Equilibrium for max-plus payoff

TL;DR

This paper explores equilibrium concepts in non-cooperative games under uncertainty using non-additive measures and max-plus integrals, establishing existence results for these equilibria.

Contribution

It introduces equilibrium notions based on capacities and max-plus integrals, extending classical game theory to non-additive belief frameworks.

Findings

Existence of Nash equilibrium in capacities established.

Existence of equilibrium under uncertainty proven.

Uses abstract convexity and fixed point theorems for proofs.

Abstract

We study equilibrium concepts in non-cooperative games under uncertainty where both beliefs and mixed strategies are represented by non-additive measures (capacities). In contrast to the classical Nash framework based on additive probabilities and linear convexity, we employ capacities and max-plus integrals to model qualitative and idempotent decision criteria. Two equilibrium notions are investigated: Nash equilibrium in mixed strategies expressed by capacities, and equilibrium under uncertainty in the sense of Dow and Werlang, where players choose pure strategies but evaluate payoffs with respect to non-additive beliefs. For games with compact strategy spaces and continuous payoffs, we establish existence results for both equilibrium concepts using abstract convexity techniques and a Kakutani-type fixed point theorem.