Constructing quantum games from symmetric non-factorizable joint probabilities

We construct quantum games from a table of non-factorizable joint probabilities, coupled with a symmetry constraint, requiring symmetrical payoffs between the players. We give the general result for a Nash equilibrium and payoff relations for a game based on non-factorizable joint probabilities, which embeds the classical game. We study a quantum version of Prisoners' Dilemma, Stag Hunt, and the Chicken game constructed from a given table of non-factorizable joint probabilities to find new outcomes in these games. We show that this approach provides a general framework for both classical and quantum games without recourse to the formalism of quantum mechanics.

[1]  Jens Eisert,et al.  Quantum games , 2000 .

[2]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[3]  Azhar Iqbal,et al.  Constructing multi-player quantum games from non-factorizable joint probabilities , 2007, SPIE Micro + Nano Materials, Devices, and Applications.

[4]  Luca Marinatto,et al.  A quantum approach to static games of complete information , 2000 .

[5]  Derek Abbott,et al.  Quantum games with decoherence , 2004 .

[6]  Howard E. Brandt,et al.  Qubit devices and the issue of quantum decoherence , 1999 .

[7]  S. J. van Enk,et al.  Classical rules in quantum games , 2002 .

[8]  Jiangfeng Du,et al.  Experimental realization of quantum games on a quantum computer. , 2001, Physical Review Letters.

[9]  Wolfgang Leiniger,et al.  Games and information: An introduction to game theory: Eric Rasmusen, (Basil Blackwell, Oxford, 1989) , 1991 .

[10]  D. Abbott,et al.  Probabilistic analysis of three-player symmetric quantum games played using the Einstein–Podolsky–Rosen–Bohm setting , 2008, 0804.2304.

[11]  Taksu Cheon,et al.  Quantum game theory based on the Schmidt decomposition , 2007, quant-ph/0702167.

[12]  M. A. Lohe,et al.  An analysis of the quantum penny flip game using geometric algebra , 2009, 0902.4296.

[13]  Simon C. Benjamin,et al.  Multiplayer quantum games , 2001 .

[14]  R. Lewontin.  ‘The Selfish Gene’ , 1977, Nature.

[15]  Ken Binmore,et al.  Game theory - a very short introduction , 2007 .

[16]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[17]  A. Zeilinger,et al.  Speakable and Unspeakable in Quantum Mechanics , 1989 .

[18]  Tsubasa Ichikawa,et al.  Duality, phase structures, and dilemmas in symmetric quantum games , 2006 .

[19]  Sahin Kaya Ozdemir,et al.  A necessary and sufficient condition to play games in quantum mechanical settings , 2007 .

[20]  Stefan Weigert,et al.  Quantum correlation games , 2003 .

[21]  Neil F. Johnson.  Playing a quantum game with a corrupted source , 2001 .

[22]  C. Ross.  Found , 1869, The Dental register.

[23]  P. Hayden,et al.  Comment on "quantum games and quantum strategies". , 2000, Physical Review Letters.

[24]  N. Mermin.  Quantum mysteries revisited , 1990 .

[25]  Daniël Wedema.  Games And Information An Introduction To Game Theory 3rd Edition , 2011 .

[26]  B. S. Cirel'son.  Quantum generalizations of Bell's inequality , 1980 .

[27]  Edward W. Piotrowski,et al.  An Invitation to Quantum Game Theory , 2002, ArXiv.

[28]  Taksu Cheon,et al.  Constructing quantum games from nonfactorizable joint probabilities. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Jose L. Cereceda.  QUANTUM MECHANICAL PROBABILITIES AND GENERAL PROBABILISTIC CONSTRAINTS FOR EINSTEIN–PODOLSKY–ROSEN–BOHM EXPERIMENTS , 2000 .

[30]  Taksu Cheon,et al.  Classical and quantum contents of solvable game theory on Hilbert space , 2006 .

[31]  Azhar Iqbal,et al.  Evolutionarily stable strategies in quantum games , 2000 .

[32]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[33]  J. Eisert,et al.  Quantum Games and Quantum Strategies , 1998, quant-ph/9806088.

[34]  N. Mermin.  Quantum theory: Concepts and methods , 1997 .

[35]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[36]  Derek Abbott,et al.  AN INTRODUCTION TO QUANTUM GAME THEORY , 2002 .

[37]  Neil Johnson,et al.  Efficiency and formalism of quantum games , 2003 .

[38]  Derek Abbott,et al.  Quantum Matching Pennies Game , 2008, 0807.3599.

[39]  E. W. Piotrowski,et al.  Quantum Market Games , 2001 .

[40]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[41]  D. Meyer.  Quantum strategies , 1998, quant-ph/9804010.

[42]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[43]  L. Vaidman.  Variations on the Theme of the Greenberger-Horne-Zeilinger Proof , 1998 .

[44]  Adrian P. Flitney,et al.  Nash equilibria in quantum games with generalized two-parameter strategies , 2007 .

[45]  Sahin Kaya Ozdemir,et al.  Quantum and classical correlations between players in game theory , 2003 .

[46]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[47]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[48]  Hui Li,et al.  Entanglement enhanced multiplayer quantum games , 2002 .