The Influence of Valuation Estimation on Sealed-Bid First and Second-Price Auctions

This paper approaches first and second-price sealed-bid auctions as initial stages in a multi-stage game. In this setting, bidding reveals information about the bidder’s valuation, and could impact payoffs and revenues in future stages. This formulation provides a quantitative privacy model, where privacy cost is the economic disadvantage, to the bidder, of the information leakage. The paper studies a general twostage game in this model: Stage I is a first or second-price sealed-bid auction for item X. Stage II is a take-it-or-leaveit offer for item Y, whose valuation is a function h of the valuation of X. The paper shows that this game does not always possess a dominant strategy. While examining symmetric Nash equilibria, this paper restricts itself to the game where (a) the seller attempts to extract all the Stage II surplus, (i.e. the seller’s Stage II price is an accurate estimate of valuation, and the bidder’s payoff in Stage II is zero) and (b) equilibrium strategies are threshold strategies – monotonic increasing beyond a threshold valuation, and zero or infinitesmal below it. The paper finds that: (a) When the valuations are positively correlated, the bidder gets at least some of the Stage II surplus. (b) When they are negatively correlated, the seller gets more than the Stage II surplus. (c) If Stage II surplus is small enough, and the valuation of Y is zero when that of X is zero, the game is revenue equivalent to independent first or second-price auctions. That is, the bidder obtains all the surplus. (d) The game is better than an independent auction for the bidder when the valuations of X and Y are positively correlated, and for the seller when they are negatively correlated.