### Parrondo's paradox via redistribution of wealth

**Stewart N. Ethier**

*(University of Utah)*

**Jiyeon Lee**

*(Yeungnam University)*

#### Abstract

In Toral's games, at each turn one member of an ensemble of $N\ge2$ players is selected at random to play. He plays either game $A'$, which involves transferring one unit of capital to a second randomly chosen player, or game $B$, which is an asymmetric game of chance whose rules depend on the player's current capital, and which is fair or losing. Game $A'$ is fair (with respect to the ensemble's total profit), so the \textit{Parrondo effect} is said to be present if the random mixture $\gamma A'+(1-\gamma)B$ (i.e., play game $A'$ with probability $\gamma$ and play game $B$ otherwise) is winning. Toral demonstrated the Parrondo effect for $\gamma=1/2$ using computer simulation. We prove it, establishing a strong law of large numbers and a central limit theorem for the sequence of profits of the ensemble of players for each $\gamma\in(0,1)$. We do the same for the nonrandom pattern of games $(A')^r B^s$ for all integers $r,s\ge1$. An unexpected relationship between the random-mixture case and the nonrandom-pattern case occurs in the limit as $N\to\infty$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-21

Publication Date: March 14, 2012

DOI: 10.1214/EJP.v17-1867

#### References

- Abbott, D.: Asymmetry and disorder: A decade of Parrondo's paradox.
*Fluct. Noise Lett.*9, (2010), 129--156. - Ajdari, A. and Prost, J.: Drift induced by a spatially periodic potential of low symmetry: Pulsed dielectrophoresis.
*C. R. Acad. Sci., Série 2*315, (1992), 1635--1639. - Optimal play.
Mathematical studies of games and gambling.
Edited by Stewart N. Ethier and William R. Eadington.
*Institute for the Study of Gambling and Commercial Gaming, Reno, NV,*2007. xxvi+551 pp. ISBN: 978-0-9796873-0-3; 0-9796873-0-6 MR2555996 - Ethier, S. N.; Lee, Jiyeon. Limit theorems for Parrondo's paradox.
*Electron. J. Probab.*14 (2009), no. 62, 1827--1862. MR2540850 - Ethier, S. N. and Lee, J.: Parrondo games with spatial dependence.
*Fluct. Noise Lett.*11, (2012), to appear. ARXIVmath.PR/1202.2609 - Harmer, G. P., Abbott, D., Taylor, P. G., and Parrondo, J. M. R.: Brownian ratchets and Parrondo's games.
*Chaos*11, (2001), 705--714. - Harmer, G. P.; Abbott, D. Parrondo's paradox.
*Statist. Sci.*14 (1999), no. 2, 206--213. MR1722065 - Harmer, G. P. and Abbott, D.: A review of Parrondo's paradox.
*Fluct. Noise Lett.*2, (2002), R71--R107. - Mihailovi'c, Z. and Rajkovi'c, M.: One dimensional asynchronous cooperative Parrondo's games.
*Fluct. Noise Lett.*3, (2003), L389--L398. - Osipovitch, D. C., Barratt, C., and Schwartz, P. M.: Systems chemistry and Parrondo's paradox: Computational models of thermal cycling.
*New J. Chem.*33, (2009), 2022--2027. - Parrondo, J. M. R., Harmer, G. P., and Abbott, D.: New paradoxical games based on Brownian ratchets.
*Phys. Rev. Lett.*85, (2000), 5226--5229. - Reed, F. A.: Two-locus epistasis with sexually antagonistic selection: A genetic Parrondo's paradox.
*Genetics*176, (2007), 1923--1929. - Toral, Raúl. Cooperative Parrondo's games.
*Fluct. Noise Lett.*1 (2001), no. 1, L7--L12. MR1870080 - Toral, Raúl. Capital redistributions brings wealth by Parrondo's paradox.
Game theory and evolutionary processes: order from disorder—the role
of noise in creative processes.
*Fluct. Noise Lett.*2 (2002), no. 4, L305--L311. MR1987293 - Xie, Neng-gang; Chen, Yun; Ye, Ye; Xu, Gang; Wang, Lin-gang; Wang, Chao. Theoretical analysis and numerical simulation of Parrondo's paradox
game in space.
*Chaos Solitons Fractals*44 (2011), no. 6, 401--414. MR2803550 - Xie, N.-G., Peng, F.-R., Ye, Y., and Xu, G.: Research on evolution of cooperation among biological system based on Parrondo's paradox game.
*J. Anhui Univ. Technol.*27, (2010), 167--174.

This work is licensed under a Creative Commons Attribution 3.0 License.