New results on existence of Nash equilibria J.J. Salamanca∗ Escuela Polit´ecnica de Ingenier´ıa, Departamento de Estad´ıstica e I.O. y D.M., Universidad de Oviedo, 33071 Gij´on, Spain,
[email protected]
Abstract In this work, it is obtained a new existence result of Nash equilibrium for noncooperative symmetric bipersonal games where any pure strategy space is compact.
Keywords: non-cooperative game; Nash equilibria; compact strategy space. MSC 2010: 91A10, 91A40, 91A44 In Game Theory, there exists a current branch in analyzing games where the pure strategy spaces are not a finite set. In this context, several results have been obtained, for instance, [4], [7] or [8]. The main ingredients are some hypothesis on the continuity or differentiability of the payoff functions and some conditions on their convexity. Topology has been present even from the very origin of Game Theory; recall the Brower fixed point theorem which was used by Nash in order to assure existence of the Nash equilibria point of any game with finite pure strategy spaces. Other minimax or fixed point type theorems have been used to obtain related results. Ichiishi [3] proved Theorem 1 Let Γ be a non-cooperative n-person game where any strategy space is a nonempty, convex, compact subset of the Euclidean space. Assume also that any payoff function uj is continuous and satisfies: uj (xj , y) is quasi-concave for y ∈ Xj given any xj ∈ X − Xj . Then there exists a Nash equilibrium of the game. In the same setting, H. Nikaido and K. Isoda proved that a convex game always has at least one equilibrium point [5]. Then, a lot of work have been done to extend and analyze new games in this direction (see, for instance, [4], [7], [8] and references therein). We will see that in case of certain class of pure strategy spaces, we can avoid the assumption of convexity. However, it should be required more than continuity on the payoff functions. ∗
The author is partially supported by the Spanish MEC-FEDER Grant MTM2010-18099.
Let Γ = {X1 , . . . , Xn ; f1 , . . . , fn } be a non-cooperative n-personal game. For Player i (i = 1, . . . , n), the (non-empty) set Xi models his (pure) strategy space, and the function fi : X1 × . . . Xn → R represents his payoff function. We will focus on compact (smooth) manifold strategy spaces. Moreover, we will assume that we have two players both having the same strategy space X. Now, we regard to a geometric invariant of a compact manifold X: its Euler-Poincar´e characteristic χ(X) (see [1] and [2], for instance). This topological quantity provides an important property: a compact manifold with non-vanishing Euler-Poincar´e characteristic admits no vector field which is null at no point (see, for instance, [1, Cor. 23.6]). In particular, the topological even-dimensional spheres have non-vanishing Euler-Poincar´e characteristic. We will endow X with a metric structure g in order to get a Riemannian manifold (X, g) [6]. In this setting, recall the cut locus of p ∈ X in Tp X: it is the set of Tp X where the exponential map is a diffeomorphism [6, C. 6]. Regarding the payoff functions, we will require that the game is symmetric. That is, f1 (p, q) = f2 (q, p). Roughly speaking, Player 1 and 2 are interchangeable. However, we need an extra assumption: for a player, whatever his opponents play, there exists a unique optimum choice. Now, we are in position to state, Theorem 2 Let Γ be a non-cooperative symmetric bipersonal game where both pure strategy spaces are a compact manifold with non-vanishing Euler-Poincar´e characteristic, X. Assume that the payoff functions are differentiable and for any choice of a player, p ∈ X, there exists a unique optimum choice q ∈ X for the other player such that exp−1 p (q) lies on the cut locus of p in Tp X. Then, there exists a point pe ∈ X such that if both players choose pe, a Nash equilibrium is found. Proof. At a point p ∈ X, representing a choice of Player 1, we have q ∈ X representing the best choice for Player 2. Define exp−1 p (q) ∈ Tp X. This is possible by assumptions. This leads to a vector field on X. This vector field is C ∞ due to differentiability of the payoff functions. Since X has non-vanishing Euler-Poincar´e characteristic, this vector field must have a zero. That is, there exists a choice pe of Player 1 such that Player 2 must choose also pe. Since the game is symmetric, if Player 2 chooses pe, Player 1 must also choose this point. We easily find that a Nash equilibrium point exists when both players choose pe. Endowing the even-dimensional spheres with the usual round metric, we obtain as a consequence, Corollary 3 Let Γ be a non-cooperative symmetric bipersonal game where both pure strategy spaces are an even-dimensional sphere S2m . Assume that the payoff functions are differentiable. Moreover, for any choice of a player, p ∈ S2m , his opponent attains his best choice at a unique point which is not the antipodal point of p. Then, there exists a point pe ∈ S2m such that if both players choose pe, a Nash equilibrium is found. Remark 4 Let us illustrate that the assumptions are necessary providing several examples. 2
(a) Consider as strategy spaces S1 . Endow this space with the canonical metric, (S1 , dθ2 ). Let us consider the payoff function of Player 1: f1 (θ1 , θ2 ) = sin(θ1 − θ2 ). In order to be symmetric the game, consider the following payoff function of Player 2: f2 (θ1 , θ2 ) = f1 (θ2 , θ1 ). For the choice θ1 of Player 1, Player 2 must choose θ2 = θ1 − 3π/2 (lying in the corresponding cut locus). For the choice θ2 of Player 2, Player 1 must choose θ2 + π/2. We see that no Nash equilibrium is found. This example shows that the topological assumption is needed. (b) Consider as strategy spaces S2 . Endow this space with the canonical round metric, 2 (S , gS2 ). Consider the payoff function f1 = −f2 = dS2 (θ1 , θ2 ), where dS2 is the metric distance. Clearly, for any choice of Player 1, Player 2 must choose the antipodal point. For any choice of Player 2, Player 1 must choose the same point. In this case, we see that no Nash equilibrium is found. This example shows that the symmetry assumption is needed. Acknowledgements.- This paper is dedicated to Luis´ın, who enlighten my life.
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