Solution
The Reserve That Reshapes Bids
Show the problem again
Two risk-neutral bidders have independent private values uniform on [0, 1] in a sealed-bid first-price auction with a public reserve price of 1/2: bids below 1/2 are not accepted, and if nobody bids the object stays unsold. In the symmetric equilibrium, bidders with value below 1/2 stay out and everyone else bids according to a strictly increasing function. What does a bidder with value 1 bid? Give a reduced fraction.
Worked solution
A bidder with value exactly 1/2 bids the reserve 1/2, and a bidder with value v ≥ 1/2 who bids b(x) wins with probability x (the rival either stays out, with probability 1/2, or has a lower value). The payoff (v − b(x))·x is maximised at x = v when (v·b(v))′ = v, so v·b(v) = v²/2 + C. The boundary condition b(1/2) = 1/2 gives 1/4 = 1/8 + C, so C = 1/8. Hence b(v) = v/2 + 1/(8v). At v = 1: b = 1/2 + 1/8 = 5/8, compared with 1/2 without a reserve.
Source: Standard first-price auction with a reserve price (see Riley and Samuelson, 1981). Statement written for AxiomIQ.