Auction Theory & Contest Strategy
All-Pay Auction Bidding & Expected Revenue Formula
Calculate Baye-Kovenock-de Vries (1996) asymmetric all-pay contest abstention probability 1 - v_2/v_1, expected expenditures, and total sunk-effort dissipation (v_2/2)*(1 + v_2/v_1).
Model R&D patent races, lobbying, and architectural pitches where EVERY entrant pays sunk bids!
In an All-Pay Auction (Baye, Kovenock & de Vries 1996), the highest bidder wins the prize BUT BOTH winners and losers forfeit their sunk bids (R&D prototypes, legal briefs, lobbying budgets)! Under complete information with Leader prize valuation v_1 >= Challenger prize valuation v_2 > 0, NO pure-strategy Nash equilibrium exists! Instead, the unique mixed equilibrium has Leader randomize uniformly over [0, v_2] (Expected Bid E[b_1] = v_2 / 2), while Challenger abstains (bids 0) with exact probability 1 - (v_2 / v_1) and randomizes uniformly over (0, v_2] with probability v_2 / v_1 (Expected Bid E[b_2] = v_2^2 / (2 * v_1)), generating Total Expected Expenditure (v_2 / 2) * (1 + v_2 / v_1)!
Check by hand with Leader_Val_v1 = 100 (B2) and Challenger_Val_v2 = 60 (B3): Challenger Abstention Probability is 1 - 60 / 100 = 0.4000 (40.00%), Leader Expected Sunk Bid is 60 / 2 = 30.00, Challenger Expected Sunk Bid is 3,600 / 200 = 18.00, Total Expected Expenditure is 30 + 18 = 48.00, and Leader Expected Net Surplus is v_1 - v_2 = 40.00 (Challenger earns 0.00!). Now test symmetric rivals v_2 = 100: abstention vanishes to 0.0000 and Total Expected Expenditure surges to 100.00 (100% Full Rent Dissipation!)!
| All-Pay Contest Metric (v_1 = 100, v_2 = 60) | Formula | Verified Output |
|---|---|---|
| Challenger Zero-Bid Abstention Prob | 1 - (v_2 / v_1) | 0.4000 (40.00% Abstains) |
| Leader Expected Sunk Bid E[b_1] | v_2 / 2 | 30.00 |
| Challenger Expected Sunk Bid E[b_2] | (v_2^2) / (2 * v_1) | 18.00 |
| Total Expected Contest Expenditure | (v_2 / 2) * (1 + v_2 / v_1) | 48.00 (100.00 if v_2 = 100!) |
Challenger Strategic Abstention Probability (1 - v_2 / v_1) Formula
Validate v_1 >= v_2 > 0; notice how a stronger incumbent (higher v_1) intimidates the challenger into sitting out more often, shrinking total industry R&D/lobbying outlay!
=LET(v_1, B2, v_2, B3, IF(OR(v_2 <= 0, v_1 < v_2), "Require v_1 >= v_2 > 0", ROUND(1 - (v_2 / v_1), 4)))Total Expected All-Pay Contest Expenditure Formula
Before sponsoring an innovation prize in a shared Arte Sheets budget, test narrowing the capability gap (raising v_2 toward v_1) to see how parity maximizes total contestant effort!
=LET(v_1, B2, v_2, B3, IF(OR(v_2 <= 0, v_1 < v_2), "Require v_1 >= v_2 > 0", ROUND(0.5 * v_2 * (1 + (v_2 / v_1)), 4)))Try the example
Use the original values or change them to test your understanding.