Insights on the Theory of Robust Games

Restricting the attention to static games, we consider the problem of ambiguity generated by uncertain values of players’ payoff functions. Uncertainty is represented by a bounded set of possible realizations, and the level of uncertainty is parametrized in such a way that zero means no uncertainty, the so-called nominal counterpart game, and one the maximum uncertainty. Assuming that agents are ambiguity averse and they adopt the worst-case optimization approach to uncertainty, we employ the robust-optimization techniques to obtain a so-called robust game, see Aghassi and Bertsimas (2006). A robust game is a distribution-free model to handle ambiguity in a conservative way. The equilibria of this game are called robustoptimization equilibria and the existence is guaranteed by standard regularity conditions. The paper investigates the sensitivity to the level of uncertainty of the equilibrium outputs of a robust game. Defining the opportunity cost of uncertainty as the extra profit that a player would obtain by reducing his level of uncertainty, keeping fixed the actions of the opponents, we prove that a robust-optimization equilibrium is an e-Nash equilibrium of the nominal counterpart game where the -approximation measures the opportunity cost of uncertainty. Moreover, considering an e-Nash equilibrium of a nominal game, we prove that it is always possible to introduce uncertainty such that the e-Nash equilibrium is a robust-optimization equilibrium. Under some regularity conditions on the payoff functions, we also show that a robust-optimization equilibrium converges smoothly towards a Nash equilibrium of the nominal counterpart game, when the level of uncertainty vanishes. Despite these analogies, the equilibrium outputs of a robust game may be qualitatively different from the ones of the nominal counterpart. An example shows that a robust Cournot duopoly model can admit multiple and asymmetric robust-optimization equilibria despite the nominal counterpart is a simple symmetric game with linear-quadratic payoff functions for which only a symmetric Nash equilibrium exists