Public Choice & Game Theory
Tullock Rent-Seeking Contest Effort & Dissipation Formula
Calculate symmetric N-player Tullock contest expenditure x* = ((N - 1)*r / N^2)*V, aggregate rent dissipation ((N - 1)/N)*r, and asymmetric two-rival lottery outlays.
Measure how lobbying, patent races, and litigation dissipate prize value V
In Gordon Tullock’s (1980) contest model, N >= 2 rivals spend non-refundable outlays x_i >= 0 competing for prize V > 0 with win probability p_i = x_i^r / SUM(x_j^r), where decisiveness exponent r > 0 governs returns to scale in contest effort (r = 1 is a proportional lottery). Whenever r <= N / (N - 1), unique symmetric pure-strategy Nash expenditure per rival is x* = ((N - 1) * r / N^2) * V, burning fraction D_rat = (N * x*) / V = ((N - 1) / N) * r of the entire prize in unproductive rent-seeking and leaving each rival expected net profit (V / N) * (1 - ((N - 1) / N) * r)!
Verify on a V = 1,000 procurement contest (B2) with N = 4 bidders (B3) and proportional lottery r = 1.00 (B4): each bidder spends x* = (3 / 16) * 1,000 = 187.50, aggregate rent dissipation reaches 4 * 187.50 = 750.00 (D_rat = 0.7500 = 75.00% of the prize!), and each bidder nets 250 - 187.50 = 62.50. Raise decisiveness r to 4/3 = 1.3333333333 and dissipation hits 1.0000 (100% of V burned, net payoff = 0.00); above r > N / (N - 1), pure-strategy symmetric equilibrium vanishes in favor of mixed strategies!
| Tullock Contest Metric (V=1,000, N=4) | Lottery r = 1.00 | Threshold r = 4/3 |
|---|---|---|
| Individual Bid Outlay (x*) | 187.50 | 250.00 |
| Total Contest Outlay (N * x*) | 750.00 | 1,000.00 |
| Rent Dissipation Ratio (D_rat) | 0.7500 (75.00%) | 1.0000 (100.00% Burned!) |
| Individual Expected Net Payoff | 62.50 | 0.00 |
Symmetric Pure-Strategy Tullock Individual Outlay x* Formula
Validate V > 0, integer N >= 2, and pure-strategy existence condition 0 < r <= N / (N - 1).
=LET(v_prz, B2, n_riv, B3, r_dec, B4, IF(OR(v_prz <= 0, n_riv < 2, r_dec <= 0), "Invalid Contest Input", IF(r_dec > n_riv / (n_riv - 1) + 1E-9, "Over-Dissipation (Mixed Equilibrium Only)", ROUND(((n_riv - 1) * r_dec / (n_riv^2)) * v_prz, 4))))Aggregate Rent Dissipation Ratio ((N - 1) / N) * r and Expected Net Payoff Formula
Compute total fraction of V dissipated across all N contestants and each player’s expected net surplus.
=LET(v_prz, B2, n_riv, B3, r_dec, B4, IF(OR(v_prz <= 0, n_riv < 2, r_dec <= 0), "Invalid Contest Input", IF(r_dec > n_riv / (n_riv - 1) + 1E-9, "Over-Dissipation (Mixed Equilibrium Only)", ROUND(((n_riv - 1) / n_riv) * r_dec, 4))))Try the example
Use the original values or change them to test your understanding.