Revisiting institutional punishment in the $N$-person prisoner's dilemma

TL;DR

This paper examines how institutional punishment can promote cooperation in large groups by revisiting the $N$-person prisoner's dilemma, incorporating new mechanisms like minimum contributor requirements and cost sharing.

Contribution

It introduces modifications to the standard model, such as overpunishment removal and cost sharing, and analyzes their effects on cooperation using replicator dynamics and finite-size scaling.

Findings

Fining non-contributors can establish cooperation in infinite populations.

Demographic noise can promote fixation of cooperative strategies in finite populations.

Certain parameter settings enable contributors to resist invasion by free riders.

Abstract

The conflict between individual and collective interests makes fostering cooperation in human societies a challenging task, requiring drastic measures such as the establishment of sanctioning institutions. These institutions are costly because they have to be maintained regardless of the presence or absence of offenders. Here we revisit some improvements to the standard $N$-person prisoner's dilemma formulation with institutional punishment in a well-mixed population, namely the elimination of overpunishment, the requirement of a minimum number of contributors to establish the sanctioning institution, and the sharing of its maintenance costs once this minimum number is reached. In addition, we focus on large groups or communities for which sanctioning institutions are ubiquitous. Using the replicator equation framework for an infinite population, we find that by sufficiently fining players who fail to contribute either to the public good or to the sanctioning institution, a population of contributors immune to invasion by these free riders can be established, provided that the contributors are sufficiently numerous. In a finite population, we use finite-size scaling to show that, for some parameter settings, demographic noise helps to fixate the strategy that contributes to the public good but not to the sanctioning institution even for infinitely large populations when, somewhat counterintuitively, its proportion in the initial population vanishes with a small power of the population size.