Solution
First-Price Auction with Uniform Values
Show the problem again
Two risk-neutral bidders have private values drawn independently and uniformly from [0, 1]. In a sealed-bid first-price auction (the winner pays their own bid), each bidder bids a fixed fraction of their value in the symmetric equilibrium. What fraction?
Worked solution
Bid half your value: b(v) = v/2. Suppose your opponent bids b(v) = v/2. Bidding an amount x wins when the opponent's value is below 2x, which happens with probability 2x, so your expected profit is 2x(v − x). Maximizing over x gives x = v/2, confirming the strategy is a mutual best response. In general with n bidders, b(v) = v(n−1)/n; bids shade less as competition grows.
Source: Standard first-price sealed-bid auction result from the auction-theory literature founded by William Vickrey (1961). Statement written for AxiomIQ.