Nash Equilibria via Duality and Homological Selection

The main topological problem addressed in this paper is the following: Let X be a metric space and Sub k (X) denote the collection of subsets of X with at most k points equipped with the Hausdorff metric. Let Sym k (X) denote the k-fold symmetric product of X, also called the configuration space of k points (counted with multiplicity) in X. Given a multifunction s : X → Y , where Y is Sub k (X) or Sym k (X), does there exist a "homological" (in a sense to be made precise) selection? A related problem addressed in the paper is: Can one lift a map Z → Sub k (X) to a map Z → Sym k (X)?

This topological problem is used to establish existence of Nash equilibria by specializing X to a simplex. The property of admitting homological selection (Definition 2.5), in a similar context of establishing existence equilibria in certain games, has been already considered under the name spanning property (or property S) in [16] (see also [15,14]).

1.1. Preliminaries. We shall work in the framework of non-cooperative games with mixed strategies, where the domain for every player is a finite dimensional simplex. The cost functions (or alternately payoff functions) in traditional problems of Nash Equilibria are multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis that cost functions are multilinear. The original proof of existence of Nash Equilibria [13] uses fairly simple Algebraic Topology, namely the Brouwer or Kakutani Fixed Point Theorems. This approach has been refined in various ways [5,2,4] (see also [3] for an effective approach using a minimax technique). Our approach in this paper is quite different inasmuch as we use standard but more sophisticated tools from Algebraic Topology to establish existence of Nash Equilibria under considerably more general conditions on cost functions. The tools we use are basic intersection theory, Poincaré Duality and the Dold-Thom Theorem. We use the Dold-Thom Theorem to prove the existence of certain relative cycles contained in graphs of multifunctions. The chains thus constructed may be thought of as homological versions of selections [9,10,11]. This furnishes us with a Homological Selection Theorem 2.7, which might be of independent interest.

The conditions we use on the cost functions are detailed in Section 1.5. We give a brief sketch here for a 2-player non-cooperative game to stress the soft topological nature of the hypotheses used. Consider a game between players 1 and 2 with mixed strategy spaces A 1 , A 2 . We use the notation A -1 = A 2 and A -2 = A 1 .

A cost function r i (x, y) for player i (i = 1, 2) is said to be configurable if 1) the set of local minima of the best response multifunction R i (y) (for each y ∈ A -i ) of the function r i (x, y) (x ∈ A i ) is finite; and 2) Counted with multiplicity, the set of local minima R i (y) is continuous on A -i . A weakly configurable cost function is one whose set of local minima can be arbitrarily well-approximated by the set of local minima of a configurable function.

Two kinds of hypotheses will be relevant in this paper. (A1) For each i, the map R i is continuous. (A2) For each i, the map R i is weakly configurable.

The motivation for the assumption (A2) is given in Section 1.5. The main Theorem of this paper (Theorem 3.2) proves that assumption (A2) is sufficient to guarantee the existence of Nash equilibria. In Section 4 we shall give an example to show that assumption (A1) is not sufficient to guarantee the existence of Nash equilibria. We describe this counterexample in brief. It is easy to see that the multifunction f (z) = ± √ z from the unit disk ∆ in the complex plane to itself has no continuous selection. However the graph gr(f )(⊂ ∆ × ∆) does support a nonzero relative cycle in H 2 (∆ × ∆, ∂∆ × ∆) and hence admits a homological selection in our terminology (see Section 2). The counterexample in Section 4.1, which is a continuous map from D 2 to Sub 3 (D 2 ), shows that a continuous multifunction need not admit even a homological selection. (Here Sub 3 (D 2 ) denotes the collection of subsets of D 2 with at most 3 points equipped with the Hausdorff metric.) 1.2. Nash Equilibria. We refer to [1] for the basics of Game Theory. An Nperson non-cooperative game is determined by 2N objects (A 1 , . . . , A N , r 1 , . . . , r N ) where A i denotes the strategy space of player i and r i :

is the cost function for player i. We shall call A the total strategy space. Each A i will, for the purposes of this paper, be the space of probability measures on a finite set of cardinality (n i + 1) and hence homeomorphic to D ni . Thus, in Game Theory terminology each element of A i is a mixed strategy or equivalently a probability measure on a finite set. Thus, A i is the space of mixed strategies or player i. Vertices of the simplex A i are also referred to as pure strategies. A mixed strategy may therefore be regarded as a probability vector with (n i + 1) components. Each player i ind

This content is AI-processed based on open access ArXiv data.