Adaptive Punishment in Social Dilemmas

TL;DR

This work presents a minimal coevolutionary model in which punishment intensity $β$ adapts to the population's cooperation level $x$, thereby reshaping the effective payoff matrix and driving endogenous transitions among Harmony, Stag Hunt, Prisoner's Dilemma, and Chicken games. The authors derive coupled dynamics for $x$ and $β$, uncovering fixed points, bistability, Hopf bifurcations, and limit cycles, all governed by the sanction alignment parameter $η$ and the slope parameter $θ$. Their results show that alignment $η$ qualitatively controls whether cooperation is stabilized, oscillates, or transitions through game classes, offering a quantitative framework for self-organized enforcement. The findings have implications for institutional design in social, ecological, and microbial systems by illustrating how adaptive punishment can sustain cooperation without fixed sanctions.

Abstract

We introduce a coevolutionary framework in which punishment intensity dynamically adapts to the fraction of cooperators in the population. Unlike static models, adaptive punishment reshapes the effective payoff landscape, driving transitions among canonical games, including the Prisoner's Dilemma, Harmony, Stag Hunt, and Chicken games. Analytical results reveal rich dynamical behaviors such as coexistence, bistability, limit cycle and Hopf bifurcation. These findings highlight adaptive punishment as a robust mechanism for sustaining cooperation by the coevolutionary feedback and offer insights into institutional design, ecological interactions, and social governance.

Figures (5)

• Figure 1: Phase diagram in the $(T,S)$ plane. (a-c) $\eta = \{-0.35,0,0.35\}$.
• Figure 2: Phase plane for an initial PDG ($T=2$, $S=-0.6$) with $\theta=1$. (a-d) $\eta=\{0.2,0.4,0.6,0.8\}$.
• Figure 3: Phase plane for an initial SHG ($T=0.9$, $S=-0.6$) with $\theta=1$. (a) $\eta=0.2$; (b) $\eta=0.8$.
• Figure 4: Phase plane for an initial CHG ($T=2$, $S=0.6$) with $\theta=1$. (a-d) $\eta=\{-0.8,-0.6,-0.4,-0.2\}$.
• Figure 5: Phase plane for an initial HG ($T=0.9$, $S=0.6$) with $\theta=1$. (a) $\eta=-1$; (b) $\eta=-0.5$.