Equilibrium for max-plus payoff
2026-03-05 • Computer Science and Game Theory
Computer Science and Game Theory
AI summaryⓘ
The authors study games where players are unsure about outcomes, and both their beliefs and strategies aren't like usual probabilities but use a more flexible math tool called capacities. They look at two types of game solutions: one where players mix their strategies with capacities, and another where they pick pure strategies but use non-standard beliefs to judge payoffs. For games with certain nice properties, the authors prove these solutions exist by applying advanced math involving fixed points and generalized notions of convexity.
non-cooperative gamescapacitiesnon-additive measuresNash equilibriumDow and Werlang equilibriummax-plus integralsabstract convexityKakutani fixed point theoremcompact strategy spacescontinuous payoffs
Authors
Taras Radul
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.