Distributionally Robust Games via Coherent Risk Measures

2026-05-18

AI Summary

This work addresses the uncertainty in payoff distributions arising from limited samples in data-driven games by proposing a distributionally robust game-theoretic framework grounded in coherent risk measures—such as Conditional Value-at-Risk and mean-semideviation—that internalize risk sensitivity as players’ preferences. It establishes, for the first time, a theoretical link between risk-awareness and distributional robustness. Leveraging multilinear complementarity programming and PPAD complexity analysis, the study proves the existence of equilibria under various ambiguity sets, characterizes their computational complexity, and quantifies the utility loss induced by risk aversion. Numerical experiments demonstrate that the proposed solutions exhibit superior out-of-sample performance and robustness.

Abstract

We study strategic interaction in data-driven games where players face uncertainty about payoff distributions inferred from finite samples. To model calibrated attitudes toward such uncertainty, we formulate distributionally robust games with a special focus on coherent utility (risk) measures, including Mean-semideviation and Conditional Value-at-Risk. This framework treats risk sensitivity as a primitive feature of player preferences while retaining a formal connection to distributional robustness. We make a number of contributions that are enumerated next. (1) We use prior results for the existence of distributionally robust equilibria to show the existence of equilibria in data-driven settings for various ambiguity sets, and (2) show that these games are inherently continuous, rather than finite matrix games, which fundamentally alters equilibrium structure and precludes direct extensions of standard correlated equilibrium notions. (3) We bound the loss in expected utility that a player can expect from being risk-averse. (4) We further characterize the computational complexity of equilibrium computation, proving PPAD-completeness in general and PPAD membership for several coherent utility measure games. (5) We present multilinear complementarity program formulations for several coherent utility measure games. (6) Numerical experiments reveal the robustness and out of sample performance of the game solutions. Our results unify risk-theoretic modeling and equilibrium analysis, providing a principled foundation for risk-aware strategic decision-making in data-driven environments.

Problem

• distributionally robust games
• coherent risk measures
• strategic interaction
• distributional uncertainty
• risk-sensitive preferences

Innovation

• Distributionally Robust Games
• Coherent Risk Measures
• PPAD-completeness
• Multilinear Complementarity Programming
• Data-driven Equilibrium