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 \(N\) players may differ in their contributions, public goods multipliers \(r_i\), and selection intensities \(\beta _i\); the long-run cooperation probability admits a closed form with \(N\) terms rather than \(2^N\). 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 selection and neutrality. Mutation also regularises the strong-selection limit \(\beta _i\rightarrow \infty \), where the mutation-free dynamics degenerate. 1 Introduction The public goods game is a canonical model of cooperation in evolutionary game theory: players decide whether to contribute to a common resource, which is multiplied and redistributed, creating a tension between individual incentive and collective benefit [14]. Classical analyses assume a homogeneous population, but real groups are heterogeneous: players differ in their productivity, their contributions, and how sensitively they respond to payoff differences [10]. A natural learning rule for such settings is introspection dynamics, introduced in [3]: at each time step a player compares their current payoff to the payoff they would have received had they played an alternative action (keeping all other players fixed), and switches with a probability governed by this difference. Because the comparison is self-referential rather than imitative, introspection applies to arbitrary asymmetric games where role-model imitation is unavailable or unnatural. Introspection dynamics is closely related to the logit-response dynamics studied in economics [1, 2], in which revising players choose actions with probabilities exponential in their payoffs; the Fermi rule is the two-action logit choice. Asymmetric games have also been approached through evolutionary dynamics directly, for example in [13]. The introspection model of [3, 4], however, contains no mutation: a selected player switches action solely according to the Fermi rule. Mutation, the adoption of a random action independently of payoffs, is a standard ingredient of evolutionary and learning dynamics [7] that accounts for exploration, error, and behavioural noise [15]. In this paper we extend introspection dynamics to include per-player, per-action mutation, together with player-specific selection intensities \(\beta _i\) rather than a single global value. From the extended dynamics we rederive, and generalise, the additive-game results of [4]. There, the authors extended introspection dynamics to multiplayer asymmetric games: for a general game the transition rate for player \(i\) switching depends on the actions of all other players through the payoff function, so the stationary distribution must be computed by solving a linear system of dimension \(2^N\) (more generally \(L^N\) when each player has \(L\) actions). They identified an important exception: for additive games, in which the payoff difference a player evaluates when switching action is independent of the other players’ actions (a property also known as equal gains from switching), the stationary distribution factorises as a product measure over the players [4, Proposition 2], and for the linear public goods game it admits a closed form ( [4]; Proposition 3). We show that these results persist under mutation: the stationary distribution remains a product measure, and each player’s long-run cooperation probability has the explicit form \(p_i=\phi _i(\delta _i)(1-\mu _{iC}-\mu _{iD})+\mu _{iC}\). Specifically, we extend the product-measure result to mutation and to arbitrary finite action sets (Sect. 3), show that for symmetric rates restricting mutation to the other action is only a reparameterisation of the mutation rate (Sect. 3.3), record the structural consequences that hold for any additive game (Sect. 4), specialise them to the heterogeneous public goods game (PGG), for which additivity holds and the closed-form cooperation probability follows (Sect. 5), and explore the effect of mutation beyond additivity numerically (Sect. 6). Section 7 concludes. 2 Setup and Notation Consider \(N\) ordered players, where each player \(i\) selects from a finite action set \(A_i\) with \(L_i:=|A_i|\ge 2\). A state is a profile \(\textbf{a}=(a_1,\dots ,a_N)\), where \(a_i\in A_i\) denotes the action currently played by player \(i\), and the state space is \(S=\prod _{i=1}^{N}A_i\). Each player \(i\) has a payoff function \(f_i:S\rightarrow \mathbb {R}\) and player-specific selection intensity \(\beta _i>0\). We reserve \(b\in A_i\) for a generic alternative action of the player under consideration: we write \(\textbf{a}_{i\rightarrow b}\) for the state with player \(i\)’s action replaced by \(b\), and for the payoff difference player \(i\) evaluates at state \(\textbf{a}\) against the alternative \(b\). Following [4], a game is additive for player \(i\) if \(\Delta f_i(\textbf{a};b)\) depends only on \(a_i\) and \(b\), not on \(\textbf{a}_{-i}\), the actions of all individuals other than \(i\); equivalently, the payoff difference a player evaluates when considering a switch is independent of the co-players’ actions, a property also known as equal gains from switching. Additivity holds precisely when the payoff decomposes as for some \(g_i:A_i\rightarrow \mathbb {R}\) and \(h_i\), in which case \(\Delta f_i(\textbf{a};b)=g_i(a_i)-g_i(b)\). In words, \(g_i\) collects the part of player \(i\)’s payoff controlled by their own action and \(h_i\) the part controlled by the co-players, and the two parts do not interact: switching action changes \(g_i\) alone. Indeed, fixing a reference action \(c\in A_i\) and setting \(g_i(a_i):=\Delta f_i(\textbf{a};c)\) and \(h_i(\textbf{a}_{-i}):=f_i(\textbf{a}_{i\rightarrow c})\) exhibits the decomposition, and the converse is immediate. Our running special case is that of two actions: \(A_i=\{C,D\}\) for all \(i\), coded as \(\{1,0\}\), so that \(S=\{0,1\}^N\) with \(|S|=2^N\). Each player then has a single alternative action \(\bar{a}_i=1-a_i\), and we abbreviate \(\Delta f_i(\textbf{a}):=\Delta f_i(\textbf{a};\bar{a}_i)\). Additivity for player \(i\) becomes the existence of a constant \(\delta _i\in \mathbb {R}\) such that where \(\delta _i = g_i(D)-g_i(C) = \Delta f_i\!\big |_{a_i=0}\) is the payoff difference when player \(i\) defects; both cases of (2) follow from \(\Delta f_i(\textbf{a})=g_i(a_i)-g_i(\bar{a}_i)\). Note that not all games are additive. For example in [3] two of the two-player, two-action games considered to illustrate introspection dynamics are the Prisoner’s Dilemma \(G_1\) and the Stag Hunt \(G_2\) of (3), for \(0< c_1, c_2 < b\). For \(G_1\) we have \(\delta _1=c_1\) and \(\delta _2=c_2\). However for \(G_2\), writing \((a_1, a_2)\) for the state in which player 1 plays \(a_1\) and player 2 plays \(a_2\), \(\Delta f_1((1, 1)) = b-c_1\) but \(\Delta f_1((1, 0)) = -c_1\). The Prisoner’s Dilemma (as defined by \(G_1\)) is additive but the Stag Hunt game (as defined by \(G_2\)) is not. Further games considered in [3] are the Volunteer’s Dilemma and the Volunteer’s Timing Dilemma (a generalisation with more than 2 actions), neither of which are additive. Beyond the donation form, the general Prisoner’s Dilemma (PD) is parameterised by the reward \(R\), temptation \(T\), sucker payoff \(S\), and punishment \(P\) with \(T> R> P > S\) and the social-efficiency condition \(2R > T + S\): Additivity requires, for each player, the equality that is, the gain from defecting against a cooperator must equal the gain from defecting against a defector. The PD axioms are strict inequalities and do not force this equality; a generic PD therefore fails it. For instance, \(T=5\), \(R=3\), \(P=1\), \(S=0\) satisfies \(T>R>P>S\) and \(2R=6>5=T+S\), yet \(T-R=2\ne 1=P-S\). The donation form \(G_1\) above is the exceptional case where the equality holds by construction (for player \(i\), both differentials equal \(c_i\)). To define introspection dynamics we introduce the following two components: - Fermi function. Each player \(i\) has a personal Fermi function \(\phi _i(x)=(1+e^{\beta _i x})^{-1}\). Since \(\beta _i>0\), the exponential \(e^{\beta _i x}\) is strictly increasing in \(x\), so \(\phi _i\) is strictly decreasing with \(\phi _i(0)=\tfrac{1}{2}\) and \(\phi _i(x)+\phi _i(-x)=1\). - Mutation. Each player \(i\) has mutation probabilities \(\mu _{ib}>0\) for each action \(b\in A_i\), with \(M_i:=\sum _{b\in A_i}\mu _{ib}<1\), where \(\mu _{ib}\) is the probability that a selected player \(i\) adopts action \(b\) regardless of the payoff comparison. For two actions the two mutation probabilities are \(\mu _{iC}\), towards cooperation, and \(\mu _{iD}\), towards defection, so that \(M_i=\mu _{iC}+\mu _{iD}\). At each step, one player \(i\) is selected uniformly at random and updates their action in one of two ways. With probability \(\mu _{ib}\) they mutate to action \(b\), for each \(b\in A_i\); the target may be their current action \(a_i\), in which case the state does not change. Otherwise, with the remaining probability \(1-M_i\), they introspect: they draw an alternative action \(b\) uniformly from the \(L_i-1\) actions other than \(a_i\) and switch to it with probability \(\phi _i\bigl (\Delta f_i(\textbf{a};b)\bigr )\). The transition probability to each state \(\textbf{a}_{i\rightarrow b}\) with \(b\ne a_i\) is therefore and the state is unchanged with the remaining probability, which collects the events that the selected player mutates to their current action or introspects without switching. Since \(\phi _i\) is strictly decreasing, a larger payoff gain from keeping the current action yields a smaller switching probability. Setting \(\mu _{ib}=0\) for all possible actions b recovers the introspection dynamics of [3, 4]; the mutation terms add a payoff-independent probability of switching to each action. For two actions the alternative \(b=\bar{a}_i\) is the only choice and (4) reads where the mutation term is indexed by the target action: \(\mu _{i\bar{a}_i}\) denotes \(\mu _{iC}\) when player \(i\) currently defects, so that \(\bar{a}_i=C\), and \(\mu _{iD}\) when they cooperate. Definition 1 (Cooperation probability) For two actions, given an ergodic chain on \(S\) with stationary distribution \(\pi \), the cooperation probability is \(p_C=\pi \cdot s\), where \(s_\textbf{a}=\frac{1}{N}\sum _i a_i\) is the fraction of cooperators in state \(\textbf{a}\). 3 Decomposition of the Stationary Distribution For additive games without mutation the stationary distribution of introspection dynamics is a product measure, for any finite action sets [4, Proposition 2]. We show that this structure persists with mutation: for an additive game the next action of a selected player depends only on their current action, so each player evolves as a Markov chain on their own action set, and the joint stationary distribution is the product of the per-player stationary distributions. We prove the general result first, then specialise to two actions, where the marginals take an explicit closed form, and close with an alternative mutation scheme in which a mutating player must change action. 3.1 General Action Spaces For each player \(i\), let \(K_i\) be the \(L_i\times L_i\) stochastic matrix whose entry \(K_i(a_i,b)\) is the probability that player \(i\)’s next action is \(b\) given that they are selected whilst playing \(a_i\). In general the switching probabilities in (4) depend on the whole state \(\textbf{a}\) and no such matrix exists; for an additive game \(\Delta f_i(\textbf{a};b)=g_i(a_i)-g_i(b)\) depends only on \(a_i\) and \(b\), so \(K_i\) is well defined, with off-diagonal entries and diagonal entries holding the remaining mass. Every entry of \(K_i\) is at least \(\mu _{ib}>0\), so \(K_i\) is irreducible and aperiodic and has a unique, strictly positive stationary distribution \(\textbf{m}_i\) on \(A_i\). Theorem 1 (Stationary distribution) For any game that is additive for every player, under introspection dynamics with selection intensities \(\beta _i>0\) and mutation probabilities \(\mu _{ib}>0\) with \(M_i<1\), the introspection chain on \(S=\prod _{i}A_i\) has the unique stationary distribution where \(\textbf{m}_i\) is the unique stationary distribution of \(K_i\). Proof One step of the chain changes at most one player’s action, so the states from which \(\textbf{a}\) is reachable in one step are \(\textbf{a}\) itself and the states \(\textbf{a}_{i\rightarrow b}\) with \(b\ne a_i\), with transition probabilities The first expression follows from the holding probability (5) and the diagonal entries \(K_i(a_i,a_i)=1-\sum _{b\ne a_i}K_i(a_i,b)\). The product form gives \(\pi _{\textbf{a}_{i\rightarrow b}}\,\textbf{m}_i(a_i)=\pi _{\textbf{a}}\,\textbf{m}_i(b)\), so since \(\sum _{b} \textbf{m}_i(b)K_i(b,a_i)=\textbf{m}_i(a_i)\) by stationarity of \(\textbf{m}_i\). Every switching probability in (4) is positive, so any state can be reached from any other by at most \(N\) single-action switches, and every state has a positive holding probability; the chain is therefore irreducible and aperiodic and \(\pi \) is its unique stationary distribution. \(\square \) Theorem 1 reduces the computation of the stationary distribution from one linear system of dimension \(\prod _i L_i\) to \(N\) systems of dimension \(L_i\). 3.2 Games with Two Actions For two actions more is true: the two rows of \(K_i\) coincide. We call this the forgetful property, since the probability that a selected player’s next action is \(C\) is then the same whichever action they currently hold, and it is what gives the marginal \(\textbf{m}_i\) an explicit closed form. The two switching probabilities are complementary, by \(\phi _i(x)+\phi _i(-x)=1\), and with a single alternative this is enough to align the rows. For \(L_i\ge 3\) they always differ: identical rows would force \(g_i\) to be constant, since \(\phi _i\) is injective, and equating a diagonal entry with the off-diagonal entries in its column would then require \(L_i=2\). The forgetful property is therefore special to two actions. Theorem 2 (Individual cooperation probabilities) For any two-action additive game with constants \(\delta _i\), under introspection dynamics with player-specific selection intensity \(\beta _i\) and mutation probabilities \(\mu _{iC},\mu _{iD}>0\) with \(\mu _{iC}+\mu _{iD}<1\), the long-run probability that player \(i\) cooperates is Proof We compute the entries of \(K_i\). When \(a_i=D\) the payoff difference in (2) is \(\Delta f_i=\delta _i\), so by (7) the probability of switching to \(C\) when selected is When \(a_i=C\) the payoff difference is \(\Delta f_i=-\delta _i\), so, using \(\phi _i(-\delta _i)=1-\phi _i(\delta _i)\), the probability of switching to \(D\) is The two rows of \(K_i\) therefore coincide: ordering \(A_i\) as \((C,D)\). A stochastic matrix with identical rows sends every distribution to that row in a single step, so \(\textbf{m}_i=(p_i,\,1-p_i)\) is the unique stationary distribution of \(K_i\). Additivity decouples player \(i\) from the rest of the population: their action evolves as a two-state Markov chain in its own right, with kernel \(\bigl (1-\tfrac{1}{N}\bigr )I+\tfrac{1}{N}K_i\), which has the same stationary distribution as \(K_i\). Hence the long-run probability that player \(i\) cooperates is \(\textbf{m}_i(C)=p_i\). \(\square \) Theorem 2 gives the common value of the two rows. Additivity and the restriction to two actions together make a player’s choice forgetful: each time player \(i\) is selected they cooperate with the same probability \(p_i\), whatever their current action. This probability is the payoff-driven Fermi value \(\phi _i(\delta _i)\), scaled by the chance of no mutation and shifted by the chance of mutating to cooperation. Theorem 3 (Stationary distribution for two actions) For any two-action additive game with constants \(\delta _i\), the stationary distribution of the introspection chain on \(S=\{0,1\}^N\) is the product measure where \(p_i\) is given by (9). Proof By the proof of Theorem 2, \(\textbf{m}_i=(p_i,\,1-p_i)\), so that \(\textbf{m}_i(a_i)=p_i^{\,a_i}(1-p_i)^{1-a_i}\), and (10) is the product measure (8) of Theorem 1. \(\square \) In the long run the players’ actions are therefore independent: the group behaves as \(N\) separate biased coins, with player \(i\) showing cooperation with probability \(p_i\). Taking the expectation of the cooperator fraction \(s\) under this product measure, the cooperation probability is the mean of the individual probabilities, Setting \(\mu _{iC}=\mu _{iD}=0\) in (9) and (10) recovers the two-action case of Proposition 2 of [4]; the argument above shows that the product-measure structure is unaffected by mutation, since mutation alters only the per-player marginal \(p_i\) and not the independence between players. Remark 1 (The two-player donation game) For the Prisoner’s Dilemma \(G_1\) of (3), from [3], the payoff \(f_1(\textbf{a})=-c_1 a_1+b a_2\) is additive with \(\delta _1=c_1\), and similarly \(\delta _2=c_2\). Theorems 2 and 3 give the per-player cooperation probability and the product-measure stationary distribution Setting \(\mu _{iC}=\mu _{iD}=0\) and a common selection intensity \(\beta _i=\beta \) recovers \(p_i=(1+e^{\beta c_i})^{-1}\) and the stationary distribution obtained by [3] for this game; they observed that their two-player stationary distribution factorises precisely under the additive condition, of which the donation game is an instance. Mutation preserves the product form while shifting each marginal: with \(b=1\), \(c_1=0.6\), \(c_2=0.1\), \(\beta =5\), and asymmetric mutation \(\mu _{iC}=0.05\), \(\mu _{iD}=0.15\) we obtain \(p_1\approx 0.088\) and \(p_2\approx 0.352\), giving \(\pi _{DD}\approx 0.591\), \(\pi _{DC}\approx 0.321\), \(\pi _{CD}\approx 0.057\), and \(\pi _{CC}\approx 0.031\). 3.3 An Alternative Mutation Scheme In the dynamics of Sect. 2 a mutating player may adopt any action, including the one they already play. An alternative convention restricts mutation to the other action: for two actions, when player \(i\) is selected they mutate to the alternative \(\bar{a}_i\) with probability \(\nu _{i\bar{a}_i}\in (0,1)\), where, as in (6), \(\nu _{iC}\) and \(\nu _{iD}\) are the rates towards each target action, and otherwise apply the Fermi comparison, so that the switching probability of (6) becomes For an additive game the selection kernel \(K_i\) remains well defined, with but its two rows no longer coincide, so the forgetful property in the proof of Theorem 2 fails. Proposition 1 (Alternative mutation scheme) For any two-action additive game with constants \(\delta _i\), under the alternative scheme (12) with \(\nu _{iC},\nu _{iD}\in (0,1)\), the unique stationary distribution is the product measure (10) with marginals Proof All entries of \(K_i\) are positive, so \(K_i\) has a unique stationary distribution \(\textbf{m}_i=(p_i,\,1-p_i)\), determined by the two-state balance \(p_i\,K_i(C,D)=(1-p_i)\,K_i(D,C)\); solving gives \(p_i=K_i(D,C)/\bigl (K_i(D,C)+K_i(C,D)\bigr )\), which is (13). The proof of Theorem 1 uses only that one player updates at a time and that the selected player’s next action depends on their own current action alone; it therefore applies verbatim, and the stationary distribution is the product of the \(\textbf{m}_i\). \(\square \) Proposition 2 (Equivalence of the mutation schemes) For symmetric rates \(\nu _{iC}=\nu _{iD}=\nu _i\), the stationary distribution under the alternative scheme coincides with that of the dynamics of Sect. 2 with symmetric mutation \(\mu _{iC}=\mu _{iD}=\nu _i/(1+\nu _i)\). Proof Both stationary distributions are product measures with Bernoulli marginals, so it suffices that the marginals agree. With \(\mu _i=\nu _i/(1+\nu _i)\), (9) gives which is (13) with \(\nu _{iC}=\nu _{iD}=\nu _i\). The map \(\nu _i\mapsto \nu _i/(1+\nu _i)\) is a bijection from \((0,1)\) onto \((0,\tfrac{1}{2})\), so the correspondence runs in both directions. \(\square \) The two chains are different, and in general mix at different speeds, but their long-run behaviour is identical: for symmetric rates, whether a mutating player may re-select their current action is a reparameterisation of the mutation rate, not a modelling choice. Every symmetric-mutation result of Sects. 4 and 5 therefore holds for the alternative scheme with \(\mu \) replaced by \(\nu /(1+\nu )\); Fig. 3 illustrates the coincidence. For asymmetric rates the equivalence fails: the denominator of (13) depends on \(\phi _i(\delta _i)\), so the marginal is no longer affine in the Fermi value, whereas (9) always is, and no fixed choice of \(\mu _{iC},\mu _{iD}\) reproduces (13) across all payoff differences. Both conventions appear in the literature. Redrawing from the full action set is the convention of the exploration dynamics of [15], and payoff-independent mutations of this kind drive the equilibrium-selection analyses of adaptive dynamics [12]: a reconsidering player re-samples their options, sometimes landing on the action they already use. Mutation restricted to the other action is the bit-flip mutation of genetic algorithms [8, 11], natural when mutation is read as an error in executing a switch. We emphasise the scope of these results: Propositions 1 and 2 hold for any number of players and for player-specific rates. Since the equivalence acts marginal by marginal, a single population may even mix the conventions, with some players redrawing from the full action set and others restricted to the alternative, and the stationary distribution remains the product of the corresponding marginals. For more than two actions, mutating to a uniformly chosen other action still yields a selection kernel that depends only on the player’s own action, so Theorem 1 continues to apply there too. 4 Structural Consequences The formula (9) for the individual cooperation probability allows immediate structural conclusions that hold for any two-action additive game, before any specific payoff structure is imposed. We record four: the two limits of the selection intensity, a cooperation threshold, and the effect of mutation on aggregate cooperation. Without mutation, several reduce to properties already observed by [4]. Remark 2 (Neutral drift) If \(\beta _i=0\) for all \(i\) then \(\phi _i\equiv \tfrac{1}{2}\), so the payoff parameters \(\delta _i\) drop out of (9) and cooperation is set entirely by mutation: so \(p_{C}=\tfrac{1}{2}\) exactly when \(\sum _i(\mu _{iC}-\mu _{iD})=0\). A player insensitive to payoff differences ignores the underlying game: their long-run behaviour is set by their mutation bias alone, and group cooperation reduces to the average asymmetry between the mutation rates. Proposition 3 (Strong selection) For each player \(i\), (9) confines the cooperation probability to the open interval \((\mu _{iC},\,1-\mu _{iD})\), and the endpoints are the strong-selection limits: as \(\beta _i\rightarrow \infty \), Proof Since \(0<\phi _i(\delta _i)<1\) and \(1-\mu _{iC}-\mu _{iD}>0\), (9) gives \(\mu _{iC}0\) and \(\rightarrow 1\) when \(\delta _i<0\), so \(p_i\rightarrow \mu _{iC}\) or \(p_i\rightarrow 1-\mu _{iD}\) respectively. \(\square \) Without mutation, a unique stationary distribution requires finite \(\beta _i\) [4]: in the limit \(\beta _i\rightarrow \infty \) the payoff-driven switches become deterministic best responses, the chain is no longer ergodic, and the long-run outcome can depend on the initial state. The strong-selection limit must therefore be taken after computing the stationary distribution at finite selection intensity, as in [3]. For example, in the donation game \(G_1\) of (3) each \(\delta _i=c_i>0\), so as \(\beta \rightarrow \infty \) the mutation-free stationary distribution collapses to a point mass on the all-defect profile. Mutation removes this degeneracy: the switching probabilities in (6) remain bounded away from \(0\) and \(1\), so the chain stays ergodic at any selection intensity and the stationary distribution converges to the non-degenerate product measure with marginals \(\mu _{iC}\) or \(1-\mu _{iD}\). Mutation therefore extends the exact analysis to arbitrarily strong selection. Corollary 1 (Cooperation threshold) Player \(i\) cooperates with probability greater than \(\tfrac{1}{2}\) if and only if When \(\mu _{iC}=\mu _{iD}\) this reduces to \(\delta _i<0\). A sufficient condition for \(p_{C}>\tfrac{1}{2}\) is that the inequality holds for every \(i\). Proof Since \(1-\mu _{iC}-\mu _{iD}>0\), When \(\mu _{iC}=\mu _{iD}\), the right-hand side equals \(\tfrac{1}{2}\), and since \(\phi _i\) is strictly decreasing with \(\phi _i(0)=\tfrac{1}{2}\), this is equivalent to \(\delta _i<0\). The claim about \(p_{C}\) follows by averaging (11). \(\square \) With symmetric mutation a player cooperates the majority of the time precisely when the payoff favours cooperation, \(\delta _i<0\); mutation does not move the threshold, it only softens the transition across it. With asymmetric mutation the threshold can be met even when the payoff favours defection. Substituting \(\phi _i(\delta _i)=(1+e^{\beta _i\delta _i})^{-1}\), the condition \(p_i>\tfrac{1}{2}\) is equivalent to The right-hand side is positive precisely when the mutation is biased towards cooperation (\(\mu _{iC}>\mu _{iD}\)). Thus when the payoff favours defection (\(\delta _i>0\)) a sufficiently strong cooperative bias still yields \(p_i>\tfrac{1}{2}\); conversely, when the payoff favours cooperation (\(\delta _i<0\)) a defection bias (\(\mu _{iD}>\mu _{iC}\)) can drive \(p_i<\tfrac{1}{2}\). The logarithmic form presumes \(\mu _{iC},\mu _{iD}<\tfrac{1}{2}\). Since \(\mu _{iC}+\mu _{iD}<1\), at most one rate can reach \(\tfrac{1}{2}\), and if one does the threshold question is settled without reference to the payoffs: Proposition 3 confines \(p_i\) to \((\mu _{iC},\,1-\mu _{iD})\), so \(\mu _{iC}\ge \tfrac{1}{2}\) forces \(p_i>\tfrac{1}{2}\) and \(\mu _{iD}\ge \tfrac{1}{2}\) forces \(p_i<\tfrac{1}{2}\), whatever the game. Proposition 4 (Mutation-selection balance) Under common symmetric mutation \(\mu _{iC}=\mu _{iD}=\mu \) for all \(i\), with \(0<\mu <\tfrac{1}{2}\), the aggregate cooperation probability is affine in \(\mu \), so that \(p_{C}\rightarrow \Phi \) as \(\mu \rightarrow 0\), \(p_{C}\rightarrow \tfrac{1}{2}\) as \(\mu \rightarrow \tfrac{1}{2}\), and Mutation raises aggregate cooperation when \(\Phi <\tfrac{1}{2}\) and lowers it when \(\Phi >\tfrac{1}{2}\). Proof With \(\mu _{iC}=\mu _{iD}=\mu \), (9) gives \(p_i=(1-2\mu )\phi _i(\delta _i)+\mu \). Averaging over \(i\) using (11) gives the stated expression for \(p_{C}\); affinity in \(\mu \), the two limits, and the derivative are immediate. \(\square \) Here \(\Phi \) is the cooperation level of the mutation-free dynamics of [4]. Mutation interpolates linearly between this selection-driven value and neutrality, reaching \(p_{C}=\tfrac{1}{2}\) as \(\mu \rightarrow \tfrac{1}{2}\): noise erases the influence of the payoffs. 5 The Heterogeneous Public Goods Game The heterogeneous public goods game [10] is played by \(N\) players with player-specific contributions \(\alpha _1,\ldots ,\alpha _N>0\) and public goods multipliers \(r_1,\ldots ,r_N>1\). The payoff to player \(i\) in state \(\textbf{a}=(a_1,\ldots ,a_N)\) is The first term is a public good shared equally by all players, in which each contribution \(\alpha _j a_j\) is scaled by the contributor’s multiplier \(r_j\); the second is player \(i\)’s own contribution cost. We write \(P(\textbf{a})=\frac{1}{N}\sum _j r_j\alpha _j a_j\) for the shared pool. The linear public goods game is additive [4], and the heterogeneous version (14), with player-specific multipliers \(r_i\), is additive for the same reason. Lemma 1 (Additivity of the PGG) The payoff (14) is additive for every player \(i\), with i.e. \(\Delta f_i(\textbf{a})=(1-2a_i)\,\alpha _i(1-r_i/N)\), independent of \(\textbf{a}_{-i}\). Proof When player \(i\) switches from \(a_i\) to \(\bar{a}_i=1-a_i\), only their own term in the pool changes, so \(P(\textbf{a}_{i\rightarrow \bar{a}_i}) = P(\textbf{a})+\tfrac{1}{N}r_i\alpha _i(1-2a_i)\). Expanding both payoffs: Subtracting: The pool \(P(\textbf{a})\) cancels entirely, confirming additivity with \(\delta _i=\alpha _i(1-r_i/N)\). \(\square \) The incentive to switch therefore depends only on the player’s own contribution \(\alpha _i\) and on whether their multiplier exceeds the group size: \(\delta _i<0\), an incentive to cooperate, precisely when \(r_i>N\). The other players’ choices never enter. We call player \(i\) a net beneficiary when \(r_i>N\), since each unit they contribute returns more than a unit to them from the pool, and a net contributor when \(r_iN\), so the symmetric-mutation threshold of Corollary 1 becomes \(r_i>N\): a player cooperates the majority of the time exactly when their personal multiplier on the public good exceeds the group size, that is, when they are a net beneficiary. Without mutation this is the cooperation-dominance condition of [4] (their \(c_iN\) lying above \(p_i=\tfrac{1}{2}\) and curves at \(r_i0\)) a cooperative mutation bias \(\mu _{iC}>\mu _{iD}\) can still raise \(p_i\) above \(\tfrac{1}{2}\), as in Corollary 1. For the PGG, \(\phi _i(\delta _i)>\tfrac{1}{2}\iff r_i>N\), so the sign of \(\partial p_{C}/\partial \mu \) in Proposition 4 is set by whether players are, on average, net beneficiaries: common mutation raises aggregate cooperation when most players have \(r_iN\). The right panel of Fig. 2 shows this mutation-selection balance for a single player: \(p_i\) is affine in the symmetric mutation rate \(\mu \) (Proposition 4), interpolating between the mutation-free value \(\phi _i(\delta _i)\) at \(\mu =0\) and \(\tfrac{1}{2}\) at \(\mu =\tfrac{1}{2}\), increasing in \(\mu \) for a net contributor and decreasing for a net beneficiary. Figure 3 isolates the role of mutation for a single player. A larger selection intensity \(\beta _i\) makes the player respond more sharply to payoff differences, adopting the higher-payoff action with greater probability, whereas \(\beta _i=0\) is payoff-blind random choice. Without mutation \(p_i\) is driven to \(0\) or \(1\) as \(\beta _i\) grows, whereas mutation confines \(p_i\) to \([\mu _{iC},1-\mu _{iD}]\) (Proposition 3). The two panels show the same player (\(r_i=7\), \(\alpha _i=2\)) at \(N=5\) and \(N=200\): increasing the group size alone turns the net beneficiary (\(r_i>N\), \(p_i\) rising to \(1-\mu \)) into a net contributor (\(r_iN\) and negative when \(r_iN\) and raising it when \(r_iN\)) cooperates more as they contribute more or respond more strongly to payoff differences; a net contributor (\(r_i0\), so that the payoff to player \(i\) is with the linear game (14) with common multiplier \(r_i=r\) recovered in the limit \(w\rightarrow 1\). For unit contributions (\(\alpha _i=1\)) the pool is \((r/N)(1+w+\cdots +w^{k-1})\) with \(k=\sum _j a_j\) cooperators: this is the synergy and discounting game of [9], in which the \(j\)th contribution is worth a factor \(w\) of the \((j-1)\)th. This geometric-sum reading requires the total contribution \(X(\textbf{a})\) to be an integer, as it is in both of our examples (\(\alpha _i=1\) here and \(\alpha _i=i\) below); for non-integer contributions we regard (17) as the canonical analytic extension of the synergy and discounting model, with the normalisation \(w-1\) retained so that the linear game is recovered as \(w\rightarrow 1\). For \(w>1\) each additional contribution is worth more than the last: the benefits of cooperation are synergistically enhanced, as when collaborators complement one another. For \(w<1\) they are discounted, as when the public good saturates and further contributions bring diminishing returns. With unit contributions, switching player \(i\) to cooperation adds \((r/N)\,w^{k_{-i}}\) to their share of the pool, where \(k_{-i}\) is the number of co-operating co-players, so cooperation is individually favoured precisely when \((r/N)\,w^{k_{-i}}>1\). For \(w=1\) this is the additive condition \(r>N\) of Sect. 5; for \(w\ne 1\) the incentive depends on the co-players and the game is not additive. With discounting (\(w<1\)) cooperation is favoured only when few others cooperate; with synergy (\(w>1\)) only when enough others do. Figure 4 reports exact quantities computed from the full \(2^N\) linear system using [5], for \(N=8\). The top row takes unit contributions; the top-left and top-centre panels take \(r=4\), so that \(r0\) regardless of \(w\), whilst a defector in a fully cooperative group forgoes \((r/N)w^{N-1}-1>0\). In the strong-selection limit the mutation-free dynamics therefore remain at whichever equilibrium the initial state reaches: started from defection, no cooperation ever arises. Any positive mutation rate makes the chain ergodic, and the top-centre panel of Fig. 4 shows the outcome. For \(w\le 1\) strong selection drives cooperation down to the mutation floor \(\mu \) (Proposition 3 for \(w=1\)); for \(w=1.4\) increasing the selection intensity instead drives \(p_C\) up, towards approximately \(1-\mu \). Occasional payoff-independent switches allow the group to escape the all-defect equilibrium; once at least three co-players cooperate (\((r/N)w^{k}>1\) for \(k\ge 3\) at these parameters), introspection favours joining them, and the group is carried to full cooperation. Mutation therefore acts as an equilibrium-selection device in non-additive games, echoing the role of rare mutations in stochastic evolutionary dynamics [7]. Beyond additivity there can also be an optimal, strictly positive mutation rate. For additive games Proposition 4 rules this out: under common symmetric mutation, aggregate cooperation is affine and hence monotone in \(\mu \). The top-right panel of Fig. 4 shows a smaller multiplier with stronger synergy (\(r=2\), \(w=1.5\)), for which cooperation is individually favoured only once four co-players cooperate. At \(\beta =5\) cooperation is nearly full at weak mutation and noise only erodes it. At \(\beta =10\) and \(\beta =20\) the curve is no longer monotone: \(p_C\) peaks at \(\mu ^*\approx 0.02\) and \(\mu ^*\approx 0.03\) respectively, with the peak falling as selection strengthens. A moderate amount of noise maximises cooperation: enough mutation to seed the escape from the all-defect equilibrium, but not so much as to erode the cooperative one. Additivity is also precisely the point at which the players’ actions decouple. The bottom row of Fig. 4 shows the pairwise correlations \(\textrm{corr}(a_i,a_j)\), the Pearson correlation coefficients of the binary actions under the exact stationary distribution (for indicator variables this is the phi coefficient), for the heterogeneous group of Fig. 1 (contributions \(\alpha _i=i\), \(\beta =0.5\)), here with \(r=4\) and symmetric mutation \(\mu =0.05\); since the total contribution reaches \(36\), mild synergy and discounting (\(w=1.05\) and \(w=0.95\)) suffice. For \(w=1\) every pairwise correlation is zero, as the product measure of Theorem 1 requires, despite the heterogeneity. For \(w=0.95\) all pairs are weakly negatively correlated: under discounting a player’s incentive to cooperate falls with the contributions of co-operating co-players, favouring anti-coordination. For \(w=1.05\), where the two strict equilibria compete, the correlations are positive and substantial, and increase with both players’ contributions: the two largest contributors are the most strongly coupled (\(\textrm{corr}(a_7,a_8)\approx 0.35\)). Beyond additivity a player’s long-run behaviour is thus tied to their co-players’ parameters, in contrast to the additive case, where it depends only on their own; raising the mutation rate erodes every off-diagonal correlation towards zero. 7 Conclusion Building on the additive-game results of [4], we have shown that introspection dynamics with mutation on any additive game remains exactly solvable: in the long run the players behave independently, so the stationary distribution is a product measure, for any finite number of actions per player (Theorem 1), and its computation reduces to one small linear system per player. The product-measure structure, due to [4] for the no-mutation case, is thus preserved under mutation. For two actions, additivity makes each player’s choice forgetful: whenever a player is selected they cooperate with a fixed probability, determined by their own payoff difference, selection intensity, and mutation probabilities alone (Theorem 2), and the stationary distribution is fully explicit (Theorem 3), although this closed form is special to two actions. For the heterogeneous public goods game, the linear payoff structure implies additivity (Lemma 1), and the long-run cooperation probability has a closed form with \(N\) terms rather than \(2^N\) (Corollary 2). The exact analysis does not extend to games that are not additive: when the payoff difference a player evaluates depends on the co-players’ actions, the players’ behaviours are coupled, the stationary distribution is no longer a product measure, and the full \(2^N\) linear system must be solved in general. Games such as the Stag Hunt contain payoff cross terms that cannot be eliminated by rescaling, so no reformulation brings them within the scope of the closed forms. Our numerical exploration of the non-linear public goods game (Sect. 6) shows that mutation then takes on further roles: it selects among coexisting strict equilibria, cooperation can be maximised at a strictly positive mutation rate, and the players’ actions become correlated. Characterising this behaviour analytically remains open. Throughout, the mutation rates are exogenous: they model trembles, experimentation, and behavioural noise in the players’ decisions. On this reading our results describe how noisy self-reflection shapes long-run behaviour: noise keeps every action in play, confines each player strictly inside the pure conventions however strong selection becomes (Proposition 3), and leaves cooperation at a mutation-selection balance between the payoff-driven Fermi level and the neutral \(\tfrac{1}{2}\) (Proposition 4). Whether such noise would itself be selected for, if update rules were heritable and fitness were the expected stationary payoff [15], is a separate question that the closed forms make tractable, and a natural direction for future work. The source code, figures, and data for this paper are under version control and available at [6] and archived at https://doi.org/10.5281/zenodo.22027085, in line with best practices for research software [16]. 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PLoS Comput Biol 13(6):e1005510 Acknowledgements We thank Marta C. Couto and Saptarshi Pal for pointing out that the product-measure decomposition for additive games was already established in [4], which an earlier version of this paper had failed to acknowledge. We also thank two anonymous reviewers for their careful and constructive reports across two rounds of review, which led to substantial improvements throughout the paper. Harry Foster’s research was supported by EPSRC grant EP/Z535126/1. Author information Authors and Affiliations Contributions V.K. and H.F. came up with the original idea. V.K. and H.F. prepared the figures. V.K. wrote the main manuscript. All authors reviewed the manuscript. Corresponding author Ethics declarations Competing interests The authors declare no competing interests. Additional information Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Rights and permissions Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. About this article Cite this article Foster, H., Knight, V. & Krapohl, S. Introspection dynamics with mutation in additive games. Dyn Games Appl (2026). https://doi.org/10.1007/s13235-026-00715-0 Received: Accepted: Published: Version of record: DOI: https://doi.org/10.1007/s13235-026-00715-0