# Probabilistic sharing solves the problem of costly punishment

arXiv: https://arxiv.org/abs/1408.1945 · HTML: https://mlsanity.com/paper/1408.1945/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Aug 2014.

## Built on by (most cited first, 3 of 3)

- [Evolutionary mechanisms that promote cooperation may not promote social welfare](https://mlsanity.com/paper/2408.05373/) (2024)
- [PPO-ACT: Proximal Policy Optimization with Adversarial Curriculum Transfer for Spatial Public Goods Games](https://mlsanity.com/paper/2505.04302/) (2025)
- [TUC-PPO: Team Utility-Constrained Proximal Policy Optimization for Spatial Public Goods Games](https://mlsanity.com/paper/2507.02675/) (2025)

## Abstract (arXiv)

Cooperators that refuse to participate in sanctioning defectors create the second-order free-rider problem. Such cooperators will not be punished because they contribute to the public good, but they also eschew the costs associated with punishing defectors. Altruistic punishers - those that cooperate and punish - are at a disadvantage, and it is puzzling how such behaviour has evolved. We show that sharing the responsibility to sanction defectors rather than relying on certain individuals to do so permanently can solve the problem of costly punishment. Inspired by the fact that humans have strong but also emotional tendencies for fair play, we consider probabilistic sanctioning as the simplest way of distributing the duty. In well-mixed populations the public goods game is transformed into a coordination game with full cooperation and defection as the two stable equilibria, while in structured populations pattern formation supports additional counterintuitive solutions that are reminiscent of Parrondo's paradox.