Ex-Post Regret Learning in Games with Fixed and Random Matching: The Case of Private Values

In contexts in which players have no priors, we analyze a learning pro- cess based on ex-post regret as a guide to understand how to play games of incomplete information under private values. The conclusions depend on whether players interact within a fixed set (fixed matching) or they are ran- domly matched to play the game (random matching). The relevant long run predictions are minimal sets that are closed under “the same or better reply” operations. Under additional assumptions in each case, the prediction boils down to pure Nash equilibria, pure ex-post equilibria or pure minimax regret equilibria. These three paradigms exhibit nice robustness properties in the sense that they are independent of beliefs about the exogenous uncertainty of type spaces. The results are illustrated in second-price auctions, first-price auctions and Bertrand duopolies.