Punish, but not too hard: How costly punishment spreads in the spatial public goods game

TL;DR

This paper investigates how costly punishment strategies influence cooperation in spatial public goods games, revealing that different punishing strategies lead to distinct spreading mechanisms and non-monotonous cooperation levels as punishment fines vary.

Contribution

It distinguishes the effects of punishing cooperators and defectors in spatial public goods games, showing their different mechanisms and impacts on cooperation.

Findings

Cooperation increases with higher fines when punishing cooperators are involved.

Reentrant transition occurs in the phase diagram with defect punishing strategies as fines increase.

Punishing strategies can spread through different mechanisms depending on their cooperativeness.

Abstract

We study the evolution of cooperation in spatial public goods games where, besides the classical strategies of cooperation (C) and defection (D), we consider punishing cooperators (PC) or punishing defectors (PD) as an additional strategy. Using a minimalist modeling approach, our goal is to separately clarify and identify the consequences of the two punishing strategies. Since punishment is costly, punishing strategies loose the evolutionary competition in case of well-mixed interactions. When spatial interactions are taken into account, however, the outcome can be strikingly different, and cooperation may spread. The underlying mechanism depends on the character of the punishment strategy. In case of cooperating punishers, increasing the fine results in a rising cooperation level. In contrast, in the presence of the PD strategy, the phase diagram exhibits a reentrant transition as the fine is increased. Accordingly, the level of cooperation shows a non-monotonous dependence on the fine. Remarkably, punishing strategies can spread in both cases, but based on largely different mechanisms, which depend on the cooperativeness (or not) of punishers.