A Study of Quantum Strategies for Newcomb's Paradox

Author(s)
Takashi Mihara

ABSTRACT

Newcomb’s problem is a game between two players, one of who has an ability to predict the future: let Bob have an ability to predict Alice’s will. Now, Bob prepares two boxes, Box1 and Box2, and Alice can select either Box2 or both boxes. Box1 contains $1. Box2 contains $1,000 only if Alice selects only Box2; otherwise Box2 is empty($0). Which is better for Alice? Since Alice cannot decide which one is better in general, this problem is called Newcomb’s paradox. In this paper, we propose quantum strategies for this paradox by Bob having quantum ability. Many other results including quantum strategies put emphasis on finding out equilibrium points. On the other hand, our results put emphasis on whether a player can predict another player’s will. Then, we show some positive solutions for this problem.

Newcomb’s problem is a game between two players, one of who has an ability to predict the future: let Bob have an ability to predict Alice’s will. Now, Bob prepares two boxes, Box1 and Box2, and Alice can select either Box2 or both boxes. Box1 contains $1. Box2 contains $1,000 only if Alice selects only Box2; otherwise Box2 is empty($0). Which is better for Alice? Since Alice cannot decide which one is better in general, this problem is called Newcomb’s paradox. In this paper, we propose quantum strategies for this paradox by Bob having quantum ability. Many other results including quantum strategies put emphasis on finding out equilibrium points. On the other hand, our results put emphasis on whether a player can predict another player’s will. Then, we show some positive solutions for this problem.

Cite this paper

nullT. Mihara, "A Study of Quantum Strategies for Newcomb's Paradox,"*iBusiness*, Vol. 2 No. 1, 2010, pp. 42-50. doi: 10.4236/ib.2010.21004.

nullT. Mihara, "A Study of Quantum Strategies for Newcomb's Paradox,"

References

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[1] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM Journal of Computing, Vol. 26, pp. 1484– 1509, 1997.

[2] L. K. Grover, “A fast quantum mechanical algorithm for database search,” Proceedings of the 28th ACM Symposium on Theory of Computing, pp. 212–219, 1996.

[3] J. von Neumann and O. Morgenstern, “Theory of games and economic behavior, third edition,” Princeton University Press, Princeton, 1953.

[4] D. A. Meyer, “Quantum strategies,” Physical Review Letters, Vol. 82, pp. 1052–1055, 1999.

[5] J. Eisert, M. Wilkens, and M. Lewenstein, “Quantum games and quantum strategies,” Physical Review Letters, Vol. 83, pp. 3077–3080, 1999.

[6] J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental realization of quantum games on a quantum computer,” Physical Review Letters, Vol. 88, 137902, 2002.

[7] J. Du, H. Li, X. Xu, X. Zhou, and R. Han, “Entanglement enhanced multiplayer quantum games,” Physics Letters A, Vol. 302, pp. 229–233, 2002.

[8] J. Eisert and M. Wilkens, “Quantum games,” Journal of Modern Optics, Vol. 47, pp. 2543–2556, 2000.

[9] A. Iqbal and A. H. Toor, “Evolutionarily stable strategies in quantum games,” Physics Letters A, Vol. 280, pp. 249–256, 2001.

[10] L. Marinatto and T. Weber, “A quantum approach to static games of complete information,” Physics Letters A, Vol. 272, pp. 291–303, 2000.

[11] M. D’Ariano, R. Gill, M. Keyl, R. Werner, B. Kümmerer, and H. Maassen, “The quantum Monty Hall problem,” Quantum Information and Computing, Vol. 2, pp. 355–366, 2002.

[12] A. P. Flitney and D. Abbott, “Quantum version of the Monty Hall problem,” Physical Review A, Vol. 65, 2002.

[13] C. F. Li, Y. S. Zhang, Y. F. Huang, and G. C. Guo, “Quantum strategies of quantum measurements,” Physics Letters A, Vol. 280, pp. 257–260, 2001.

[14] A. P. Flitney, J. Ng, and D. Abbott, “Quantum Parrondo’s games,” Physica A, Vol. 314, pp. 35–42, 2002.

[15] E. W. Piotrowski and J. S?adkowski, “Quantum-like approach to financial risk: Quantum anthropic principle,” Acta Physica Polonica B, Vol. 32, pp. 3873–3879, 2001.

[16] E. W. Piotrowski and J. S?adkowski, “Quantum bargaining games,” Physica A, Vol. 308, 391–401, 2002.

[17] E. W. Piotrowski and J. S?adkowski, “Quantum market games,” Physica A, Vol. 312, pp. 208–216, 2002.

[18] E. W. Piotrowski and J. S?adkowski, “Quantum solution to the Newcomb’s paradox,” International Journal of Quantum Information, Vol. 1, pp. 395–402, 2003.

[19] H. Buhrman, R. Cleve, and W. van Dam, “Quantum entanglement and communication complexity,” SIAM Journal of Computing, Vol. 30, pp. 1829–1841, 2000.

[20] H. Buhrman, W. van Dam, P. Hoyer, and A. Tapp, “Multiparty quantum communication complexity,” Physical Review A, Vol. 60, pp. 2737–2741, 1999.

[21] R. Cleve and H. Buhrman, “Substituting quantum entanglement for communication,” Physical Review A, Vol. 56, pp. 1201–1204, 1997.

[22] R. B. Myerson, “Game theory,” Harvard University Press, Cambridge, 1991.

[23] M. J. Osborne and A. Rubinstein, “A course in game theory,” MIT Press, Cambridge, 1994.