[Paper Review] Distributionally Robust Game Theory

Nicolas Loizou|arXiv (Cornell University)|Dec 10, 2015

Economic theories and models45 references3 citations

TL;DR

This paper introduces Distributionally Robust Game Theory, a novel framework that generalizes Nash, Bayesian, and robust games by modeling payoff uncertainty through an ambiguity set of probability distributions. It proves that equilibria can be computed as the component-wise projection of solutions to a multi-linear system, enabling a unified treatment of incomplete-information games with varying risk attitudes via Conditional Value-at-Risk (CVaR).

ABSTRACT

The classical, complete-information two-player games assume that the problem data (in particular the payoff matrix) is known exactly by both players. In a now famous result, Nash has shown that any such game has an equilibrium in mixed strategies. This result was later extended to a class of incomplete-information two-player games by Harsanyi, who assumed that the payoff matrix is not known exactly but rather represents a random variable that is governed by a probability distribution known to both players. In 2006, Bertsimas and Aghassi proposed a new class of distribution-free two-player games where the payoff matrix is only known to belong to a given uncertainty set. This model relaxes the distributional assumptions of Harsanyi's Bayesian games, and it gives rise to an alternative distribution-free equilibrium concept. In this thesis we present a new model of incomplete information games without private information in which the players use a distributionally robust optimization approach to cope with the payoff uncertainty. With some specific restrictions, we show that our "Distributionally Robust Game" constitutes a true generalization of the three aforementioned finite games (Nash games, Bayesian Games and Robust Games). Subsequently, we prove that the set of equilibria of an arbitrary distributionally robust game with specified ambiguity set can be computed as the component-wise projection of the solution set of a multi-linear system of equations and inequalities. Finally, we demonstrate the applicability of our new model of games and highlight its importance.

Motivation & Objective

• To unify classical game-theoretic models—Nash, Bayesian, and robust games—under a single distributionally robust framework.
• To model payoff uncertainty not by assuming a known distribution (as in Bayesian games) or a fixed worst-case scenario (as in robust games), but via an ambiguity set of plausible distributions.
• To incorporate heterogeneous risk attitudes among players using Conditional Value-at-Risk (CVaR), making the model more realistic for practical applications.
• To establish a computational method for identifying equilibria in such games through a multi-linear system of equations and inequalities.
• To demonstrate the model’s applicability through two concrete examples: the Free Rider Game and the Inspection Game.

Proposed method

• Formulates a distributionally robust game model where players minimize their worst-case CVaR over an ambiguity set of payoff distributions.
• Defines the ambiguity set using known moment information (e.g., mean and support), ensuring all distributions in the set are consistent with observed data.
• Uses a multi-linear system of equations and inequalities to characterize the set of equilibria, with equilibria obtained as component-wise projections of the solution set.
• Applies the CVaR risk measure to allow players to express different levels of risk aversion, parameterized by a risk level ε.
• Employs numerical algorithms to approximately compute equilibria, with convergence and performance evaluated on test cases.
• Extends the robust game framework by allowing distributional uncertainty rather than just parameter uncertainty, enabling richer modeling of ambiguity.

Experimental results

Research questions

• RQ1 Can a single game-theoretic framework unify Nash, Bayesian, and robust games under a common distributionally robust optimization approach?
• RQ2 How do varying risk attitudes—modeled via CVaR—impact equilibrium outcomes in games with payoff uncertainty?
• RQ3 Under what conditions does the distributionally robust game model reduce to classical game-theoretic models (e.g., Nash or Bayesian games)?
• RQ4 Can the set of equilibria in a distributionally robust game be systematically computed using a well-defined mathematical structure?
• RQ5 How do changes in the ambiguity set’s parameters (e.g., mean vector, maximum distance) affect equilibrium existence and payoffs?

Key findings

• The distributionally robust game model generalizes Nash, Bayesian, and robust games, meaning any such game can be represented as a special case of the proposed framework.
• Equilibria in distributionally robust games can be computed as the component-wise projection of the solution set of a multi-linear system of equations and inequalities.
• For specific ambiguity sets (e.g., support with a single point), the model reduces to a complete-information finite game, confirming consistency with classical Nash games.
• In the Distributionally Robust Free Rider Game, equilibria and payoffs remain invariant across different risk levels when the ambiguity set is constrained (e.g., s=0), indicating robustness to risk attitude changes.
• In the Distributionally Robust Inspection Game, equilibria and payoffs are invariant across risk levels, suggesting that risk preferences do not alter outcomes under certain ambiguity set configurations.
• Numerical experiments show that varying risk levels (ε) affects equilibria and payoffs in general cases, confirming the model’s sensitivity to risk preferences when ambiguity is non-degenerate.