Insights on the Theory of Robust Games

TL;DR

This paper explores the sensitivity of robust-optimization equilibria in robust games, showing their relation to epsilon-Nash equilibria and how uncertainty can induce multiple equilibria, with implications for economic models.

Contribution

It establishes the connection between robust-optimization equilibria and epsilon-Nash equilibria, and demonstrates how uncertainty can generate multiple equilibria in robust games.

Findings

Robust-optimization equilibrium is an epsilon-Nash equilibrium of the nominal game.

Any epsilon-Nash equilibrium can be realized as a robust-optimization equilibrium with appropriate uncertainty.

A robust Cournot duopoly can have multiple asymmetric equilibria despite symmetric Nash equilibrium in the nominal game.

Abstract

A robust game is a distribution-free model to handle ambiguity generated by a bounded set of possible realizations of the values of players' payoff functions. The players are worst-case optimizers and a solution, called robust-optimization equilibrium, is guaranteed by standard regularity conditions. The paper investigates the sensitivity to the level of uncertainty of this equilibrium. Specifically, we prove that it is an epsilon-Nash equilibrium of the nominal counterpart game, where the epsilon-approximation measures the extra profit that a player would obtain by reducing his level of uncertainty. Moreover, given an epsilon-Nash equilibrium of a nominal game, we prove that it is always possible to introduce uncertainty such that the epsilon-Nash equilibrium is a robust-optimization equilibrium. An example shows that a robust Cournot duopoly model can admit multiple and asymmetric robust-optimization equilibria despite only a symmetric Nash equilibrium exists for the nominal counterpart game.