Solution
A Prize Worth Ten and Six
Show the problem again
One prize is auctioned in a sealed-bid all-pay auction (both bidders pay their own bid; the higher bid wins; a tie is broken by a fair coin). Bidder 1 values the prize at 10 and bidder 2 values it at 6, and these values are common knowledge. Bids are real numbers at least 0. In the unique Nash equilibrium, what is the seller's expected total revenue? Give a reduced fraction.
Worked solution
The equilibrium is in mixed strategies. The high-value bidder randomises uniformly on [0, 6]. The low-value bidder bids 0 with probability 1 − 6/10 = 2/5 and otherwise randomises uniformly on (0, 6]. The low bidder is indifferent: their expected payoff is 0, so E[b₂] = 6·P(bidder 2 wins) = 6·(6/20) = 9/5. The high bidder earns 10 − 6 = 4, so E[b₁] = 10·(1 − 3/10) − 4 = 3. Revenue is 9/5 + 3 = 24/5. Notice that the revenue is less than the lower valuation 6, and the stronger bidder keeps a surplus equal to the valuation gap.
Source: Complete-information all-pay auction analysed by Baye, Kovenock and de Vries (1996). Statement written for AxiomIQ.