Parrondo effect and its maximization in discrete-time dependent random walks†
Jiyeon Lee1
1Department of Statistics, Yeungnam University
Correspondence to: † This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1F1A1A01064244).
1 Professor, Department of Statistics, Yeungnam University, Kyeongsan 38541, Korea. E-mail: leejy@yu.ac.kr
1 Professor, Department of Statistics, Yeungnam University, Kyeongsan 38541, Korea. E-mail: leejy@yu.ac.kr
Received June 28, 2022; Revised September 19, 2022; Accepted September 19, 2022.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Abstract
- Parrondo games are fair if played separately, but there is a paradoxical feature of winning or losing if two games are randomly selected and played. Since each state of the Parrondo game is determined by the state of the previous point in time, even if the asymptotic mean value of each game is close to 0, if two games are arbitrarily mixed, then the asymptotic mean value can be positive or negative. For the original Parrondo game, this state dependency was established and analyzed with a Markov chain model of simple random walks in discrete time. In this paper, we extend to the dependent random walk with the jump size of a general discrete probability distribution and the dependent random walk with the jump size of a normal distribution. We find the conditions for the Parrondo effect to exist and we also find the mixed weight probability that maximizes its effectiveness.
- Keywords : Asymptotic mean, dependent random walks, jump size, Markov chains, Parrondo effect, stationary distriburions.
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