Abstract
The paper generalizes the models of the standard first- and second-price auctions (FPA and SPA) by using the Cobb-Douglas function to evaluate bidders’ two conflicting bidding preferences for the probability of winning the item (Cpwin) and the profit conditional on winning (Cprofit). Compared with the popular expected profit function, this function has the advantage of allowing the bidders to equally or differently value Cpwin and Cprofit from the point of view of multiple criteria decision making, and then can capture bidders’ bidding behavior in reality. Introduction of the Cobb-Douglas function into the FPA allows us to provide new interpretations for overbidding (underbidding) behavior and the dependence of the optimal reserve price on the number of bidders. Our new interpretations are all attributed to the bidders’ different preferences for Cpwin and Cprofit. In contrast, this introduction into the SPA does not matter. Our findings suggest that the seller should prefer the FPA (SPA) over the SPA (FPA) and set a lower (higher) optimal reserve price if he conjectures that the bidders have a stronger desire to win an auction (to obtain a profit). The above results help the seller to select between the two auctions and set a reserve price in practice.
Highlights
It is well known that auctions have been applied to discover price and resource allocation in practice
It is worth mentioning that we mainly investigate the respective effects of the two weights of α and β on bidding behavior, properties of optimal reserve prices, and the seller’s expected revenue instead of the related effects of parameter 1 − d in the expected utility maximization model (EUMax) model or parameter c in the probability weighting (PW) model
Part (ii) shows that the optimal reserve price rI is dependent on the number of bidders when the two weights are different and the weight placed on Cpwin does not equal n/(2n − 1)
Summary
It is well known that auctions have been applied to discover price and resource allocation in practice. We aim to compare the optimal reserve price and the seller’s expected revenue across four kinds of auctions: the standard FPA and SPA and the generalized FPA and SPA. The generalized necessary condition shows that the reason that the optimal reserve price is independent of the number of the bidders is due to the bidders’ same preference between Cpwin and Cprofit , i.e., α = β. (3) We show that as the weight assigned to the Cpwin (Cprofit ) or α (β) increases, the optimal reserve price decreases (increases) and the seller’s (maximum) expected revenue increases (decreases) in the FPA.
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