Insights on the Theory of Robust Games

Abstract

AbstractA 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 focusing on robust games with no private information. Specifically, we prove that a robust-optimization equilibrium is an $$\epsilon $$ ϵ -Nash equilibrium of the nominal counterpart game, where $$\epsilon $$ ϵ measures the extra profit that a player would obtain by reducing his level of uncertainty. Moreover, given an $$\epsilon $$ ϵ -Nash equilibrium of a nominal game, we prove that it is always possible to introduce uncertainty such that the $$\epsilon $$ ϵ -Nash equilibrium is a robust-optimization equilibrium. These theoretical insights increase our understanding on how uncertainty impacts on the solutions of a robust game. Solutions that can be extremely sensitive to the level of uncertainty as the worst-case approach introduces non-linearity in the payoff functions. 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.