Evolutionary dynamics of cooperation in a population with probabilistic corrupt enforcers and violators

TL;DR

This paper investigates how cooperation persists in a population with corrupt enforcers and violators within a public goods game, revealing conditions for stable coexistence or cyclic dominance of strategies through mathematical analysis and simulations.

Contribution

It introduces a novel model incorporating probabilistic corruption in enforcement and violation, providing new insights into the evolution of cooperation under corruption.

Findings

Cooperation can be maintained through coexistence of strategies.

Strategies can form a cyclic dominance similar to rock-scissors-paper.

Mathematical conditions for stable coexistence or cyclic dominance are identified.

Abstract

Pro-social punishment is a key driver of harmonious and stable society. However, this institution is vulnerable to corruption since law-violators can avoid sanctioning by paying bribes to corrupt law-enforcers. Consequently, to understand how altruistic behavior survives in a corrupt environment is an open question. To reveal potential explanations here we introduce corrupt enforcers and violators into the public goods game with pool punishment, and assume that punishers, as corrupt enforcers, may select defectors probabilistically to take a bribe from, and meanwhile defectors, as corrupt violators, may select punishers stochastically to be corrupted. By means of mathematical analysis, we aim to study the necessary conditions for the evolution of cooperation in such corrupt environment. We find that cooperation can be maintained in the population in two distinct ways. First, cooperators, defectors, and punishers can coexist by all keeping a steady fraction of the population. Second, these three strategies can form a cyclic dominance that resembles a rock-scissors-paper cycle or a heteroclinic cycle. We theoretically identify conditions when the competing strategies coexist in a stationary way or they dominate each other in a cyclic way. These predictions are confirmed numerically.