Bidding in first-price auctions crucially depends on the beliefs of the bidders. We analyze bidding behavior in a first-price auction in which the knowledge of the bidders about the distribution of the values of their competitors is restricted to the range and the mean. To model this situation, we assume that under such uncertainty a bidder will expect to face the distribution of values that minimizes her expected payoff, given her bid is an optimal reaction to the bids of her competitors induced by this distribution. This introduces a novel way to endogenize beliefs in games of incomplete information. We find that for a bidder with a given valuation her worst-case belief just puts sufficient probability on lower valuations of her competitors to induce a high bid. The rest of the probability is distributed between a higher valuation and zero in a way that keeps the mean constant and minimizes the winning probability of the bidder. This implies that even though the worst case beliefs are type dependent in a non-monotonic way, an efficient equilibrium of the first-price auction exists.
Gretschko, Vitali and Helene Mass (forthcoming), Endogenous worst-case beliefs in first-price auctions,