A Convergence Theorem for Competitive Bidding with Differential Information

Abstract

IN THIS PAPER we investigate the properties of the winning bid in a sealed bid tender auction where each player has private information. We find that it is possible for the winning bid to converge in probability to the true value of the object at auction, even though no bidder knows the true value. Necessary and sufficient conditions for this phenomenon are derived, extending and generalizing certain of Wilson's results [3]. We study an auction in which a seller offers to sell at the highest bid an item of unknown value V. The kth bidder receives a private signal Sk (for k = 1, 2,.. .) and submits a bid without knowledge of the other signals. A finitely additive probability measure P reflects the bidders' unanimous beliefs about V and the signals. Conditional on V, the signals are independent and identically distributed. The signals take their values in some space &'. With n bidders, a bidding strategy for k is a function Pnk: 9' -> R. k's strategy specifies that upon receiving the signal Sk, he shall bid Pnk(Sk).2 Thus the winning

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