Introspection dynamics with mutation in additive games

Abstract Cooperation in heterogeneous groups, where individuals differ in resources, productivity, and behavioural responsiveness, underpins collective action across social and biological systems. Introspection dynamics is a learning rule suited to such asymmetric settings: at each time step a randomly selected player compares their current payoff to the payoff an alternative action would have given, and switches with a probability increasing in the difference. We extend introspection dynamics to include mutation , where a selected player adopts an action with payoff-independent, player-specific probabilities, together with player-specific selection intensities. From the extended dynamics we rederive and generalise the results of Couto and Pal 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 remains a product measure, and each player’s long-run cooperation probability has an explicit form in terms of their own payoffs, selection intensity, and mutation probabilities alone. We consider the heterogeneous public goods game, where $$N$$ N players may differ in their contributions, public goods multipliers $$r_i$$ r i , and selection intensities $$\beta _i$$ β i ; the long-run cooperation probability admits a closed form with $$N$$ N terms rather than $$2^N$$ 2 N . Several structural consequences follow: a player-specific cooperation threshold at $$r_i = N$$ 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 selection and neutrality. Mutation also regularises the strong-selection limit $$\beta _i\rightarrow \infty $$ β i → ∞ , where the mutation-free dynamics degenerate.