Title:

# Parrondo's Paradox for Games with Three Players

Ejlali, Nasim, Pezeshk, Hamid, Chaubey, Yogendra P. ORCID: https://orcid.org/0000-0002-0234-1429 and Sadeghi, Mehdi (2018) Parrondo's Paradox for Games with Three Players. Technical Report. Concordia University. Department of Mathematics and Statistics, Montreal, Quebec.

## Abstract

Parrondo’s paradox appears in game theory which asserts that playing two losing games, A and B (say) randomly or periodically may result in a winning expectation. In the original paradox the strategy of game B was capital-dependent. Some extended versions of the original Parrondo’s game as history dependent game, cooperative Parrondo’s game and others have been introduced. In all of these methods, games are played by two players. In this paper, we introduce a generalized version of this paradox by considering three players. In our extension, two games are played among three players by throwing a three-sided dice. Each player will be in one of three places in the game. We set up the conditions for parameters under which player one is in the third place in two games A and B. Then paradoxical property is obtained by combining these two games periodically and chaotically and (s)he will be in the first place when (s)he plays the games in one of the mentioned fashions. Mathematical analysis of the generalized strategy is presented and the results are also justified by computer simulations.

Divisions: Concordia University > Faculty of Arts and Science > Mathematics and Statistics Monograph (Technical Report) No Ejlali, Nasim and Pezeshk, Hamid and Chaubey, Yogendra P. and Sadeghi, Mehdi Department of Mathematics & Statistics.Technical Report No. 1/18 Concordia University. Department of Mathematics and Statistics Concordia University 26 April 2018 Natural Sciences and Engineering Research Council Parrondo’s paradox, Combined game, Game theory. 983832 Yogen Chaubey 07 May 2018 13:24 07 May 2018 13:29

## References:

[1] Ajdari, A. and Prost, J. (1992). Drift induced by a spatially periodic potential of low symmetry-pulsed dielectrophoresis. CR Acad. Sci. Paris II, 315:1635–1639.
[2] Almeida, J., Peralta-Salas, D., and Romera, M. (2005). Can two chaotic systems give rise to order? Physica D: Nonlinear Phenomena, 200(1):124–132.
[3] Amengual, P., Allison, A., Toral, R., and Abbott, D. (2004). Discrete–time ratchets, the Fokker–Planck equation and Parrondo’s paradox. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 460, pages 2269–2284. The Royal Society.
[4] Arena, P., Fazzino, S., Fortuna, L., and Maniscalco, P. (2003). Game theory and non-linear dynamics: the Parrondo paradox case study. Chaos, Solitons & Fractals, 17(2):545–555.
[5] Astumian, R. D. and Bier, M. (1994). Fluctuation driven ratchets: molecular motors. Physical Review Letters, 72(11):1766.
[6] Boyarsky, A., G´ora, P., and Islam, M. S. (2005). Randomly chosen chaotic maps can give rise to nearly ordered behavior. Physica D: Nonlinear Phenomena, 210(3):284–294.
[7] Buceta, J., Lindenberg, K., and Parrondo, J. (2001). Stationary and oscillatory spatial patterns induced by global periodic switching. Physical review letters, 88(2):024103.
[8] Challet, D. and Johnson, N. F. (2002). Optimal combinations of imperfect objects. Physical Review Letters, 89(2):028701.
[9] Chandrashekar, C. and Banerjee, S. (2011). Parrondo’s game using a discrete-time quantum walk. Physics Letters A, 375(14):1553–1558.
[10] Cheong, K. H. and Soo, W. W. M. (2013). Construction of novel stochastic matrices for analysis of Parrondo’s paradox. Physica A: Statistical Mechanics and its Applications, 392(20):4727–4738.
[11] Comai, L. (2005). The advantages and disadvantages of being polyploid. Nature reviews genetics, 6(11):836–846.
[12] Danca, M.-F. (2013). Convergence of a parameter switching algorithm for a class of nonlinear continuous systems and a generalization of Parrondo’s paradox. Communications in Nonlinear Science and Numerical Simulation, 18(3):500–510.
[13] Danca, M.-F. (2016). Chaos control and anticontrol of complex systems via Parrondo’s game. In Complex Systems and Networks, pages 263–282. Springer.
[14] Danca, M.-F., Feˇckan, M., and Romera, M. (2014). Generalized form of Parrondo’s paradoxical game with applications to chaos control. International Journal of Bifurcation and Chaos, 24(01):1450008.
[15] Danca, M.-F. and Tang, W. K. (2015). Parrondo’s paradox for chaos control and anticontrol of fractional-order systems. Chinese Physics B, 25(1):010505.
[16] Di Crescenzo, A. (2006). A Parrondo paradox in reliability theory. arXiv preprint math/0602308. [17] Ethier, S. N. and Lee, J. (2012). Parrondo games with spatial dependence and a related spin system, ii. arXiv preprint arXiv:1206.6567.
[18] Ethier, S. N. and Lee, J. (2015). Parrondo games with spatial dependence, iii. Fluctuation and Noise Letters, 14(04):1550039.
[19] Felsenstein, J. (1974). The evolutionary advantage of recombination. Genetics, 78(2):737–756.
[20] Flitney, A. P., Ng, J., and Abbott, D. (2002). Quantum Parrondo’s games. Physica A: Statistical Mechanics and its Applications, 314(1):35–42.
[21] Fotoohinasab, A., Fatemizadeh, E., Pezeshk, H., and Sadeghi, M. (2018). Denoising of genetic switches based on Parrondo’s paradox. Physica A: Statistical Mechanics and its Applications, 493:410–420.
[22] Gr¨unbaum, F. A. and Pejic, M. (2016). Maximal Parrondo’s paradox for classical and quantum markov chains. Letters in Mathematical Physics, 106(2):251–267.
[23] Harmer, G. P. and Abbott, D. (1999). Parrondo’s paradox. Statistical Science, 14(2):206–213.
[24] Harmer, G. P., Abbott, D., Taylor, P. G., and Parrondo, J. M. (2001). Brownian ratchets and Parrondo’s games. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(3):705–714.
[25] Heath, D., Kinderlehrer, D., and Kowalczyk, M. (2002). Discrete and continuous ratchets: from coin toss to molecular motor. Discrete and Continuous Dynamical Systems Series B, 2(2):153–168.
[26] Karlin, S. and Taylor, H. E. (1981). A Second Course in Stochastic Processes. Elsevier.
[27] Key, E. (1987). Computable examples of the maximal lyapunov exponent. Probability Theory and Related Fields, 75(1):97–107.
[28] Kocarev, L. and Tasev, Z. (2002). Lyapunov exponents, noise-induced synchronization, and Parrondo’s paradox. Physical Review E, 65(4):046215.
[29] Lee, C. F. and Johnson, N. F. (2002). Exploiting randomness in quantum information processing. Physics Letters A, 301(5):343–349.
[30] Lee, C. F., Johnson, N. F., Rodriguez, F., and Quiroga, L. (2002). Quantum coherence, correlated noise and Parrondo games. Fluctuation and Noise Letters, 2(04):L293–L298.
[31] Masuda, N. and Konno, N. (2004). Subcritical behavior in the alternating supercritical domany-kinzel dynamics. The European Physical Journal B-Condensed Matter and Complex Systems, 40(3):313–319.
[32] Meyer, D. A. and Blumer, H. (2002). Parrondo games as lattice gas automata. Journal of statistical physics, 107(1-2):225–239.
[33] Mihailovi´c, Z. and Rajkovi´c, M. (2006). Cooperative Parrondo’s games on a two-dimensional lattice. Physica A: Statistical Mechanics and its Applications, 365(1):244–251.
[34] Moraal, H. (2000). Counterintuitive behaviour in games based on spin models. Journal of Physics A: Mathematical and General, 33(23):L203.
[35] Moran, N. A. (1996). Accelerated evolution and muller’s rachet in endosymbiotic bacteria. Proceedings of the National Academy of Sciences, 93(7):2873–2878.
[36] Muller, H. J. (1964). The relation of recombination to mutational advance. Mutation Research/Fundamental and Molecular Mechanisms of Mutagenesis, 1(1):2–9.
[37] Parrondo, J. (1996). How to cheat a bad mathematician. EEC HC&M Network on Complexity and Chaos.
[38] Parrondo, J. M., Harmer, G. P., and Abbott, D. (2000). New paradoxical games based on brownian ratchets. Physical Review Letters, 85(24):5226.
[39] Pawela, Ł. and Sładkowski, J. (2013). Cooperative quantum Parrondo’s games. Physica D: Nonlinear Phenomena, 256:51–57.
[40] Pejic, M. (2015). Quantum Bayesian networks with application to games displaying Parrondo’s paradox. arXiv preprint, arXiv:1503.08868.
[41] Pinsky, R. and Scheutzow, M. (1992). Some remarks and examples concerning the transience and recurrence of random diffusions. In Annales de l’IHP Probabilit´es et statistiques, volume 28, pages 519–536.
[42] Reed, F. A. (2007). Two-locus epistasis with sexually antagonistic selection: a genetic Parrondo’s paradox. Genetics, 176(3):1923–1929.
[43] Spurgin, R. and Tamarkin, M. (2005). Switching investments can be a bad idea when Parrondo’s paradox applies. The Journal of Behavioral Finance, 6(1):15–18.
[44] Tang, T.W., Allison, A. G., and Abbott, D. (2004). Parrondo’s games with chaotic switching. In Second International Symposium on Fluctuations and Noise, pages 520–530. International Society for Optics and Photonics.
[45] Toral, R. (2001). Cooperative Parrondo’s games. Fluctuation and Noise Letters, 1(01):L7–L12.
[46] Wolf, D. M., Vazirani, V. V., and Arkin, A. P. (2005). Diversity in times of adversity: probabilistic strategies in microbial survival games. Journal of Theoretical Biology, 234(2):227–253.
[47] Wu, D. and Szeto, K. Y. (2014). Extended Parrondo’s game and brownian ratchets: Strong and weak Parrondo effect. Physical Review E, 89(2):022142.
[48] Ye, Y., Wang, L., and Xie, N.-G. (2013a). Parrondo’s games based on complex networks and the paradoxical effect. PloS One, 8(7):e67924.
[49] Ye, Y., Xie, N.-G., Wang, L., and Cen, Y.-W. (2013b). The multi-agent Parrondo’s model based on the network evolution. Physica A: Statistical Mechanics and its Applications, 392(21):5414–5421
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