| Description |
Parrondo games with spatial dependence have been studied by Ethier and Lee. More precisely, they studied Toral's Parrondo games with N players arranged in a circle. The players play either game A or game B. In game A, a randomly chosen player wins or loses one unit according to the toss of a fair coin. In game B, which depends on parameters p0; p1; p2; p3 2 [0; 1], a randomly chosen player wins or loses one unit according to the toss of a pm-coin, where m 2 f0; 1; 2; 3g depends on the winning or losing status of the player's two nearest neighbors. In this dissertation, we study a spatially dependent game A, which we call game A0, introduced by Xie and others and considered by Ethier and Lee. Noting that game A0 is fair, we say that the Parrondo e#11;ect occurs if game B is losing or fair and the random mixture C0 := A0 + (1 -y )B [respectively, the nonrandom periodic pattern C0 := (A0)rBs] is winning. With p1 = p2 and the parameter space being the unit cube, we investigate numerically the region in which the Parrondo e#11;ect appears. We give su#14;cient conditions for the ergodicity of an interacting particle system in f0; 1gZ corresponding to the random mixture C0 := A0+(1-y )B by applying a theorem of Liggett, and also by means of \annihilating duality". We also show that limN!1 #22;N ( ;1-y )0 and limN!1 #22;N [r;s]0 exist under certain conditions, where #22;N ( ;1-y )0 denotes the mean pro#12;t per turn at equilibrium to the N players playing the random mixture C0 := A0 + (1-y )B, and #22;N [r;s]0 denotes the mean pro#12;t per turn at equilibrium to the N players playing the nonrandom periodic pattern C0 := (A0)rBs. |
