Nonlinear Dynamics of Evolutionary Public Goods Games with Consistent- and Inconsistent-Moral-Standard Exclusive Sanctions

Abstract

This paper investigates the evolution of public cooperation within a four-strategy public goods game that incorporates both consistently and inconsistently moralistic exclusion mechanisms. Using replicator dynamics in an infinite well-mixed population, we demonstrate that the presence of Inconsistent Moralists (IMs), i.e., non-contributors who hypocritically exclude other defectors, fundamentally reshapes the dynamical structure of the multi-player social dilemma game. While the system admits no interior fixed point and the IM strategy itself is evolutionarily unstable, IM acts as a critical catalyst by destabilizing pure defection and redirecting evolutionary trajectories toward exclusion-based cooperation. Ultimately, these findings reveal that diverse enforcement strategies can qualitatively alter evolutionary outcomes by providing a previously overlooked indirect pathway for cooperation to emerge and persist in social dilemmas.

1. Introduction

Cooperation is a fundamental phenomenon widely observed in biological, ecological, and socio-economic systems (Nowak & Highfield, 2011). From microbial communities to human societies, cooperative behaviors enable individuals to achieve collective benefits that could not otherwise be achieved through individual actions (Özkaya et al., 2017; Perc et al., 2017). However, the persistence of cooperation is constantly threatened by the presence of free riders who benefit from the public goods produced by others but without contributing to the common pool. This conflict between individual rationality and collective welfare constitutes a typical social dilemma and is commonly described by the classical “Tragedy of the Commons” (Hardin, 1968). One of the ubiquitous cooperative actions in the microbial communities is the synergetic production of invertase. This enzyme is necessary for Saccharomyces cerevisiae to break down the sugar sucrose into monosaccharides. Or else, the budding yeast cannot directly access the energy. In this case, some yeast cells, i.e., cooperators, consume their own energy to secrete invertase, i.e., the common goods, into the periplasm, whereas other ones, i.e., cheaters, do not produce it. A cooperative cell merely obtains about $1\%$ of the glucose and fructose it helps to create, while the other $99\%$ diffuses away into the shared environment, where monosaccharides become freely available to any cells nearby. This instance fits well the classic game-theoretical framework of social dilemmas. In this respect, the public goods game has become one of the most widely used frameworks for studying the evolution of collective cooperation in the context of multi-player interactions (Sigmund, 2010). In a typical public goods game, cooperators contribute a fixed cost to a common pool, whereas defectors refrain from contributing but still share the resulting benefits. The total contributions are multiplied by a synergy factor and then equally redistributed among all group members (Santos et al., 2008). Although mutual cooperation maximizes the collective payoff, individual rationality favors defection, which leads to the collapse of cooperation in both game and classic evolutionary game settings (Hauert et al., 2002; Stewart & Plotkin, 2014). The paradox between empirical observations and theoretical predictions has stimulated extensive studies in the field of evolutionary game theory with the aim of identifying mechanisms capable of sustaining cooperation (Nowak, 2006; Rand & Nowak, 2013). These mechanisms include direct reciprocity (Hilbe et al., 2018; Nowak & Sigmund, 1992; Rossetti & Hilbe, 2024), indirect reciprocity (Fu et al., 2008; Nowak & Sigmund, 2005; Tkadlec et al., 2023; J. Wang & Xia, 2023), network reciprocity (Allen et al., 2017; Fu et al., 2009; A. Li et al., 2020; Ohtsuki et al., 2006; Sheng et al., 2024; Su et al., 2022), collective risks (Du et al., 2012; Pacheco & Santos, 2024; Santos & Pacheco, 2011; J. Wang et al., 2009; T. Wu et al., 2013), various forms of incentive mechanisms (Forsyth & Hauert, 2011; Hauert et al., 2007; Sasaki & Uchida, 2014; Sigmund et al., 2010, 2001; Szolnoki & Perc, 2010, 2017; J. Wu et al., 2022; C.-L. Yang et al., 2018; H.-X. Yang et al., 2015; Zhang et al., 2017), and so on (Jusup et al., 2022; Nowak, 2012; Perc et al., 2013; Szabó & Fath, 2007). Among them, negative incentive mechanisms have received considerable attention (Perc et al., 2017; Sigmund, 2007). Generally speaking, these mechanisms can effectively reduce the payoff advantage of defectors and thus help maintain cooperative behaviors within groups by imposing additional costs or restricting access to the shared benefits.

In particular, social exclusion has been recognized as an efficient mechanism for sustaining cooperation. Instead of directly punishing defectors through monetary penalties, exclusive behaviors are frequently modeled as a sanction way to prevent defectors from sharing the benefits generated by cooperative individuals in both experimental and theoretical studies (Cinyabuguma et al., 2005; Sasaki & Uchida, 2013). As a matter of fact, such a class of exclusive mechanisms has been observed in many real-world systems, including social groups and economic organizations (Salop & Scheffman, 1983; Williams, 2007). Furthermore, lots of theoretical studies have predicted that social exclusion can significantly alter the evolutionary dynamics of public goods games. For example, Sasaki and Uchida introduced a public goods game model with social exclusion and showed that excluders can become evolutionarily stable against the first-order free-riding by defectors as well as the second-order exploitation by pure cooperators under reasonable conditions (Sasaki & Uchida, 2013). Whenever the exclusion cost is sufficiently low and the synergy factor is sufficiently high, the strategy of excluding defectors can dominate the population and effectively suppress both types of free-riding behavior. Although previous studies have significantly improved our understanding of exclusion mechanisms in public goods games, most existing models assume that exclusion behaviors either are attached to cooperative individuals or belong to the class of antisocial behaviors (K. Li et al., 2015; Liu et al., 2019; Quan et al., 2019; Sasaki & Uchida, 2013; Sun et al., 2023; X. Wang & Perc, 2022). However, such an assumption overlooks the kinds of prosocial behaviors of exclusion by other classes of actors observed in real-world systems. For instance, in many competitive environments, individuals who do not contribute to the collective may still attempt to exclude their competitive counterparts in order to maximize their own benefits. Examples of such behaviors can be found in economic monopolies, ecological competition, and social conflicts, where individuals attempt to eliminate rivals while still exploiting available resources. The heuristic reasoning and empirical observations stated above thus suggest that it is necessary to model not only contributors but also free-riders to follow the prosocial norm of exclusive sanctions.

Until now, the evolutionary impacts of such hybrid exclusion strategies in the social dilemma games remain insufficiently understood. In particular, when both cooperative and defective excluders coexist in a population, the resulting strategic interactions may lead to complex evolutionary dynamics that differ fundamentally from traditional models of social exclusion. Motivated by this consideration, we extend the classical public goods game by introducing a four-strategy framework that incorporates different forms of exclusion behaviors. Specifically, the population consists of four types of individuals adopting four different strategies: pure cooperators (Cs); pure defectors (Ds); consistent moralist (CMs), i.e., cooperators who exclude free-riding group members from sharing the common goods; and inconsistent moralists (IMs), i.e., defectors who exclude other counterparts in the same group despite being non-cooperative themselves. While CMs contribute to the common pool and pay an additional cost to exclude defective participants from sharing the benefits, IMs do not contribute to the common pool but still invest in peer exclusion against other free-riders. In this sense, prosocial exclusion can be interpreted as an access-control mechanism by which contributors restrict non-contributors from sharing collective benefits, as may occur in social groups, economic organizations, or ecological communities with shared resources. The incorporation of IMs and CMs in our model introduces a richer competitive structure of strategic interactions, which enables us to explore how different classes of exclusive behaviors influence the evolutionary dynamics of public cooperation.

In this work, we focus on the evolutionary dynamics of this four-strategy system in an infinitely well-mixed population. By employing methods from replicator dynamics, we systematically investigate the existence and stability properties of equilibria and analyze the evolutionary trajectories in the state space of the system. In particular, we examine the dynamical behaviors of the three-dimensional dynamical system and identify the critical conditions under which various strategies become stable, unstable, or coexist with each other.

2. Model

In this study, we investigate the evolutionary dynamics of a public goods game with prosocial exclusion in an infinitely well-mixed population. In each round of the game, N individuals are randomly selected from the population to form a group and participate in a public goods game. Cs and CMs contribute a fixed cost $c=1$, without loss of generality, to the common pool, while Ds and IMs do not contribute. The total contribution in the group is multiplied by a synergy factor r ($1<r<N$) and then equally distributed among all eligible participants. In addition to the stage of the typical public goods game, IMs and CMs also participate in another phase of uncoordinated sanctions by executing peer exclusion. Specifically, CMs pay an additional cost to exclude each non-contributing individual from sharing the common goods. Meanwhile, IMs also pay a cost to exclude all other non-contributors. Operationally, exclusion is applied after contribution decisions and before payoff allocation: CMs exclude non-contributors from sharing the public good, whereas IMs exclude other non-contributors, with each excluded target imposing a cost $c_e$ on the excluder. For simplicity but without loss of generality, we assume the action of exclusion to be perfect in the sense that such a form of active sanction does not fail as well as assume the efficiency of CMs and that of IMs to be equal by setting the same cost of exclusion, $c_e\ge 0$, for CMs and IMs such that no bias is given to each class of excluders throughout this study. Let x, y, z, and w denote the frequencies of Cs, Ds, IMs, and CMs in a population, respectively, which should satisfy $x+y+z+w=1$ as well as $x,y,z,w\ge 0$. Based on the game rules described above, one can derive the expected payoff of Cs, Ds, IMs, and CMs, i.e., $P_{\mathrm{C}},P_{\mathrm{D}}$, $P_{\mathrm{IM}}$, and $P_{\mathrm{CM}}$ in an infinitely well-mixed population as

$$\left\{\begin{array}{cc}{P}_{\mathrm{C}}\hfill & =2rz{\displaystyle \frac{y^N+{(x+y)}^{N-1}[Nx-(x+y)]}{N{x}^2}}+{\displaystyle \frac{r(Nx+y){(x+y)}^{N-2}}{N}}-1\hfill \\ & \phantom{=}-r(N-1){(x+y)}^{N-2}(1-w)+r(N-2){(x+y)}^{N-1}+r,\hfill \\ P_{\mathrm{D}}\hfill & ={\displaystyle \frac{r(N-1)x}{N(1-w-z)}}{(1-w-z)}^{N-1},\hfill \\ P_{\mathrm{IM}}\hfill & =r{\displaystyle \frac{y^N+{(x+y)}^{N-1}[(N-1)x-y]}{Nx}}-c_e(N-1)(y+z),\hfill \\ P_{\mathrm{CM}}\hfill & =r-1-c_e(y+z)(N-1).\hfill \end{array}\right.$$

(1)

For completeness, the detailed derivation of the expected payoffs in Equation (1) is provided in Appendix A. Without loss of generality, we assume that the evolutionary dynamics of the four types of players are governed by the following replicator equations (Taylor & Jonker, 1978):

$$\left\{\begin{array}{cc}\hfill \dot{x}& =x\left(P_{\mathrm{C}}-\overline{P}\right),\hfill \\ \hfill \dot{y}& =y\left(P_{\mathrm{D}}-\overline{P}\right),\hfill \\ \hfill \dot{z}& =z\left(P_{\mathrm{IM}}-\overline{P}\right),\hfill \\ \hfill \dot{w}& =w\left(P_{\mathrm{CM}}-\overline{P}\right),\hfill \end{array}\right.$$

(2)

where $\overline{P}=x{P}_{\mathrm{C}}+y{P}_{\mathrm{D}}+z{P}_{\mathrm{IM}}+w{P}_{\mathrm{CM}}$ denotes the average payoff of the population.

3. Results

In this section, we will systematically study nonlinear dynamics of evolutionary public goods games with consistent- and inconsistent-moral-standard exclusive sanctions by first analyzing the evolutionary dynamics between any two strategies of the full strategy set, then examining the existence and stability of fixed points on the four faces of the simplex, and finally characterizing the evolutionary flows as well as inferring the long-term evolutionary outcomes of strategic interactions starting from any interior points of the state space.

3.1. Evolutionary Outcomes on the Edges of the Simplex

On the C–D edge, the expected payoff difference between Cs and Ds is $P_{\mathrm{C}}-P_{\mathrm{D}}={\displaystyle \frac{r}{N}}-1$, which is negative due to the constraint condition for the enhancement factor $1<r<N$. Hence, consistent with the results of the standard public goods game, no interior fixed point exists along the C–D edge and the direction of evolutionary flows along it always points from C to D. On the C–CM edge, one can obtain $P_{\mathrm{C}}=P_{\mathrm{CM}}=r-1$, which means that each state along the C–CM edge constitutes a fixed point. On the D–IM edge, the expected payoffs of Ds and IMs are given by $P_{\mathrm{D}}=0,\mathrm{and}{P}_{\mathrm{IM}}=-(N-1)c_e$, respectively. This indicates that the evolutionary trajectories starting from any initial point always flow from IM to D except if the exclusion cost $c_e=0$, in which case each state along the D–IM edge is an equilibrium point. On the IM–CM edge, the expected payoff difference between CMs and IMs becomes $P_{\mathrm{CM}}-P_{\mathrm{IM}}=r-1>0$, which implies that CMs strictly dominate IMs. On the D–CM edge, we find that the dynamics of the reduced system are governed by the following differential equation:

$$\dot{w}=w(1-w)\left[r-1-(N-1)c_e(1-w)\right],$$

(3)

which yields the unique interior equilibrium located at

$$w^{*}=1-{\displaystyle \frac{r-1}{(N-1)c_e}},$$

(4)

which lies in $(0,1)$ if and only if $1<r<(N-1)c_e+1$. On the C–IM edge, the expected payoffs of Cs and IMs are simplified to

$$P_{\mathrm{C}}=r-1-{\displaystyle \frac{r(N-1)(N-2)}{N}}(1-x)x^{N-2},$$

(5)

and

$$P_{\mathrm{IM}}={\displaystyle \frac{r(N-1)}{N}}{x}^{N-1}-c_e(N-1)(1-x),$$

(6)

respectively. In this case, we find that there must be a unique interior fixed point on the C–IM edge (see Appendix B). Next, let us study the stability of the fixed points on the boundary of the simplex by analyzing the eigenvalues of the reduced three-dimensional Jacobian matrix on the simplex. Firstly, we consider the stability of the four trivial fixed points: $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, and $(0,0,0,1)$. At the fixed point $(1,0,0,0)$, the eigenvalues of the corresponding Jacobian matrix are given by ${\lambda }_1={\lambda }_2=1-{\displaystyle \frac{r}{N}}$, and ${\lambda }_3=0$. Under the condition $r<N$, the pure C state is unstable. At the fixed point $(0,1,0,0)$, the corresponding eigenvalues become ${\lambda }_1={\displaystyle \frac{r}{N}}-1$, ${\lambda }_2=-(N-1)c_e$, and ${\lambda }_3=r-1-(N-1)c_e$. Therefore, the pure D state is locally asymptotically stable if $r<1+(N-1)c_e$, but is unstable if $r>1+(N-1)c_e$. At the fixed point $(0,0,1,0)$, the eigenvalues of the corresponding Jacobian matrix are given by ${\lambda }_1=r-1+(N-1)c_e$, ${\lambda }_2=(N-1)c_e$ and ${\lambda }_3=r-1$. Thus, for $r>1$, the pure IM state is always unstable. Finally, at the fixed point $(0,0,0,1)$, the corresponding eigenvalues are given by ${\lambda }_1=0$, and ${\lambda }_2={\lambda }_3=1-r$. Thus, the pure CM state is non-hyperbolic: it is neutrally stable along the C–CM edge and transversely stable for $r>1$. Secondly, let us turn to analyze the stability of the nontrivial fixed points on the edges C–CM, D–CM, and C–IM. For any interior point $(x,0,0,1-x)$ on the C–CM edge with $x\in (0,1)$, the eigenvalues of the corresponding Jacobian matrix are given by ${\lambda }_1=0$, and ${\lambda }_2={\lambda }_3={\displaystyle \frac{r(N-1)}{N}}{x}^{N-1}-(r-1)$. Let $x_c={\left[{\displaystyle \frac{N(r-1)}{r(N-1)}}\right]}^{{\displaystyle \frac{1}{N-1}}}$, we find that this equilibrium is non-hyperbolic: it is neutrally stable along the C–CM edge, transversely stable for $x<x_c$, transversely unstable for $x>x_c$, and critical at $x=x_c$. At the unique interior equilibrium $(0,1-w^{*},0,w^{*})$ on the D–CM edge, where $w^{*}=1-{\displaystyle \frac{r-1}{(N-1)c_e}}$, the corresponding eigenvalues are given by ${\lambda }_1=w^{*}(1-w^{*})(N-1)c_e$, ${\lambda }_2=r-1-{\displaystyle \frac{r(N-1)}{N}}{(1-w^{*})}^{N-1}$, and ${\lambda }_3=-(N-1)c_e(1-w^{*})=-(r-1)$. This equilibrium exists only when $1<r<1+(N-1)c_e$. Since ${\lambda }_1>0$ and ${\lambda }_3<0$, the equilibrium is always unstable. More precisely, it is a saddle point if $1-w^{*}>x_c$, unstable with a two-dimensional unstable manifold if $1-w^{*}<x_c$, and critical if $1-w^{*}=x_c$. In addition, we have shown that the unique interior equilibrium on the C–IM edge is unstable (see Appendix B).

3.2. Evolutionary Dynamics on the Faces of the Simplex

Firstly, we focus on the nonlinear dynamics on the C–IM–CM face. Figure 1a shows a typical scenario for the replicator dynamics among these three strategies. Under the condition of the reasonable parameter settings: $N=5$, $r=3$, and $c_e=0.3$, one can observe that CMs out-compete IMs on the IM–CM, and that CMs and Cs are the same in terms of evolutionary performance, which can be verified by the theoretical analysis given in Section 3.1. Furthermore, we find that all interior trajectories in Figure 1a converge toward the C–CM edge, implying that no interior equilibrium exists on this face for the present parameter values. Such an observation is confirmed by a simple proof by contradiction in Appendix C. We next analyze the evolutionary dynamics on the D-IM-CM face of the simplex. The corresponding phase portrait for the typical case $N=5$, $r=1.6$, and $c_e=0.25$ is shown in Figure 1b. Along the boundaries, D dominates IM on the D–IM edge, while CM dominates IM on the IM–CM edge. On the D–CM edge, a boundary equilibrium appears, which is unstable and separates the boundary line into two. These findings can be confirmed by the theoretical analysis given in Section 3.1. Besides, we further discover that all interior trajectories point to either the vertex D or CM, but there exists no stationary point in the interior region of the face D–IM–CM due to the fact that $P_{\mathrm{CM}}-P_{\mathrm{IM}}>0$ always holds. Thirdly, along the C–D edge, the evolutionary direction is pointing from C to D, starting from any initial composition of population. On the D–IM edge, the flow is directed from IM to D, implying that Ds outperform IMs along this boundary. Along the C–IM edge, an unstable boundary equilibrium exists. Furthermore, two interior equilibria, i.e., a stable one and an unstable one, emerge within the face. In addition, we observe that the flows in this face evolve into either the pure defection state or the interior fixed point depending on the initial composition of the population (see Figure 1c). Finally, we determine evolutionary dynamics on the C–D–CM face of the simplex. Figure 1d presents a classical example of numerical simulation on the C–D–CM face of the simplex when $N=5$, $r=3$, and $c_e=0.03$. As shown in Figure 1d, the replicator dynamics on the C–CM edge admit a continuum of boundary equilibria. As for their stability, only a small region near the C vertex is unstable, whereas the remaining equilibria along the edge are stable. These results indicate that the long-term outcomes are attracted to the stable segment of the C–CM edge.

3.3. Evolutionary Competition in the Full Strategy Set

In the interior region of the simplex, we have proved that there exists no fixed point for any combination of the relevant parameters (see Appendix D), which is also supported by the simulation results of numerical examples as shown in Figure 2a,b. In particular, the coexistence of the $\mathrm{C}$–$\mathrm{CM}$ and $\mathrm{D}$–$\mathrm{IM}$ continua of equilibria in Figure 2b is a neutral boundary phenomenon specific to the costless-exclusion limit $c_e=0$. The $\mathrm{C}$–$\mathrm{CM}$ edge is stationary because no non-contributors are present and $P_{\mathrm{C}}=P_{\mathrm{CM}}=r-1$. The $\mathrm{D}$–$\mathrm{IM}$ edge is stationary because no public good is produced, and when $c_e=0$, IMs bear no exclusion cost, yielding $P_{\mathrm{D}}=P_{\mathrm{IM}}=0$. Although no interior equilibrium exists, the evolutionary dynamics in the interior of the simplex exhibit a well-defined directional structure rather than arbitrary motions. In particular, the trajectories are governed by a monotonic quantity that determines the global flow direction. To determine the direction of the flow inside the simplex, we examine the time evolution of the ratio $z/w$ between the frequencies of IMs and CMs. It then follows that ${\displaystyle \frac{d}{dt}}ln\left({\displaystyle \frac{z}{w}}\right)=P_{\mathrm{IM}}-P_{\mathrm{CM}}$. Because $P_{\mathrm{IM}}<P_{\mathrm{CM}}$ (see Appendix D), we have ${\displaystyle \frac{d}{dt}}ln\left({\displaystyle \frac{z}{w}}\right)<0$, which implies that the ratio $z/w$ strictly decreases along any interior trajectory. This result indicates that the evolutionary dynamics always proceed in a direction that suppresses IMs and favors CMs (see Figure 2a,b). This monotonicity provides a strong characterization of the interior flow. Moreover, this monotonicity also rules out periodic orbits in the interior of the full four-strategy simplex. Indeed, if an interior periodic orbit with period T existed, then $ln(z/w)(t+T)=ln(z/w)\left(t\right)$ would have to hold. However, since $P_{\mathrm{IM}}-P_{\mathrm{CM}}<0$ along any interior trajectory, $ln(z/w)$ is strictly decreasing, which yields a contradiction. Therefore, the cooperation-promoting role of IM should be understood as a transient directional pathway rather than as a recurrent cooperation–defection limit cycle.

3.4. Effects of Prosocial Exclusion on Public Cooperation

As shown in Figure 3b, the $(c_e,r)$ parameter space is partitioned into distinct regions, each corresponding to qualitatively different evolutionary outcomes. In particular, the phase diagram characterizes how the long-term stationary states of the system depend on the synergy factor r and the exclusion cost $c_e$ starting from random initial conditions. For small values of r, the system tends to converge to a pure state dominated by Ds, where the pure defection state acts as a stable attractor. In this regime, the benefits generated by cooperation are insufficient to compensate its cost, and exclusion mechanisms are ineffective in preventing the spread of defectors. As r increases, the system enters a regime in which cooperation can be sustained. Here, r should be interpreted as the enhancement factor of the public good, rather than as a direct temptation-to-defect parameter. On the $\mathrm{C}$–$\mathrm{D}$ edge, $P_{\mathrm{C}}-P_{\mathrm{D}}=r/N-1<0$ for $1<r<N$, so defectors still have a payoff advantage. Increasing r only weakens this disadvantage of cooperation. Meanwhile, a larger exclusion cost $c_e$ raises the threshold value of r required to sustain $\mathrm{CM}$-supported cooperation. In this region, trajectories are typically attracted toward the C–CM edge, indicating stable coexistence between Cs and CMs. Importantly, the phase boundary separating defection and cooperation-supporting regimes depends systematically on the exclusion cost $c_e$. As $c_e$ increases, the critical threshold of r required to sustain cooperation also increases, indicating that higher exclusion costs weaken the effectiveness of exclusion. Although IM is unstable, this does not imply that it is irrelevant to the evolutionary dynamics of public cooperation. On the contrary, IM plays a crucial functional role in shaping the evolutionary pathway of the system towards a more efficient state. The key feature of IM lies in its ability to exclude other non-contributing individuals. Unlike pure defectors, IMs actively impose exclusion on both defectors and other counterparts. As a result, the payoff structure within the defective population is fundamentally altered: Ds can no longer freely exploit one another, and mutual exclusion among non-contributors leads to a significant reduction in their own effective payoffs. This suppression effect weakens or even reverses the relative advantage of defective strategies over cooperative strategies, and thus destabilizes the defection-dominated state. In particular, the presence of IMs disrupts the payoff symmetry among defectors, which thus drives the system to escape from the trap of a stable free-riding environment. Consequently, cooperative strategies gain a relative advantage. In this sense, IMs act as a transitional or intermediate strategy that facilitates a directional evolutionary process. Rather than serving as a stable endpoint, IMs reshape the competitive landscape and enable an additional pathway from defection to cooperation. The evolutionary dynamics may therefore follow a transient directional pathway in which Ds are first replaced by IMs, which are subsequently invaded by CMs, ultimately leading to the stable portion of the $\mathrm{C}$–$\mathrm{CM}$ edge rather than to a recurrent cycle. Since the replicator dynamics preserve the support of the initial state, extinct Ds cannot spontaneously reemerge without mutation or external perturbations. We note that the pure $\mathrm{C}$ state is indeed vulnerable to Ds, whereas the CM-rich part of the $\mathrm{C}$–$\mathrm{CM}$ edge is transversely stable because CMs exclude non-contributors. Therefore, although the IM pure state is evolutionarily unstable, it promotes cooperation indirectly by destabilizing defection and enabling the emergence of more effective cooperative strategies.

4. Discussion and Conclusions

Our study demonstrates that exclusion mechanisms do not merely enhance the level of cooperation in a quantitative sense but fundamentally alter the qualitative structure of evolutionary dynamics. Although the phase portraits are illustrated using representative parameter values, the main qualitative conclusions are not restricted to these numerical examples. The edge dynamics, the absence of interior equilibria in the full four-strategy system, and the monotonic decrease in the ratio between IMs and CMs are obtained analytically under the social-dilemma condition $1<r<N$. Thus, the numerical examples mainly serve to visualize the corresponding dynamical regimes, whereas the precise phase boundaries depend on N, r, and $c_e$. A key insight of this work is the nontrivial role of IMs. Although IM itself is not evolutionarily stable, it modifies the payoff structure within the defective population by introducing a mutual suppression mechanism among non-contributors. This effect eliminates the neutrality typically among defective strategies and destabilizes the defection-dominated state. As a consequence, the evolutionary dynamics exhibit a directional pathway from defection to cooperation mediated by IM. This indirect mechanism highlights that strategies need not be stable in order to play a decisive role in shaping evolutionary outcomes. A related variant is an inconsistent moralist who excludes pure defectors but does not sanction other IMs. Such a modification would change the mechanism considered in the present model. In our setting, IMs impose exclusion on both Ds and other IMs, which generates mutual suppression among non-contributors and thereby helps destabilize the defection-dominated state. If IMs did not exclude their IM counterparts, the exclusion cost borne by IMs would be reduced, and defective excluders might become less self-limiting and persist over a wider parameter range. However, the removal of exclusion among IMs would also weaken the mutual suppression mechanism within the defective subpopulation, which is central to the indirect pathway from defection to CM-supported cooperation identified here. Therefore, this modified strategy may lead to qualitatively different boundary equilibria or coexistence patterns, and its full dynamical consequences require a separate analysis of the corresponding payoff structure. We leave this extension for future work. More broadly, our findings suggest that heterogeneous enforcement mechanisms can give rise to qualitatively new dynamical regimes. These results underscore the importance of incorporating strategic diversity in exclusive behaviors when studying the evolution of cooperation in complex systems. In this work, we investigated the evolutionary dynamics of a public goods game with prosocial exclusion. By virtue of replicator dynamics, we analyzed the existence and stability structure of the evolutionary system across from its subsystems to the full system. Our analysis shows that the introduction of prosocial exclusion fundamentally reshapes the evolutionary landscape. Furthermore, we identify a novel dynamical role of defective exclusion that destabilizes defection and enables an indirect pathway toward cooperation. These findings provide a new perspective on the role of exclusion in evolutionary games, and suggest that the diversity of enforcement strategies is crucial for understanding the emergence of cooperation in social dilemmas.