on the existence of mixed strategy equilibria of two ...

ON THE EXISTENCE OF MIXED STRATEGY EQUILIBRIA OF TWO PERSON GAMES WITH INCOMPLETE INFORMATION Hiroshi Tateishi School of Economics, Nagoya University Keywords: games with incomplete information, mixed strategy equilibrium, Young measure, fiber product JEL classification: C62, C72 Abstract:

This paper is concerned with the existence of mixed strategy

equilibria of games with incomplete information. Since the work (Milgrom and Weber 1985), it is common to assume that the regularity condition with respect to the prior that the prior is absolutely continuous with respect to the product of marginal probabilities. We offer an alternative set of assumptions without the regularity condition which ensure the existence of mixed strategy equilibria. 1. Introduction. Although the Glicksberg’s mixed strategy existence theorem (Glicksberg 1952) which is a generalization of the theorem due to (Nash 1950) to a Hausdorff locally convex topological space, is useful for establishing the existence of equilibria of non-cooperative games, it can not apply directly to the games with incomplete information. The stumbling block is the continuity of each players’ payoff functions. In the non-cooperative games with complete information, each player plays independently and this induces the continuity property of the payoff functions. But, in the setting of incomplete information, each player’s strategies can be correlated to each other. Thus, without additional constraints, the continuity property of payoff functions does not necessarily hold. Consequently, in the work (Milgrom and Weber 1985), the so-called regularity condition on the prior was assumed. This condition states that the prior is absolutely continuous with respect to

the product of marginal probabilities. Although this condition on the prior makes each player’s payoff function continuous, there exist cases where this does not hold. The most simple example is the case of perfect correlation, that is the case that the nature informs each player the identical type signal. So it is natural to consider the existence of equilibria without having recourse to the condition of Milgrom and Weber. The aim of this note is to establish the existence of equilibria of games with incomplete information without the regularity assumption. Instead impose the condition on the one of the two players’ type set that it is denumerable. 2. Preliminaries. In this preliminaries, we shall explain the property of the so-called Young measures to the extent that we will make use of them in the following section to establish the existence of equilibria of games with incomplete information (for the systematic exposition of the concept of Young measures, see, e.g., (Valadier 1990) or (Warga 1972)). Let 𝛺, ℱ, 𝜇 be a probability measure space and 𝕊, 𝑑 a compact metric space. We denote by ℳ ! 𝕊 the space of bounded measures on 𝕊 and by ℳ!! 𝕊 the space of probability measures. The set ℳ ! 𝕊 and ℳ!! 𝕊 are endowed with the weak topology, that is, the coarsest topology which makes the maps continuous: 𝜈 ↦ 𝜈 𝑓 : 𝑓 ∈ 𝐶 𝕊 . A Young measure 𝜈 is a measurable family of probabilities on 𝕊, that is: (i) for each 𝐴 ∈ ℱ, the map 𝜔 ↦ 𝜈! 𝐴 is measurable, and (ii) for each 𝜔 ∈ 𝛺, 𝜈! ∈ ℳ!! 𝕊 . We denote by 𝑌 𝛺, ℝ the set of Young measures on 𝛺×𝕊 and the set of equivalence class of 𝜇-almost equal Young measures by 𝒴 𝛺, 𝕊 . A Caratheodory integrand is a measurable function 𝑓: 𝛺×𝕊 → ℝ such that (i) for each 𝜔 ∈ 𝛺, the map 𝑠 ↦ 𝑓(𝜔, 𝑠) is continuous, and (ii) there exists an integrable function 𝜑 such that 𝑓 𝜔, 𝑠

≤ 𝜑(𝜔) for all 𝜔, 𝑠 ∈ 𝛺×𝕊. We denote by 𝒢! (𝛺, 𝕊) the set of

all Caratheodory integrands. Remark that, since 𝕊 is compact, 𝒢! (𝛺, 𝕊) is isometrically isomorphi to the space 𝐿! = 𝐿! (𝛺, 𝐶 𝕊 ) of Bochner integrable functions defined on 𝛺 into 𝐶(𝕊) of continuous functions on 𝕊. The dual is

𝐿∞ = 𝐿∞ (𝛺, ℳ ! 𝕊 𝐶 𝕊 ) of bounded scalarly measurable functions defined on 𝛺 into ℳ ! (𝕊). Remark that the set 𝑌(𝛺, 𝕊) of Young measures can be considered as the subet of 𝐿∞ . Let us call the topology on 𝒴(𝛺, 𝕊) induced by the duality 𝜎(𝐿∞ , 𝐿! ) a

weak topology. This topology also can be defined by the coarsest topology such that all the maps continuous: 𝜈 ↦ 𝜈 𝑓 : 𝑓 ∈ 𝒢! (𝛺, 𝕊). Remark that the weak topology makes the set 𝒴(𝛺, 𝕊) a Hausdorff topological space. Furthermore, since 𝕊 is compact by assumption, the set 𝒴(𝛺; 𝕊) is also compact with respect to the weak topology. On the set 𝑌 𝛺, 𝕊 , we consider the topology of pointwise convergence. This topology is induced by the family of semidistences 𝜑 !,! : 𝒮! → ℝ! ; 𝜔 ∈ 𝛺, 𝑓 ∈ 𝐶(𝕊) defined by 𝜑 !,! : 𝜈 ↦ |𝜈! 𝑓 |. Then, the following theorem holds true. Theorem 1.

Let 𝛺 be denumerable and 𝕊 a compact metric space. Then,

the set of Young measures 𝑌(𝛺, 𝕊) is compact with respect to the topology of pointwise convergence. Proof.

The compactness of ℳ!! (𝕊) with respect to the weak topology

implies that the set 𝐷 = 𝛱!∈! ℳ!! (𝕊) is compact with respect to the produt topology by virtue of the Tychonoff’s theorem. 𝐷 is the set of all functions defined on 𝛺 into ℳ!! (𝕊) and 𝑌 𝛺, 𝕊 ⊂ 𝐷. Thus, it is sufficient to prove that 𝑌(𝛺, 𝕊) is a closed subset of 𝐷 with respect to the product topology. Remark that, since the set 𝛺 is denumerable, the topology of pointwise convergence is metrizable and the limit of the pointwise convergence of a sequence of measurable functions is also measurable. That is, when a sequence 𝜈 ! ∈ 𝑌(𝛺, 𝕊) converges pointwisely to 𝜈 ∗ ∈ 𝐿, the map 𝜔 ↦ 𝜈!∗ : 𝛺 → ℳ!! (𝕊) is measurable. Hence 𝜈 ∗ ∈ 𝑌(𝛺, 𝕊) and this completes the proof of Theorem 1. ! The following propositions are crucial to establish the existence of equilibria of games with incomplete information. The first is a well-known fixed point theorem due to (Fan 1952) and (Glicksberg 1952).

Proposition 2 (Fan-Glicksberg) Let 𝑆 be a compact and convex subet of a Hausdorff locally convex topological vector space. Let 𝛤: 𝑆 ↠ 𝑆 be a compact and convex-valued correspondence (=multi-valued map) with closed graph. Then there exists a point 𝑠 ∈ 𝑆 such that 𝑠 ∈ 𝛤 𝑠 . The second is the so-called the fiber product lemma of Young measures. The following version appeared in (Tateishi 2002). Proposition 3 (Tateishi) Let (𝛺, ℱ, 𝜇) be a completely finite positive measure space, 𝕊! , 𝕊! be complete separable metric spaces and 𝜆!! ∈ 𝒴 𝛺, 𝕊! , 𝜆!! ∈ 𝒴 𝛺, 𝕊! 𝑛 ∈ ℕ ∪ ∞

. Let us define 𝜃 ∈ 𝒴(𝛺, 𝕊! ×𝕊! ) by ∞

𝜃 ! 𝜔 = 𝜆!! 𝜔 ⊗ 𝜆!! 𝜔 ; 𝜔 ∈ 𝛺 . Assume that 𝜆!! 𝜔 → 𝜆! (𝜔) weakly in ∞

ℳ!! (𝕊! ) for all 𝜔 ∈ 𝛺 and 𝜆!! → 𝜆!

weakly in 𝒴 𝛺, 𝕊! . Then 𝜃 ! → 𝜃 ∞

weakly in 𝒴(𝛺, 𝕊! ×𝕊! ). 3. Existence of mixed strategy equilibria This section is devoted to the existence of mixed strategy equilibria of two person games with incomplete information. The characters of the game 𝐺 is as follows: (a) 𝑇! is a set of types for player 𝑖 (𝑖 = 1,2). We assume that 𝑇! is denumerable and 𝑇! is equipped with the 𝜎-algebra 𝒯! . We set 𝑇 = 𝑇! ×𝑇! and 𝒯 = 𝒯! ⊗ 𝒯! . (b) 𝜇 is a prior probability on the measurable set (𝑇, 𝒯). Denote by 𝜇! the marginal probability of 𝜇 on 𝑇! . (c) 𝕊! is a compact metric space of actions for player 𝑖. (d) 𝑢! : 𝑇×𝕊! ×𝕊! → ℝ is a utility function for player 𝑖 satisfying the following conditions: (i) 𝑢! is 𝒯 ⊗ ℬ 𝕊! ⊗ ℬ(𝕊! )-measurable, (ii) for each 𝑡 ∈ 𝑇, the map 𝑠! , 𝑠! ↦ 𝑢! (𝑡, 𝑠! , 𝑠! ) is continuous, and (iii) there exists 𝜑: 𝑇 → ℝ such that 𝑢! 𝑡, 𝑠! , 𝑠!

≤ 𝜑(𝑡) for

all 𝑡, 𝑠! , 𝑠! ∈ 𝑇×𝕊! ×𝕊! . Remark that we do not assume that the regularity condition on 𝜇 that 𝜇 is absolutely continuous with respect to 𝜇! ⊗ 𝜇! . Instead of that, we assume that the type set for either one of two player is

denumerable. The (mixed) strategy et for player 1 is 𝑆! = 𝑌(𝑇! , 𝕊! ) and the strategy set for player 2 is 𝒮! = 𝒴(𝑇! , 𝕊! ). We define the payoff function 𝐸! : 𝑆! ×𝒮! → ℝ for player 𝑖  (= 1,2) as follows: 𝐸! 𝜈! , 𝜈! =

!

!! ×!!

𝑢! 𝑡, 𝑠! , 𝑠! 𝜈! 𝑡! , 𝑑𝑠! 𝜈! 𝑡! , 𝑑𝑠!

𝜇 𝑑𝑡 .

Then, we have the following theorem: Theorem 4

The game 𝐺 admits a mixed strategy equilibrium.

Proof. The proof makes use of the usual methods to establish the existence of Nash equilibria. Define the reaction function 𝛤!  (𝑖 = 1,2) as follows: 𝛤! 𝜈! = 𝜈! ∈ 𝑆! : 𝐸! 𝜈! , 𝜈! = max 𝐸! (𝜆! , 𝜈! ) !! ∈!!

𝛤! 𝜈! = 𝜈! ∈ 𝒮! : 𝐸! 𝜈! , 𝜈! = max 𝐸! (𝜈! , 𝜆! ) . !! ∈𝒮!

Then, the payoff function 𝐸! is continuous with respect to the topology of pointwise convergence on 𝑆! and the weak topology on 𝒮! thanks to Proposition 3. Furthermore, both of 𝑆! and 𝒮! are compact and convex subsets of Hausdorff locally convex spaces with the respective topology thanks to Theorem 1 and the properties of Young measures. Thus, the map 𝛤: 𝑆! ×𝒮! ¥↠ 𝑆! ×𝒮! defined by 𝛤 𝜈! , 𝜈! = 𝛤! 𝜈! ×𝛤! (𝜈! ) has a closed graph by virtue of Berge’s maximum theorem. It is certainly convex and compact-valued. Thus, we have a fixed point of 𝛤 thanks to Proposition 2. It is easily verified that the fixed point satisfies the assertion of Theorem 4. References Fan, Ky. 1952. “Fixed-Point and Minimax Theorems in Locally Convex Topological Linear Spaces.” Proceedings of the National Academy of

Sciences of the United States of America 38(2): 121–26.

Glicksberg, I L. 1952. “A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium Points.” Proceedings of

the American Mathematical Society 3(1): 170–74. Milgrom, Paul R, and Robert J Weber. 1985. “Distributional Strategies for Games with Incomplete Information.” Mathematics of Operations

Research 10(4): 619–32. Nash, John F. 1950. “Equilibrium Points in N-Person Games.” Proceedings

of the National Academy of Sciences of the United States of America 36(1): 48–49. Tateishi, H. 2002. “On the Existence of Equilibria of Equicontinuous Games with Incomplete Information.” Advances in Mathematical Economics 4: 41–59. Valadier, M. 1990. “Young Measures.” In Methods of Nonconvex Analysis, Berlin: Springer-Verlag, 154–88. Warga, Jack. 1972. Optimal Control of Differential and Functional

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