Solution
The Optimal Seller
Show the problem again
Two bidders have independent private values, each exponentially distributed with mean 1 (density e^(−v) for v ≥ 0). The seller values the item at 0 and uses a revenue-maximising auction (for example a second-price auction with the optimal reserve price). Bidders are risk neutral and bid optimally. What is the seller's expected revenue? Give an exact expression; a decimal accurate to three places is also accepted.
Worked solution
For the exponential distribution, the virtual value is ψ(v) = v − (1 − F(v))/f(v) = v − 1, so the optimal reserve is r = 1 (where ψ = 0). A second-price auction with reserve 1 is optimal. The revenue equals E[max(0, ψ(highest value))]. The maximum M has density 2e^(−v)(1 − e^(−v)), so the revenue is ∫₁^∞ (v − 1)·2e^(−v)(1 − e^(−v)) dv = 2∫₁^∞ (v − 1)e^(−v) dv − 2∫₁^∞ (v − 1)e^(−2v) dv = 2/e − 2·e^(−2)/4 = 2/e − 1/(2e²) ≈ 0.668. Without a reserve, the revenue would be E[min] = 1/2, so the reserve helps considerably even though it sometimes forgoes a sale.
Source: Application of Roger Myerson's optimal auction design (1981). Statement written for AxiomIQ.