Mechanism with unique learnable equilibria

Abstract

The existence of a unique equilibrium is the classic tool for ensuring predictiveness of game theory. Typical uniqueness results, however, are for Nash and Bayes-Nash equilibria and do not guarantee that natural game playing dynamic converges to this equilibrium. In fact, there are well known examples in which the equilibrium is unique, yet natural learning behavior does not converge to it. Motivated by this, we strive for stronger uniqueness results. We do not only require that there is a unique equilibrium, but also that this equilibrium must be learnable. We adopt correlated equilibrium as our solution concept, as simple and natural learning algorithms guarantee that the empirical distribution of play converges to the space of correlated equilibria. Our main result is to show uniqueness of correlated equilibria in a large class of single-parameter mechanisms with matroid structure. We also show that our uniqueness result extends to problems with polymatroid structure under some conditions. Our model includes a number of special cases interesting on their own right, such as procurement auctions and Bertrand competitions. An interesting feature of our model is that we do not need to assume that the players have quasi-linear utilities, and hence can incorporate models with risk averse players and certain forms of externalities.