Parrondo games with spatial dependence, III

TL;DR

This paper investigates spatially dependent Parrondo games with multiple players, visualizes parameter regions, and critiques a recent state reduction method, highlighting inconsistencies with the original model.

Contribution

It provides graphical analysis of Parrondo regions for 3 to 9 players and challenges the validity of a recent state space reduction approach.

Findings

Parrondo and anti-Parrondo regions identified for N=3 to 9 players.

The state reduction method by Li et al. is shown to be inconsistent with Xie et al.'s model.

Visualizations clarify the parameter dependencies in spatial Parrondo games.

Abstract

We study Toral's Parrondo games with $N$ players and one-dimensional spatial dependence as modified by Xie et al. Specifically, we use computer graphics to sketch the Parrondo and anti-Parrondo regions for $3\le N\le 9$. Our work was motivated by a recent paper of Li et al., who applied a state space reduction method to this model, reducing the number of states from $2^N$ to $N+1$. We show that their reduced Markov chains are inconsistent with the model of Xie et al.