On Rationalizable Outcomes in Private-Value First-Price Discrete Auctions

In this paper, we extend the result of Dekel and Wollinsky ("Rationalizable Outcomes of Large Private-Value First-Price Discrete Auctions" Games and Economic Behavior, 2003) on rationalizable outcomes in first-price auctions. Dekel and Wollinsky show that under certain conditions, each player chooses a unique bid conditional on her valuation. Their result however depends on the assumption that the number of players is sufficiently large (relative to the number of available bids). We first provide a different set of sufficient conditions for the uniqueness result. We then show that for the independent (possibly asymmetric) private value case, (i) the result holds if the distributions are such that the inverse hazard rate is sufficiently high for each valuation, implying that auctions need not necessarily be large, and (ii) if the distributions satisfy the conditions of Dekel and Wollinsky, they always satisfy ours