Paradox in collective history-dependent Parrondo games

집단 과거 의존 파론도 게임의 역설

  • Lee, Ji-Yeon (Department of Statistics, Yeungnam University)
  • Received : 2011.05.20
  • Accepted : 2011.06.22
  • Published : 2011.08.01

Abstract

We consider a history-dependent Parrondo game in which the winning probability of the present trial depends on the results of the last two trials in the past. When a fraction of an infinite number of players are allowed to choose between two fair Parrondo games at each turn, we compare the blind strategy such as a random sequence of choices with the short-range optimization strategy. In this paper, we show that the random sequence of choices yields a steady increase of average profit. However, if we choose the game that gives the higher expected profit at each turn, surprisingly we are not supposed to get a long-run positive profit for some parameter values.

Acknowledgement

Supported by : 한국연구재단

References

  1. 김혜경, 박준표 (2009). 일반화된 분수 지배게임에 대한 균형성. <한국데이터정보과학회지>, 20, 49-55.
  2. 오창혁 (2010). 윷놀이와 확률. <한국데이터정보과학회지>, 21, 719-727.
  3. 이지연 (2009). 집단 파론도 게임의 최적 전략. <한국데이터정보과학회지>, 20, 973-982.
  4. Abbott, D. (2010). Asymmetry and disorder: A decade of Parrondo's paradox. Fluctuation Noise Letters, 9, 129-156. https://doi.org/10.1142/S0219477510000010
  5. Dinis, L. and Parrondo, J. M. R. (2003). Optimal strategies in collective Parrondo games. Europhysics Letters, 63, 319-325. https://doi.org/10.1209/epl/i2003-00461-5
  6. Dinis, L. and Parrondo, J. M. R. (2004). Inefficiency of voting in Parrondo games. Physica A, 343, 701-711. https://doi.org/10.1016/j.physa.2004.06.076
  7. Epstein, R. A. (2007). Parrondo's principle: An overview. In Optimal Play: Mathematical Studies of Games and Gambling, edited by S. N. Ethier and W. R. Eadington, 471-492, Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno.
  8. Ethier, S. N. (2007). Markov Chains and Parrondo's Paradox. In Optimal Play: Mathematical Studies of Games and Gambling, edited by S. N. Ethier and W. R. Eadington, 493-506, Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno.
  9. Ethier, S. N. and Lee, J. (2009). Limit theorems for Parrondo's paradox. Electronic Journal of Probability, 14, 1827-1862. https://doi.org/10.1214/EJP.v14-684
  10. Ethier, S. N. and Lee, J. (2011). A discrete dynamical system for the greedy strategy at collective Parrondo games, Dynamical Systems: An International Journal, (to appear).
  11. Kay, R. J. and Johnson, N. F. (2003). Winning combinations of history-dependent games, Physical Review E, 67, 056128. https://doi.org/10.1103/PhysRevE.67.056128
  12. Kim, H. K. and Lee, D.-S. (2007). Characterizations of the cores of integer total domination games. Journal of the Korean Data & Information Science Society, 18, 1115-1121.
  13. Parrondo, J. M. R. (1996). How to cheat a bad mathematician. In EEC HC&M Network on Complexity and Chaos, Institute for Scientific Interchange Foundation, Torino, Italy.
  14. Parrondo, J. M. R., Harmer, G. P. and Abbott, D. (2000). New paradoxical games based on Brownian ratchets. Physical Review Letters, 85, 5226-5229. https://doi.org/10.1103/PhysRevLett.85.5226
  15. Parrondo, J. M. R., Dinis, L., Garcia-Torano, E. and Sotillo, B. (2007). Collective decision making and paradoxical games. European Physical Journal Special Topics, 143, 39-46. https://doi.org/10.1140/epjst/e2007-00068-0
  16. Van den Broeck, C. and Cleuren, B. (2004). Parrondo games with strategy. In Noise in Complex Systems and Stochastic Dynamics II. SPIE, edited by Z. Gingl, J. M. Sancho, L. Schimansky-Geier and J. Kertesz, 109-118, Bellingham, WA.