Figure 4.

(a–c) Shapes of the mixed strategy equilibria in biological all-pay auctions with two rewards depend on the relative value of the second reward v₂ to the first reward v₁. Full lines are derived from equation (2.3). Dotted lines correspond to the average strategy distribution obtained through stochastic computer simulations of 10⁷ generations. Average strategy distributions converge very well to the calculated mixed equilibrium. Differences for high strategy values are because of the discrete strategies in the computer simulations. Parameter values: value of highest reward v₁ = 1, population size N = 1000, auctions per generations K = 1000, number of strategies ℓ = 101 in [0, v₁] and mutation probability u = 0.5%.