Auction Theory & Corporate M&A
Winner’s Curse & Common-Value Bid Shading Formula
Quantify the Winner’s Curse conditional value E[V | Win] = s_i - eps*(N - 1)/(N + 1) and Kagel-Levin (1986) symmetric Nash bid shading B*(s_i) = s_i - eps in common-value auctions.
Why bidding your unbiased engineering estimate guarantees expected losses in oil-lease and M&A auctions!
In offshore oil-tract, spectrum, and corporate acquisition auctions (Capen, Clapp & Campbell 1971; Kagel & Levin 1986), all N bidders compete for an asset of common unknown value V, each observing an unbiased geological/valuation estimate s_i ~ Uniform[V - eps, V + eps] (mean V, error bound eps > 0). Why does bidding your unbiased estimate b = s_i lose money on average? Because YOU WIN ONLY WHEN YOUR ESTIMATE IS THE HIGHEST OF ALL N ESTIMATES! Conditional on s_i = max_j s_j, expected true value drops to E[V | Win] = s_i - eps * ((N - 1) / (N + 1))! Furthermore, in a first-price sealed-bid auction, adding strategic bid shading (2 * eps) / (N + 1) yields the exact Kagel-Levin Symmetric Nash Bid B*(s_i) = s_i - eps and positive Winner Expected Profit (2 * eps) / (N + 1)!
Check by hand with Signal_Estimate_si = 100 (B2), Estimation_Error_eps = 15 (B3), and Bidders_N = 4 (B4): although s_i = 100 is unconditionally unbiased, winning among N = 4 bidders implies E[V | Win] = 100 - 15 * (3 / 5) = 91.00 — so a naive unshaded bid of 100.00 suffers an immediate -9.00 Winner’s Curse loss! Shading to the Nash bid B*(100) = 100 - 15 = 85.00 earns expected profit 91.00 - 85.00 = +6.00 (= 30 / 5) upon winning!
| Common-Value Metric (s_i = 100, eps = 15, N = 4) | Formula | Verified Output |
|---|---|---|
| Winner-Conditional Value E[V | Win] | s_i - eps * (N - 1) / (N + 1) | 91.00 (-9.00 Below Signal!) |
| Naive Unshaded Bid Profit (b = s_i) | E[V | Win] - s_i | -9.00 (Winner’s Curse Loss!) |
| Kagel-Levin Nash Shaded Bid B*(s_i) | s_i - eps | 85.00 |
| Equilibrium Winner Profit | E[V | Win] - B*(s_i) = 2*eps / (N + 1) | +6.00 |
Conditional Asset Value Upon Winning E[V | s_i = max s_j] Formula
Notice how adding competitors (raising N from 2 to 9) leaves each bidder’s own signal accuracy unchanged yet worsens the Winner’s Curse discount eps * (N - 1) / (N + 1) from 0.3333*eps to 0.8000*eps!
=LET(s_est, B2, ep_bd, B3, n_bd, B4, IF(OR(ep_bd < 0, s_est <= ep_bd, n_bd < 2, INT(n_bd) <> n_bd), "Require s_i > eps >= 0 and Integer N >= 2", ROUND(s_est - ep_bd * ((n_bd - 1) / (n_bd + 1)), 4)))Naive Unshaded Bid Loss vs Nash Winner Profit Formula
In a shared Arte Sheets M&A deal model, have a suggester test doubling N from 4 to 9: conditional value drops to 88.00 and Nash winner profit shrinks from +6.00 to +3.00!
=LET(s_est, B2, ep_bd, B3, n_bd, B4, IF(OR(ep_bd < 0, s_est <= ep_bd, n_bd < 2, INT(n_bd) <> n_bd), "Require s_i > eps >= 0 and Integer N >= 2", ROUND(-ep_bd * ((n_bd - 1) / (n_bd + 1)), 4)))Try the example
Use the original values or change them to test your understanding.