Mechanism Design & Auction Theory
Myerson Optimal Auction Virtual Valuation and Reserve Price Formula
Calculate Myerson’s (1981) virtual valuation psi(v) = 2*v - H, optimal bidder-count-invariant reserve price r* = (v_0 + H) / 2, and expected Vickrey auction seller payoff.
Why the revenue-maximizing auction reserve price r* does NOT depend on bidder count N!
Suppose a seller with retention value v_0 auctions an asset to N >= 1 symmetric bidders whose valuations are drawn independently from Uniform [0, H]. Setting reserve price r = v_0 leaves surplus on the table whenever only one bidder exceeds v_0! Myerson (1981) proved that expected revenue equals expected virtual valuation psi(v) = v - (1 - F(v)) / f(v) = 2*v - H of the winner, so optimal reserve solves psi(r*) = v_0 -> r* = (v_0 + H) / 2 — completely independent of N!
Verify on Uniform [0, 100] (H = 100 in B3) with seller salvage value v_0 = 20 (B2) and N = 2 bidders (B4): optimal reserve is r* = (20 + 100) / 2 = 60.0000! Under a Vickrey second-price auction with reserve r* = 60, both bidders fall below 60 with probability 0.36 (value 7.20), exactly one exceeds 60 with probability 0.48 (paying reserve 60 -> 28.80), and both exceed 60 with probability 0.16 (paying E[min] = 220/3 -> 11.7333), totaling 47.7333 (beating naive reserve r = 20’s 37.0667 by +28.8%)!
| Bidder Count N (v_0 = 20, H = 100) | Optimal Reserve r* = (v_0 + H) / 2 | Expected Seller Total Payoff |
|---|---|---|
| Single Buyer Monopoly (N = 1) | 60.0000 (Unchanged!) | 36.0000 (0.4*60 + 0.6*20) |
| Duopoly Vickrey Auction (N = 2) | 60.0000 (Unchanged!) | 47.7333 (vs 37.0667 at r = 20) |
| Five-Bidder Auction (N = 5) | 60.0000 (Unchanged!) | 67.4790 |
Myerson Optimal Reserve Price r* = (v_0 + H) / 2 Formula (Uniform [0, H])
Validate 0 <= v_0 < H and integer bidder count N >= 1.
=IF(OR(B2 < 0, B3 <= B2, B4 < 1), "Invalid Auction Input", ROUND(0.5 * (B2 + B3), 4))Exact Expected Seller Payoff (Revenue + Retained Salvage) for Any N >= 1 Formula
Integrate Myerson’s virtual surplus v_0 + INTEGRAL_{r*}^H 2*(v - r*) * N*v^(N-1)/H^N dv in closed form for any N >= 1.
=LET(v_0, B2, h_top, B3, n_bid, B4, IF(OR(v_0 < 0, h_top <= v_0, n_bid < 1), "Invalid Auction Input", LET(r_opt, 0.5 * (v_0 + h_top), h_n, POWER(h_top, n_bid), r_n, POWER(r_opt, n_bid), term_1, (n_bid / (n_bid + 1)) * (h_n * h_top - r_n * r_opt), term_2, r_opt * (h_n - r_n), ROUND(v_0 + (2 / h_n) * (term_1 - term_2), 4))))Try the example
Use the original values or change them to test your understanding.