How strong can the Parrondo effect be? II

TL;DR

This paper demonstrates that the Parrondo effect, where combined losing strategies produce winning outcomes, can be maximized to nearly 100% profit even with random game sequences, extending previous periodic sequence results.

Contribution

It proves that the Parrondo effect can reach near-maximum profit with random game sequences, generalizing prior work on periodic sequences.

Findings

The Parrondo effect can achieve arbitrarily high profit close to 100%.

Random sequences of games can produce the same maximal effect as periodic sequences.

The result applies under general fairness constraints for the games.

Abstract

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 $\gamma$ and game $B$ is played otherwise, where $\gamma\in(0,1)$ is arbitrary.