Indirect Reciprocity with Environmental Feedback

TL;DR

This work extends indirect reciprocity by integrating environmental feedback into eco-evolutionary game theory, allowing the payoff structure to coevolve with cooperation levels via a dynamic environment $n$ that modulates payoff matrices $A(n)$. Using a nested model of pairwise interactions, reputation updates under multiple norms, and replicator dynamics for strategies and norms, the authors show that environmental feedback lowers cooperation thresholds and can lock the population into high cooperation via DISC, even from diverse initial conditions. The results reveal a hierarchy among norms—Stern Judging and Simple Standing often dominate, while Scoring and Shunning risk unstable dynamics—emphasizing discriminative punishment as essential for resilience in dynamic settings. The study provides a theoretical framework for understanding how reputation and governance co-evolve with ecological and economic resources, with implications for collective action and the governance of shared resources under environmental variability.

Abstract

Indirect reciprocity maintains cooperation in stranger societies by mapping individual behaviors onto reputation signals via social norms. Existing theoretical frameworks assume static environments with constant resources and fixed payoff structures. However, in real-world systems, individuals' strategic behaviors not only shape their reputation but also induce collective-level resource changes in ecological, economic, or other external environments, which in turn reshape the incentives governing future individual actions. To overcome this limitation, we establish a co-evolutionary framework that couples moral assessment, strategy updating, and environmental dynamics, allowing the payoff structure to dynamically adjust in response to the ecological consequences of collective actions. We find that this environmental feedback mechanism helps lower the threshold for the emergence of cooperation, enabling the system to spontaneously transition from a low-cooperation state to a stable high-cooperation regime, thereby reducing the dependence on specific initial conditions. Furthermore, while lenient norms demonstrate adaptability in static environments, norms with strict discrimination are shown to be crucial for curbing opportunism and maintaining evolutionary resilience in dynamic settings. Our results reveal the evolutionary dynamics of coupled systems involving reputation institutions and environmental constraints, offering a new theoretical perspective for understanding collective cooperation and social governance in complex environments.

Figures (6)

• Figure 1: Schematic framework of the coevolutionary model of strategies, norms, and environment. The timeline at the bottom illustrates the nested dynamics of the model: within each generation interval, repeated individual interactions occur first, followed by system-level intergenerational updates. In the intra-generational interaction phase (left boxes), individuals engage in pairwise games based on their own strategies and the reputation of their opponents; these actions are then evaluated according to the norms of their respective groups, leading to reputation updates. This "game-reputation" loop repeats until the reputation of the population reaches a quasi-steady state. The collective outcomes of these interactions drive changes in the environmental state, thereby altering the payoff context for subsequent games. At the end of a generation, based on accumulated fitness, the population undergoes strategy updates or norm updates through imitation mechanisms, determining the state of the next generation.
• Figure 2: Two-strategy evolutionary dynamics in static and dynamic environments. (A-D) Strategies are distinguished by color: green for DISC, red for ALLD, blue for ALLC, and yellow for coexistence. Solid and open circles denote stable and unstable equilibria, respectively.(A) Bistability between DISC and ALLD in a static environment ($n > 1/2$), where environmental scarcity facilitates DISC invasion. (B-D) DISC vs. ALLC in a static environment. SJ leads to bistability in poor environments (B), whereas SC supports stable coexistence in rich environments (C). Extreme parameters prevent DISC maintenance under SJ at low $n$ (D). (E-H) Impact of environmental feedback. The black solid dot indicates the stable equilibrium of the eco-evolutionary system. Feedback generally breaks the bistability in DISC vs. ALLD, promoting DISC dominance (E) and (F). In DISC vs. ALLC, feedback allows coexistence under SC (G) but triggers oscillations under SH (H). Parameters: $b=1.01$ in (D) and $b=2$ in all other panels; $p=0$, $q=1$ in (B) and (D); $p=1$, $q=0$ in (C); $q=0$, $\theta=2$ in (E); $q=0$, $\theta=5$ in (F); $p=1$, $q=0$, $\theta=2$ in (G); $p=0$, $q=0$, $\theta=0.5$ in (H).
• Figure 3: Evolutionary dynamics of three strategies in static environments. The simplexes illustrate evolutionary trajectories under two social norms (rows) and varying static environments ($n$, columns). Solid circles denote stable equilibria, and open circles denote unstable equilibria. (A–D) Under Stern Judging, the system transitions from a bistable configuration of ALLC and DISC in poor environments ($n < 1/2$) to a bistability between ALLD and DISC in rich environments ($n > 1/2$). (E–H) Under Scoring, ALLC is globally stable in poor environments. In rich environments, the system exhibits a unique bistability between a pure ALLD equilibrium and a stable boundary coexistence of DISC and ALLC. Parameter: b=2.
• Figure 4: Impact of environmental feedback on three-strategy evolutionary dynamics. (A–C) Evolutionary trajectories on the simplex under Stern Judging in static poor ($n=0$, A), static rich ($n=1$, B), and dynamic environments (C), illustrating how feedback alters the basins of attraction. (D) Comparison of steady-state strategy frequencies across four norms under static ($n=0, n=1$) and dynamic conditions. (E) Corresponding average cooperation levels for the scenarios in (D). (F–G) Temporal evolution of strategy frequencies and environmental state ($n$). The numerical solutions of the replicator dynamics (F) show excellent agreement with agent-based Monte Carlo simulations (G). Parameters: $b=3, \theta=3, \epsilon=0.1$.
• Figure 5: Norm competition in static environments. The panels plot the time derivative of Group 1’s fraction, $\dot{\nu}_1$ (vertical axis), against its current fraction, $\nu_1$ (horizontal axis), under different static environmental conditions $n$. The dynamics typically exhibit bistability, where the intersection with the horizontal axis, denoted as $\nu_1^*$, represents the unstable threshold separating the basins of attraction for the two competing norms. The relative competitiveness of a norm is visually determined by the position of this threshold: if $\nu_1^* < 1/2$, the norm adopted by Group 1 possesses a larger basin of attraction and can take over the population even from a minority size. Parameter: $b=2$.