Parrondo games with spatial dependence

TL;DR

This paper investigates the Parrondo effect in spatially dependent games with multiple players arranged in a circle, providing explicit formulas and numerical evidence for the effect's persistence as the number of players grows.

Contribution

It offers explicit formulas for small numbers of players and numerical analysis suggesting the Parrondo effect persists in large populations with spatial dependence.

Findings

Explicit formulas for N=3 to 6 players

Exact computations for N=7 to 19 players

Numerical evidence of nonzero Parrondo region as N approaches infinity

Abstract

Toral introduced so-called cooperative Parrondo games, in which there are N players (3 or more) arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair and the random mixture (1/2)(A+B) is winning. With the parameter space being the unit cube, we investigate the region in which the Parrondo effect appears. Explicit formulas can be found if N=3,4,5,6 and exact computations can be carried out if N=7,8,9,...,19, at least. We provide numerical evidence suggesting that the Parrondo region has nonzero volume in the limit as N approaches infinity.