Game Theory & Competitive Strategy
War of Attrition Quit Hazard & Duration Formula
Calculate the Maynard Smith / Bishop-Cannings War of Attrition mixed equilibrium quit hazard h* = c / ((N - 1)*V), expected standoff duration, and survival probability EXP(-N*h**t).
Model price wars, standards battles, and strike standoffs via the War of Attrition!
In John Maynard Smith’s (1974) and D. T. Bishop & C. Cannings’s (1978) War of Attrition, N >= 2 symmetric rivals endure continuous cash burn c > 0 per period until N - 1 rivals exit, leaving monopoly prize V > 0! For an N = 2 duopoly price war, keeping the rival indifferent between exiting immediately (0) and burning c * dt for an instant yield probability h* * dt requires V * h* = c — giving Individual Equilibrium Quit Hazard h* = c / V, Combined Price-War Resolution Rate 2 * c / V (generally (N / (N - 1)) * (c / V) for first exit among N players where each quits at c / ((N - 1) * V)!), Expected Duopoly Standoff Duration E[T] = V / (2 * c), and Probability the Price War Exceeds Horizon t equal to EXP(-(N / (N - 1)) * (c / V) * t)!
Check by hand for a Duopoly Price War (N = 2 in B4) over Market_Prize_V = 120 (B2), Period_Burn_Rate_c = 6 (B3), and Horizon_t = 10 periods (B5): each firm quits at memoryless exponential hazard h* = 6 / 120 = 0.0500 (5.00%/period), first-exit resolution hazard is 2 * 0.0500 = 0.1000, Expected Price-War Duration is 1 / 0.1000 = 10.0000 periods, and probability BOTH firms are still bleeding cash past t = 10 periods is EXP(-1.0) = 0.3679 (36.79%)!
| War of Attrition Metric (V=120, c=6, N=2, t=10) | Formula | Verified Output |
|---|---|---|
| Individual Rival Quit Hazard (h*) | c / ((N - 1) * V) | 0.0500 / period |
| First-Exit Standoff Resolution Rate | N * h* = (N / (N - 1)) * (c / V) | 0.1000 / period |
| Expected Time to First Exit E[T_1] | ((N - 1) * V) / (N * c) | 10.0000 periods |
| Standoff Survival Prob Past t = 10 | EXP(-N * h* * t) | 0.3679 (36.79% Still Active!) |
Symmetric N-Player War of Attrition Expected First-Exit Time Formula
Validate V > 0, c > 0, integer N >= 2, and t >= 0.
=LET(v_pz, B2, c_bn, B3, n_pl, B4, IF(OR(v_pz <= 0, c_bn <= 0, n_pl < 2, INT(n_pl) <> n_pl), "Require V > 0, c > 0, and Integer N >= 2", ROUND(((n_pl - 1) * v_pz) / (n_pl * c_bn), 4)))Probability Standoff Persists Past Horizon t Formula
Because exponential attrition is memoryless, surviving 10 months of a price war gives ZERO advantage over month 1 unless cash reserves are finite!
=LET(v_pz, B2, c_bn, B3, n_pl, B4, t_hz, B5, IF(OR(v_pz <= 0, c_bn <= 0, n_pl < 2, INT(n_pl) <> n_pl, t_hz < 0), "Require V > 0, c > 0, and Integer N >= 2", LET(h_res, (n_pl / (n_pl - 1)) * (c_bn / v_pz), ROUND(EXP(-h_res * t_hz), 4))))Try the example
Use the original values or change them to test your understanding.