Nash Equilibrium and Bisimulation Invariance

TL;DR

This paper investigates how bisimilarity affects Nash equilibria in concurrent games, revealing that standard models do not preserve equilibrium existence under behavioral equivalence, and proposes alternative strategy models that do.

Contribution

The paper identifies the non-preservation of Nash equilibria under bisimilarity in standard models and introduces new strategy models that ensure preservation, enhancing the semantic foundations of strategic logics.

Findings

Bisimilarity does not preserve Nash equilibria in standard strategy models.

Alternative strategy models can ensure equilibrium preservation under bisimilarity.

New semantic frameworks for strategic reasoning are proposed.

Abstract

Game theory provides a well-established framework for the analysis of concurrent and multi-agent systems. The basic idea is that concurrent processes (agents) can be understood as corresponding to players in a game; plays represent the possible computation runs of the system; and strategies define the behaviour of agents. Typically, strategies are modelled as functions from sequences of system states to player actions. Analysing a system in such a setting involves computing the set of (Nash) equilibria in the concurrent game. However, we show that, with respect to the above model of strategies (arguably, the "standard" model in the computer science literature), bisimilarity does not preserve the existence of Nash equilibria. Thus, two concurrent games which are behaviourally equivalent from a semantic perspective, and which from a logical perspective satisfy the same temporal logic formulae, may nevertheless have fundamentally different properties (solutions) from a game theoretic perspective. Our aim in this paper is to explore the issues raised by this discovery. After illustrating the issue by way of a motivating example, we present three models of strategies with respect to which the existence of Nash equilibria is preserved under bisimilarity. We use some of these models of strategies to provide new semantic foundations for logics for strategic reasoning, and investigate restricted scenarios where bisimilarity can be shown to preserve the existence of Nash equilibria with respect to the conventional model of strategies in the computer science literature.