Distributionally Robust Games via Coherent Risk Measures

TL;DR

This paper introduces a framework for analyzing strategic interactions under uncertainty using distributionally robust games with coherent risk measures, highlighting existence, complexity, and computational methods.

Contribution

It develops a unified approach to risk-aware game modeling, proving existence of equilibria, analyzing complexity, and providing computational formulations for data-driven environments.

Findings

Existence of equilibria in distributionally robust games is established.

Games are inherently continuous, affecting equilibrium concepts.

The computational complexity of finding equilibria is PPAD-complete.

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.