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Abstract
1- Introduction
2- Model
3- Equilibrium of the first price auction
4- Discussion of the results
5- Extensions
6- Concluding remarks
References
Abstract
I solve a first-price auction for two bidders with asymmetric budget distributions and known valuations for one object. I show that in any equilibrium, the expected utilities and bid distributions of both bidders are unique. If budgets are sufficiently low, the bidders will bid their entire budget in any equilibrium. For sufficiently high budgets, mass points in the equilibrium strategies arise. A less restrictive budget distribution could make both bidders strictly worse off. If the budget distribution of one bidder is dominated by the budget distribution of the other bidder in the reverse-hazard-rate order, the weaker bidder will bid more aggressively than the stronger bidder. In contrast to existing results for symmetric budget distributions, with asymmetric budget distributions, a second-price auction can yield a strictly higher revenue than a first-price auction. Under an additional assumption, I derive the unique equilibrium utilities and bid distributions of both bidders in an all-pay auction.
Introduction
Auctions are a widely used method of allocating objects, property rights and procurement contracts. If bidders in an auction are budget constrained, this will influence their bidding strategies, break the revenue equivalence of standard auctions, and lower revenues. Budget constraints can arise due to credit limits and imperfect capital markets, such that bidders’ willingness to pay might exceed their ability to pay. The existing research on standard auctions with budget constrained bidders concentrates on identical budget distributions. Yet, there are scenarios where bidders have asymmetric budget distributions. In a narrow market with a few players, e.g., a telecommunications sector, bidders hold noisy information about the other bidders and their budgets. This information might stem from previous interactions or from publicly available information, such as annual budget reports. Moreover, an auctioneer can contribute to this asymmetry by revealing the identities of the participants before the auction via a participation register. In the spectrum auction of the U.S. Federal Communications Commission, 30 bidders registered for the auction (Salant, 1997). Assessing the budget constraint of rival bidders was a major part of the preparation before the auction (Salant, 1997). GTE was one of the largest telecommunication firms in the U.S. It is reasonable to expect that the expectations of GTE about the budget of a smaller bidder, such as Poka Lambro, differed from the expectation of the smaller bidder about the financial resources of GTE.