Cooperation in heterogeneous groups, where individuals differ in resources, productivity, and behavioural responsiveness, underpins collective action across many social and biological systems. Introspection dynamics, in which each player compares their payoff to what they would have received under the alternative action, provides a natural learning rule for such asymmetric settings. We study introspection dynamics on multiplayer games in which the payoff difference $Δf_i$ evaluated by a player when considering a strategy switch is independent of all other players' current actions, a property we call state-independence. Under this condition the introspection Markov chain decomposes as a random-scan product of $N$ independent two-state chains, one per player, and the stationary distribution is a product measure. As our main application we consider the heterogeneous public goods game, where $N$ players may differ in their contributions $α_i$, public goods multipliers $r_i$, and selection intensities $β_i$. We prove that the linear payoff structure implies state-independence, and the long-run cooperation probability admits the exact closed form $$ p_C = \frac{1}{N}\sum_{i=1}^{N} \left[\frac{1-μ_{i0}-μ_{i1}}{1+e^{\,β_iα_i(1-r_i/N)}}+μ_{i0}\right], $$ with no asymptotic approximation. Several structural consequences follow immediately: a player-specific cooperation threshold at $r_i = N$ (under symmetric mutation), payoff-neutrality under zero selection intensity, and the sign of each player's sensitivity to their own parameters.

翻译：异质群体中的合作——个体在资源、生产力和行为反应性上存在差异——支撑着众多社会与生物系统中的集体行动。内省动力学中，每个参与者将自身收益与选择替代行动时可获得的收益进行比较，为这种非对称环境提供了一种自然的学习规则。我们研究了多人博弈中的内省动力学，其中参与者在考虑策略转换时评估的收益差$Δf_i$独立于其他所有参与者的当前行动，我们将这一性质称为状态独立性。在此条件下，内省马尔可夫链可分解为$N$个独立两态链（每个参与者对应一条链）的随机扫描乘积，其平稳分布为乘积测度。作为主要应用，我们考虑了异质公共品博弈：$N$个参与者在贡献$α_i$、公共品乘数$r_i$和选择强度$β_i$上可能存在差异。我们证明线性收益结构可推导出状态独立性，且长期合作概率具有精确闭式表达式：$$ p_C = \frac{1}{N}\sum_{i=1}^{N} \left[\frac{1-μ_{i0}-μ_{i1}}{1+e^{\,β_iα_i(1-r_i/N)}}+μ_{i0}\right], $$ 无需渐近近似。由此可直接推导出若干结构性推论：在对称突变条件下每个参与者的合作阈值为$r_i = N$、零选择强度下的收益中性，以及每个参与者对自身参数的敏感性符号。