Solution
The Gap Between the Top Two
Show the problem again
Three bidders have independent private values uniform on [0, 1] and play the dominant-strategy equilibrium of a sealed-bid second-price auction. What is the winner's expected profit (their value minus the price they pay)? Give a reduced fraction.
Worked solution
Bidding one's value is a dominant strategy, so the winner is the highest-value bidder and pays the second-highest value. The profit is the gap between the top two order statistics. For three uniform values, E[max] = 3/4 and E[second highest] = 2/4 = 1/2, so the expected gap is 1/4. (The gaps between consecutive order statistics of n uniform values each have mean 1/(n + 1) = 1/4.)
Source: Original problem written for AxiomIQ.