论文标题
Parrondo效应能有多强？ ii
How strong can the Parrondo effect be? II
论文作者
论文摘要
帕伦多（Parrondo）的投币游戏包括两款游戏，$ a $ and $ b $。游戏$ a $的结果取决于一枚公平硬币的折腾。游戏$ b $的结果是由$ p_0 $ -coin的折腾确定的，如果资本是$ r $的倍数，否则，否则$ p_1 $ -coin的折腾。在任何一场比赛中，玩家都赢得了一个单位，并用尾巴损失一个单位。如果$（1-P_0）（1-P_1）^{R-1} = P_0 \，P_1^{R-1} $，则游戏$ B $是公平的。在上一篇论文中，我们表明，如果$ b $的参数，即$ r $，$ p_0 $和$ p_1 $，则可以受到任意约束的任意，并且两个（公平的游戏$ a $ a $ a $ a $ b $ the the the the the the Pripity of Parter the Pars the Parry of Parrys declience n i i i i。 100％）。在这里，我们证明了这两个游戏的随机序列而不是周期性的序列，也就是说，在每个回合游戏中，$ a $都会使用概率$γ$，并且游戏$ b $否则会播放，其中$γ\ in（0,1）$是任意的。
Parrondo's coin-tossing games comprise two games, $A$ and $B$. The result of game $A$ is determined by the toss of a fair coin. The result of game $B$ is determined by the toss of a $p_0$-coin if capital is a multiple of $r$, and by the toss of a $p_1$-coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails. Game $B$ is fair if $(1-p_0)(1-p_1)^{r-1}=p_0\,p_1^{r-1}$. In a previous paper we showed that, if the parameters of game $B$, namely $r$, $p_0$, and $p_1$, are allowed to be arbitrary, subject to the fairness constraint, and if the two (fair) games $A$ and $B$ are played in an arbitrary periodic sequence, then the rate of profit can not only be positive (the so-called Parrondo effect), but also be arbitrarily close to 1 (i.e., 100%). Here we prove the same conclusion for a random sequence of the two games instead of a periodic one, that is, at each turn game $A$ is played with probability $γ$ and game $B$ is played otherwise, where $γ\in(0,1)$ is arbitrary.
