Skip to main content Accesibility Help
×
×
Home

On Parrondo's paradox: how to construct unfair games by composing fair games

  • E. S. Key (a1), M. M. Kłosek (a2) and D. Abbott (a3)
Abstract

We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player and yet the new combined game is favourable—this is a counter-intuitive phenomenon known as Parrondo's paradox. We observe that all of the games in question are random walks in periodic environments (RWPE) when viewed on the proper time scale. Consequently, we use RWPE techniques to derive conditions under which Parrondo's paradox occurs.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On Parrondo's paradox: how to construct unfair games by composing fair games
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On Parrondo's paradox: how to construct unfair games by composing fair games
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On Parrondo's paradox: how to construct unfair games by composing fair games
      Available formats
      ×
Copyright
References
Hide All
[1]Allison, A. and Abbott, D., “Stochastically switched control systems”, in Proc. 2nd Int. Conf. on Unsolved Problems of Noise (UPoN'99), Adelaide, Australia, 11–15th July 1999 (eds. Abbott, D. and Kish, L. B.), (American Institute of Physics Conference Proceedings 511, 2000) 249254.
[2]Doering, C. R., “Randomly rattled ratchets”, Nuovo Cimento 17D (1995) 685–679.
[3]Eisert, J., Wilkens, M. and Lewenstein, M., “Quantum games and quantum strategies”, Phys. Rev. Len. 83 (1999) 30773080.
[4]Gargamo, L. and Vaccaro, U., “Efficient generation of fair dice with few biased coins”, IEEE Trans. info. Theory. 45 (1999) 16001606.
[5]Goldenberg, L., Vaidman, L. and Wiesner, S., “Quantum gambling”, Phys. Rev. Lett. 82 (1999) 33563359.
[6]Harmer, G. P. and Abbott, D., “Losing strategies can win by Parrondo's paradox”, Nature 402 (1999) 846.
[7]Harmer, G. P. and Abbott, D., “Parrondo's paradox”, Statistical Science 14 (1999) 206213.
[8]Harmer, G. P., Abbott, D. and Taylor, P., “The paradox of Parrondo's games”, Proc. R. Soc. Lond. A. 456 (2000) 247259.
[9]Harmer, G. P., Abbott, D., Taylor, P. and Parrondo, J. M. R., “Parrondo's paradoxical games and the discrete Brownian ratchet”, in Proc. 2nd Int. Conf. on Unsolved Problems of Noise (UPoN'99), Adelaide, Australia, 11–15th July 1999 (eds. Abbott, D. and Kish, L. B.), (American Institute of Physics Conference Proceedings 511, 2000) 189200.
[10]Harmer, G. P., Abbott, D., Taylor, P., Pearce, C. E. M. and Parrondo, J. M. R., “Information entropy and Parrondo's discrete-time ratchet”, in Proc. Stochaos, Ambleside, UK, 16–20 August 1999 (eds. Broornhead, D. S., Luchinskaya, E. A., McClintock, P. V. E. and Mullin, T.), (American Institute of Physics Conference Proceedings 502, 2000) 544549.
[11]Key, E. S., “Recurrence and transience criteria for random walk in a random environment”, Ann. Prob. 12 (1984) 529560.
[12]Key, E. S., “Computable examples of the maximal Lyapunov exponent”, Probab. Th. Rel. Fields 75 (1987) 97107.
[13]Key, E. S. and Volkmer, H., “Eigenvalue multiplicities of products of companion matrices”, Electronic J. Lin. Algebra 11 (2004) 396409.
[14]Klosek, M. M. and Cox, R. W., “Steady-state currents in sharp stochastic ratchets”, in Proc. Srochaos, Ambleside, UK, 16–20 August 1999 (eds. Broomhead, D. S., Luchinskaya, E. A., McClintock, P. V. E. and Mullin, T.), (American Institute of Physics Conference Proceedings 502, 2000) 325330.
[15]Lee, Y., Abbott, D. and Stanley, H. E., “Minimal Parrondian ratchet: an exactly solvable model”, Phys. Rev. Lett. 91 (2003) 220601.
[16]Linke, H., Humphrey, T. E., Löfgren, A., Sushkov, A. O., Newbury, R., Taylor, R. P. and Omling, P., “Experimental tunneling ratchets”, Science 286 (1999) 23142317.
[17]Maslov, S. and Zhang, Y., “Optimal investment strategy for risky assets”, Int. J. Theor Appl. Fin. 1 (1998) 377387.
[18]McClintock, P. V. E., “Unsolved problems of noise”, Nature 401 (1999) 2325.
[19]Smith, J. Maynard, “personal communication”, 1999.
[20]Meyer, D. A., “Quantum strategies”, Phys. Rev. Lett. 82 (1998) 10521055.
[21]Parrondo, J. M. R., Harmer, G. P. and Abbott, D., “New paradoxical games based on Brownian ratchets”, Phys. Rev. Lett. 85 (2000) 52265229.
[22]Pearce, C. E. M., “On Parrondo's paradoxical games”, in Proc. 2nd Int. Conf. on Unsolved Problems of Noise (UPoN'99), Adelaide, Australia, 11–15th July 1999 (eds. Abbott, D. and Kish, L. B.), (American Institute of Physics Conference Proceedings 511, 2000) 201206.
[23]Pearce, C. E. M., “Entropy, Markov information sources and Parrondo games”, in Proc. 2nd Int. Conf. on Unsolved Problems of Noise (UPoN'99), Adelaide, Australia, 11–15th July 1999 (eds. Abbott, D. and Kish, L. B.), (American Institute of Physics Conference Proceedings 511, 2000) 207212.
[24]Pinsky, R. and Scheutzow, M., “Some remarks and examples concerning transience and recurrence of random diffusions”, Ann. Inst. H. Poincaré Probab. Statist. 28 (1992) 519536.
[25]Plaskota, L., “How to benefit from noise”, J. Complexity 12 (1996) 175184.
[26]Rosato, A., Strandburg, K. J., Prinz, F. and Swendsen, R. H., “Why the Brazil nuts are on top: size segregation of particulate matter shaking”, Phys. Rev. Lett. 58 (1987) 10381040.
[27]Sarmiento, A., Reigada, R., Romero, A. H. and Lindenberg, K., “Enhanced pulse propagation in non-linear arrays of oscillators”, Phys. Rev. E 60 (1999) 53175326.
[28]Seigman, A. E., “personal communication”, 1999.
[29]von Neumann, J., “Various techniques used in connection with random digits, notes by G. E. Forsythe, National Bureau of Standards”, Appl. Math. Ser. 12 (1951) 3638.
[30]Westerhoff, H. V., Tsong, T. Y., Chock, P. B., Chen, Y. and Astumian, R. D., “How enzymes can capture and transmit fee energy from an oscillating electric field”, Proc. Natl. Acad. Sci. 83 (1986) 47344738.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed