Parrondo's Paradox for Games with Three Players

Parrondo’s paradox appears in game theory which asserts that playing two losing games, A and B (say) randomly or periodically may result in a winning expectation. In the original paradox the strategy of game B was capital-dependent. Some extended versions of the original Parrondo’s game as history dependent game, cooperative Parrondo’s game and others have been introduced. In all of these methods, games are played by two players. In this paper, we introduce a generalized version of this paradox by considering three players. In our extension, two games are played among three players by throwing a three-sided dice. Each player will be in one of three places in the game. We set up the conditions for parameters under which player one is in the third place in two games A and B. Then paradoxical property is obtained by combining these two games periodically and chaotically and (s)he will be in the first place when (s)he plays the games in one of the mentioned fashions. Mathematical analysis of the generalized strategy is presented and the results are also justified by computer simulations

Doctor of Philosophy

dissertationParrondo 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