Auction Theory & Mechanism Design
Revenue Equivalence: First-Price vs All-Pay Bid Formula
Compare symmetric Bayes-Nash equilibrium bids across Second-Price/English, First-Price/Dutch, and All-Pay auctions on [0, V_max] and verify William Vickrey (1961) and Roger Myerson’s (1981) Revenue Equivalence identity.
See how wildly different bidding rules generate identical interim payments and seller revenue!
William Vickrey (1961) and Roger Myerson (1981) proved the crown jewel of auction theory — the Revenue Equivalence Theorem: any auction among N symmetric risk-neutral bidders with independent private values on [0, V_max] that awards the item to the highest-valuation bidder and gives zero surplus to type v = 0 yields the EXACT SAME interim expected payment m(v) = ((N - 1) / N) * (v^N / V_max^(N - 1)) for every bidder type v and the EXACT SAME Expected Seller Revenue E[R] = ((N - 1) / (N + 1)) * V_max!
Verify on N_Bidders = 4 (B2), Valuation_Ceiling V_max = 100.00 (B3), and Focal_Bidder_Value v_i = 80.00 (B4 -> Win Probability G(v) = (80 / 100)^3 = 0.5120): (1) Second-Price / English Dominant Bid is b_2(v) = 80.00 (pays expected rival max conditional on winning = 60.00 * 0.5120 = 30.72); (2) First-Price / Dutch Shaded Bid is b_1(v) = ((4 - 1) / 4) * 80.00 = 60.00 (paid only when winning with prob 0.5120 -> interim expected payment 0.5120 * 60.00 = 30.72!); (3) All-Pay Auction Bid (paid unconditionally win or lose!) is b_ap(v) = ((4 - 1) / 4) * (80^4 / 100^3) = 0.75 * 40.96 = 30.72; and Expected Seller Revenue across all four formats is ((4 - 1) / (4 + 1)) * 100 = 60.00!
| Auction Format Equilibrium Metric (N=4, V_max=100, Bidder v=80) | Equilibrium Formula | Verified Output |
|---|---|---|
| Second-Price / English Bid b_2(v) | v | 80.00 (Zero Shading) |
| First-Price / Dutch Shaded Bid b_1(v) | ((N - 1) / N) * v | 60.00 (25.0% Shading) |
| All-Pay Unconditional Bid b_ap(v) = m(v) | ((N - 1) / N) * (v^N / V_max^(N - 1)) | 30.72 (= 0.5120 * 60.00) |
| Expected Seller Revenue E[R] (All Formats) | ((N - 1) / (N + 1)) * V_max | 60.00 |
All-Pay Bid & Universal Interim Expected Payment m(v) Formula
Validate integer N >= 2, V_max > 0, and 0 <= v_i <= V_max!
=LET(n_bd, B2, vm_x, B3, vi_b, B4, IF(OR(n_bd < 2, INT(n_bd) <> n_bd, vm_x <= 0, vi_b < 0, vi_b > vm_x), "Invalid Auction Input", ROUND(((n_bd - 1) / n_bd) * ((vi_b ^ n_bd) / (vm_x ^ (n_bd - 1))), 4)))Revenue-Equivalent Expected Seller Revenue Formula
Raise N_Bidders from 4 to 9 in cell B2: First-Price bid shading shrinks from 25% to 11.1% (b_1 = 71.1111) and Expected Seller Revenue climbs from 60.00 to (8 / 10) * 100 = 80.00!
=IF(ISTEXT(E4), E4, ROUND(((B2 - 1) / (B2 + 1)) * B3, 4))Try the example
Use the original values or change them to test your understanding.