Auction Theory & M&A Strategy
Bulow-Klemperer Extra-Bidder Auction Revenue Formula
Verify Jeremy Bulow & Paul Klemperer’s (1996) theorem that an (N + 1)-bidder zero-reserve auction (N/(N + 2))*V_max strictly beats an N-bidder Myerson optimal-reserve auction!
Why M&A advisors should recruit ONE extra buyer rather than haggling over clever reserve prices!
Jeremy Bulow and Paul Klemperer (1996) proved one of the most practical theorems in M&A and auction design: under regular independent private valuations (such as v_i ~ Uniform[0, V_max]), a simple zero-reserve English auction with N + 1 symmetric bidders generates STRICTLY HIGHER expected seller revenue than ANY mathematically optimal Myerson mechanism (optimal reserve price r* = V_max / 2, lock-ups, or take-it-or-leave-it bargaining) with N bidders! On [0, V_max], N + 1 zero-reserve bidders yield E[Rev_NoRes(N + 1)] = (N / (N + 2)) * V_max, whereas N bidders with optimal reserve r* = V_max / 2 yield E[Rev_Opt(N)] = V_max * (((N - 1) / (N + 1)) + (1 / ((N + 1) * (2^N))))!
Check by hand with Max_Valuation_Vmax = 120 (B2) and Current_Bidders_N = 2 (B3): a 2-bidder zero-reserve auction yields (1 / 3) * 120 = 40.00; adding Myerson’s optimal reserve r* = 60 lifts 2-bidder expected revenue to 120 * (1/3 + 1/12) = 50.00; yet simply inviting ONE extra bidder (N + 1 = 3) with ZERO reserve price yields (2 / 4) * 120 = 60.00 — beating optimal 2-bidder negotiation by +10.00 (+20.00%!) with zero risk of a failed no-sale!
| Seller Mechanism (Uniform [0, 120], Current N = 2) | Formula | Expected Revenue |
|---|---|---|
| N-Bidder Zero-Reserve Auction (N = 2) | ((N - 1) / (N + 1)) * V_max | 40.00 |
| N-Bidder Myerson Optimal Reserve (r* = 60) | V_max * ((N - 1)/(N + 1) + 1/((N + 1)*2^N)) | 50.00 |
| (N + 1)-Bidder Zero-Reserve Auction (3 Bidders) | (N / (N + 2)) * V_max | 60.00 (+10.00 Bulow-Klemperer Gain!) |
| Single-Buyer Case (N = 1: Monopoly vs 2-Bidder) | 40.00 vs 30.00 | +10.00 Gain Over Monopoly Pricing! |
N-Bidder Myerson Optimal-Reserve Expected Revenue Formula
Validate V_max > 0 and integer bidder count N >= 1 (INT(N) = N).
=LET(v_mx, B2, n_bd, B3, IF(OR(v_mx <= 0, n_bd < 1, INT(n_bd) <> n_bd), "Require V_max > 0 and Integer N >= 1", ROUND(v_mx * (((n_bd - 1) / (n_bd + 1)) + (1 / ((n_bd + 1) * (2^n_bd)))), 4)))Bulow-Klemperer Extra-Bidder Revenue Advantage Formula
Compute E[Rev_NoRes(N + 1)] - E[Rev_Opt(N)] to prove to deal committees how much value exclusivity clauses destroy when they block a (N + 1)-th bidder.
=LET(v_mx, B2, n_bd, B3, IF(OR(v_mx <= 0, n_bd < 1, INT(n_bd) <> n_bd), "Require V_max > 0 and Integer N >= 1", LET(rv_n1, (n_bd / (n_bd + 2)) * v_mx, rv_op, v_mx * (((n_bd - 1) / (n_bd + 1)) + (1 / ((n_bd + 1) * (2^n_bd)))), ROUND(rv_n1 - rv_op, 4))))Try the example
Use the original values or change them to test your understanding.