Introspection Dynamics with Mutation in Additive Games
Cooperation in heterogeneous groups, where individuals differ in resources, productivity, and behavioural responsiveness, underpins collective action across social and biological systems. Introspection dynamics, in which each player compares their payoff to their payoff under the alternative action, provides a natural learning rule for such asymmetric settings. Couto and Pal showed that for additive games, those in which the payoff difference a player evaluates when considering a switch is independent of the other players' actions, the stationary distribution of introspection dynamics is a product measure. We extend this result to introspection dynamics with mutation, where a selected player switches to a random action with some probability independent of payoffs, and with player-specific selection intensities. We show that the product structure is preserved, and we obtain the explicit per-player cooperation probability $p_i=\phi_i(\delta_i)(1-\mu_{i0}-\mu_{i1})+\mu_{i0}$. We consider the heterogeneous public goods game, where $N$ players may differ in their contributions $\alpha_i$, public goods multipliers $r_i$, and selection intensities $\beta_i$; the long-run cooperation probability admits the closed form $$ p_C = \frac{1}{N}\sum_{i=1}^{N} \left[\frac{1-\mu_{i0}-\mu_{i1}}{1+e^{\,\beta_i\alpha_i(1-r_i/N)}}+\mu_{i0}\right]. $$ Several structural consequences follow: a player-specific cooperation threshold at $r_i = N$ under symmetric mutation, a neutral-drift regime in which cooperation is governed entirely by mutation bias, and a mutation-selection balance in which aggregate cooperation is affine in the mutation rate, interpolating between the selection-driven level and neutrality. Mutation also regularises the strong-selection limit, so the closed form holds as $\beta_i\to\infty$, where the mutation-free dynamics degenerate.