Myopic equilibria, the spanning property, and subgame bundles

TL;DR

This paper introduces the concept of myopic equilibria and demonstrates their existence under the spanning property, extending the Kohberg-Mertens Structure Theorem with applications in game theory and evolutionary models.

Contribution

It extends the theory of equilibria by establishing the existence of myopic equilibria under the spanning property and explores their applications in various game-theoretic contexts.

Findings

Myopic equilibria exist when payoffs satisfy the spanning property.

The spanning property is preserved under relevant operations on set-valued functions.

Applications include analysis of repeated games, behavior strategies, and evolutionary game theory.

Abstract

For a set-valued function $F$ on a compact subset $W$ of a manifold, spanning is a topological property that implies that $F(x) \ne 0$ for interior points $x$ of $W$. A myopic equilibrium applies when for each action there is a payoff whose functional value is not necessarily affine in the strategy space. We show that if the payoffs satisfy the spanning property, then there exist a myopic equilibrium (though not necessarily a Nash equilibrium). Furthermore, given a parametrized collection of games and the spanning property to the structure of payoffs in that collection, the resulting myopic equilibria and their payoffs have the spanning property with respect to that parametrization. This is a far reaching extension of the Kohberg-Mertens Structure Theorem. There are at least four useful applications, when payoffs are exogenous to a finite game tree (for example a finitely repeated game followed by an infinitely repeated game), when one wants to understand a game strategically entirely with behaviour strategies, when one wants to extends the subgame concept to subsets of a game tree that are known in common, and for evolutionary game theory. The proofs involve new topological results asserting that spanning is preserved by relevant operations on set-valued functions.