Online auctions with a fixed end-time often experience a sharp increase in bidding towards the end despite using a proxy-bidding format. We provide a novel explanation of this phenomenon under private values. We study a correlated private values environment in which the seller bids in her own auction (shill bidding). Bidders selected randomly from some large set arrive randomly in an auction, then decide when to bid (possibly multiple times) over a continuous time interval. A submitted bid arrives over a continuous time interval according to some stochastic distribution. The auction is a continuous-time game where the set of players is not commonly known, a natural setting for online auctions. Our results are robust with respect to the seller’s and the bidders’ priors regarding the set of bidders arriving at the auction. We show that there is a late-bidding equilibrium in which bids are delayed to the latest instance involving no sacrifice of probability of bid arrival, but shill bids fail to arrive with positive probability, and in this sense optimal late bidding serves to snipe the shill bids. We show conditions under which the equilibrium outcome is unique. Further, if these conditions do not hold, and there are any equilibria with a different outcome, they are necessarily characterized by early bidding. Any such equilibria are Pareto dominated for the bidders compared to the late-bidding equilibrium. Finally, our results suggest that under private values, the case against shill-bidding might be weak.
