Mechanism Design with Set-Theoretic Beliefs

In settings of incomplete information, we put forward (1) a very conservative —indeed, purely set-theoretic — model of the beliefs (including totally wrong ones) that each player may have about the payoff types of his opponents, and (2) a new and robust solution concept, based on mutual belief of rationality, capable of leveraging such conservative beliefs. We exemplify the applicability of our new approach for single-good auctions. In particular we show that, under our solution concept, there exists a simple normal-form mechanism, which always sells the good, always has non-negative revenue, and guarantees (up to an arbitrarily small, additive constant) a revenue benchmark that is always greater than or equal to the second-highest valuation, and sometimes much greater. By contrast, we also prove that the same benchmark cannot even be approximated within any positive factor, under classical solution concepts. ∗An earlier version of this work has been presented at the Symposium on Foundations of Computer Science, 2011. The present version is under review at the Journal of Economic Theory (special issue dedicated to the interface between economics and computer science). We would like to thank Gabriel Carroll, Robert Kleinberg, and Ronald Rivest for discussions that motivated us to prove results stronger than the ones we originally had. We also would like to thank Amos Fiat and Anna Karlin for helping us improve the presentation of our results. Many thanks also to Andrés Perea for helping us understand beliefs in economic settings, to Paul Milgrom for helping us clarify the fragility of implementation at equilibria, and to Elchanan Ben-Porath, Sergiu Hart, and Philip Reny for helping us clarify our connections to ex-post equilibria. (Any remaining lack of clarity is solely due to us!) This work is supported in part by ONR Grant No. N00014-09-1-0597.