Abstract: The Milgrom-Weber model is extended by assuming that the number of bidders is a random variable obeying the Poisson distribution, and the method of static game is used with incomplete information to obtain the equilibrium bidding function of bidders in the first price all-pay auction, which overcomes the difficulty in determining the number of bidders in practice. Besides, the objective function of bidders is modified, two parameters are introduced to reflect the  preference of bidders for winning probability and net income, and the corresponding equilibrium bidding function of bidders is obtained, which can better conform to the actual situation. Moreover, the analytic expression of equilibrium bidding function is further deduced, and a scientific reference is provided for bidders to make decisions.

Key words: all-pay auction, affiliated value, number of bidders, objective function, equilibrium bidding

摘要： 对Milgrom-Weber模型进行了拓展，假定投标人数是服从Poisson分布的随机变量，利用不完全信息静态博弈的方法，得到了第一价格全支付拍卖中投标人的均衡报价函数，解决了实际中难以确定投标人数的困难。并在此基础上对投标人的目标函数进行了修正，引入两个参数反映投标人对获胜概率和净收益的偏好程度，得到了相应的投标人均衡报价函数，能够更好地符合实际情况。同时，对均衡报价函数的解析表达式做了更进一步的推导，为投标人做决策提供了科学的参考依据。

关键词: 全支付拍卖, 关联价值, 投标人数, 目标函数, 均衡报价