Replicator-mutator dynamics for public goods games with institutional incentives

TL;DR

This work analyzes the replicator-mutator dynamics for public goods games with and without institutional incentives, incorporating both additive and multiplicative mutations. It derives explicit equilibrium conditions, establishes upper bounds on the number of equilibria, and characterizes stability and bifurcations, with particular emphasis on how incentives shift cooperation levels. A key finding is that strong per-capita incentives can guarantee a stable, cooperation-dominant equilibrium (x^* > 1/2), while additive and multiplicative mutations generally suppress cooperation; punishment often outperforms reward in promoting cooperation. The large-group-size limit reveals tractable asymptotics where equilibria converge to at most two, clarifying how group size shapes evolutionary outcomes. Overall, the paper provides rigorous, quantitative insights into the design of institutions to sustain cooperation in public goods contexts, and it identifies rich bifurcation structures arising from combined mutation and incentive mechanisms.

Abstract

Understanding the emergence and stability of cooperation in public goods games is important due to its applications in fields such as biology, economics, and social science. However, a gap remains in comprehending how mutations, both additive and multiplicative, as well as institutional incentives, influence these dynamics. In this paper, we study the replicator-mutator dynamics, with combined additive and multiplicative mutations, for public goods games both in the absence or presence of institutional incentives. For each model, we identify the possible number of (stable) equilibria, demonstrate their attainability, as well as analyse their stability properties. We also characterise the dependence of these equilibria on the model's parameters via bifurcation analysis and asymptotic behaviour. Our results offer rigorous and quantitative insights into the role of institutional incentives and the effect of combined additive and multiplicative mutations on the evolution of cooperation in the context of public goods games.

Figures (17)

• Figure 1: $x_1$ as a function of $\mu$ and $q$, for fixed values of $d, \ r$, and $c$.
• Figure 2: Numerical solutions for PGGs without institutional incentives.
• Figure 3: Numerical solutions of $\dot{x}=G(x)$. The equilibria are given by dashed orange lines, and the non-constant solutions by purple lines.
• Figure 4: Different possible numbers of solutions of the equation $G(x)=0$, in the presence of institutional incentives.
• Figure 5: Number of solutions for randomly chosen parameters, different values of $d$ and numbers of iterations

Theorems & Definitions (32)

• Proposition 3.1
• Proposition 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Proposition 3.6
• Lemma 3.7
• Theorem 3.8
• Lemma 3.9
• proof : Proof of Proposition \ref{['st: equilibria delta zero']}