# Cooperation on the monte carlo rule Prison's dilemma game on the grid

## TL;DR

This paper explores the use of a Monte Carlo rule in the prisoner's dilemma game on a grid, demonstrating higher cooperation and robustness compared to other rules, with a smooth transition in cooperation levels characterized by trigonometric behavior.

## Contribution

It introduces a Monte Carlo rule for the prisoner's dilemma game that improves cooperation levels and robustness, providing new insights into equilibrium behavior and payoff dynamics.

## Key findings

- Higher cooperation levels achieved with Monte Carlo rule
- Smooth transition in cooperator density characterized by trigonometric behavior
- Cooperators' average returns remain nearly constant across temptation ranges

## Abstract

In this paper, we investigate the prison's dilemma game with monte carlo rule in the view of the idea of the classic Monte Carlo method on the grid. Monte carlo rule is an organic combination of the current dynamic rules of individual policy adjustment, which not only makes full use of information but also reflects the individual's bounded rational behavior and the ambivalence between the pursuit of high returns and high risks. In addition, it also reflects the individual's behavioral execution preferences. The implementation of monte carlo rule brings an extremely good result, higher cooperation level and stronger robustness are achieved by comparing with the unconditional imitation rule, replicator dynamics rule and fermi rule. When analyse the equilibrium density of cooperators as a function of the temptation to defect, it appears a smooth transition between the mixed state of coexistence of cooperators and defectors and the pure state of defectors when enhancing the temptation, which can be perfectly characterized by the trigonometric behavior instead of the power-law behavior discovered in the pioneer's work. When discuss the relationship between the temptation to defect and the average returns of cooperators and defectors, it is found that cooperators' average returns is almost a constant throughout the whole temptation parameter ranges while defectors' decreases as the growth of temptation. Additionally, the insensitivity of cooperation level to the initial density of cooperators and the sensitivity to the social population have been both demonstrated.