Epistemic Implementation and The Arbitrary-Belief Auction

In settings of incomplete information we put forward an epistemic framework for designing mechanisms that successfully leverage the players’ arbitrary higher-order beliefs, even when such beliefs are totally wrong, and even when the players are rational in a very weak sense. Following Aumann [5], we consider a player i rational if he uses a pure strategy si such that no alternative pure strategy si performs better than si in every world i considers possible, and consider him order-k rational if he is rational and believes that all other players are order-(k − 1) rational. We then introduce an iterative deletion procedure of dominated strategies and use it to precisely characterize the strategies consistent with the players being order-k rational. We exemplify the power of our framework in single-good auctions by introducing and achieving a new class of revenue benchmarks, defined over the players’ arbitrary beliefs, that can be much higher than classical ones, and are unattainable by traditional mechanisms. Namely, we exhibit a mechanism that, for every k ≥ 0 and ε > 0 and whenever the players are order-(k+1) rational, guarantees revenue ≥ Gk−ε, where Gk is the second highest belief about belief about . . . (k times) about the highest valuation of some player, even when such a player’s identity is not precisely known. Importantly, our mechanism is possibilistic interim individually rational. Essentially this means that, based on his beliefs, a player’s utility is non-negative not in expectation, but in each world he believes possible. We finally show that our benchmark Gk is so demanding that it separates the revenue achievable with order-k rational players from that achievable with order-(k+1) rational ones. That is, no possibilistic interim individually rational mechanism can guarantee revenue ≥ Gk − c, for any constant c > 0, when the players are only order-k rational.