finite number of points Q = {q1 , . . . , qm }. Let {Υ ! j } be the possible loops which meet the complement of Bt . The set {Υj } \ Bt is contained in Γ and can be expressed as the union of compact arcs (not necessarily disjoint) which lie outside of B t except for their two end points which lie in Q. Since Γ cannot contain simple closed curves, two different arcs cannot share the same end points, and there can only be finitely many such arcs. Hence, there exists a ball Bδ which contains all the loops Υ, and D ⊂ Bδ . !

We shall show that D ∪ Γ \ Γ is bounded. Without loss of generality, we may suppose that D is a closed ball. Since Γ is connected, the hull ! " r , where Kr is the connected component of !

D ∪ Γ is equal to r≥δ K

Approximation on arcs going to infinity

17

!r = K ! δ ∪ Kr , for B r ∩ (D ∪ Γ) which contains D. We can prove that K every r ≥ δ, using Alexander’s original argument. The following lemma is a literal translation of Lemma 1.(a) of [3], to our context. !r = K ! δ ∪ τr where τr = Kr \ Kδ .

Lemma 3.11 For every r ≥ δ, K

Since the notation is quite complicated and different from Alexander’s, and we need to invoke Theorem 2.1.B, we shall include the proof of Lemma 3.11, but first we conclude the proof of the theorem. !r = K ! δ ∪τr = !

By Lemma 3.11, the set D ∪ Γ\Γ is bounded because K ! δ ∪ Γ. Moreover, the equality Γ ! = Γ holds as ! δ ∪ Kr and D ! ∪Γ = K K well because each compact subset of Γ is contained in a dendrite of finite length and is polynomially convex (see Lemma 3.10 and Alexander’s work [2]), so we can deduce from Proposition 3.8 that Γ is a set of tangential approximation. ✷ ! δ ∪ τr be the set on the right Proof: [Proof of Lemma 3.11] Let Tr = K ! r ⊂ T!r (the hand side of the asserted equality. Clearly, we have Tr ⊂ K second inclusion is in fact equality). Thus it suffices to show that Tr is polynomially convex. Arguing by contradiction, we suppose otherwise.

By Theorem 2.1.B, T!r \ Tr is a 1-dimensional analytic subvariety of Cn \ T r .

Let V be a non-empty irreducible analytic component of T!r \ Tr . We ! δ ∪τr , claim that V \Kr is an analytic subvariety of Cn \Kr . Since Tr = K it suffices to verify this locally at a point x ∈ V ∩ Q where ! δ \ Kδ .

Q=K

! r and Q are analytic near x, where near

By Theorem 2.1.B, both K x refers to the intersection of sets with small enough neighbourhoods !r, V ⊂ K ! r \ Q and of x, here and below. Furthermore, near x, V ⊂ K ! r . Thus, near x, Q is a union of some analytic components of Q⊂K !

Kr . It follows that near x, V is just a union of some of the other local ! r at x; in fact, near x, V = V ∪ {x}. Put analytic components of K W = V \ Kr .

Then W is an irreducible analytic subset of Cn \ Kr and moreover, W \ W ⊂ Kδ ∪ τr = Kr . ! r by the maximum principle.

Thus W ⊂ K

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Paul M. Gauthier and E. S. Zeron

! δ and therefore Fix a point p ∈ V ⊂ W . Since p ̸∈ Tr , we have p ̸∈ K ! δ . By there exists a polynomial h such that h(p) = 0 and ℜh < 0 on K the open mapping theorem, either h(W ) is an open neighbourhood of 0 or h ≡ 0 on W . In the latter case, h ≡ 0 on W and so W \ W is disjoint from Kδ . This implies that W \ W ⊂ τˆr so W ⊂ τˆr . We have a contradiction because τr is contained in a dendrite of finite length and is polynomially convex (see Lemma 3.10 and Alexander’s work [2]), and moreover, a dendrite cannot contain a 1-dimensional analytic set.

Hence, the former case holds. Since h(τr ) is nowhere dense in the plane (recall that it is of finite length), there is a small complex number α ∈ h(W ) such that α ̸∈ h(τr ). Now put g = h − α. If α is sufficiently small, ! δ , (ii) g(q) = 0 for some q ∈ W and we conclude that (i) ℜg < 0 on K (iii) 0 ̸∈ g(τr ).

Now (i) implies that the polynomial g has a continuous logarithm ! δ and so, by restriction, on Kδ . We can extend this logarithm of g on K on Kδ to a continuous logarithm of g on Kr because of (iii), since the ball Bδ was chosen such that every simple closed curve (loop) Υ ⊂ Kr is contained in Bδ and hence in Kδ . But Kr contains W \ W . Applying the argument principle [21, p. 271] to g on the analytic set W gives a contradiction to (ii). ✷ We remark that the condition of having zero area is essential in Propositions 3.4 and 3.8, as the following example (inspired by [8]) shows.

Example 3.12 Let I be the closed" unit interval [0, # 1] of the real line 1 1 1 and K ⊂ I the compact set K = 0, 1, 2 , 3 , 4 , . . . . It is easy to see that the (2 + ϵ)-dimensional Hausdorff measure of the closed connected set Y = (I × {0}) ∪ (K × C) in C2 is equal to zero for every ϵ > 0; moreover, the equality Y! = Y holds. However, the following continuous function f ∈ C(Y ) cannot be uniformly approximated by holomorphic functions in O(Y ): $ z if w = 1 f (w, z) = 0 otherwise Suppose there exists a real number ϵ > 0 and a holomorphic function g ∈ O(Y ) such that |f −g| < ϵ on Y . We automatically have that g(w, z) is bounded, holomorphic and constant on each complex line { 1j } × C,

j = 2, 3, . . .. Hence, the holomorphic function ∂g ∂z vanishes on each 1 complex line { j } × C, j = 2, 3, . . . as well. Since the zero set of ∂g ∂z is an

Approximation on arcs going to infinity

19

analytic set, this derivative must be zero in a neighbourhood of {0} × C and hence on the connected set Y . The last statement is a contradiction to the fact that |g(1, z) − z| < ϵ for every z ∈ C. " On the other hand, to see that Y! = Y , notice that Y = r>0 Yr , where Yr = (I × {0}) ∪ (K × ∆r ) and ∆r ⊂ C are closed disks of radius r. The set K × ∆r is polynomially convex because it is the Cartesian product of two polynomially convex sets in C; and so Yr is polynomially convex because of Theorem 2.1.

Although connectivity, as we have emphasized, plays a crucial role in this paper, similar results can be obtained for sets whose connected components form a locally finite family. Finally, we remark that, on a Stein manifold, analogous results also hold by simply embedding the Stein manifold into some Cn . A possible exception is Proposition 3.2, since ℜ(z) is not well-defined on a manifold.

4

Historical notes

In this section we recapitulate and supplement some of the historical remarks which are dispersed throughout this paper.

Of course, the foundation of approximation theory is the Weierstrass theorem (1885), which affirms that each closed interval is a set of uniform approximation by polynomials. This is essentially a real result.

In the complex setting, the most beautiful approximation theorem is a deep theorem of Walsh [24, in 1926] which lifts the Weierstrass theorem to the complex domain by asserting that each Jordan arc (homeomorphic image of a closed interval) in the complex plane is a set of uniform approximation by polynomials. For a survey on this result of Walsh and its impact, see [12].

Just as Walsh’s theorem is the most beautiful result of uniform approximation in the complex plane C, the outstanding open problem in complex approximation is to extend Walsh’s theorem to higher dimensions (Cn ). Any compact set firstly needs to be polynomially convex (see [21]) in order to be a set of uniform approximation by polynomials. In high dimensions, Wermer [25, in 1955] and Rudin [19, in 1956] gave examples of Jordan arcs which are not polynomially convex, and hence they are not sets of uniform polynomial approximation. The main problem can be then formulated more precisely: Is it true that each polynomially convex Jordan arc is a set of polynomial approximation?

This problem has remained open for over half a century.

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Paul M. Gauthier and E. S. Zeron

The following Jordan arcs are known to be sets of uniform approximation: analytic arcs (Wermer [26], [27] and [28] in 1958), C 1 -smooth arcs (Stolzenberg [22] in 1966), rectifiable arcs (Alexander [2] in 1971); and in the present paper we allow arcs which are of finite length at each point, except perhaps at a finite set of points.

One can also consider rational approximation and; here again, any compact set firstly needs to be rationally convex in order to be a set of uniform rational approximation. It is known that any compact set of area zero is a set of rational approximation. Bagby and one of the authors [6] have given an example of an arc of finite area which is not rationally convex and, a fortiori, it is not a set of rational approximation.

We have seen, on one side, that we have polynomial approximation on Jordan arcs whose length is locally finite except perhaps at a finite subset of points. On the other hand, we do not have rational approximation on a certain Jordan arc of finite area. It is quite natural to ask about the intermediate cases, namely, Jordan arcs whose dimension lies between 1 and 2. This question was in fact posed by Gamelin [11].

As mentioned earlier in this paper, the Weierstrass theorem was also extended in a different way by Carleman [7, in 1927], who showed that the real-line in C is a set of tangential approximation by entire functions in C. This result was also generalized to several complex variables in two ways. First of all, Scheinberg [20, in 1976] showed that the real part of Cn is a set of Carleman approximation by entire functions in Cn .

Secondly, and this is the generalization which concerns us in the present paper, Carleman himself had conjectured and Keldysh proved that each unbounded Jordan arc in C is a set of tangential approximation by entire functions as well. This result was extended by Alexander [2, in 1979] to unbounded Jordan arcs in Cn which are piecewise C 1 -smooth. We have shown that Alexander’s result also holds for unbounded Jordan arcs which are of locally finite length. This had been conjectured by Aupetit [1, in 1978] and announced by Alexander [2]. But we showed a stronger result, by allowing a discrete subset of exceptional points, and also by allowing more general sets than only unbounded arcs.

Paul M. Gauthier D´epartement de math´ematiques et de statistique et Centre de rech`erches math´ematiques, Universit´e de Montr´eal, CP 6128 Centre Ville, Montr´eal, H3C 3J7, Canada. gauthier@dms.umontreal.ca

Eduardo Santillan Zeron Departamento de Matem´ aticas, CINVESTAV - I.P.N., A. Postal 14-740, M´exico D.F. 07000, MEXICO. eszeron@math.cinvestav.mx

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21

References [1] Aupetit, B., L’approximation enti`ere sur les arcs allant ` a l’infini dans Cn , Complex approximation (Proceedings, Qu´ebec, 1978), 93-102. Progress in Mathematics 4, Birkh¨auser, Boston Mass.

1980.

[2] Alexander, H., Polynomial approximation and hulls in sets of finite linear measure in Cn , Amer. J. Math., 93 (1971), 65–74.

[3] Alexander, H., A Carleman theorem for curves in Cn , Math.

Scand., 45 (1979), no. 1, 70–76.

[4] Alexander, H., The polynomial hull of a set of finite linear measure in Cn , J. Analyse Math., 47 (1986), 238–242.

[5] Arakelian, N. U., Certain questions of approximation theory and the theory of entire functions (Russian), Doctoral Dissertation.

Mat. Inst. Steklov., Moscow 1970.

[6] Bagby, T.; Gauthier, P. M., An arc of finite 2-measure that is not rationally convex, Proc. Amer. Math. Soc., 114 (1992), no. 4, 1033–1034.

[7] Carleman, T., Sur un th´eor`eme de Weierstrass, Ark. f¨ or Math.

Astr. Fys., 20 (1927), 1–5.

[8] Chacrone, S.; Gauthier, P. M.; Nersessian, A., Carleman approximation on products of Riemann surfaces, Complex Variables Theory Appl., 37 (1998), no. 1-4, 97–111.

[9] Gaier, D., Lectures on complex approximation, Translated from the German by R. McLaughlin. Birkh¨auser, Boston Mass. 1987.

[10] Gamelin, T. W., Uniform algebras. Prentice-Hall, Englewood Cliffs N.J. 1969.

[11] Gamelin, T. W., Polynomial approximation on thin sets, Symposium on several complex variables (Park City, Utah 1970), 50–78.

Lecture Notes in Math. 184, Springer-Verlag, Berlin 1971.

[12] Gauthier, P. M., Commentary, Walsh, Joseph L., Selected papers, 247–253. Edited by Theodore J. Rivlin and Edward B. Saff, Springer-Verlag, New York 2000.

22

Paul M. Gauthier and E. S. Zeron

[13] Gauthier, P. M.; Sabidussi, G., Complex potential theory. Proceedings of the NATO Advanced Study Institute and the S´eminaire de Math´ematiques Superi´eures held in Montreal, Quebec. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 439. Kluwer, Dordrecht, 1994.

[14] Gauthier, P. M.; Zeron, E. S., Approximation on arcs and dendrites going to infinity in Cn , Can. Math. Bull., 45 (2002), No. 1, 80–85.

[15] Glicksberg, I., Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc., 105 (1962), 415–453.

[16] Hocking, J. G.; Young, S. G., Topology. Dover Publications, New York 1988.

[17] Kuratowski, K., Topology Vol. II. Academic Press, New York and London 1968.

[18] Rado, T.; Reichelderfer, P., Continuous transformations in analysis. Springer-Verlag 1955.

[19] Rudin, W., Subalgebras of spaces of continuous functions, Proc.

Amer. Math. Soc., 7 (1956), 825–830.

[20] Scheinberg, S., Uniform approximation by entire functions, J. Analyse Math., 29 (1976), 16–18.

[21] Stolzenberg, G., Polynomially and rationally convex sets, Acta Math., 109 (1963), 259–289.

[22] Stolzenberg, G., Uniform approximation on smooth curves, Acta Math., 115 (1966), 185–198.

[23] Stout, E. L., The theory of uniform algebras. Bogden & Quigley, Tarrytown-on-Hudson NY. 1971. ¨ [24] Walsh, J. L., Uber die Entwicklung einer Funktion einer komplexen Ver¨ anderlichen nach Polynomien, Math. Ann., 96 (1926).

[25] Wermer, J., Polynomial approximation on an arc in C 3 , Ann of Math., 62 (1955), 269–270.

[26] Wermer, J., Function rings and Riemann surfaces, Ann. Math., 67 (1958), 45–71.

Approximation on arcs going to infinity

23

[27] Wermer, J., Rings of analytic functions, Ann. Math., 67 (1958), 497â&#x20AC;&#x201C;516.

[28] Wermer, J., The hull of a curve in C n , Ann. Math., 68 (1958), 550â&#x20AC;&#x201C;561.

[29] Wermer, J., Banach algebras and several complex variables. Graduate Texts in Mathematics No. 35, Springer-Verlag, New YorkHeidelberg 1976.

Morfismos, Vol. 6, No. 2, 2002, pp. 25–41

Bayesian procedures for pricing contingent claims: Prior information on volatility ∗ Francisco Venegas-Mart´ınez

1

Abstract This paper develops a Bayesian model for pricing derivative securities with prior information on volatility. Prior information is given in terms of expected values of levels and rates of precision: the inverse of volatility. We provide several approximate formulas, for valuing European call options, on the basis of asymptotic and polynomial approximations of Bessel functions.

2000 Mathematics Subject Classification: 62F15, 60G15.

Keywords and phrases: Bayesian inference, options pricing, stochastic volatility, numerical methods.

1

Introduction

When studying market volatility, the standard procedure is to analyze data. For example, we may explore plots and frequency histograms, or even examine how observations were collected. However, there is another approach to study market volatility before data is observed, which is based on previous practical experience and understanding, the Bayesian approach. In such a case, the parameters of a sampling model are regarded as random variables, and all judgements are made in terms of the degree of belief on potential values of the parameters. In this framework, a prior distribution is used to describe initial knowledge of the possible values of the parameters of a sampling model. For example, we may feel, based on earlier experience, that our degree of belief ∗ 1

Invited article.

Centro de Investigacio ´n en Finanzas, Tecnolo ´gico de Monterrey, M´exico.

25

26

Francisco Venegas-Mart´ınez

about the values of a given parameter may be expressed by a specific probability distribution, which describes initial knowledge.

In pricing contingent claims it is of particular interest to draw inferences about unknown volatility or uncertain volatility parameters of the underlying asset on the basis of prior information. Considering initial information before data is observed is not just a sophisticated extension but an essential issue to be taken into account for the theory and practice of derivatives. In this paper, we present a new Bayesian method to price derivative securities when there is prior information on uncertain and changing volatility. In our proposal, investors are rational in the sense that they use efficiently prior information by choosing a prior distribution that maximizes logarithmic utility among all admissible distributions describing available information. After all, the core of finance theory (mathematical or empirical) is the study of the rational behavior of investors in an uncertain environment. A study for the behavior of rational agents in the Mexican case can be seen in Venegas-Mart´ınez [20].

This paper is organized as follows. In the next section, we mention some of the limitations of the stochastic volatility approach, and discuss the need of considering prior information in pricing derivatives. In section 3, we review the Bayesian inference framework and its relationship with information theory. In section 4, we develop a Bayesian model to price derivative securities and exploit its relationship with Bessel functions. In sections 5 and 6, we examine some asymptotic and polynomial approximations of the basic Bayesian valuation problem. Through section 7, we carry out a comparison of our approach with other models available in the literature. Finally, In section 8, we draw conclusions, acknowledge limitations, and make suggestions for further research.

2

Limitations of the stochastic volatility approach

The most common set-up of the stochastic volatility model consists in a geometric Brownian motion correlated with a mean-reverting OrsteinUhlenbeck process. This approach for pricing derivatives has been widely studied with a remarkable theoretical progress; see, for instance, Ball and Roma [3], Heston [10], Renault and Touzi [18], Stein and Stein

Bayesian Procedures

27

[19], Wiggins [22], and Avellaneda et al. [2]. In particular, the stochastic volatility models allow us to reproduce in a more realistic way asset returns, specially in the presence of fat tails (Wilmott [23]), asymmetry in the distribution (Fouque, Papanicolaou, and Sircar [7]), and the smile effect (Hull and White [11]). However, there is a set of empirical regularities (or stylized facts) that still need to be explained. In particular, the existing models do not explain how investors, ranging from non corporate individual to large trading institution, choose the best patterns of investment (rational behavior) if there is prior information on volatility, and its implications when valuing derivatives.

3

The Bayesian approach to price derivative securities

In the real world, volatility is neither constant nor directly observed.

Hence, it is natural to think of volatility as a non-negative random variable with some initial knowledge coming from practical experience and understanding before data is observed. This is just the Bayesian way of thinking about prior information. Under this approach, prior information is described in terms of a probability distribution (subjective beliefs) of the potential values of volatility. It is common in Bayesian inference, instead of studying volatility, σ > 0, to study precision, which is defined as the inverse of the variance, h = σ −2 ; see, for instance, Leonard and Hsu [12], and Berger [4]. Thus, the lower the variance, the higher the precision. More precisely, from the Bayesian point of view, we have a distribution, Ph , h > 0, describing prior information.

We shall assume that Ph is absolutely continuous with respect to the Lebesgue measure ν, so that the Radon-Nykodim derivative provides a prior density, π(h), i.e., dPh / ν(h) = π(h) for all h > 0. Then, we may write !

Ph {h ∈ A} = π(h)dν(h) A

for all Borel sets A.

3.1

Maximum entropy priors

There are several well-known methods reported in the Bayesian literature to construct densities that incorporate prior information by max-

28

Francisco Venegas-Mart´ınez

imizing a criterion functional subject to a set of constraints in terms of expected values. Some of such methods are: non-informative priors (Jeffreys [14]); maximal data information priors (Zellner [24]); maximum entropy priors (Jaynes [13]); minimum cross-entropy priors, also known as relative entropy priors (Kullback [16]); reference priors (Good [8] and Bernardo [5])); and controlled priors (Venegas-Martnez et al.

[21]). We shall specialize in this paper in Jaynes’ maximum entropy for pragmatic and theoretical reasons that will appear later.

Let us suppose that there is initial information on volatility in terms ! ¯k , k = 0, 1, 2, ..., N, of expected values, say ak (h)π(h)I{h>0} dν(h) = a where the functions ak (h) are Lebesgue-mesurable known functions and all the constants a ¯k are known, as well. The maximum entropy principle states that from all densities satisfying the given information (constraints) we should choose the one that maximizes H[π(θ)] = −

"

ln[π(h)]π(h)dν(h). h>0

¯0 = 1 to ensure that the solution is indeed a We define a0 (h) ≡ 1 and a proper density. Hence, we are interested in finding π(h) that solves the following variational problem: max H[π(θ)] = − π

subject to C :

"

"

ln[π(h)]π(h)dν(h), h>0

ak (h)π(h)I{h>0} dν(h) = a ¯k ,

k = 0, 1, 2, ..., N.

In the sequel, we shall assume that the set of the constraints, C, form a convex and compact set on π. Since H[π(h)] is strictly concave in π(h), the solution exists and is unique. In such a case, the necessary condition for π(h) to be a maximum, is also sufficient. By using standard necessary conditions derived from calculus of variations (see, for instance, Chiang [6]), we found that if π(h) is optimal, then

(1)

π(h) = e

1+λ0

exp

#N $

k=1

%

λk ak (h) ,

where λk , k = 0, 1, 2, ..., N , are the Lagrange multipliers associated with the constraints C.

29

Bayesian Procedures

3.2

Relative entropy

Another useful inference method to estimate an unknown probability density, π(h), when there is an initial estimate p(h) of π(h), and information about precision h in terms of expectations, is based on determining π(h) that solves the following variational problem: min π

!

subject to:

"

!

!

π(h) ln h>0

π(h) dν(h), p(h)

π(θ)I{h>0} dν(h) = 1, ak (h)π(h)I{h>0} dν(h) = a ¯k ,

k = 1, 2, ..., N.

The quantity h>0 π(h) ln(π(h)/p(h))dν(h) is called the relative entropy between π(h) and p(h), and satisfies a set of axioms of consistency: uniqueness of the final estimate; invariance under one-to-one coordinate transformations; system independence; and subset independence. In this case, if π(h) is optimal, we have that π(h) = p(h)e

1+λ0

exp

#N $

%

λk ak (h) ,

k=1

where λk , k = 0, 1, 2, ..., N , are the Lagrange multipliers associated with the constraints. Observe that when the initial estimate is a uniform density, then relative entropy becomes entropy, as defined in section 3.1. Finally, it is important to mention the work of Avellaneda, Levy, and Par´as [1] on derivative securities when modeling potential volatility values occurring within an open interval using relative entropy.

3.3

Examples of priors on precision

Suppose that prior information on precision is given in terms of expected values of levels and rates. That is, prior knowledge is expressed as: !

(2) and (3)

!

h>0

hπ(h)dν(h) = h>0

β , α

ln(h)π(h)dν(h) = ψ(α) − ln(β),

30

Francisco Venegas-Mart´ınez

where α > 0, β > 0, ψ(α) = dΓ(α)/dα, and Γ(·) is the Gamma function.

Notice that for given expected values on levels and rates, equations (2) and (3) become a nonlinear system in the variables α and β. Since entropy is strictly concave and the Gamma distribution is the unique distribution that satisfies (2) and (3), we find that

π(h|α, β) =

(4)

hα−1 β α e−βh , Γ(α)

h > 0, α > 0,

and

β > 0,

solves the maximum entropy problem. Another priors of interest, after some changes of variable, could be: # ! " hα+1 β α e−βh 1 "" , π "α, β = h" Γ(α)

(5)

(6) π and

!

"

1 " √ ""α, β h" π

#

!

h > 0, α > 0,

and

β > 0,

1

2hα+ 2 β α e−βh , = Γ(α)

h > 0, α > 0,

and

β > 0,

# $ % "" − ln(1/h) −ln(1/h) 1 " β α e−βe , ln "α, β = h " Γ(α)

h > 0, α > 0, and, β > 0, which stand, respectively, for prior distributions of σ 2 , σ, and ln(σ 2 ). In any case, the best choice should reflect what has been learned from previous practical experience.

4

Statement of the basic Bayesian valuation problem

Let us consider a Wiener process (Wt )t≥0 defined on some fixed filtered probability space (Ω, F, (Ft )t≥0 , IP), and a European call option on an underlying asset whose price at time t, St , is driven by a geometric Brownian motion accordingly to dSt = rSt dt + h−1/2 St dWt , that is, (Wt )t≥0 is defined on a risk neutral probability measure IP. Notice that the stochastic differential equation driving the price of the

31

Bayesian Procedures

underlyng asset depends only on the risk-free rate of interest. The drift is independent of risk preferences about the expected return on the asset.

In this case, investors do not require a premium as long as volatility remains constant. Girsanov’s theorem can be used to remove a drift with risk preferences by providing an equivalent risk neutral probability measure (see, for instance, Fouque, Papanicolaou, and Sircar [7].). The option is issued at t0 = 0 and matures at T > 0 with strike price X. Under the Bayesian framework, we have that the price, at time t0 = 0, of the contingent claim when there is prior information on volatility, as expressed in (4), is given by: c(S0 , T, X, r|α, β) = e−rT E(π) {E [max(ST − X, 0)|S0 ]} = e

(7)

−rT

!

h>0

"!

s>X

#

(s − X)fST |S0 (s)ds π(h)dν(h),

where the conditional density of ST given S0 satisfies $

h h1/2 fST |S0 (s) = √ exp − 2T s 2πT and G(s) = ln

%

%

s S0 e−rT

T G(s) + 2h

&

&2 '

,

.

If we assume that the required conditions to apply Fubinis’ theorem are satisfied, so we can guarantee that integrals can be interchanged, then (7) becomes e−rT β α c= √ 2πT Γ(α)

(8)

!

s>X

%

X 1− s

&

I(s|α, β)ds,

where (9)

$

!

h exp − I(s|α, β) = 2T h>0

%

T G(s) + 2h

&2 '

1

hα− 2 e−βh dν(h).

Notice now that (9) can be, in turn, rewritten as

(10)

I(s|α, β) = exp

"

−G(s) 2

#!

h>0

"

exp −A(s)h −

B h

#

hδ−1 dν(h),

32

Francisco Venegas-Mart´ınez

where A(s) =

!

G(s)2 +β 2T

"

> 0,

B=

T > 0, 8

and

δ =α+

1 > 0.

2

The integral in (10) satisfies (see, for instance, Gradshteyn and Ryzhik [9]) #

h>0

$

exp −A(s)h −

B h

%

hδ−1 dν(h) = 2

&

B A(s)

'δ

2

& (

'

Kδ 2 BA(s) ,

(11) ) where Kδ (x), x = 2 BA(s), is the modified Bessel function of order δ, which is solution of the second-order ordinary differential equation (see, for instance, Redheffer [17]) !

1 δ2 y + y′ − 1 + 2 x x ′′

(12)

"

y = 0,

x > 0.

We also have that Kδ (x) is always positive, and Kδ (x) → 0 as x → ∞. Equation (11) is of noticeable importance since it says that all the additional information on volatility provided by the prior distribution and the relevant information on the process driving the dynamics of the underlying asset are now contained in Kδ .

4.1

Constant elasticity of return variance

In this section, we deal with the constant elasticity instantaneous variance case. Let us assume the underlying asset, St , evolves according to b/2

dSt = rSt dt + h−1/2 St

dWt ,

where the elasticity of return variance with respect to the price is defined as b − 2. If b = 2, then the elasticity is zero and asset prices are lognormally distributed. In this section, we are concerned with the case b < 2. After computing the Jacobian for transforming W . t ∼ N (0, t.) into ST , we find that the conditional density of ST given St satisfies fST |S0 (s) = where

&

'

( h D[U V (s)1−2b ]1/(4−2b) e−h[U +V (s)] Iδ 2h U V (s) , δ

33

Bayesian Procedures

δ = 1/(2 − b), !

D = U

#

=

2r

(2 − b)[er(2−b)T − 1]

DS0 erT

V (s) = (Ds)2−b ,

$2−b

"1/(2−b)

,

,

%

and Iδ (x), x = 2h U V (s), is the modified Bessel function of the first kind of order δ. If we assume that prior distribution is described by a Gamma density, then De−rT β α c= √ δ 2πT Γ(α) where J (s|α, β) =

&

&

s>X

'

(s − X) U V (s)1−2b α −h[β+U +V (s)]

h e h>0

)

(1/(4−2b)

J (s|α, β)ds,

*

Iδ 2h U V (s)

+

dν(h)

which is related with the non-central chi-square density function. Moreover, ) * + ∞ , [U V (s)]k+(δ/2) .

Iδ 2h U V (s) = hδ+2k Γ(k + 1)Γ(δ + k + 1) k=0

Hence,

∞ ,

[U V (s)]k+(δ/2) J (s|α, β) = Γ(k + 1)Γ(δ + k + 1) k=0

&

h>0

hα+δ+2k e−h[β+U +V (s)] dν(h)

, [U V (s)]k Γ [α + δ + 2k + 1] [U V (s)] . ∞ (β + U + V (s))α k=0 Γ(k + 1)Γ(δ + k + 1)

=

δ/2

In the particular case that there is not prior information, the solution of maximizing H[π(θ)], subject only to the normalizing constraint, will lead to an improper uniform prior distribution, say π(h) ≡ 1 almost everywhere with respect to ν, then if z = [V (s)/D]1/(2−b) , equivalently V (s) = Dz 2−b , we have c = S0

& ∞

hDX 2−b

· Iδ

& ∞

-

Dz 2−b U

h(U +Dz 2−b )

e hDX 2−b $ √ 2h U Dz 2−b dz,

+ Xe #

−rT

e

h(U +Dz 2−b )

.1/(4−2b)

)

U Dz 2−b

# √ $ Iδ 2h U Dz 2−b dz

+1/(4−2b)

34

Francisco Venegas-Mart´ınez

where the following identity holds ! ∞

hDX 2−b

+

5

%

U Dz 2−b

e

h(U +Dz 2−b )

&1/(4−2b) '

"

#

Dz 2−b U

$1/(4−2b)

( √ ) Iδ 2h U Dz 2−b dz = 1.

Asymptotic approximations for the basic Bayesian valuation problem

In this section, we find an asymptotic approximate formula for pricing vanilla contingent claims according to equation (8)-(11). In order to use asymptotic approximations for equation (11), we have to make some assumption on the strike price, X. Note first, that if the strike price X is large, then x is large. In such a case, we may use the following approximation (see, for instance, Gradshteyn and Ryzhik [9]): *δ (x) = Kδ (x) ∼ K

+

#

$

π −x 4δ 2 − 1 e 1+ , 2x 8x

which, in practice, performs well. In this case, we have the estimate price *1 (S0 , T, X, r|α, β) − e−rT X M *2 (S0 , T, X, r|α, β), c, = S0 M

where

√ α % &δ ! ∞ 2 1 2β T *1 = *δ (2 BA(s))ds, √ M e−( 2 G(s)+r) K 8A(s) S0 πT Γ(α) X and

√

α *2 = √ 2β M

πT Γ(α)

% ! ∞ 1 − 1 G(s) e 2 X

s

T 8A(s)

&δ

2

-

*δ (2 BA(s))ds.

K

*1 and M *2 can be approximated with simple procedures The integrals M in MATLAB by using a large enough upper limit in the integral. The *1 and M *2 are taken large enough so that upper limits of the integrals M *1 and M *2 have no sustantial change when larger uper the values of M limits are used (within an error of 0.0001). Figure 1 shows the values of c, as a function of α (δ = α + 12 ) and β, with S0 = 42.00, X = 41.00, r = 0.11, and T = 0.25.

35

Bayesian Procedures

Figure 1. Values of c! as a function of α and β.

6

Polynomial approximations for the basic Bayesian valuation problem

Polynomial approximations, for the basic Bayesian valuation problem stated in (8)-(11), can be done only for some numerical values of the parameters. In this case, we apply the Frobenius’ method to obtain an approximate polynomial of finite order. Let us consider the particular case α = 0.5, i.e., δ = 1, in equation (9). The following polynomial approximation is based on Frobenius’ method of convergent power-series expansion: "

# $

6 % x 1 x ln I1 (x) + ak (13) K1 (x) = x 2 k=0

# $2k

x 2

&

+ϵ ,

0 < x ≤ 2,

where a0 = 1, a1 = 0.15443144, a2 = −0.67278579, a3 = −0.18156897, a4 = −0.01919402, a5 = −0.0110404, a6 = −0.00004686, and

36

Francisco Venegas-Mart´ınez

I1 (x) = x

! 6 "

bk

k=0

#

4x 5

$k

%

0<x≤

+ϵ ,

15 , 4

where b0 = 1/2, b1 = 0.878900594, b2 = 0.51498869, b3 = 0.15084934, b4 = 0.02658733, b5 = 0.00301532, b6 = 0.00032411, and ϵ < 8 × 10−9 .

The complementary polynomial are given by

(14) K1 (x) = √

# $

6 " x 1 a ¯k I (x) + ln 1 xex 2 k=0

# $−2k x

2

+ ϵ¯,

x > 2,

where a ¯0 = 1.25331414, a ¯1 = 0.23498619, a ¯2 = −0.03655620, a ¯3 = 0.01504268, a ¯4 = −0.00780353, a ¯5 = 0.00325614, a ¯6 = −0.00068245, and I1 (x) = x

! 8 "

k=0

¯bk

#

4x 5

$−k

%

+ ϵ¯ ,

x>

15 , 4

where ¯b0 = 39894228, ¯b1 = −0.03988024, ¯b2 = −0.00362018, ¯b3 = 0.00163801, ¯b4 = −0.01031555, ¯b5 = 0.02282967, ¯b6 = −0.02895312, ¯b7 = 0.01787654, ¯b8 = −0.00420059, and ϵ¯ < 2.2 × 10−7 . It is important to point out that K1 (x) and I1 (x) are linearly independent modified Bessel functions, thus they determine a unique solution of Bessel differ(ϵ) ential equation. If we denote by K1 (x) the polynomial approximation in (13) and (14), we get from (8)-(11) the following call option price: c(ϵ) = S0 M1 (S0 , T, X, r|α = 0.5, β) − e−rT XM2 (S0 , T, X, r|α = 0.5, β), where 1 2

M1 =

β 2S0 π

(ϵ) M2

β2 = 2π

(ϵ)

& ∞ X

1

e−( 2 G(s)+r) [A(s)]

and 1

− 21

⎛)

(ϵ) K1 ⎝

⎞

T A(s) ⎠ ds, 2

⎛) ⎞ & ∞ 1 1 − 1 G(s) T A(s) (ϵ) − ⎠ ds. e 2 [A(s)] 2 K1 ⎝ X

(ϵ)

s

2

(ϵ)

As before, integrals M1 and M2 can be approximated by using simple procedures in MATLAB. Figure 2 shows the values of c(ϵ) as a function of β with α = 0.5, S0 = 42.00, X = 41.00, r = 0.11, and T = 0.25.

Bayesian Procedures

37

Figure 2. Values of c!(ϵ) as a function of β.

7

Comparison with other models available in the literature

In the Mexican case, there is not an exchange for trading options, and the over-the-counter market on options is an incipient market, so data is poor in both quantity and quality. Hence, it is impossible to carry out a reliable empirical analysis to compare market option prices with our theoretical prices. However, we work out an interesting numerical experiment. In this experiment, we compare our prices with two other prices from models available in the literature. In figure 3, the case of the classical Black and Scholes’ price, as a function of the strike price, is considered as a benchmark with parameter values S0 = 100, T = 0.5, r = 0.05, and σ = 0.2, and is represented by the solid line. The Korn and Wilmott’s [15] price with subjective beliefs on future behavior of stock prices is represented by the dashed line. The parameter values in the Korn and Wilmott’s [15] model are µ = 0.1, α = 0.33, β = 3.33, and γ = 0.1. Finally, the doted line shows our price c!(ϵ) with prior information on levels and rates. We examined, in this experiment,

38

Francisco Venegas-Mart´ınez

about 800 different combinations of the parameter values α and β, with parameter values β = 17 and α = 0.5. Notice that option prices with prior information on levels are higher than option prices with only prior information on future prices. As expected, Black and Scholes prices are smaller than option prices with any prior information.

Figure 3. Option values as a function of the strike price.

8

Summary and conclusions

Prior information is a subjective issue, that is, different individuals have different initial beliefs. It is difficult to accept that all individuals participating in a specific market can describe their initial knowledge with the same functional form for the prior distribution, and it is still more difficult to recognize as being true that all of such distributions have the same parameters. The existence of a prior distribution is useful to describe initial beliefs in much more complex markets than those in a naive Black-Scholes. In a richer stochastic environment, we have developed a Bayesian procedure to value a European call option when there is prior information on uncertain or changing volatility. In conclusion, the existence of a prior distribution is useful to describe initial beliefs

Bayesian Procedures

39

in much more complex markets than those in a naive Black-Scholes.

Needless to say, Monte Carlo methods should be developed and applied in our proposed framework, and that will be our next goal.

Francisco Venegas-Mart´ınez C. de Investigaci´ on en Finanzas, Tecnol´ ogico de Monterrey, 14380 M´exico D.F., MEXICO, fvenegas@itesm.mx

References [1] Avellaneda, M.; Levy, A.; Par´ as, A., Pricing and Hedging Derivative Securities in Markets with Uncertain Volatilities, Appl. Math.

Finance, 2 (1995), 73-88.

[2] Avellaneda, M.; Friedman, C.; Holmes, R.; Samperi, D., Calibrating Volatility Surfaces Relative-Entropy Minimization, Appl.

Math. Finance, 4 (1996), 37-64.

[3] Ball, C.; Roma, A., Stochastic Volatility Option Prices, J. Financial and Quantitative Analysis, 24 (1994), 589-607.

[4] Berger, J. O., Statistical Decision Theory and Bayesian Analysis, Second edition, Springer-Verlag, New York, 1985.

[5] Bernardo, J. M., Reference Posterior Distributions for Bayesian Inference, J. Roy. Statist. Soc., B41 (1979), 113-147.

[6] Chiang, A. C., Elements of Dynamic Optimization, McGraw-Hill Inc, 1992.

[7] Fouque, J.; Papanicolaou, G.; Sircar, K. R., Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2000.

[8] Good, I. J., What is the Use of a Distribution? In Multivariate Analysis, (Krishnaia, ed.), Vol. II, 183-203, Academic Press, New York, 1969.

[9] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products, Sixth Edition, Academic Press, 2000.

40

Francisco Venegas-Mart´ınez

[10] Heston, S., A Closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies, 6 (1993), 327-343.

[11] Hull, J.; White, A., The Pricing of Option on Assets with Stochastic Volatility, J. Finance, 62 (1987), 281-300.

[12] Leonard, T.; Hsu, J. S. J., Bayesian Methods: An Analysis for Statisticians and Interdisciplinary Researchers, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 1999.

[13] Jaynes, E. T., Prior Probabilities, IEEE Transactions on Systems Science and Cybernetics, SSC-4 (1968), 227-241.

[14] Jeffreys, H., Theory of Probability, Third edition, Oxford University Press, London, 1961.

[15] Korn, R.; Wilmott, P., Option Prices and Subjective Probabilities, Working paper, Mathematical Finance Group, Mathematical Institute, Oxford University, 1996.

[16] Kullback, S., An Application of Information Theory to Multivariate Analysis, Ann. Math. Statistics, 27 (1956), 122-146.

[17] Redheffer, R., Differential Equations: Theory and Applications, Jones and Bartlett Publishers, Boston, MA, 1991.

[18] Renault, E.; Touzi, N., Option Hedging and Implied Volatility in a Stochastic Volatility Model, Math. Finance, 6 (1996), 279-302.

[19] Stein, E.; Stein, J., Stock Price Distribution with Stochastic Volatility: An Analytic Approach, Review of Financial Studies, 4 (1991), 727-752.

[20] Venegas-Mart´ınez, F., Temporary Stabilization: A Stochastic Analysis, Journal of Economic Dynamics and Control, 25(9) (2001), 1429-1449.

[21] Venegas-Mart´ınez, F.; de Alba, E.; Ordorica-Mellado, M., On Information, Priors, Econometrics, and Economic Modeling, Estudios Econmicos, 14 (1999), 53-86.

[22] Wiggins, J., Option Values under Stochastic Volatility, J. of Financial Economics, 9(2) (1996), 351-372.

Bayesian Procedures

41

[23] Wilmott, P., Derivatives, The Theory and Practice of Financial Derivatives, John Wiley & Sons Ltd, 1998.

[24] Zellner, A., Maximal Data Information Prior Distributions. In New Methods in the Applications of Bayesian Methods, A Aykac, and C. Brumat (Eds), North-Holland, 1977.

Morfismos, Vol. 6, No. 2, 2002, pp. 43–65

Existence of Nash equilibria in discounted nonzero-sum stochastic games with additive structure ∗ Heriberto Hern´andez–Herna´ndez

Abstract This work considers two-person nonzero-sum dynamic stochastic games when the state and action sets are Borel spaces, with possibly unbounded (immediate) cost functions, and discounted cost criteria. The aim is to prove, under suitable assumptions, the existence of a Nash equilibrium in stationary strategies. One of those assumptions is that the transition law and the cost functions have an additive (or separable) structure.

2000 Mathematics Subject Classification: 91A15, 91A10.

Keywords and phrases: Nonzero-sum stochastic games, additive structure, discounted cost, Nash equilibria.

1

Introduction

In this work we study two–person nonzero–sum stochastic games when the state and action sets are uncountable Borel spaces. For such a class of games, with either a discounted or an average cost criterion, the existence of stationary Nash equilibria is still an open problem. Here we give a solution to this problem for games with discounted cost criteria in the particular case when the cost functions and the transition law have an additive structure. ∗ Research partially supported by a CONACyT scholarship. This paper is part of the author’s M. Sc. Thesis presented at the Department of Mathematics of CINVESTAV-IPN.

43

44

Heriberto Hern´ andez–Hern´andez

The existence of Nash equilibria in stationary strategies for discounted stochastic games with additive (also known as “separable”) structure was first studied by Himmelberg et al. [7], Theorem 1, under the assumption that the action spaces are finite sets and the state space is Borel. They showed that, given a probability distribution p of the initial state, there is a pair of stationary strategies which form a Nash equilibrium p–a.e. (a so-called p–equilibrium). This result was strengthened by Parthasarathy [15], Theorem 4, who showed that such games have a Nash equilibrium in stationary strategies. An extension of Parthasarathy’s result in [15] was given in Nowak [12], Theorem 1.1.

In the latter work the state set is a measurable space with a countably generated σ–algebra. Nowak also considered compact metric action spaces and bounded reward functions. Our main result, Theorem 4.4, is also an improvement of Parthasarathy’s Theorem 4 in [15], because the state space and the action spaces that we consider are Borel spaces and the cost functions are possibly unbounded. Moreover, our approach to prove Theorem 4.4 is different from Nowak’s in [12]. Here we follow the approach developed in several works, via a fixed-point theorem for multifunctions, as in Ghosh and Bagchi [5] and Parthasarathy [15], for instance.

Other works dealing with an additive structure include [3], [8], [10], [16]. Examples may be found in [3] and [5]. A good survey of the existing literature for both discounted and average criteria is given in [13].

The work is organized as follows. Section 2 presents standard material on stochastic games. The discounted optimality criteria we are interested in is introduced in section 3. Our assumptions and main result, Theorem 4.4, are stated in section 4. Sections 5 and 6 are devoted to prove our main result. Section 7 presents some concluding remarks.

2

The stochastic game model

In this section we introduce the game model we are interested in. We start with the following remark on terminology and notation —for further details see Bertsekas and Shreve [1], chapter 7, for instance.

Remark 2.1 (a) A Borel subset X of a complete and separable metrizable space is called a Borel space, and its Borel σ–algebra is denoted

by B(X). We only deal with Borel spaces, and so “measurable” (for either sets or functions) always means “Borel measurable”. Given a Borel

Existence of Nash equilibria

45

space X, we denote by P(X) the family of all probability measures on X, endowed with the weak topology. In this case, P(X) is a Borel space.

Moreover, if X is compact, then so is P(X). (b) Let X and Y be Borel spaces. A measurable function φ : Y → P(X) is called a stochastic kernel on X given Y , and we denote by P(X|Y ) the family of all those stochastic kernels. If φ is in P(X|Y ), then we write its values either as φ(y)(C) or as φ(C|y), for all y ∈ Y and C ∈ B(X).

The game model. We shall consider the two-person nonzero-sum stochastic game model (X, A, B, KA , KB , Q, c1 , c2 ) where X is the state space, and A and B are the action spaces for players 1 and 2, respectively. These spaces are all assumed to be Borel spaces.

The sets KA ∈ B(X × A) and KB ∈ B(X × B) are the constraint sets.

That is, for each state x ∈ X the x–section in KA , namely A(x) := {a ∈ A|(x, a) ∈ KA }, represents the set of admissible actions for player 1 in the state x. Similarly, the x–section in KB , B(x) := {b ∈ B|(x, b) ∈ KB }, stands for the set of admissible actions for player 2 in the state x. Let K := {(x, a, b)|x ∈ X, a ∈ A(x), b ∈ B(x)}, which is a Borel subset of X × A × B (see Lemma 1.1 in Nowak [11], for instance). Then Q ∈ P(X|K) is the game’s transition law, and, finally, ci : K → R, i = 1, 2, is a measurable function that represents the one-stage cost function for player i.

The game is played as follows. At each stage (or time) t = 0, 1, 2, . . . , the players 1 and 2 observe the current state x ∈ X of the system, and then independently choose actions a ∈ A(x) and b ∈ B(x), respectively.

As a result of this two things happen: (1) player i (i = 1, 2) incurs a cost ci (x, a, b); and (2) the system moves to a new state according to the probability distribution Q(·|x, a, b). Cost accumulates throughout the course of the game, and the goal of each player is to minimize his/her cost.

46

Heriberto Hern´ andez–Hern´andez

Strategies. Let H0 := X, and Ht := Kt × X for t = 1, 2, . . . . For each t, an element ht = (x0 , a0 , b0 , . . . , xt−1 , at−1 , bt−1 , xt ) of Ht represents a “history” of the game up to time t. A strategy for player 1 is then defined as a sequence π 1 = {πt1 , t = 0, 1, . . .} of stochastic kernels πt1 ∈ P(A|Ht ) such that πt1 (A(xt )|ht ) = 1 ∀ht ∈ Ht , t = 0, 1, . . . .

The set of all strategies for player 1 is denoted by Π1 . Let S1 be the set of stochastic kernels φ ∈ P(A|X) with the property φ(A(x)|x) = 1 ∀x ∈ X.

A strategy π 1 = (πt1 ) is called stationary if there exists φ1 ∈ S1 such that πt1 (·|ht ) = φ1 (·|xt ) ∀ht ∈ Ht , t = 0, 1, . . . .

We shall identify S1 with the family of all stationary strategies for player 1. Let F1 be the set of measurable functions f : X → A such that f (x) is in A(x) for all x. Identifying f (x) with the Dirac measure δf (x) (·) concentrated at f (x) we see that F1 ⊂ S1 .

A stationary strategy φ ∈ S1 for player 1 is said to be deterministic if there exists f ∈ F1 such that φ(·|x) = δf (x) (·)

∀x ∈ X.

The sets of strategies Π2 , S2 and F2 for player 2 are defined similarly.

Let (Ω, F) be the canonical measurable space that consists of the sample space Ω = (X × A × B)∞ and its product σ–algebra F. Then for each pair of strategies (π 1 , π 2 ) ∈ Π1 ×Π2 and each “initial state” x ∈ X, there exists, by a theorem of C. Ionescu-Tulcea (see, for example, Bert1 2 sekas and Shreve [1], pp. 140-141), a unique probability measure Pxπ ,π and a stochastic process {(xt , at , bt ), t = 0, 1, . . .} defined on (Ω, F) in a canonical way, where xt , at , and bt represent the state and the actions of players 1 and 2, respectively, at each stage t = 0, 1, . . . . The expectation 1 2 1 2 operator with respect to Pxπ ,π is denoted by Exπ ,π .

3

The discounted cost

We now introduce the optimality criteria we are concerned with. Let α be a number in (0, 1), a so-called “discount factor”, to be fixed throughout this work.

47

Existence of Nash equilibria

Definition 3.1 Let (π 1 , π 2 ) ∈ Π1 × Π2 and x ∈ X. The total expected α–discounted cost for player i, i = 1, 2, when the players use the strategies π 1 , π 2 , given the initial state x0 = x is defined as ! ∞ # " 1 2 αt ci (xt , at , bt ) .

V i (π 1 , π 2 , x) := Exπ ,π t=0

Definition 3.2 A pair of strategies (π ∗1 , π ∗2 ) ∈ Π1 × Π2 is called a Nash equilibrium for the α–discounted cost criterion if, for each x ∈ X, we have V 1 (π ∗1 , π ∗2 , x) ≤ V 1 (π 1 , π ∗2 , x) ∀π 1 ∈ Π1 , V 2 (π ∗1 , π ∗2 , x) ≤ V 2 (π ∗1 , π 2 , x)

∀π 2 ∈ Π2 .

We shall establish, under certain assumptions, the existence of a Nash equilibrium in S1 × S2 .

Before proceeding we give some notation. For a given measurable function f : K → R and probability measures φ ∈ P(A(x)) and ψ ∈ P(B(x)), let $ $ f (x, φ, ψ) := f (x, a, b)ψ(db)φ(da), A(x)

B(x)

whenever the integrals are well defined. In particular, $ $ ci (x, φ, ψ) := ci (x, a, b)ψ(db)φ(da), A(x)

Q(·|x, φ, ψ) :=

4

$

A(x)

B(x)

$

Q(·|x, a, b)ψ(db)φ(da).

B(x)

Existence of Nash equilibria

In this section we state our main result, Theorem 4.4, which requires the following assumptions.

Assumption 4.1 (a) For each x ∈ X, the sets A(x) and B(x) are compact. (b) There exists a measurable function w : X → R, with w(·) ≥ 1, such that $ $ w(y)Q(dy|x, ·, b) and w(y)Q(dy|x, a, ·) X

X

48

Heriberto Hern´ andez–Hern´andez

are continuous on the sets A(x) and B(x), respectively. (c) There exists a constant 1 ≤ β < 1/α such that ! w(y)Q(dy|x, a, b) ≤ βw(x) ∀(x, a, b) ∈ K. X

The following assumption concerns the game’s additive (or “separable”) structure.

Assumption 4.2 (a) For i = 1, 2, there exist measurable functions ci1 : KA → R and ci2 : KB → R such that ci (x, a, b) = ci1 (x, a) + ci2 (x, b)

∀(x, a, b) ∈ K.

Moreover, the functions ci1 (x, ·) and ci2 (x, ·) are continuous on A(x) and B(x), respectively, and max |ci1 (x, a)| ≤ w(x),

a∈A(x)

max |ci2 (x, b)| ≤ w(x)

b∈B(x)

∀x ∈ X,

where w(·) is the function in Assumption 4.1. (b) There exist substochastic kernels Q1 ∈ P(X|KA ), and Q2 ∈ P(X|KB ), such that Q(·|x, a, b) = Q1 (·|x, a) + Q2 (·|x, b)

∀(x, a, b) ∈ K.

Further, Q1 (C|x, ·) and Q2 (C|x, ·) are continuous on A(x) and B(x), respectively, for each C ∈ B(X).

Now we give our last assumption, in which we use again the function w(·) in Assumption 4.1.

Assumption 4.3 There exists a probability measure µ ∈ P(X) and a density function z on K × X such that for each (x, a, b) ∈ K, !

Q(C|x, a, b) = z(x, a, b, y)µ(dy) ∀C ∈ B(X).

C

Moreover, for each x ∈ X, ! |z(x, an , bn , y) − z(x, a0 , b0 , y)|w(y)µ(dy) = 0 lim n→∞ X

whenever an → a0 in A(x) and bn → b0 in B(x). It is also assumed that w ∈ L1 (µ).

Existence of Nash equilibria

49

We now state our main result.

Theorem 4.4 Under the Assumptions 4.1, 4.2 and 4.3, there exists a Nash equilibrium in S1 × S2 .

Theorem 4.4 is proved in Section 6, after some technical preliminaries in Section 5.

5

Technical preliminaries

The w–norm. We denote by M(X) the family of real–valued measurable functions on X, and by B(X) the subfamily of bounded functions in M(X).

If u ∈ M(X) we define its w–norm as ∥u∥w := sup

x∈X

|u(x)| , w(x)

and u is w–bounded if ∥u∥w < ∞. Let Bw (X) be the Banach space of all w–bounded functions in M(X). Thus Bw (X) contains B(X).

Lemma 5.1 Let x ∈ X, π 1 ∈ Π1 , and π 2 ∈ Π2 be arbitrary. Then |V i (π 1 , π 2 , x)| ≤ w(x)

2 1 − αβ

i = 1, 2.

Proof: Assumption 4.1(c) implies Exπ

1 ,π 2

[w(xt )] ≤ β t w(x)

∀t = 0, 1, . . . .

Also, by Assumption 4.2(a) we have, for i = 1, 2, |ci (xt , at , bt )| ≤ 2w(xt ) Hence Exπ

1 ,π 2

|ci (xt , at , bt )| ≤ 2Exπ

∀t = 0, 1, 2, . . . .

1 ,π 2

and it follows that

|V i (π 1 , π 2 , x)| ≤ ≤

∞ !

t=0 ∞ !

αt Exπ

[w(xt )] ≤ 2β t w(x),

1 ,π 2

|ci (xt , at , bt )|

αt 2β t w(x)

t=0

= w(x)

2 .!

1 − αβ

50

Heriberto Hern´ andez–Hern´andez

Dynamic programming. We next develop the dynamic programming results needed to prove Theorem 4.4.

If one of the players, say player 2, fixes a strategy φ2 ∈ S2 , then we have a Markov control process for player 1, with control model (1)

(X, A, KA , Qφ2 , c1φ2 ),

where Qφ2 (·|x, a) = Q(·|x, a, φ2 (x)), and c1φ2 (x, a) = c1 (x, a, φ2 (x)).

Let Vφ12 (·) be the α–discounted value function associated to (1). Note that every (deterministic) stationary strategy for player 1 is a (deterministic) stationary policy for (1) and vice versa. Also, the corresponding α–discounted costs coincide. This fact, together with Lemma 5.1 and Theorem 8.3.6(b) in Hern´ andez-Lerma and Lasserre [6], p. 47, implies (2)

||Vφ12 ||w ≤ 2/(1 − αβ)

∀φ2 ∈ S2 .

We say that π ∗1 ∈ Π1 is an optimal response to φ2 if V 1 (π ∗1 , φ2 , x) = inf V 1 (π 1 , φ2 , x) π 1 ∈Π1

∀x ∈ X.

Similar considerations apply for each fixed φ1 ∈ S1 .

Lemma 5.2 Let v : KA → R be a measurable function such that v(x, ·) is continuous on A(x) for each x ∈ X. Then there exists f ∈ F1 such that, for each x ∈ X, (3)

v ∗ (x) := min v(x, a) = v(x, f (x)), a∈A(x)

and v ∗ is a measurable function. Moreover, for each x ∈ X, (4)

v ∗ (x) =

min

λ∈P(A(x))

v(x, λ).

Proof: The existence of f ∈ F1 that satisfies (3) follows, for instance, from Lemma 8.3.8(a) in Hern´ andez–Lerma and Lasserre [6], p. 50. On the other hand, identifying each a ∈ A(x) with the Dirac measure δa (·) ∈ P(A(x)) we get (5)

v ∗ (x) = min v(x, a) ≥ a∈A(x)

min

λ∈P(A(x))

v(x, λ).

Existence of Nash equilibria

51

On the other hand, for each λ ∈ P(A(x)) we have ! v(x, a)λ(da) ≥ v ∗ (x) ∀x ∈ X. v(x, λ) = A(x)

Hence (6)

min

λ∈P(A(x))

v(x, λ) ≥ v ∗ (x).

From (6) and (5) we obtain (4). !

Proposition 5.3 Let φ2 ∈ S2 be fixed. Then: (a) There exists f1 ∈ F1 such that f1 is an optimal response of player 1 to φ2 . (b) The function Vφ12 is the unique solution in Bw (X) to the dynamic programming equation " # !

(7) Vφ12 (x) =

min

λ∈P(A(x))

c1 (x, λ, φ2 (x))+α

X

Vφ12 (y)Q(dy|x, λ, φ2 (x)) .

(c) If φ∗1 ∈ S1 is an optimal response of player 1 to φ2 then, for each x ∈ X, !

1 ∗ Vφ12 (y)Q(dy|x, φ∗1 (x), φ2 (x)).

(8) Vφ2 (x) = c1 (x, φ1 (x), φ2 (x)) + α X

Similar results hold for each fixed φ1 ∈ S1 .

Proof: (a),(b). If f1 ∈ F1 is an optimal policy for (1) then f1 is an optimal response to φ2 . This follows as in Theorem 3.1 in Maitra and Parthasarathy [9], p. 295. Hence, to prove (a) it suffices to show the existence of f1 ∈ F1 optimal for (1). To this end, let ! v(x, a) := c1 (x, a, φ2 (x)) + α Vφ12 (y)Q(dy|x, a, φ2 (x)).

X

The function c1 (x,$·, φ2 (x)) is continuous on A(x) by Assumption 4.2(a).

Also the function X Vφ12 (y)Q(dy|x, ·, φ2 (x)) is continuous on A(x) by (2) and Lemma 8.3.7 in Hern´ andez–Lerma and Lasserre [6], p. 48. Hence,

by Lemma 5.2 there exists f1 ∈ F1 such that for each x ∈ X, v ∗ (x) = min v(x, a) = v(x, f (x)). a∈A(x)

52

Heriberto Hern´ andez–Hern´andez

Therefore, by Theorem 8.3.6 in Hern´andez–Lerma and Lasserre [6], p.

47, the function f1 is an optimal policy for (1). This proves (a). The same theorem gives that, for each x ∈ X, (9)

Vφ12 (x) = v(x, f (x)) = v ∗ (x).

From (9) and (4) we obtain (7). This proves (b). (c) As in Theorem 3.1 in Maitra and Parthasarathy [9], p. 295, it follows that (10)

V 1 (φ∗1 , φ2 , x) = Vφ12 (x)

∀x ∈ X.

On the other hand, as φ∗1 is a stationary strategy, it follows as in Remark 8.3.10 in Hern´ andez-Lerma and Lasserre [6], p. 54, that V 1 (φ∗1 , φ2 , ·) is the unique solution in Bw (X) of the equation (11) ! u(x) = c1 (x, φ∗1 (x), φ2 (x)) + α

X

u(y)Q(dy|x, φ∗1 (x), φ2 (x))

∀x ∈ X.

This fact and (10) give (8). !

The proof of Proposition 5.3 also shows that, for each φ2 ∈ S2 , Vφ12 (x) = (12)

6

=

inf V 1 (φ1 , φ2 , x)

φ1 ∈S1

inf V 1 (π 1 , φ2 , x).

π 1 ∈Π1

Proof of Theorem 4.4

The proof of Theorem 4.4 is based on a standard procedure; see Ghosh and Bagchi [5] or Parthasarathy [15], for example. This procedure consists of two steps: (i) Topologize S1 and S2 so that they become metrizable and compact spaces. (ii) Show that the multifunction τ : S1 × S2 → 2S1 ×S2 defined by τ (φ1 , φ2 ) := {(φ∗1 , φ∗2 )|φ∗1 is an optimal response to φ2 ,

and φ∗2 is an optimal response to φ1 }

has a fixed point. That is, there exists (φ∗1 , φ∗2 ) ∈ S1 × S2 such that (φ∗1 , φ∗2 ) ∈ τ (φ∗1 , φ∗2 ). In the latter case, we clearly have that (φ∗1 , φ∗2 ) is a Nash equilibrium.

53

Existence of Nash equilibria

We next consider step (i). We follow Parthasarathy [15] to topologize S1 and S2 with the topology of relaxed controls introduced by Warga [17], chapter 4. Let B1 be the set of all measurable functions h : KA → R such that h(x, ·) is continuous on A(x) for each x ∈ X, and maxa∈A(x) |h(x, a)| is a µ–integrable function on X. Here µ is the probability measure in Assumption 4.3. Then B1 becomes a Banach space if we define the norm of h ∈ B1 as ! max |h(x, a)|µ(dx). ∥h∥ := X a∈A(x)

We identify two stationary strategies φ, ψ ∈ S1 if φ(x) = ψ(x) µ–a.e.

Further, we will identify each φ ∈ S1 with the bounded linear functional Λφ : B1 → R given by Λφ (h) := =

! ! !X

h(x, a)φ(da|x)µ(dx) A

h(x, φ(x))µ(dx)

X

∀h ∈ B1 .

In this way we can view S1 as a subset of B∗1 . Equip S1 with the weak∗ topology of B∗1 . So a sequence (φn ) in S1 converges to φ ∈ S1 if and only if Λφn (h) → Λφ (h) ∀h ∈ B1 , or, more explicitly, for each h ∈ B1 ! ! ! ! h(x, a)φn (da|x)µ(dx) → h(x, a)φ(da|x)µ(dx) X

A

X

A

which is the same as ! !

(1) h(x, φn (x))µ(dx) → h(x, φ(x))µ(dx) X

X

∀h ∈ B1 .

Thus, it follows from Theorem IV.3.11 in Warga [17], p. 287, that S1 is metrizable and compact. Similarly, we define B2 to obtain that S2 is metrizable and compact. In the remainder of this work, convergence in S1 or in S2 is understood with respect to the topology just described.

Before proceeding with step (ii), we establish a useful lemma.

54

Heriberto Hern´ andez–Hern´andez

Lemma 6.1 Suppose that (φ1n , φ2n ) → (φ1 , φ2 ) in S1 × S2 . Let h : K → R be such that h(x, a, b) = h1 (x, a) + h2 (x, b)

∀(x, a, b) ∈ K,

for functions h1 ∈ B1 , h2 ∈ B2 such that max h1 (x, a), max h2 (x, b) ∈ Bw (X).

a∈A(x)

b∈B(x)

Then, as n → ∞, h(x, φ1n (x), φ2n (x)) → h(x, φ1 (x), φ2 (x))

µ–a.e.

Proof: Choose an arbitrary f ∈ L1 (µ). It is clear that f (·)

h1 (·, ·) h2 (·, ·) ∈ B1 and f (·) ∈ B2 . w(·) w(·)

Therefore, by (1), ! ! h1 (x, φ1n (x)) h1 (x, φ1 (x)) µ(dx) → µ(dx) f (x) f (x) w(x) w(x) X X and, similarly, ! ! h2 (x, φ2n (x)) h2 (x, φ2 (x)) µ(dx) → µ(dx). f (x) f (x) w(x) w(x) X X If we add these two expressions we obtain ! ! h(x, φ1n (x), φ2n (x)) h(x, φ1 (x), φ2 (x)) µ(dx) → µ(dx). f (x) f (x) w(x) w(x) X X Therefore, by a Riesz’s representation theorem (see, for example, Folland [4], p. 182), the desired conclusion follows. !

To carry out step (ii) we next show that the multifunction τ is upper-semicontinuous and then we will apply Fan’s fixed-point theorem (see Theorem 1 in Fan [2]). To prove that τ is upper-semicontinuous, suppose that (2) and that

(φ1n , φ2n ) → (φ1 , φ2 ) in S1 × S2 ,

Existence of Nash equilibria

(3)

55

(φ∗1n , φ∗2n ) → (φ∗1 , φ∗2 ) in S1 × S2

where, for each n ∈ N, (4)

(φ∗1n , φ∗2n ) ∈ τ (φ1n , φ2n );

then we have to show that (5)

(φ∗1 , φ∗2 ) ∈ τ (φ1 , φ2 ).

To this end, we only prove that φ∗1 is an optimal response to φ2 . The proof that φ∗2 is an optimal response to φ1 is similar. As the proof is a little bit long, we will split it into several results, the most important being Proposition 6.7 and Proposition 6.10.

Lemma 6.2 The following holds µ–a.e. as n → ∞: c1 (x, φ∗1n (x), φ2n (x)) → c1 (x, φ∗1 (x), φ2 (x)).

Proof: This result is a consequence of Assumption 4.2 and Lemma 6.1. !

For notational ease, let un (·) := Vφ12n (·) and u ˜n (·) :=

un (·) , w(·)

and define m := 2/(1 − αβ). By (2) we have |˜ un (·)| ≤ m

∀n ∈ N.

Lemma 6.3 There exists a subsequence (unk ) of (un ) and a function u0 ∈ Bw (X) such that (unk ) converges µ–a.e. to u0 . Therefore, without loss of generality we may assume that (un ) converges µ–a.e. to u0 .

Proof: Identify two functions in M(X) if they are equal µ–a.e., and define U := {u ∈ M(X) : |u(·)| ≤ m µ–a.e.} ⊂ L∞ (µ).

By a Riesz’s representation theorem (see, for example, Folland [4], p.

182), we can view U as a subset of [L1 (µ)]∗ . More precisely, we may identify U with the set U = {u ∈ [L1 (µ)]∗ : ||u|| ≤ m},

56

Heriberto Hern´ andez–Hern´andez

where ||u|| is the norm of u in [L1 (µ)]∗ . Hence, by Alaoglu’s Theorem (see, for example, Folland [4], p. 162), U is a metrizable and compact un ) is subset of [L1 (µ)]∗ in the corresponding weak* topology. Then, as (˜ un ) and a function a sequence in U, there exists a subsequence (˜ unk ) of (˜ unk ) converges to u ˜0 in the weak* sense of [L1 (µ)]∗ . u ˜0 ∈ U such that (˜ This implies that (˜ unk ) converges to u ˜0 in L∞ (µ). Then, letting u0 (·) := u ˜0 (·)w(·) we have u0 ∈ Bw (X) and (unk ) converges µ–a.e. to u0 . !

Lemma 6.4 For each x ∈ X it holds that, as n → ∞, !" ! ! ! ! !

(6) max ! [un (y) − u0 (y)]Q(dy|x, λ, γ)! → 0. ! λ∈P(A(x)),γ∈P(B(x)) ! X

Proof: [Nowak [14], pp. 413-414]. Clearly, (6) will follow if we prove that for each x ∈ X !" ! ! ! ! !

(7) fn (x) := max ! [un (y) − u0 (y)]Q(dy|x, a, b)! → 0. ! a∈A(x),b∈B(x) ! X Before doing this, note that, for n ∈ N, we can write “max” in (6) because ! !# ! [un (y) − u0 (y)]Q(dy|x, ·, ·)! is continuous on the compact set X P(A(x)) × P(B(x)) (see Hern´ andez–Lerma and Lasserre [6], p. 48, for the continuity, and Remark 2.1(a) for the compactness of P(A(x)) × P(B(x))). Now pick an arbitrary x ∈ X, and let fn (x) be as in (7).

For each n ∈ N, let (an , bn ) be a point in A(x) × B(x) at which the maximum in (7) is attained; such a point exists by Assumption 4.1(a).

By the latter assumption we may assume without loss of generality that an → a0 ∈ A(x) and bn → b0 ∈ B(x).

We obviously have !" ! ! ! ! ! fn (x) = ! [un (y) − u0 (y)]Q(dy|x, an , bn )! ! X ! !" " ! ! = ! [un (y) − u0 (y)]Q(dy|x, a0 , b0 ) + un (y)Q(dy|x, an , bn ) ! X X " " − un (y)Q(dy|x, a0 , b0 ) − u0 (y)Q(dy|x, an , bn ) X X ! " ! ! + u0 (y)Q(dy|x, a0 , b0 ) ! !

X ≤ gn (x) + hn (x) + kn (x),

Existence of Nash equilibria

57

where

!" ! ! ! ! ! gn (x) := ! [un (y) − u0 (y)]Q(dy|x, a0 , b0 )!, ! X ! !" ! " ! ! ! ! hn (x) := ! un (y)Q(dy|x, an , bn ) − un (y)Q(dy|x, a0 , b0 )!, ! X !

X !" ! " ! ! ! ! kn (x) := ! u0 (y)Q(dy|x, an , bn ) − u0 (y)Q(dy|x, a0 , b0 )!. ! X !

X

Assumptions 4.3 and 4.1(c) imply that " " w(y)z(x, a0 , b0 , y)µ(dy) = w(y)Q(dy|x, a0 , b0 ) ≤ βw(x).

X

X

Hence, w(·)z(x, a0 , b0 , ·) is in L1 (µ). Therefore, by Assumption 4.3 and because we may assume that (˜ un ) converges to u ˜0 in the weak* sense of 1 ∗ [L (µ)] (see the proof of Lemma 6.3), we have !" ! ! ! ! ! gn (x) = ! [˜ un (y) − u ˜0 (y)]w(y)Q(dy|x, a0 , b0 )! ! X ! !" ! ! ! ! ! = ! [˜ un (y) − u ˜0 (y)]w(y)z(x, a0 , b0 , y)µ(dy)! → 0 ! X ! as n → ∞. Next, we have !" ! ! u ˜ (y)w(y)z(x, an , bn , y)µ(dy) hn (x) = ! ! X n ! " ! ! − u ˜n (y)w(y)z(x, a0 , b0 , y)µ(dy)! !

X " ≤ m |z(x, an , bn , y) − z(x, a0 , b0 , y)|w(y)µ(dy), X

and so hn (x) → 0 by Assumption 4.3. Since |˜ u0 (·)| ≤ m, we conclude in the same manner that kn (x) → 0. Thus we have proved that fn (x) → 0 as n → ∞, i.e. (7) holds. !

Lemma 6.5 For each fixed u ∈ Bw (X) the following holds µ–a.e.: " " ∗ u(y)Q(dy|x, φ1n (x), φ2n (x)) → u(y)Q(dy|x, φ∗1 (x), φ2 (x)).

X

X

58

Heriberto Hern´ andez–Hern´andez

!

Proof: Let u ∈ Bw (X). We first show that X u(y)Q1 (dy|·, ·) ∈ B1 .

This function is measurable on KA (see, for example, Bertsekas and Shreve [1], p. 144), and continuous on A(x) for each x (see Hern´andez– Lerma and Lasserre [6], p. 48). Also, for x ∈ X and a ∈ A(x), "# " # " " " u(y)Q1 (dy|x, a)" ≤ |u(y)|Q1 (dy|x, a) " " X X # ≤ ||u||w w(y)Q1 (dy|x, a) X # $ ≤ ||u||w w(y)Q1 (dy|x, a) X # % + w(y)Q2 (dy|x, b) X # = ||u||w w(y)Q(dy|x, a, b) ∀b ∈ B(x) X

≤ ||u||w βw(x)

and so

by Assumption 4.1(c),

"# " " " " " max " u(y)Q1 (dy|x, a)" ≤ β||u||w w(x). " a∈A(x) " X

Therefore, as w(·) ∈ L1 (µ), we get &# ' # u(y)Q1 (dy|·, ·) ∈ B1 and max u(y)Q1 (dy|x, a) ∈ Bw (X). a∈A(x)

X

X

Similarly #

X

u(y)Q2 (dy|·, ·) ∈ B2 and

max

b∈B(x)

&#

X

' u(y)Q2 (dy|x, b) ∈ Bw (X).

Thus, by Lemma 6.1, the proof is complete. !

Lemma 6.6 For µ–almost all x ∈ X we have (8) # # ∗ un (y)Q(dy|x, φ1n (x), φ2n (x)) = u0 (y)Q(dy|x, φ∗1 (x), φ2 (x)). lim n→∞ X

X

Proof: For any x ∈ X we have "# " # " " " " un (y)Q(dy|x, φ∗1n (x), φ2n (x)) − u0 (y)Q(dy|x, φ∗1 (x), φ2 (x))" " " X " X ≤ fn (x) + gn (x) + hn (x),

Existence of Nash equilibria

where

!" ! ! ! ! ! ∗ fn (x) := ! [un (y) − u0 (y)]Q(dy|x, φ1n (x), φ2n (x))!, ! X ! !" ! ! ! ! ! ∗ gn (x) := ! [un (y) − u0 (y)]Q(dy|x, φ1 (x), φ2 (x))!, ! X ! !" ! ! hn (x) := ! u (y)Q(dy|x, φ∗1n (x), φ2n (x)) ! X 0 ! " ! ! − un (y)Q(dy|x, φ∗1 (x), φ2 (x))!. !

X

By Lemma 6.4, both fn (x) and gn (x) tend to zero as n → ∞. It remains to show that hn (x) → 0 µ–a.e. as n → ∞.

(9)

Lemma 6.5 implies that the following holds µ–a.e.: (10) "

X

u0 (y)Q(dy|x, φ∗1n (x), φ2n (x))

→

"

X

u0 (y)Q(dy|x, φ∗1 (x), φ2 (x)).

Also, the fact that |un (·)| = |˜ un (·)w(·)| ≤ mw(·), together with Assumption 4.1(c), Lemma 6.3 and Lebesgue’s Dominated Convergence Theorem imply that, for each x ∈ X, " " ∗ (11) un (y)Q(dy|x, φ1 (x), φ2 (x)) → u0 (y)Q(dy|x, φ∗1 (x), φ2 (x)).

X

X

Thus, (10) and (11) imply (9). !

The preceding lemmas are summarized in the following proposition.

Proposition 6.7 The following equality holds µ–a.e.: " ∗ (12) u0 (x) = c1 (x, φ1 (x), φ2 (x)) + α u0 (y)Q(dy|x, φ∗1 (x), φ2 (x)).

X

Proof: By Proposition 5.3(c) and (4) we have, for each n ∈ N, (13) " ∗ un (x) = c1 (x, φ1n (x), φ2n (x)) + α un (y)Q(dy|x, φ∗1n (x), φ2n (x)).

X

59

60

Heriberto Hern´ andez–Hern´andez

Then, letting n → ∞ in (13), by Lemmas 6.2, 6.3 and 6.6 we obtain (12) µ–a.e. !

Similar arguments yield the following result.

Lemma 6.8 There exists a set C ∈ B(X) such that µ(C) = 1, and for each x ∈ C and λ ∈ P(A(x)) we have !

X

c1 (x, λ, φ2n (x)) → c1 (x, λ, φ2 (x)), ! un (y)Q(dy|x, λ, φ2n (x)) → u0 (y)Q(dy|x, λ, φ2 (x)).

X

We also need the following simple fact.

Lemma 6.9 Let fn , f : X → R be given functions, with f continuous, where X is a compact metrizable space. If fn → f pointwise, then lim sup [inf fn (x)] ≤ min f (x). x

n→∞

x

Proof: Let x∗ ∈ X be such that f (x∗ ) = min f (x). x

Thus fn (x∗ ) → f (x∗ ) implies lim sup [inf fn (x)] ≤ lim fn (x∗ ) = min f (x). ! n→∞

x

n→∞

x

Proposition 6.10 The following equality is satisfied µ–a.e.: " # ! c1 (x, λ, φ2 (x)) + α u0 (y)Q(dy|x, λ, φ2 (x)) .

(14) u0 (x) = min λ∈P(A(x))

X

Proof: Choose an arbitrary x ∈ C, where C is the set in Lemma 6.8.

Define for λ ∈ P(A(x)) and n ∈ N ! fn (λ) := c1 (x, λ, φ2n (x)) + α un (y)Q(dy|x, λ, φ2n (x)).

X

on P(A(x)) because so are the functions The function fn is continuous $ c1 (x, ·, φ2n (x)) and X un (y)Q(dy|x, ·, φ2n (x)), by Assumption 4.2 and

Existence of Nash equilibria

61

Lemma 8.3.7 in Hern´ andez-Lerma and Lasserre [6], p. 48, respectively.

By the same argument we have that the function ! u0 (y)Q(dy|x, λ, φ2 (x)), f (λ) := c1 (x, λ, φ2 (x)) + α X

is continuous on P(A(x)). We also have, by Lemma 6.8, that fn → f pointwise. Proposition 5.3(b) imply that un (x) =

min

λ∈P(A(x))

fn (λ).

Therefore, because P(A(x)) is compact (see Remark 2.1(a)), we deduce from Lemmas 6.3 and 6.9 that u0 (x) = = ≤ =

lim un (x) " lim min

n→∞ n→∞

λ∈P(A(x))

min

λ∈P(A(x))

min

λ∈P(A(x))

fn (λ)

#

f (λ) " # ! c1 (x, λ, φ2 (x)) + α u0 (y)Q(dy|x, λ, φ2 (x)) .

X

Thus, since x ∈ C was arbitrary, the following holds µ–a.e.: " # ! u0 (x) ≤ min c1 (x, λ, φ2 (x)) + α u0 (y)Q(dy|x, λ, φ2 (x)) . λ∈P(A(x))

X

The reverse inequality follows from Proposition 6.7. !

To conclude the proof of Theorem 4.4 we need to eliminate the qualifier “µ–a.e.” in Proposition 6.7 and Proposition 6.10.

Let D ∈ B(X) be such that µ(D) = 1 and such that (12) and (14) are satisfied in D. Define v0 : X → R such that v0 (x) := u0 (x) for x ∈ D, whereas for x ∈ Dc (the complement of D) $ % ! v0 (x) :=

min

λ∈P(A(x))

c1 (x, λ, φ2 (x)) + α

u0 (y)Q(dy|x, λ, φ2 (x)) .

X

Clearly v0 ∈ Bw (X). Since µ(Dc ) = 0, by Assumption 4.3 we have ! c z(x, a, b, y)µ(dy) = 0 ∀(x, a, b) ∈ K. Q(D |x, a, b) = Dc

62

Heriberto Hern´ andez–Hern´andez

Therefore, for each (x, a, b) ∈ K, ! !

(15) v0 (y)Q(dy|x, a, b) = u0 (y)Q(dy|x, a, b).

X

X

Thus, from (14) and (15) we have, for all x ∈ X, # " ! v0 (y)Q(dy|x, λ, φ2 (x)) .

(16) v0 (x) = min c1 (x, λ, φ2 (x)) + α λ∈P(A(x))

X

From Proposition 5.3(b) and (16) we obtain (17)

v0 (x) = Vφ12 (x)

∀x ∈ X.

Then, it follows from Proposition 5.3(a),(c) that there exists f1 ∈ F1 such that (18) ! v0 (x) = c1 (x, f1 (x), φ2 (x)) + α v0 (y)Q(dy|x, f1 (x), φ2 (x)) ∀x ∈ X. X

Define ψ1∗ (x)

:=

$

φ∗1 (x) if x ∈ D, f1 (x) if x ∈ Dc .

Clearly ψ1∗ (x) = φ∗1 (x) µ–a.e. and ψ1∗ ∈ S1 . By Proposition 6.7, (15) and (18) we have (19) ! ∗ v0 (x) = c1 (x, ψ1 (x), φ2 (x)) + α v0 (y)Q(dy|x, ψ1∗ (x), φ2 (x)) ∀x ∈ X. X

As we are identifying two elements in S1 if they are equal µ–a.e. then all of the results in this section that involve φ∗1 hold if we replace φ∗1 by ψ1∗ . Hence, by (19) we may assume without loss of generality that (20) ! v0 (x) = c1 (x, φ∗1 (x), φ2 (x)) + α

X

v0 (y)Q(dy|x, φ∗1 (x), φ2 (x))

Therefore (as in (11)), from (20) we get (21)

v0 (x) = V 1 (φ∗1 , φ2 , x)

∀x ∈ X.

From (17) and (21) we obtain (22)

Vφ12 (x) = V 1 (φ∗1 , φ2 , x)

∀x ∈ X.

∀x ∈ X.

Existence of Nash equilibria

63

Thus, by (12), φ∗1 is an optimal response to φ2 . In a similar way it can be seen that φ∗2 is an optimal response to φ1 . This establishes (5), and so the multifunction τ is upper-semicontinuous. Finally, by Fan’s fixedpoint theorem (see Theorem 1 in Fan [2]), we conclude that there exists a Nash equilibrium in stationary strategies. This completes the proof of Theorem 4.4. !

7

Concluding remarks

In this work we have imposed an additive structure on the cost functions and the transition law to establish the existence of a Nash equilibrium in stationary strategies. An interesting and challenging open problem is to establish a similar existence result without such an additive structure.

We have dealt with a two-person game for notational convenience.

The result can easily be extended to an N –person game, for any finite N > 2.

Acknowledgement The author thanks to Dr. On´esimo Hern´andez-Lerma for his comments.

Heriberto Hern´ andez-Hern´ andez Departamento de Matem´ aticas, CINVESTAV-IPN, A. Postal 14–740, 07000 M´exico D.F., MEXICO. hhdez@math.cinvestav.mx

References [1] Bertsekas, D. P.; Shreve, S. E., Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York, 1978.

[2] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S.A., 38 (1952), 121-126.

[3] Filar, J.; Vrieze, K., Competitive Markov Decision Processes, Springer–Verlag, New York, 1997.

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Heriberto Hern´ andez–Hern´andez

[4] Folland, G. B., Real Analysis: Modern Techniques and Their Applications, Wiley, New York, 1984.

[5] Ghosh, M. K.; Bagchi, A., Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.

[6] Hern´ andez-Lerma, O.; Lasserre, J. B., Further Topics on Discrete– Time Markov Control Processes, Springer–Verlag, New York, 1999.

[7] Himmelberg, C.; Parthasarathy, T.; Raghavan, T. E. S.; Van Vleck, F., Existence of p–equilibrium and optimal stationary strategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976), 245–251.

[8] K¨ uenle, H.–U., Equilibrium strategies in stochastic games with additive cost and transition structure and Borel state and action spaces, Internat. Game Theory Review, 1 (1999), 131-147.

[9] Maitra, A.; Parthasarathy, T., On stochastic games, J. Optim.

Theory Appl. 5 (1970), 289-300.

[10] Mohan, S. R.; Neogy, S. K.; Parthasarathy, T.; Sinha, S., Vertical linear complementarity and discounted zero–sum stochastic games with ARAT structure, Math. Program., Ser. A, 86 (1999), 637648.

[11] Nowak, A. S., Measurable selection theorems for minimax stochastic optimization problems, SIAM J. Control Optim., 23 (1985), 466-476.

[12] Nowak, A. S., Nonrandomized strategy equilibria in noncooperative stochastic games with additive transition and reward structure, J. Optim. Theory Appl., 52 (1987), 429-441.

[13] Nowak A. S.; Szajowski, K., Nonzero–sum stochastic games, Ann. Internat. Soc. Dynamic Games, 4 (1999), 297-342.

[14] Nowak, A. S., Optimal strategies in a class of zero-sum ergodic stochastic games, Math. Meth. Oper. Res., 50 (1999), 399-419.

[15] Parthasarathy, T., Existence of equilibrium stationary strategies in discounted stochastic games, Shankhya Series A, 44 (1982), 114-127.

Existence of Nash equilibria

65

[16] Raghavan, T. E. S.; Tijs, S. H.; Vrieze, O. J., On stochastic games with additive reward and transition structure, J. Optim. Theory Appl., 47 (1985), 451-464.

[17] Warga, J., Optimal Control of DiďŹ&#x20AC;erential and Functional Equations, Academic Press, New York, 1972.

Morfismos, Vol. 6, No. 2, 2002, pp. 67–87

Existence of Nash equilibria in some Markov games with discounted payoff ∗ Carlos Gabriel Pacheco Gonz´alez

Abstract This work considers N −person stochastic game models with a discounted payoff criterion, under two different structures. First, we consider games with finite state and action spaces, and infinite horizon. Second, we consider games with Borel state space, compact action sets, and finite horizon. For each of these games, we give conditions that ensure the existence of a Nash equilibrium, which is a stationary strategy in the former case, and a Markovian strategy in the latter.

2000 Mathematics Subject Classification: 91A15, 91A25, 91A50 Keywords and phrases: Stochastic game, Nash equilibrium.

1

Introduction

In this paper we study the existence of Nash equilibria for stochastic games with a discounted payoff criterion. First we consider a game with finite state and action spaces, and infinite horizon. In a more general framework, we study games with a Borel state space, compact action sets, and finite horizon. The purpose of this work is to present in a clear, self-contained manner the proofs of these results. Our main source was the paper by Dutta and Sundaram [7].

The first studies of games in the economics literature were the papers

by Cournot [6], Bertrand [3], and Edgeworth [8] on oligopoly pricing and ∗ Research partially supported by a CONACyT scholarship. This paper is part of the author’s M. Sc. Thesis presented at the Department of Mathematics of CINVESTAV-IPN.

67

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Carlos Gabriel Pacheco Gonz´alez

production. The idea of a general theory of games was introduced by John von Neumann and Oskar Morgenstern in their famous 1944 book Theory of Games and Economic Behavior [25], which proposed that most economic questions should be analyzed as games. Nash [15] introduced what came to be known as ”Nash equilibrium” as a way of extending game-theoretic analyses to nonzero-sum games. Stochastic games with discounted payoffs have been widely studied. This class includes the two-person zero-sum stochastic games, for which Nash equilibria are known to exist under a variety of assumptions; see, for instance, Filar and Vrieze [10], Nowak [16] or Ram´ırez-Reyes [19].

In this work we study nonzero-sum games under two different sets of hypotheses. The first result (for games with finite state spaces) was proved by Rogers [21] and Sobel [24], with an extension to countable state spaces by Parthasarathy [17]. The second result was proved by Rieder [20] (as an approximation to what he calls an ε−equilibrium) under some special assumptions on the structure of the game. The general result (that is, games with Borel state and action spaces, and infinite horizon) is an open problem, even with compact action sets. However, it has been solved imposing an additive structure in the reward functions and the transition law; see Hern´ andez-Hern´andez [12] or Parthasarathy and Sinha [17], for instance.

The remainder of this work is organized as follows. Section 2 presents standard material on stochastic games, including the discounted optimality criteria and the definition of a Nash equilibrium. Sections 3 and 4 are devoted to proving the two main results, Theorems 3.1 and 4.1 respectively, that is, the existence of Nash equilibria for the games mentioned in the first paragraph. An appendix is included with some useful facts needed in the proofs of Theorems 3.1 and 4.1.

2

The stochastic game model

In this section we introduce the N − person stochastic game model. We start with the following remark on terminology and notation (for further details see Bertsekas and Shreve [4], chapter 7).

Remark 2.1 a) A Borel subset X of a complete and separable metric space is called a Borel space, and its Borel σ−algebra is denoted by B(X). A Borel subset of a Borel space is itself a Borel space.

Nash Equilibria

69

b) Let X and Y be Borel spaces. A stochastic kernel on X given Y , is a function P (· | ·) such that b.1) P (· | y) is a probability measure on X for each fixed y ∈ Y , and b.2) P (D | ·) is a measurable function on Y for each fixed D ∈ B(X). c) The set of all stochastic kernels on X given Y is denoted by P(X | Y ). Moreover, P(X) denotes the set of probability measures on X. Definition 2.2 A stochastic game model is described by

(1)

GM := {N , S, (Ai , Φi , ri )i∈N , Q, T } ,

where: (1) N = {1, ..., N } is the finite set of players.

(2) S is the state space, a Borel space.

(3) Each player i ∈ N is characterized by three objects (Ai , Φi , ri ), where: (a) Ai , a Borel space, is the action space of player i. Let A = A1 × ... × AN and denote by a a generic element of A. (b) Φi , a multifunction from S to Ai , defines for each s ∈ S the set Φi (s) of feasible actions for player i at state s. Let Φ(s) = Φ1 (s) × ... × ΦN (s) and K = {(s, a) : s ∈ S, a ∈ Φ(s)} . (c) ri , a bounded measurable function from K to R, specifies (for each state s and action a ∈ Φ(s) taken by the players at s) a reward ri (s, a) for player i.

(4) Q, a stochastic kernel in P(S | K), specifies the game transition law.

(5) T ∈ {0, 1, 2, ...} ∪ ∞ is the horizon of the game.

If T = 1, the game is static, and it is denoted by {N , S, (Ai , Φi , ri )i∈N } .

The game is played as follows. At each time t = 0, 1, ..., each player observes the current state s ∈ S of the system, and, independently of the other players, chooses an action ai ∈ Φi (s). Then each player i ∈ N obtains a reward ri (s, a), and the system moves to a new state according

70

Carlos Gabriel Pacheco Gonz´alez

to the probability distribution Q(· | s, a). The objective of each player is to win as much as possible.

Histories. A t-history of the game is a complete description of the evolution of the game up to the beginning of period t. Thus, a t-history specifies the state sr that occurred in each previous period r ∈ {0, 1, ..., t − 1} , the actions ar = (a1,r , ..., aN,r ) taken by the players in those periods (ai,r denotes de action taken by player i at period r), and the state st in the period t. Let Ht be the set of all possible t−histories, with ht denoting a typical element of Ht , i.e.

(2)

ht = (s0 , a0 , s1 , a1 , ..., st−1 , at−1 , st )

with ar ∈ Φ(sr ).

Note that H0 = S and Ht = K × Ht−1 for t = 1, 2, .... −1 Strategies. A strategy πi for player i is a vector {πit }Tt=0 (or sequence if T = ∞) of stochastic kernels πit ∈ P(Ai | Ht ), where for each t and each t−history ht up to t, πit specifies the action πit (ht ) ∈ P(Ai ) such that

πit (Φi (st ) | ht ) = 1 ∀ht ∈ Ht , t = 0, 1, ....

A strategy is also called a mixed or randomized strategy, which means that the player chooses an action in a random manner. The set of mixed strategies includes the pure strategies, when the player chooses the actions in a deterministic way.

Let Πi denote the set of all strategies for player i, and let Π: = Π1 × ... × ΠN . A generic element of Π is denoted by π, and it is said −1 for player i is called to be a multistrategy. A strategy πi = {πit }Tt=0 Markov if πit ∈ P(Ai | S) for each t = 0, 1, ..., T − 1, meaning that each πit depends only on the current state st of the system. The set of all Markov strategies of player i will be denoted ΠiM . A Markov −1 strategy πi = {πit }Tt=0 is said to be stationary if πit = πi0 for each t = 0, 1, ..., T − 1, where πi0 ∈ P(Ai | S). We denote by ΠiS the set of all stationary strategies of player i. We have ΠiS ⊂ ΠiM ⊂ Πi .

In a similar manner ΠS ⊂ ΠM ⊂ Π,

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Nash Equilibria

where ΠS := Π1S × ... × ΠN S is the set of stationary multistrategies, and ΠM := Π1M × ... × ΠN M is the set of Markov multistrategies.

Optimality criteria. Let δ be a fixed number in (0, 1), and define the δ−discounted expected payoff function for player i as

(3)

Ji,δ (s, π) :=

Esπ

!

∞ " t=0

t

δ ri (st , at )

#

for each multistrategy π and each initial state s. It represents the expected present value of the rewards of player i under the multistrategy π. The number δ is called a ”discount factor”.

Definition 2.3 For n = 1, 2, ..., we define the T -stage expected discounted payoff function for player i as !T −1 # " π t Ji,δ,T (s, π) := Es δ ri (st , at ) , t=0

where 0 < δ < 1 is a discount factor. If T = ∞, we write Ji,δ,T (s, π) as Ji,δ (s, π); see (3).

Now we are in position to define a Nash equilibrium. As usual in the literature, the vector (π i , π−i ) will signify the multistrategy π with its strategy πi replaced by π i .

Definition 2.4 A multistrategy π is a Nash Equilibrium of the T stage game GM if Ji,δ,T (s, π) ≥ Ji,δ,T (s, (π i , π−i )) for all s ∈ S, π i ∈ Πi , i ∈ N . $ Before proceeding we give some notation.

First note that A means $ $ % % % A means A1 ...

AN . Let ν : K → R be a A1 ... AN and that measurable function, π0 ∈ P(Φ1 (s))×...×P(ΦN (s)) and π i0 ∈ P(Φi (s)) for some s ∈ S and some i ∈ N , then & ν(s, π0 ) := ν(s, a)π0 (da) A

(4)

=

&

ν(s, a)π1,0 (da1 )...πi0 (dai )...πN,0 (daN ) A

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Carlos Gabriel Pacheco Gonz´alez

and (5)

ν(s, (π i0 , π−i0 )) :=

!

ν(s, a)π1,0 (da1 )...π i0 (dai )...πN,0 (daN ) A

In particular ri (s, π0 ) := =

!

!

ri (s, a)π0 (da) A

ri (s, a)π1,0 (da1 )...πi0 (dai )...πN,0 (daN ),

A

ri (s, (π i0 , π−i0 )) :=

!

ri (s, a)π1,0 (da1 )...π i0 (dai )...πN,0 (daN ), A

Q(· | s, π0 ) := =

!

A

!

A

Q(· | s, a)π0 (da)

Q(· | s, a)π1,0 (da1 )...πi0 (dai )...πN,0 (daN )

and Q(· | s, (π i0 , π−i0 )) :=

!

A

Q(· | s, a)π1,0 (da1 )...π i0 (dai )...πN,0 (daN ).

Remark 2.5 If A1 , ..., AN (and hence A) are finite or countable sets, then the integrals are replaced with summations.

The next two sections are devoted to proving the existence of a Nash equilibrium in games with 1) finite state and action spaces, and infinite horizon (Theorem 3.1); and 2) games with Borel state space, compact action sets, and finite horizon (Theorem 4.1).

The proofs are based on a standard procedure; see, for instance, Dutta and Sundaram [7]. The procedure is to introduce a multifunction which is a K−mapping (see Definition 5.5) on a nonempty compact convex set. Then we use Kakutani´s or Glicksberg´s fixed-point theorem to ensure the existence of a fixed point (Definition 5.6), which yields a Nash equilibrium.

Nash Equilibria

3

73

The equilibrium existence in finite spaces

Theorem 3.1 Suppose S and Ai are finite spaces for each i, and T = ∞. Then the stochastic game model GM has a Nash equilibrium in stationary strategies.

Proof: As was already mentioned, the idea is to prove the existence of a fixed point of a certain multifunction; this fixed point is an equilibrium.

To prove the existence of a fixed point of the multifunction, we use Kakutani´s theorem (Theorem 5.7).

The proof is organized as follows: In Step 0 we define the multifunction. In Step 1 we prove that such a multifunction is defined on a nonempty compact convex subset of Rn .

In Step 2 we prove that the multifunction maps points to a nonempty convex set, and, finally, in Step 3, we prove that the multifunction is u.s.c. (see Definition 5.5). The last two steps prove that the multifunction is a K−mapping (Definition 5.5). Since Steps 1, 2 and 3 verify the hypotheses of Kakutani´s theorem, a fixed point exists.

Step 0: Definition of the multifunction BR. Let π := (π1 , ..., πn ) ∈ ΠS , and let BRi (π) be the set of best responses of player i to π, that is " !

BRi (π) :=

π i ∈ ΠiS : Ji,δ (s, (π i , π−i )) = sup Ji,δ (s, (α, π−i )) ∀s ∈ S α∈ΠiS

Let BR = BR1 × .... × BRN . Note that BR : ΠS ! ΠS .

Step1: The set ΠS is a nonempty compact convex subset of a normed space.

Because Ai is a finite set, the set P(Ai ) is simply the positive unit simplex of dimension |Ai | − 1 (|·| means the cardinality). Note that P(Ai ) ⊂ R|Ai |−1 . Since S is also a finite set, the |S| −fold P(Ai | S) = P(Ai )|S| , the Cartesian product of this simplex, is a compact convex subset of a finite-dimensional Euclidean space. Now every stationary strategy for player i can be associated in the obvious way with a unique sequence of elements of I |Φ(s1 )|−1 × ... × I |Φ(s|S| )|−1 , where I := [0, 1] . Hence ΠiS is a nonempty compact convex subset of Rn for some n, and so is ΠS = Π1S × ... × ΠN S .

Step 2: BR(π) is a nonempty convex subset of ΠS for each π ∈ ΠS .

.

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Carlos Gabriel Pacheco Gonz´alez

Given π ∈ ΠS , the best-response problem faced by player i (that is, finding BRi (·)) is a discounted Markov decision problem " !

M DP := S, Ai , Φi , ri′ , Q′ , T = ∞

where S is the state space and Ai is the action space. Furthermore, for each s ∈ S, Φi (s) is the set of feasible actions in state s. The reward function is ri′ (s, ai ) := ri (s, (ai , π−i ))

(6)

Similarly, the transition law is (7)

Q′ (s′ | s, ai ) := Q(s′ | s, (ai , π−i ))

By Remark 2.5, (6) and (7) represent finite sums. It is well known that there exists a nonempty set BRi (π) of optimal stationary strategies in response to π; see Filar and Vrieze [10], Theorem 2.3.1, for example.

Denote the value function of the M DP as Ji∗ (s). To prove that BRi (π) is convex, let α, β ∈ BRi (π) and 0 ≤ λ ≤ 1; then we want to prove that µ := λα + (1 − λ)β is in BRi (π).

Since α, β ∈ BRi (π), we have that the Bellman equations # ∗ ′ (8) Ji (s) = ri (s, α) + δ Ji∗ (s′ )Q(ds′ | s, α) S

and (9)

Ji∗ (s)

=

ri′ (s, β)

+δ

#

S

Ji∗ (s′ )Q(ds′ | s, β)

hold. Therefore, by (8) and (9), # ′ ri (s, µ) + δ Ji∗ (s′ )Q(ds′ | s, µ), S

=λ

$

ri′ (s, α)

+(1 − λ)

$

+δ

ri′ (s, β)

#

S

+δ

Ji∗ (s′ )Q(ds′ #

S

| s, α)

Ji∗ (s′ )Q(ds′

%

| s, β)

= λJi∗ (s) + (1 − λ)Ji∗ (s) = Ji∗ (s).

That is ri′ (s, µ) + δ

#

S

Ji∗ (s′ )Q(ds′ | s, µ) = Ji∗ (s),

%

Nash Equilibria

75

and so µ is in BRi (π). This shows that BR (π) is a nonempty convex set for each π ∈ ΠS .

Step 3: BR is an u.s.c. multifunction on ΠS , and BR (π) is compact for each π ∈ ΠS .

This will be established if we show that it holds for each BRi . So fix i. Suppose that πn := (π1n , ..., πN n ) → π := (π1 , ..., πN ), and αin ∈ BRi (πn ) is such that αin → αi ∈ ΠiS . We are going to show that αi ∈ BRi (π). ∗ (s) denote the value function of the player i in a best response Let Ji,n ∗ (s) is uniformly bounded by to πn . Since δ < 1, the sequence Ji,n M = (1 − δ)−1 max {|ri (s, a)| : (s, a) ∈ S × A} . ∗ (s) ∈ [−M, M ] for each s ∈ S and for each n, J ∗ So, because Ji,n i,n converges pointwise (perhaps through a subsequence) to a limit Ji∗ . ∗ (s) satisfies the Bellman equaOn the other hand, we know that Ji,n tion ! ∗ ∗ (s′ )Q (ds´| s, (αin , π−i )) , (10) Ji,n (s) = ri (s, (αin , π−i )) + δ Ji,n S

for each n and s. Moreover, for any β ∈ P(Φi (s)), we have ! ∗ ∗ (s′ )Q (ds´| s, (β, π−i )) .

(11) Ji,n (s) ≥ ri (s, (β, π−i )) + δ Ji,n S

Since the integrals are finite sums (recall our Remark 2.5), when n → ∞ we have !

S

ri (s, (αin , π−i )) → ri (s, (αi , π−i )), ! ∗ Ji,n (s′ )Q (ds´| s, (αin , π−i )) → Ji∗ (s′ )Q (ds´| s, (αi , π−i )) , S

and, similarly, ! ! ∗ ′ Ji,n (s )Q (ds´| s, (β, π−i )) → Ji∗ (s′ )Q (ds´| s, (β, π−i )) .

S

S

Hence, letting n → ∞ in (10) and (11) we get ! ∗ Ji (s) = ri (s, (αi , π−i , )) + δ Ji∗ (s′ )Q (ds´| s, (αi , π−i )) S

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Carlos Gabriel Pacheco Gonz´alez

and Ji∗ (s) ≥ ri (s, (β, π−i )) + δ

!

S

Ji∗ (s′ )Q (ds´| s, (β, π−i )) ,

respectively. These expressions establish precisely that Ji∗ (s) is the value function in a best response of i to π, and that αi is a stationary bestresponse, that is, αi ∈ BRi (π) . Thus, BRi is an u.s.c. multifunction for each i, and so is BR. Also note that the u.s.c. shows that BRi (π) is a closed set in ΠiS .

Thus, since ΠiS is compact, BRi (π) is also compact.

Summarizing, we have that for each π, BR (π) is a nonempty compact convex subset, and that BR(·) is u.s.c.; hence BR(·) is a K−mapping.

An appeal to Kakutani´s fixed-point theorem (Theorem 5.7) yields the existence of π ∗ ∈ ΠS such that π ∗ ∈ BR (π ∗ ) , completing the proof of Theorem 3.1. !

4

The equilibrium existence in Borel spaces

Consider the stochastic game model GM in (1) with the following assumptions.

Assumption 0 S and Ai are Borel spaces, and Ai is compact for each i.

Assumption 1 For all i, Φi : S " Ai is a compact–valued multifunction on S. Assumption 2 For all i, ri is bounded and jointly measurable in (s, a), and it is continuous in a for each fixed s ∈ S. Assumption 3 For each Borel subset B of S, Q(B | s, a) is jointly measurable in (s, a), and setwise continuous in a for each fixed s; that is, if an → a then Q(B | s, an ) converges to Q(B | s, a).

Theorem 4.1 Suppose the assumptions 0,1,2,3 hold and T is finite.

Then the stochastic game model GM has a Nash equilibrium in Markovian strategies (possibly nonstationary).

The result is an easy consequence of the following lemmata. The first lemma essentially shows that the theorem is true for T = 1; the combination of the lemmata, together with a selection theorem establishes the result for general T < ∞ through an induction argument.

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Nash Equilibria

Lemma 4.2 For some fixed s ∈ S, consider a N -player stochastic game model in which the action sets are the compact metric spaces Φ1 (s) , ..., ΦN (s) and the reward functions are r1 (s, ·), ..., rN (s, ·) defined on Φ(s) = Φ1 (s) × ... × ΦN (s). If ri (s, ·) is continuous on Φ(s) for each i ∈ N , the stochastic game model {N , {s} , (Φi (s), ri )i∈N } ∗ ) ∈ admits a Nash equilibrium. That is, there exist π ∗ := (π1∗ , ..., πN P(Φ1 (s)) × ... × P(ΦN (s)) such that, for each i, ∗ ri (s, π ∗ ) ≥ ri (s, (πi , π−i )) ∀πi ∈ P(Φi (s)).

In the following proof we consider Π(s) := P(Φ1 (s)) × ... × P(ΦN (s)).

Proof: The idea is to prove the existence of a fixed point of a certain multifunction; this fixed point is an equilibrium. To prove the existence of such a fixed point, we use Glicksberg´s theorem (Theorem 5.8).

The proof is organized as follows: In Step 0 we define a multifunction BR. In Step 1 we prove that BR is defined on a nonempty compact convex subset of a locally convex Hausdorff space. In Step 2 and Step 3 we prove that BR is a K−mapping (Definition 5.5). By the Steps 1, 2 and 3 and Glicksberg’s theorem, a fixed point exists.

Step 0: We define the multifunction BR as in the Step 0 of the proof of the Theorem 3.1.

For each π := (π1 , ..., πN ) ∈ Π(s) , let BRi (π) be the set of best responses of player i to π, that is " !

BRi (π) :=

π i ∈ P(Φi (s)) : ri (s, (π i , π−i )) =

sup

µi ∈P(Φi (s))

ri (s, (µi , π−i )) .

Let BR := BR1 × .... × BRN . Note that BR : Π(s) ! Π(s).

Step1: Π(s) is a nonempty compact convex space.

Convexity is obvious. Morever, by Theorem 5.4, P(Φi (s)) equipped with the topology of weak convergence is a compact metric space; hence so is Π(s).

Step 2: BR(π) is a nonempty convex set for each π ∈ Π(s).

Given π := (π1 , ..., πN ) ∈ Π(s), the best-response problem faced by player i is a discounted Markov decision problem M DP := {{s} , Φi , ri′ } where {s} is the state space, Φi (s) is the action space, and (12)

ri′ (s, ai ) := ri (s, (ai , π−i )),

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Carlos Gabriel Pacheco Gonz´alez

which is continuous in ai . It is well known that the set BRi (π) is nonempty (see Puterman [18], Theorem 6.2.10).

Let ri∗ (s) be the value function of the M DP and let µ1 , µ2 ∈ BRi (π), 0 ≤ λ ≤ 1 and µ = λµ1 + (1 − λ)µ2 (i.e. µ(B) = λµ1 (B) + (1 − λ)µ2 (B) for each B ∈ B(Φi (s))). Then (13)

′

′

′

ri (s, µ) = λri (s, µ1 ) + (1 − λ)ri (s, µ2 ).

Since µ1 , µ2 ∈ BRi (π), the expression (13) is the same as λri∗ (s) + (1 − λ)ri∗ (s) = ri∗ (s), i.e.

′

ri (s, µ) = ri∗ (s).

Therefore, µ ∈ BRi (π), and so BRi (π) is convex.

Step 3: BR is u.s.c. and BR(π) is compact for each π ∈ Π(s).

Let πn := (π1n , ..., πN n ) ∈ BR(π) such that (14)

πn → π := (π1 , ..., πN )

in the weak topology, and αin ∈ BRi (πn ) with (15)

αin → αi ∈ P(Φi (s)).

We are going to show that αi ∈ BRi (π). With this in mind, note that αin ∈ BRi (πn ) gives (16)

ri (s, (αin , π−in )) ≥ ri (s, (β, π−in )) ∀β ∈ P(Φi (s)) and n.

Since ri (s, a) is continuous in a, by Corollary 5.3, as n → ∞ (14) and (15) give ri (s, (αin , π−in )) → ri (s, (αi , π−i )) and ri (s, (β, π−in )) → ri (s, (β, π−i )) ∀β ∈ P(Φi (s)).

Then, as n → ∞, the inequality (16) yields ri (s, (αi , π−i )) ≥ ri (s, (β, π−i )) ∀β ∈ P(Φi (s)); hence αi ∈ BRi (π), which shows that BRi is u.s.c.

The u.s.c. proves that BRi (π) is a closed set in P(Φi (s)). Thus, since P(Φi (s)) is compact, BRi (π) is compact.

Nash Equilibria

79

We have that for each π, BR (π) is a compact convex nonempty set and BR(·) is u.s.c., so BR(·) is a K−mapping. Finally, Glicksberg´s fixed-point theorem implies the existence of π ∗ ∈ Π(s) such that π ∗ ∈ BR (π ∗ ) , completing the proof of Lemma 4.2, because π ∗ is a Nash equilibrium. !

Lemma 4.3 For i = 1, ..., N, let vi : S → RN be a bounded measurable function. For each s ∈ S, i ∈ N , a ∈ Φ1 (s) × ... × ΦN (s), define !

Hi (s, a) := ri (s, a) + δ vi (s′ )Q(ds′ | s, a).

S

Then the stochastic game model {N , S, (Ai , Φi (s), Hi (s, ·))i∈N } admits a ∗ ) ∈ P(Φ (s)) × Nash equilibrium. That is, there exists π ∗ := (π1∗ , ..., πN 1 ... × P(ΦN (s)) such that, for each i ∈ N , ∗ Hi (s, π ∗ ) ≥ Hi (s, (πi , π−i )) ∀πi ∈ P(Φi (s)), s ∈ S.

In the following proof we consider Π := P(A1 ) × ... × P(AN ) and Π(s) := P(Φ1 (s)) × ... × P(ΦN (s)).

Proof: The idea is, using Lemma 4.2, to prove the existence of Nash equilibria for each s ∈ S (Step 0), and then use a selection theorem to get a Nash equilibrium (Step 1).

Step 0: It is easy to show that thet continuity condition in Assumption 3 is equivalent to the following: If an → a, then ! ! ′ ′ f (s )Q(ds | s, an ) → f (s′ )Q(ds′ | s, a) S

S

for each bounded measurable function f . Then Hi (s, an ) → Hi (s, a) if an → a. This gives that Hi (s, a) is continuous in a. By Lemma 4.2, for each s ∈ S the stochastic game model {N , {s} , (Φi (s), Hi (s, ·))i∈N } has a Nash equilibrium.

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Carlos Gabriel Pacheco Gonz´alez

Since it is possible to find equilibria for each s ∈ S , consider the multifunction Θ : S ! Π which assigns the set of equilibria points to each s ∈ S, i.e.

(17)

∗ Θ(s) := {π ∗ ∈ Π(s) : Hi (s, π ∗ ) ≥ Hi (s, (πi , π−i ))

∀ πi ∈ P(Φi (s)), i = 1, ..., N } or equivalently (18)

Θ(s) := {π ∗ ∈ Π(s) : Hi (s, π ∗ ) =

sup πi ∈P(Φi (s))

∗ )), Hi (s, (πi , π−i

i = 1, ..., N } Before proceeding with the next step, we make the following remark: without loss of generality we may assume that ri (s, a) = θ − 1 if a ∈ / Φ(s) for each i ∈ N , ∗ ) ∈ Θ(s) then where θ := inf i,s,a ri (s, a). So, if π ∗ := (π1∗ , ..., πN

(19)

∗ ri (s, π ∗ ) ≥ ri (s, (µi , π−i )) ∀ µi ∈ P(Ai ),

and, moreover, if θ = min(inf i,s,a ri (s, a), inf i,s vi (s)), then (20)

∗ Hi (s, π ∗ ) ≥ Hi (s, (µi , π−i )) ∀ µi ∈ P(Ai ),

even if support(µi )∩Φi (s)c ̸= ∅. So Hi (s, π) is well defined for every π ∈ Π.

We use this remark in the next step.

Step 1: There is a measurable selector for Θ, i.e. a measurable function ξ : S → Π, such that ξ(s) ∈ Θ(s) for each s ∈ S. We want to use Theorem 5.11to show the existence of a measurable selection. To use such theorem, we need to prove that (i) Π satisfies the property S (Remark 5.10), (ii) Θ(s) is compact for each s ∈ S, and (iii) Θ−1 (F ) is a Borel set in S for every closed set F in Π.

Since Π is separable metric space (by Theorem 5.4), it is easy to see (i) holds; it suffices to take a countable dense subset of Π and the family of closed balls with radius a rational number and center an element of the countable dense set.

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Nash Equilibria

∗ , ..., π ∗ )) In order to prove (ii), let {πn∗ } ⊂ Θ(s) (where πn∗ := (π1n Nn ∗ ∗ ∗ be such that πn → π . Since πn ∈ Θ(s), for each n we have ∗ Hi (s, πn∗ ) ≥ Hi (s, (µi , π−in ))

(21)

∀ µi ∈ P(Ai ),

and, therefore, by Corollary 5.3, as n → ∞ we obtain ∗ Hi (s, π ∗ ) ≥ Hi (s, (µi , π−i ))

(22)

∀ µi ∈ P(Ai ).

It follows that π ∗ ∈ Θ(s), and so Θ(s) is a closed subset of the compact space Π(s). Hence Θ is a compact-valued multifunction.

To prove (iii), we use (18).

First note that Hi (·, ·) and supπi ∈P(Φi (s)) Hi (·, (πi , ·)) are jointly measurable, then the function F : S × Π → RN ,

(23) defined by F (s, π) :=

!

Hi (s, π) −

sup µi ∈P(Φi (s))

Hi (s, (µi , π−i ))

"

i=1,...,N

is jointly measurable.

Then, the set F −1 ((0, ..., 0)), with (0, ..., 0) ∈ RN is a Borel set. Finally, note that F −1 ((0, ..., 0)) = Gr(Θ); hence, by Theorem 5.9, (iii) holds.

Since the assumptions of Theorem 5.11 hold, a measurable selector for Θ exists. !

Proof: [Proof of Theorem 4.1] Consider T = 1. By assumption, ri (s, ·) is continuous on Φ(s) for each s, and ri is measurable on S × A. Thus,

by Lemma 4.3 , there exists a strategy π :=(π1 , ..., πN ) ∈ Π such that for each s ∈ S, ri (s, π) ≥ ri (s, (β, π−i )) ∀β ∈ P(Φi (s)) and

i ∈ N.

Denote π by π1 , and let νi (s) = ri (s, π1 ) (i = 1, ..., N ). Then π1 is a Nash equilibrium of the one period game {N , S, (Ai , Φi , ri )}, with νi

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Carlos Gabriel Pacheco Gonz´alez

(i = 1, ..., N ) the corresponding equilibrium payoffs. Now, let Hi (s, a) as in Lemma 4.3, then the stochastic game {N , S, (Ai , Φi (s), Hi (s, ·))i∈N } has an equilibrium. Denote π2 the equilibrium of this game, then (π1 , π2 ) is the equilibrium of {N , S, (Ai , Φi , ri )i∈N , Q, 2} .

We now proceed by induction. Suppose the existence of a Nash equilibrium strategy (πT −1 , ..., π1 ) of the stochastic game {N , S, (Ai , Φi , ri )i∈N , Q, T − 1} and let νi (s) be the corresponding discounted equilibrium payoff of player i (i = 1, ..., N ) and initial state s. By Lemma 4.3, the game {N , S, (Ai , Φi (s), Hi (s, ·))i∈N } admits an equilibrium πT ∈ Π. It follows that (πT , πT −1 , ..., π1 ) specifies an equilibrium of the stochastic game {N , S, (Ai , Φi , ri )i∈N , Q, T }, and that the discounted equilibrium payoff is given by !

Hi (s, a)πT (da | s), i = 1, ..., N. νi (s, πT ) = A

Note that if π = (πT , πT −1 , ..., π1 ), then νi (s, πT ) = Ji,δ,T (s, π).

Also note that π ∈ ΠM . This completes the proof. !

5

Appendix

In this section we summarize some facts used in the proof of Theorems 3.1 and 4.1.

The topology of weak convergence Definition 5.1 Let P(X) be the set of all probability measures on (X, B(X)) where X is a general metric space. The topology of weak

83

Nash Equilibria

convergence is the topology in the space P(X) which has the following basic neighborhoods of any element µ ∈ P(X), " "# $ ! # " " " " fi dµ" < ϵ, i = 1, ..., k Uϵ (µ, {f1 , ..., fk }) := λ ∈ P(X) : " fi dλ − S

S

where ε is positive and f1 , ..., fk are elements of C(X) (the space of continuous bounded functions on X).

The following are some useful results.

Lemma 5.2 Let (X, d) be a metric space. Let {µn } ⊂ P(X) and µ ∈ % µn → µ in the topology of weak convergence if and only if %P(X). Then f dµn → f dµ for all f ∈ C(X).

For a proof of Lemma 10 see, for instance, Proposition 7.21 in Bertsekas and Shreve [10].

Corollary 5.3 Let X1 , ..., XN be metric spaces. Let {µin } be a sequence in P(Xi ) and µi ∈ P(Xi ) for i = 1, ..., N. Each P(Xi ) has the topology of weak convergence. If µin → µi for i = 1, ..., N, then #

f dµ1n ...dµN n →

#

f dµ1 ...dµN ∀f ∈ C(X1 × ... × XN ).

Next, we have a result used in sections 3 and 4; for a proof see Proposition 7.22 in Bertsekas and Shreve [4].

Theorem 5.4 If (X, d) is a compact separable metric space, then the topology of weak convergence in P(X) is compact, separable and metrizable.

Kakutani´s Theorem If each point x of a space X is mapped into a nonempty set T (x) of a space Y , we call T a set-valued mapping, also known as a multifunction or correspondence. We write T : X ! Y to specify that it is a multifunction.

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Definition 5.5 Let T be a multifunction from a topological space X to a topological space Y. We say that T is a K−mapping of X into Y if i) for each x in X, T (x) ⊂ Y is a compact convex set; and ii) the graph of T, which defined as Gr(T ) = {(x, y) : y ∈ T (x)} , is closed in X × Y. If i) holds, then the condition ii) is equivalent to the upper semicontinuity (u.s.c.) condition, i.e. if xn → x in X, yn ∈ T (xn ) and yn → y, then y ∈ T (x).

Definition 5.6 A fixed point for a multifunction T : X ! X is a point x such that x ∈ T (x).

Theorem 5.7 (Kakutani theorem) If T : X ! X is a K−mapping, where X is a nonempty compact convex subset of Rm , then T has a fixed point.

See Smart [23], Chap. 9 for further details. Kakutani´s theorem was extended to Banach spaces by Bohnenblust and Karlin [5], and to locally convex spaces by Ky Fan [9] and Glicksberg [11].

Theorem 5.8 (Glicksberg theorem) Let T : X ! X be a K-mapping where X is a nonempty compact convex subset of a locally convex Hausdorff space. Then T has a fixed point.

The proof is in Corollary 16.51 in Aliprantis and Border [1].

Correspondence with measurable graph Let T : X ! Y be a multifunction and A ⊂ Y , then the lower inverse T −1 is defined by T −1 (A) := {x ∈ X : T (x) ∩ A ̸= ∅} .

Theorem 5.9 Let X and Y be nonempty Borel spaces. Let T : X ! Y be a compact-valued multifunction, then the following statements are equivalent: a) Gr(T ) is a Borel subset of X × Y ; b) T −1 (F ) is a Borel subset of X for every closed set F ⊂ Y. The proof is in Aliprantis and Border [1] Theorem 14.84 (see Proposition D.4 in Hernandez-Lerma and Lasserre [13] for reference).

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A selection theorem Remark 5.10 A topological space X is said to satisfy condition S if there is a countable family {Fn } of closed sets which separates the points of X, that is, if x and y are any distinct points of X, there is a set Fn which contains one of them but not both.

Condition S is trivially satisfied when X is a Borel space.

Theorem 5.11 (Selection theorem) Let (S, A) be a measurable space, and X be a topological space which satisfies condition S. If Θ : S ! X is a multifunction such that Θ(s) is a nonempty compact set of X and Θ−1 (F ) ∈ A for every closed set F in X, then there is a measurable selector ξ for Θ (that is, a measurable function from S to X with ξ(s) ∈ Θ(s) for each s ∈ S).

See Leese [14] for the proof. For further details see Wagner [26].

Carlos Gabriel Pacheco Gonz´ alez Departamento de Mathem´ aticas, CINVESTAV-IPN, Apartado Postal 14–740, 07000 M´exico D. F., MEXICO cpacheco@math.cinvestav.mx

References [1] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis, Springer-Verlag, Heidelberg, 1999.

[2] Amir, R., On stochastic games with uncountable state and action spaces, in Stochastic Games and Related Topics (Raghavan, T. E. S.; Ferguson, T. S.; Parthasarathy, T.; Vrieze, O., eds.), Kluwer, Boston, 1991.

[3] Bertrand, J., Th´eorie math´ematique de la richesse sociale, Journal des Savants (1883), 499-508.

[4] Bertsekas, D. P.; Shreve, S. E., Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York, 1978.

[5] Bohnenblust, H. F.; Karlin, S., On a theorem of Ville, in Contributions to the Theory of Games (Kuhn H. W.; Tuker, A. W., eds.), Vol. I, 155-60, Princeton University Press, 1950.

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[6] Cournot, A., Recherches sur les Principes Mathematiques de la Theorie des Richesses. English edition (Bacon, N., ed.): Researches into the Mathematical Principles of the Theory of Wealth, Macmillan, 1897.

[7] Dutta, P. K.; Sundaram, R. K., The equilibrium existence problem in general markovian games, in Organizations with Incomplete Information: Essays in Economics Analysis (M. Majumdar, ed.), Cambridge University Press, 1998.

[8] Edgeworth, F., La teoria pura del monopolio, Giornale degli Economisti (1897), 13-31.

[9] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-6.

[10] Filar, J.; Vrieze, K., Competitive Markov Decision Processes, Springer-Verlag, New York, 1997.

[11] Glicksberg, I., A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer.

Math. Soc., 3 (1952), 170-174.

[12] Hern´andez Hern´ andez, H., Existence of Nash equilibria in discounted nonzero-sum stochastic games with additive structure, Tesis de Maestria, CINVESTAV-IPN, M´exico D.F, 2002.

[13] Hern´andez-Lerma, O.; Lasserre, J. B., Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996.

[14] Leese, S. J., Measurable selections and the uniformization of Souslin sets, Amer. J. Math., 100 (1978), no. 1, 19-41.

[15] Nash, J., Equilibrium points in N-person games, Proc. Nat. Acad.

Sci. U.S.A., 36 (1950), 48-49.

[16] Nowak, A. S., Optimal strategies in a class of zero-sum ergodic stochastic games, Math. Meth. Oper. Res., 50 (1999), 399-419.

[17] Parthasarathy, T.; Sinha, S., Existence of equilibrium in discounted stochastic games, Shankhya Series A, 44 (1982), 114-127.

[18] Puterman, M. L., Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley & Sons, New York, 1994.

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[19] Ram´ırez Reyes, F., Existence of optimal strategies for zero-sum stochastic games with discounted payoff, Morfismos, Vol. 5, No. 1 (2001), 63-83.

[20] Rieder, U., Equilibrium plans for non-zero sum Markov games, in Seminar on Game Theory and Related Topics (Moeschlin, O.; Pallaschke, D., eds.), Springer, Berlin, 1979.

[21] Rogers, P. D., Non-zero sum stochastic games, ORC Report no. 698, Operations Research Center, Univesity of California, Berkeley, 1969.

[22] Royden, H. L., Real Analysis, Macmillan, New York, 1988.

[23] Smart, D. R., Fixed Point Theorems, Cambridge University Press, 1974.

[24] Sobel, M. J., Non-cooperative stochastic games, Ann. Math. Stat., 42 (1971), 1930-1935.

[25] Von Neumann J.; Morgenstern, O., Theory of Games and Economic Behavior, Princeton University Press, 1944.

[26] Wagner, D., Survey of measurable selection theorems, SIAM J. Control Optim., 15 (1977), 859-1003.

MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matem´aticas del CINVESTAV, se termin´ o de imprimir en el mes de enero de 2003 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contenido Approximation on arcs and dendrites going to infinity in C n (Extended version) Paul M. Gauthier and E. S. Zeron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Bayesian procedures for pricing contingent claims: Prior information on volatility Francisco Venegas-Mart´ınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Existence of Nash equilibria in discounted nonzero-sum stochastic games with additive structure Heriberto Herna ´ndez-Herna ´ndez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Existence of Nash equilibria in some Markov games with discounted payoff Carlos Gabriel Pacheco Gonz´ alez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67