Monotonicity and rationalizability in a large first price auction

This paper proves that the monotonicity of bidding strategies together with the rationality of bidders implies that the winning bid in a first price auction converges to the competitive equilibrium price as the number of bidders increases (Wilson, 1977). Instead of analysing the symmetric Nash equilibrium, we examine rationalizable strategies (Bernheim (1984), Pearce (1984)) among the set of monotonic bidding strategies to prove that any monotonic rationalizable bidding strategy must be within a small neighbourhood of the "truthful" valuation of the object, conditioned on the signal received by the bidder. We obtain an information aggregation result similar to that of Wilson (1977), while dispensing with almost all symmetric assumptions and using a milder solution concept than the Nash equilibrium. In particular, if every bidder is ex ante identical, then any rationalizable bidding strategy must be within a small neighbourhood of the symmetric Nash equilibrium. In a symmetric first price auction, the symmetry of outcomes is implied rather than assumed.