PARRONDO GAMES WITH SPATIAL DEPENDENCE

摘要

Toral introduced so-called cooperative Parrondo games, in which there are N ≥ 3 playersnarranged in a circle. At each turn one player is randomly chosen to play. He plays eitherngame A or game B. Game A results in a win or loss of one unit based on the toss ofna fair coin. Game B results in a win or loss of one unit based on the toss of a biasedncoin, with the amount of the bias depending on whether none, one, or two of the player’sntwo nearest neighbors have won their most recent games. Game A is fair, so the gamesnare said to exhibit the Parrondo effect if game B is losing or fair and the randomnmixture (1/2)(A + B) is winning. With the parameter space being the unit cube, weninvestigate the region in which the Parrondo effect appears. Explicit formulas can benfound if 3 ≤ N ≤ 6 and exact computations can be carried out if 7 ≤ N ≤ 19, at least.nWe provide numerical evidence suggesting that the Parrondo region has nonzero volumenin the limit as N →∞.