Individuals are distributed over groups. The interaction structure of the population is given by a two-colored graph: from the point of view of a focal individual, its encounters are either in-group or out-group. Individuals may use different strategies for in-group and out-group interactions, and even different games can be played within and between groups. Individuals interact with everyone else according to their prescribed behavioral strategies. Both individual strategies and group memberships are updated proportional to payoff.

Shown are the stationary distributions of strategies over the unit square with respect to varying (a) cost of cooperation, c, (b) number of groups, M, (c) strategy mutation rate, u, and (d) migration rate between groups, v. The color bar indicates the equilibrium abundance of strategies: red means high while blue low. The straight lines, below which strategies are favored by natural selection, represent our analytical predictions for weak selection. A positive slope suggests that the lower-right corner (1, 0), maximum in-group favoritism, is most favored. We find excellent agreement between our analytical theory and computer simulations. Increasing c and u values each disfavor the emergence of in-group favoritism, while increasing M helps its establishment. An intermediate optimal migration rate v most promotes the evolution of in-group favoritism. Parameters: N = 100, β = 0.005, b = 1, (a) M = 4, u = 0.04, v = 0.06, (b) c = 0.1, u = 0.04, v = 0.06, (c) c = 0.08, M = 4, v = 0.06, (d) c = 0.08, M = 4, u = 0.06. Results are averaged over T = 2 × 10⁹ time steps.

(a)–(d) show how the population average p and q values are determined by model parameters. Strengthened selection can cause the most favorable strategy to move from the lower-right corner, maximum in-group favoritism, towards the lower-left corner, defection. Under some circumstances (K > 0), as shown here, increasing selection pressure β first enhances in-group favoritism, but opposes it when β exceeds a certain threshold. The greater the number of groups, the more biased individuals become: They maintain in-group helping at high levels, but shun helping the out-group. Increasing the strategy mutation (or experimentation) rate, u, leads to reduced in-group and out-group helping. An intermediate migration rate between groups most enhances in-group helping while most reducing out-group helping. Parameters: (a) N = 100, M = 100, b = 1, c = 0.1, u = 0.002, v = 0.1, (b) N = 100, β = 0.01, b = 1, c = 0.2, u = 0.01, v = 0.1, (c) N = 100, β = 0.01, M = 10, b = 1, c = 0.1, v = 0.15, (d) N = 100, β = 0.01, M = 15, b = 1, c = 0.15, u = 0.01. Results are averaged over T = 2 × 10⁹ time steps.