Solution
Three Bidders, All Pay
Show the problem again
Three risk-neutral bidders have independent private values uniform on [0, 1]. In a sealed-bid all-pay auction, every bidder pays their own bid and the highest bid wins the object. In the symmetric equilibrium the bid is a strictly increasing function of the value. What does a bidder with value 3/4 bid? Give a reduced fraction.
Worked solution
Let b(v) be the increasing equilibrium bid. A bidder of value v who mimics type x wins with probability x² (both rivals must have lower values) and pays b(x) regardless, so the payoff is v x² − b(x). The first-order condition at x = v gives b′(v) = 2v², and b(0) = 0, so b(v) = 2v³/3. At v = 3/4: b = (2/3)(27/64) = 9/32. Expected revenue is 3·E[2V³/3] = 2·(1/4) = 1/2, which agrees with the revenue of a second-price auction with three bidders (revenue equivalence).
Source: Standard all-pay auction equilibrium from auction theory (e.g. Krishna, Auction Theory). Statement written for AxiomIQ.