# Evolutionary dynamics in the public goods games with switching between punishment and exclusion

## TL;DR

This paper investigates how switching between punishment and exclusion strategies influences cooperation in public goods games, revealing that intermediate thresholds stabilize cooperation and that exclusion alone can induce cyclic dynamics.

## Contribution

It introduces two novel switching strategies, peer and pool based, and analyzes their effects on cooperation dynamics in a theoretical framework.

## Key findings

- Intermediate thresholds stabilize coexistence of strategies.
- Exclusion alone can generate limit cycle attractors.
- Switching strategies promote cooperation regardless of type.

## Abstract

Pro-social punishment and exclusion are common means to elevate the level of cooperation among unrelated individuals. Indeed, it is worth pointing out that the combined use of these two strategies is quite common across human societies. However, it is still not known how a combined strategy where punishment and exclusion are switched can promote cooperation from the theoretical perspective. In this paper, we thus propose two different switching strategies, namely peer switching that is based on peer punishment and peer exclusion, and pool switching that is based on pool punishment and pool exclusion. Individuals adopting the switching strategy will punish defectors when their numbers are below a threshold and exclude them otherwise. We study how the two switching strategies influence the evolutionary dynamics in the public goods game. We show that an intermediate value of the threshold leads to a stable coexistence of cooperators, defectors and players adopting the switching strategy in a well-mixed population, and this regardless of whether the pool-based or the peer-based switching strategy is introduced. Moreover, we show that the pure exclusion strategy alone is able to evoke a limit cycle attractor in the evolutionary dynamics, such that cooperation can coexist with other strategies.