PARRONDO GAMES WITH SPATIAL DEPENDENCE

Toral introduced so-called cooperative Parrondo games, in which there are N >= 3 players 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 3 <= N <= 6 and exact computations can be carried out if 7 <= N <= 19, at least. We provide numerical evidence suggesting that the Parrondo region has nonzero volume in the limit as N -> infinity.