It has been extensively shown in past literature that Bayesian game theory and quantum non-locality have strong ties between them. Pure entangled states have been used, in both common and conflict interest games, to gain advantageous payoffs, both at the individual and social level. In this paper, we construct a game for a mixed entangled state such that this state gives higher payoffs than classically possible, both at the individual level and the social level. Also, we use the I-3322 inequality so that states that aren’t useful advice for the Bell-CHSH1 inequality can also be used. Finally, the measurement setting we use is a restricted social welfare strategy (given this particular state).
References
[1]
Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47, 777-780. https://doi.org/10.1103/PhysRev.47.777
[2]
Bohr, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 48, 696-702. https://doi.org/10.1103/PhysRev.48.696
[3]
Bell, J.S. (1987) Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. Cambridge University Press, Cambridge.
[4]
Clauser, J.F., Horne, M.A., Shimony, A. and Holt, R.A. (1969) Proposed Experiment to Test Local Hidden-Variable Theories. Physical Review Letters, 23, 880-884. https://doi.org/10.1103/PhysRevLett.23.880
[5]
Collins, D. and Gisin, N. (2003) A Relevant Two Qubit Bell Inequality Equivalent to the CHSH Inequality. Journal of Physics A, 37, 6. https://doi.org/10.1088/0305-4470/37/5/021
[6]
Gibbons, R. (1992) Game Theory for Applied Economists. Princeton University Press, Princeton. https://doi.org/10.1515/9781400835881
[7]
Ordeshook, P. (1986) Game Theory and Political Theory: An Introduction. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511666742
[8]
Colman, A. (1995) Game Theory and Its Applications: In the Social and Biological Sciences. Psychology Press, East Sussex.
[9]
Osborne, M.J. (2003) An Introduction to Game Theory. Oxford University Press, Oxford.
[10]
Neumann, V. and Morgenstern, O. (1994) Theory of Games and Economic Behavior. Princeton University Press, Princeton.
John, C.H. (1968) Games with Incomplete Information Played by “Bayesian” Players, I-III. Part II. Bayesian Equilibrium. Management Science, 14, 320-334. https://doi.org/10.1287/mnsc.14.5.320
[13]
Robert, J.A. (1974) Subjectivity and Correlation in Randomized Strategies. Journal of Mathematical Economics, 1, 67-96. https://doi.org/10.1016/0304-4068(74)90037-8
[14]
Nicolas, B. and Noah, L. (2013) Bell Nonlocality and Bayesian Game Theory. Nature Communications, 4, 2057. https://doi.org/10.1038/ncomms3057
[15]
Taksu, C. and Azhar, I. (2008) Bayesian Nash Equilibria and Bell Inequalities. Journal of the Physical Society of Japan, 77, Article ID: 024801. https://doi.org/10.1143/JPSJ.77.024801
[16]
Mermin, N.D. (1990) Quantum Mysteries Revisited. American Journal of Physics, 58, 731-734. https://doi.org/10.1119/1.16503
[17]
Mermin, N.D. (1990) Simple Unified Form for the Major No-Hidden-Variables Theorems. Physical Review Letters, 65, 3373-3376. https://doi.org/10.1103/PhysRevLett.65.3373
[18]
Asher, P. (1990) Incompatible Results of Quantum Measurements. Physics Letters A, 151, 107-108. https://doi.org/10.1016/0375-9601(90)90172-K
[19]
Zu, C., Wang, Y.-X., Chang, X.-Y., Wei, Z.-H., Zhang, S.-Y. and Duan, L.-M. (2012) Experimental Demonstration of Quantum Gain in a Zero-Sum Game. New Journal of Physics, 3, Article ID: 033002. https://doi.org/10.1088/1367-2630/14/3/033002
[20]
Pappa, A., Kumar, N., Lawson, T., Santha, M., Zhang, S., Diamanti, E. and Kerenidis, I. (2015) Nonlocality and Conflicting Interest Games. Physical Review Letters, 114, Article ID: 020401. https://doi.org/10.1103/PhysRevLett.114.020401
[21]
Roy, A., Mukherjee, A., Guha, T., Ghosh, S., Bhattacharya, S. and Banik, M. (2016) Nonlocal Correlations: Fair and Unfair Strategies in Bayesian Games. Physical Review A, 94, Article ID: 032120. https://doi.org/10.1103/PhysRevA.94.032120
[22]
Banik, M., Bhattacharya, S., Ganguly, N., Guha, T., Ghosh, S., Mukherjee, A., Rai, A. and Roy, A. (2019) Bayesian Games, Social Welfare Solutions and Quantum Entanglement. Quantum, 3, 185. https://doi.org/10.22331/q-2019-09-09-185
[23]
Pal, K.F. and Vertesi, T. (2010) Maximal Violation of the I3322 Inequality Using Infinite Dimensional Quantum Systems. Physical Review A, 82, Article ID: 022116. https://doi.org/10.1103/PhysRevA.82.022116
