🤖 AI Summary
本文通过将问题转化为分位数空间的等价形式，提供了一个显式解来解决两个竞标者全支付拍卖中任意类型分布下的均衡问题。
📝 Abstract
This paper provides an explicit characterization of the (unique) equilibrium of the two-bidder all-pay auction with asymmetrically distributed, independent private values. The characterization is achieved through recasting the problem into its equivalence in the quantile space and thereby obtaining equilibrium bid functions that involve only integrals and inverses of the primitives. No regularity assumption is required: the value distributions can be continuous, discrete, singular continuous, or any mixture of these. The explicit formula makes comparative statics transparent. Raising one bidder's value distribution in first-order stochastic dominance raises that bidder's equilibrium bid distribution in the same order. Perturbing a symmetric-bidder model has different revenue effects depending on the direction of the perturbation: whereas slightly weakening one of the two bidders reduces expected revenue, slightly strengthening one of the two has an ambiguous effect. The explicit formula also turns some involved arguments in the literature into direct corollaries.