Punishment in bipartite societies

TL;DR

This paper investigates how punishment that operates between distinct groups affects cooperation in bipartite populations. By comparing a uniform punishment scheme to a cross-subset (intergroup) punishment model, the authors derive and analyze replicator dynamics for both settings, including symmetric and asymmetric intergroup enforcement. They find that symmetric intergroup punishment can robustly promote cooperation and improve social welfare, even at low punishment intensity, while breaking symmetry undermines cooperative outcomes. The results highlight symmetry as a unifying principle across social and biological systems and offer insights for designing cross-group accountability in organizations and communities.

Abstract

From ant-acacia mutualism to performative conflict resolution among Inuit, dedicated punishments between distinct subsets of a population are widespread and can reshape the evolutionary trajectory of cooperation. Existing studies have focused on punishments within a homogeneous population, paying little attention to cooperative dynamics in a situation where belonging to a subset is equally important to the actual strategy represented by an actor. To fill this gap, we here study a bipartite population where cooperator agents in a public goods game penalize exclusively those defectors who belong to the alternative subset. We find that cooperation can emerge and remain stable under symmetric intergroup punishment. In particular, at low punishment intensity and at a small value of the enhancement factor of the dilemma game, intergroup punishment promotes cooperation more effectively than a uniformly applied punishment. Moreover, intergroup punishment in bipartite populations tends to be more favorable for overall social welfare. When this incentive is balanced, cooperators can collectively restrain defectors of the alternative set via aggregate interactions in a randomly formed working group, offering a more effective incentive. Conversely, breaking the symmetry of intergroup punishment inhibits cooperation, as the imbalance creates an Achilles' heel in the enforcement structure. Our work, thus, reveals symmetry in intergroup punishment as a unifying principle behind cooperation across human and biological systems.

Figures (9)

• Figure 1: Illustration of punishment mechanisms within a uniform population and a bipartite population in the framework of public goods game. (a) In a uniform population, cooperators punish all defectors within their working group. (b) In a bipartite population, cooperators penalize only those defectors who are from the alternative subsets. Green and blue colors mark the different subsets and arrows denote the actual punishment act between group members.
• Figure 2: Evolutionary dynamics in infinite mixed subgroups. Panels (a)-(c) show ${\dot{x}_C}$ as a function of ${x_{C}}$ for different enhancement factors $r$ and punishment fine $\gamma$. Solid lines indicate the dynamics, arrows show the direction of evolution, and filled circles mark stable equilibria, while open circles mark unstable equilibria. The parameters are set as $N=5, \lambda=0.1$, (a) $r=2, \gamma=0.05$, (b) $r=2, \gamma=0.3$, (c) $r=4.8, \gamma=0.05$.
• Figure 3: Stable distribution of the cooperation level ${x}_C$ under different initial conditions $x_0$ and punishment fine $\lambda$. Panels (a–c) correspond to $\lambda=0.1$, and panels (d–f) correspond to $\lambda=0.5$, with initial cooperation levels $x_0=0.2,0.4,0.8$, respectively. The group size is $N=5$ in all panels.
• Figure 4: Phase flow of subset-$A$ and subset-$B$ cooperators under intergroup punishment. Arrows show evolutionary trajectories in the ${x_C^{[A]}},{x_C^{[B]}}$ plane. Blue curves denote stable manifolds separating two basins of attraction. Red open circles mark saddle points, and black closed circles mark stable equilibria. Panels (a–c) correspond to $r=2$, and (d–f) to $r=3.5$. The punishment fine and cost vary as $(\gamma_{A,B}, \lambda_{A,B})=(0.2, 0.2), (0.5,0.2), (0.2,0.1)$, respectively.
• Figure 5: Saddle point curves and cooperator fractions of subset-$A$ and subset-$B$ under symmetric punishment between subsets, plotted as functions of the multiplication factor $r$. Panel (a) illustrates the saddle points $x{_C^{[A]}}^*$ and $x{_C^{[B]}}^*$ as $r$ varies. $E_1=(0,0)$ and $E_4=(\alpha,1-\alpha)$ are two stable equilibria. Panel (b) shows the fractions of cooperators $x_C^{[A]}$ and $x_C^{[B]}$. The vertical dashed line marks the threshold $r_{th}$, above which a stable internal equilibrium emerges. Parameters: (b) $x_0=(0.4,0.47)$, $\alpha=0.4$, $\lambda_{A,B}=0.2$, $\gamma_{A,B}=0.4$ and $N=5$.