We study the revenue maximizing allocation of m units among n symmetric agents that have unit demand and convex preferences over the probability of receiving an object. Such preferences are naturally induced by a game where the agents take costly actions that affect their values before participating in the mechanism. Both the uniform m + 1 price auction and the discriminatory pay-your-bid auction with reserve prices constitute symmetric revenue maximizing mechanisms. Contrasting the case with linear preferences, the optimal reserve price reacts to both demand and supply, i.e., it depends both on the number of objects m and on number of agents n. The main tool in our analysis is an integral inequality involving majorization, super-modularity and convexity due to Fan and Lorentz (1954).

中文翻译：

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具有内生价值的拍卖理论

我们研究了在 n 个对称代理之间分配 m 个单位的收入最大化分配，这些代理具有单位需求和对接收对象概率的凸偏好。这种偏好自然是由一个游戏引起的，在这个游戏中，代理在参与机制之前采取影响其价值观的代价高昂的行动。统一的 m + 1 价格拍卖和带有底价的歧视性付费拍卖构成了对称的收入最大化机制。与线性偏好的情况相比，最优底价对需求和供应都有反应，即它取决于对象的数量 m 和代理的数量 n。我们分析中的主要工具是由 Fan 和 Lorentz (1954) 提出的涉及专业化、超模性和凸性的积分不平等。