Quantum games: a review of the history, current state, and interpretation

  • Faisal Shah KhanEmail author
  • Neal Solmeyer
  • Radhakrishnan Balu
  • Travis S. Humble


We review both theoretical and experimental developments in the area of quantum games since the inception of the subject circa 1999. We will also offer a narrative on the controversy that surrounded the subject in its early days, and how this controversy has affected the development of the subject.


Quantum games Abstract economies Physical implementations of quantum games Stochastic games Bayesian games Bell’s inequalities 



This material is based upon work supported by the US Department of Energy, Office of Science Advanced Scientific Computing Research and Early Career Research programs.


  1. 1.
    Wiesner, S.: Conjugate coding. ACM SIGACT News Spec. Issue Cryptogr. 15, 77–78 (1983)zbMATHGoogle Scholar
  2. 2.
    Ingarden, R.S.: Quantum information theory. Rep. Math. Phys. 10(1), 43–72 (1976)ADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press, Cambridge (2000)Google Scholar
  4. 4.
    Feynman, R.: Simulating physics with computers. Int. J. Theoret. Phys. 21, 467–488 (1982)MathSciNetGoogle Scholar
  5. 5.
    Deutsch, D.: Quantum theory, the Chruch–Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97–117 (1985)ADSzbMATHGoogle Scholar
  6. 6.
    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)ADSGoogle Scholar
  7. 7.
    Bullock, S.S., O’Leary, D.P., Brennen, G.K.: Asymptotically optimal quantum circuits for \(d\)-level systems. Phys. Rev. Lett. 94, 230502 (2005)ADSGoogle Scholar
  8. 8.
    Charles Bennett, G.B.: Quantum cryptography: public key distributions and coin tossing. Theor. Comput. Sci. 560, 7–11 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Barnett, S.: Quantum Information. Oxford Master Series in Physics. Oxford University Press, Oxford (2009)Google Scholar
  10. 10.
    Wilde, M.: Quantum Information Theory. Quantum Information Theory. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  11. 11.
    Miyake, A., Wadati, M.: Geometric strategy for the optimal quantum search. Phys. Rev. A 64, 042317 (2001)ADSGoogle Scholar
  12. 12.
    Tzu, S.: The Art of War: The Denma Translation. Shambhala Library, Shambhala (2002)Google Scholar
  13. 13.
    Nash, J.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36, 48–49 (1950)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8(3), 457–459 (1941)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Glicksberg, I.L.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. In: Proceedings of the American Mathemtical Society, vol. 3 (1952)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Binmore, K.: Playing for Real: A Text on Game Theory. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  17. 17.
    Myerson, R.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)zbMATHGoogle Scholar
  18. 18.
    Blaquiere, A.: Wave mechanics as a two-player game. In: Blaquire, M.A., Fer F. (eds.) Dynamical Systems and Microphysics. International Centre for Mechanical Sciences (Courses and Lectures), pp. 33–69, Springer, Berlin (1980)Google Scholar
  19. 19.
    Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052–1055 (1999)ADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077–3080 (1999)ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Bleiler, S.: Quantized poker, preprint: arXiv:0902.2196
  22. 22.
    Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272(5), 291–303 (2000)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Khan, F.S., Phoenix, S.: Mini-maximizing two qubit quantum computations. Quant. Inf. Process. 12, 3807–3819 (2013)ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    Sutton, B.: Computing the complete cs decomposition. Numer. Algorithm. 50, 33–65 (2009)ADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Khan, F.S., Humble, T.S.: Nash embedding and equilibrium in pure quantum states, arXiv:1801.02053 [quant-ph] (2018)
  26. 26.
    Frackiewicz, P.: A new model for quantum games based on the marinattoweber approach. J. Phys. A Math. Theoret. 46(27), 275301 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Deng, X., Deng, Y., Liu, Q., Shi, L., Wang, Z.: Quantum games of opinion formation based on the marinatto-weber quantum game scheme. EPL (Europhys. Lett.) 114(5), 50012 (2016)ADSGoogle Scholar
  28. 28.
    Samadi, A.H., Montakhab, A., Marzban, H., Owjimehr, S.: Quantum barrogordon game in monetary economics. Phys. A Stat. Mech. Appl. 489, 94–101 (2018)Google Scholar
  29. 29.
    Khan, F.S., Phoenix, S.: Gaming the quantum. Quant. Inf. Comput. 13, 231–244 (2013)MathSciNetGoogle Scholar
  30. 30.
    van Enk, S.J., Pike, R.: Classical rules in quantum games. Phys. Rev. A 66, 024306 (2002)ADSMathSciNetGoogle Scholar
  31. 31.
    Aumann, R.J.: Subjectivity and correlation in randomized strategies. J. Math. Econom. 1(1), 67–96 (1974)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Benjamin, S.C., Hayden, P.M.: Comment on quantum games and quantum strategies. Phys. Rev. Lett. 87, 069801 (2001)ADSGoogle Scholar
  33. 33.
    Eisert, J., Wilkens, M.: Quantum games. J. Mod. Opt. 47(14–15), 2543–2556 (2000)ADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Benjamin, S.C., Hayden, P.M.: Multiplayer quantum games. Phys. Rev. A 64, 030301 (2001)ADSGoogle Scholar
  35. 35.
    Johnson, N.F.: Playing a quantum game with a corrupted source. Phys. Rev. A 63, 020302 (2001)ADSMathSciNetGoogle Scholar
  36. 36.
    Iqbal, A., Toor, A.: Evolutionarily stable strategies in quantum games. Phys. Lett. A 280(5–6), 249–256 (2001)ADSMathSciNetzbMATHGoogle Scholar
  37. 37.
    Flitney, A.P., Abbott, D.: Quantum version of the Monty Hall problem. Phys. Rev. A 65, 062318 (2002)ADSGoogle Scholar
  38. 38.
    Iqbal, A., Toor, A.H.: Quantum mechanics gives stability to a Nash equilibrium. Phys. Rev. A 65, 022306 (2002)ADSMathSciNetGoogle Scholar
  39. 39.
    Du, J., Li, H., Xu, X., Zhou, X., Han, R.: Entanglement enhanced multiplayer quantum games. Phys. Lett. A 302(5), 229–233 (2002)ADSMathSciNetzbMATHGoogle Scholar
  40. 40.
    Flitney, A.P., Abbott, D.: Quantum games with decoherence. J. Phys. A Math. Gen. 38(2), 449 (2005)ADSMathSciNetzbMATHGoogle Scholar
  41. 41.
    Iqbal, A., Weigert, S.: Quantum correlation games. J. Phys. A Math. Gen. 37(22), 5873 (2004)ADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Chen, J.-L., Kwek, L.C., Oh, C.H.: Noisy quantum game. Phys. Rev. A 65, 052320 (2002)ADSGoogle Scholar
  43. 43.
    Nayak, A., Shor, P.: Bit-commitment-based quantum coin flipping. Phys. Rev. A 67, 012304 (2003)ADSGoogle Scholar
  44. 44.
    Cleve, R., Hoyer, P., Toner, B., Watrous, J.: Consequences and limits of nonlocal strategies. In: 19th IEEE Annual Conference on Computational Complexity, 2004. Proceedings, pp. 236–249, IEEE (2004)Google Scholar
  45. 45.
    Fitzi, M., Gisin, N., Maurer, U.: Quantum solution to the Byzantine agreement problem. Phys. Rev. Lett. 87, 217901 (2001)ADSGoogle Scholar
  46. 46.
    Kempe, J., Kobayashi, H., Matsumoto, K., Toner, B., Vidick, T.: Entangled games are hard to approximate. In: Proceedings of 49th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 447–456 (2008)Google Scholar
  47. 47.
    Aharonov, D., Ta-Shma, A., Vazirani, U.V., Yao, A.C.: Quantum bit escrow. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC ’00 (New York, NY, USA), pp. 705–714, ACM (2000)Google Scholar
  48. 48.
    Marriott, C., Watrous, J.: Quantum Arthur-Merlin games. Comput. Complex. 14, 122–152 (2005)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Chi-Chih Yao, A.: Quantum circuit complexity. In: Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science, SFCS ’93 (Washington, DC, USA), pp. 352–361, IEEE Computer Society (1993)Google Scholar
  50. 50.
    Aumann, R.: Game Theory. Palgrave MacMillan, Basingstoke (1989)Google Scholar
  51. 51.
    Brandenburger, A.: Cooperative Game Theory, Lecture NotesGoogle Scholar
  52. 52.
    Piotrowski, E., Sadkowski, J.: Quantum market games. Phys. A Stat. Mech. Appl. 312(1), 208–216 (2002)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Piotrowski, E.W., Sadkowski, J., Syska, J.: Interference of quantum market strategies. Phys. A Stat. Mech. Appl. 318(3), 516–528 (2003)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Mariantoni, M., Wang, H., Bialczak, R.C., Lenander, M., Lucero, E., Neeley, M., O’Connell, A.D., Sank, D., Weides, M., Wenner, J., Yamamoto, T., Yin, Y., Zhao, J., Martinis, J.M., Cleland, A.N.: Photon shell game in three-resonator circuit quantum electrodynamics. Nat. Phys. 7(4), 287–293 (2011)Google Scholar
  55. 55.
    Quantum game theory. Accessed 12 March 2018
  56. 56.
    Guo, H., Zhang, J., Koehler, G.J.: A survey of quantum games. Decis. Support Syst. 46(1), 318–332 (2008)Google Scholar
  57. 57.
    Shimamura, J., Zdemir, A.K., Morikoshi, F., Imoto, N.: Quantum and classical correlations between players in game theory. Int. J. Quant. Inf. 02(01), 79–89 (2004)zbMATHGoogle Scholar
  58. 58.
    Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38, 886–893 (1952)ADSMathSciNetzbMATHGoogle Scholar
  59. 59.
    Partha Sarathi Dasgupta, E.M.: Commentary—physical sciences—mathematics: Debreus social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 112, 15769–16770 (2015)Google Scholar
  60. 60.
    Piotrowski, E.W., Sładkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Zhang, P.E.A.: Quantum gambling based on Nash-equilibrium. NPJ Quant. Inf. 3, 24 (2017)ADSGoogle Scholar
  62. 62.
    Bouyer, P., Brenguier, R., Markey, N., Ummels, M.: Pure Nash equilibria in concurrent deterministic games. Log. Methods Comput. Sci. 11(2) (2015)Google Scholar
  63. 63.
    de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. Theor. Comput. Sci. 386, 188–217 (2007)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Zabaleta, O., Arizmendi, C.: Quantum games based communication protocols. J. Adv. Appl. Comput. Math. 4, 35–39 (2017)Google Scholar
  65. 65.
    Houshmand, M., Houshmand, M., Mashhadi, H.R.: Game theory based view to the quantum key distribution bb84 protocol. In: 2010 Third International Symposium on Intelligent Information Technology and Security Informatics, pp. 332–336 (2010)Google Scholar
  66. 66.
    Giannakis, K., Papalitsas, C., Kastampolidou, K., Singh, A., Andronikos, T.: Dominant strategies of quantum games on quantum periodic automata. Computation 3(4), 586–599 (2015)Google Scholar
  67. 67.
    Anand, N., Benjamin, C.: Do quantum strategies always win. Quant. Inf. Process. 14, 4027–4038 (2015)ADSMathSciNetzbMATHGoogle Scholar
  68. 68.
    Mishima, H.: Non-abelian strategies in quantum penny flip game. Progr. Theor. Exp. Phys. 2018(1), 013A04 (2018)MathSciNetGoogle Scholar
  69. 69.
    Bao, N., Yunger Halpern, N.: Quantum voting and violation of arrow’s impossibility theorem. Phys. Rev. A 95, 062306 (2017)ADSGoogle Scholar
  70. 70.
    Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.: On a network creation game. In: Proceedings of the Twenty-second Annual Symposium on Principles of Distributed Computing, PODC ’03 (New York, NY, USA), pp. 347–351, ACM (2003)Google Scholar
  71. 71.
    Demaine, E.D., Hajiaghayi, M., Mahini, H., Zadimoghaddam, M.: The price of anarchy in network creation games. ACM Trans. Algorithms 8(2), 13:1–13:13 (2012)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Scarpa, G.: Network games with quantum strategies. In: Quantum Communication and Quantum Networking. QuantumComm 2009. Lecture Notes of the Institute for Computer Sciences, Engineering, vol, Social Informatics and Telecommunications, vol. 36Google Scholar
  73. 73.
    Khan, F., Elsokkary, N., Humble, T.: arXiv:1808.06926v2 [cs.GT], 2018
  74. 74.
    Rai, A., Paul, G.: Strong quantum solutions in conflicting-interest bayesian games. Phys. Rev. A 96, 042340 (2017)ADSGoogle Scholar
  75. 75.
    Brunner, N., Linden, N.: Connection between Bell nonlocality and Bayesian game theory. Nat. Commun. 4, 2057 (2013)ADSGoogle Scholar
  76. 76.
    Harsanyi, J.C.: Games with incomplete information played by Bayesian players. Mgt. Sci. 14, 159–182 (1967)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Cheon, T., Iqbal, A.: Bayesian Nash equilibria and Bell inequalities. J. Phys. Soc. Jpn. 77, 024801 (2008)ADSGoogle Scholar
  78. 78.
    Fine, A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23, 1306–1310 (1982)ADSMathSciNetGoogle Scholar
  79. 79.
    Silman, J., Machnes, S., Aharon, N.: On the relation between Bell’s inequalites and nonlocal games. Phys. Lett. A 372, 3796–3800 (2008)ADSMathSciNetzbMATHGoogle Scholar
  80. 80.
    Flitney, A., Schlosshauer, M., Chmid, C., Laskowski, W., Hollenberg, L.: Equivalence between Bell inequalities and quantum minority games. Phys. Lett. A 373, 521–524 (2009)ADSMathSciNetzbMATHGoogle Scholar
  81. 81.
    Iqbal, A., Abbott, D.: Equivalence between Bell inequalities and quantum minority games. Phys. Lett. A 374, 3155–3163 (2010)ADSMathSciNetzbMATHGoogle Scholar
  82. 82.
    LA Mura, P.: Correlated equilibria of classical strategic games with quantum signals. Int. J. Quant. Inf. 03(01), 183–188 (2005)zbMATHGoogle Scholar
  83. 83.
    Brandenburger, A., Mura, P.L.: Team decision problems with classical and quantum signals. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 374, 2058 (2016)Google Scholar
  84. 84.
    Zhang, S.: Quantum strategic game theory. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS ’12 (New York, NY, USA), pp. 39–59, ACM (2012)Google Scholar
  85. 85.
    Auletta, V., Ferraioli, D., Rai, A., Scarpa, G., Winter, A.: Belief-invariant equilibria in games with incomplete information, CoRR, arXiv:1605.07896 (2016)
  86. 86.
    Pappa, A., Kumar, N., Lawson, T., Santha, M., Zhang, S., Diamanti, E., Kerenidis, I.: Nonlocality and conflicting interest games. Phys. Rev. Lett. 114, 020401 (2015)ADSGoogle Scholar
  87. 87.
    Melo-Luna, C., Susa, C., Ducuara, A., Barreiro, A., Reina, J.: Quantum locality in game strategy. Sci. Rep. 7, 44730 (2016)ADSGoogle Scholar
  88. 88.
    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24, 379–385 (1991)ADSMathSciNetGoogle Scholar
  89. 89.
    Guney, V., Hiller, M.: Bell inequalities from group actions: three parties and non-abelian groups. Phys. Rev. A 91, 052110 (2015)ADSGoogle Scholar
  90. 90.
    Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Birkhauser, Basel (1992)zbMATHGoogle Scholar
  91. 91.
    Chang, M.-H.: Quantum Stochastics. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  92. 92.
    Barry, Jennifer, Barry, Daniel T., Aaronson, S.: Quantum partially observable Markov decision processes. Phys. Rev. A 90, 032311 (2014)ADSGoogle Scholar
  93. 93.
    Shapley, L.S.: Stochastic games. Proc. Natl. Acad. Sci. 39(10), 1095–1100 (1953)ADSMathSciNetzbMATHGoogle Scholar
  94. 94.
    Solana, E., Vieille, N.: Stochastic games—perspective. In: Proceedings of the National Academy of Sciences, vol. 112 (2015)ADSMathSciNetzbMATHGoogle Scholar
  95. 95.
    Johari, R.: Lecture Notes in Game Theory. preprint. Stanford UniversityGoogle Scholar
  96. 96.
    Nayyar, A., Gupta, A., Langbort, C., Basar, T.: Common information based Markov perfect equilibria for stochastic games with asymmetric information: finite games. IEEE Trans. Autom. Control 59, 3 (2014)MathSciNetzbMATHGoogle Scholar
  97. 97.
    Blackwell, D.: Discrete dynamic programming. Ann. Math. Stat. 33(2), 719–726 (1962)MathSciNetzbMATHGoogle Scholar
  98. 98.
    Hora, A., Obata, N.: Quantum Probability and Spectral Analysis of Graphs. Springer, Berlin (2007)zbMATHGoogle Scholar
  99. 99.
    Gleason, A.: Measures on the closed subspace of a Hilbert space. J. Math. Mech. 6, 885–893 (1957)MathSciNetzbMATHGoogle Scholar
  100. 100.
    Bouten, V.B.L.M., Edward, S.: Bellman equations for optimal control of qubits. J. Phys. B At. Mol. Opt. Phys. 38(3) (2005)ADSGoogle Scholar
  101. 101.
    Kurt Jacobs, H.W., Wang, X.: Coherent feedback that beats all measurement-based feedback protocols. New J. Phys. 16, 073036 (2014)Google Scholar
  102. 102.
    DiVincenzio, D.P.: The physical implementation of quantum computation. Fortschr. Phys 48, 771–783 (2000)Google Scholar
  103. 103.
    Pfaff, W., et al.: Unconditional quantum teleportation between distant solid-state quantum bits. Science 345, 532 (2014)ADSMathSciNetzbMATHGoogle Scholar
  104. 104.
    Du, J., Li, H., Xu, X., Shi, M., Wu, J., Zhou, X., Han, R.: Experimental realization of quantum games on a quantum computer. Phys. Rev. Lett 88, 137902 (2002)ADSGoogle Scholar
  105. 105.
    Mitra, A., Sivapriya, K., Kumar, A.: Experimental implementation of a three qubit quantum game with corrupt source using nuclear magnetic resonance quantum information processor. J. Magn. Reson. 187, 306–313 (2007)ADSGoogle Scholar
  106. 106.
    Cory, D., Price, M., Havel, T.F.: Nuclear magnetic resonance spectroscopy: an experimentally accessible paradigm for quantum computing. Phys. D Nonlinear Phenom. 120(1) (1998)ADSGoogle Scholar
  107. 107.
    Braunstein, S.L., Caves, C.M., Jozsa, R., Linden, N., Popescu, S., Schack, R.: Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83(5), 1054–1057 (1999)ADSGoogle Scholar
  108. 108.
    Zhang, P., Zhang, Y.-S., Huang, Y.-F., Peng, L., Li, C.-F., Guo, G.-C.: Optical realization of quantum gambling machine. EPL 82, 30002 (2008)MathSciNetGoogle Scholar
  109. 109.
    Balthazar, W., Passos, M., Schmidt, A., Caetano, D., Huguenin, J.: Experiemntal realization of the quantum duel game using linear optical circuits. J. Phys. B Atom. Mol. Opt. Phys. 48, 165505 (2015)ADSGoogle Scholar
  110. 110.
    Pinheiro, A.R.C., Souza, C., Caetano, D., Juguenin, J., Schmidt, A., Khoury, A.: Vector vortex implementaion of a quantum game. J. Opt. Soc. Am. B 30, 3210–3214 (2013)ADSGoogle Scholar
  111. 111.
    Prevedel, R., Andre, S., Walther, P., Zeilinger, A.: Experimental realization of a quantum game on a one-way quantum computer. N. J. Phys. 9, 205 (2007)Google Scholar
  112. 112.
    Altepeter, J., Hall, M., Medic, M., Patel, M., Meyer, D., Kumar, P.: Experimental realization of a multi-player quantum game, OSA/IPNRA/NLO/SL (2009)Google Scholar
  113. 113.
    Schmid, C., Flitney, A., Wieczorek, W., Kiesel, N., Weinfurter, H., Hollenberg, L.: Experiental implementation of a four-player quantum game. N. J. Phys. 12, 063031 (2010)Google Scholar
  114. 114.
    Zu, C., Wang, Y.X., Chang, X.-Y., Wei, Z.-H., Zhang, S.-Y., Duan, L.-M.: Experimental demonstration of quantum gain in a zero-sum game. N. J. Phys. 14, 033002 (2012)Google Scholar
  115. 115.
    Buluta, I.M., Fujiwara, S.: Quantum games in ion traps. Phys. Lett. A 358, 100–104 (2006)ADSzbMATHGoogle Scholar
  116. 116.
    Shuai, C., Mao-Fa, F., Jian-Bin, L., Xin-Wen, W., Xiao-Juan, Z.: A scheme for implementing quantum game in cavity QED. Chin. Phys. B 18, 894 (2009)ADSGoogle Scholar
  117. 117.
    Debnath, S., et al.: Demonstration of a small programmable quantum computer with atomic qubits. Nature 536, 63 (2016)ADSGoogle Scholar
  118. 118.
    Hucul, D.: Modular entanglement of atomic qubits using photons and phonons. Nat. Phys. 11, 37–42 (2015)Google Scholar
  119. 119.
    Solmeyer, N., Linke, N.M., Figgatt, C., Landsman, K.A., Balu, R., Siopsis, G., Monroe, C.R.: Demonstration of Bayesian quantum game on an ion trap quantum computer. Quant. Sci. Technol. 3(4) (2018)ADSGoogle Scholar
  120. 120.
    Chen, K., Hogg, T.: How well do people play a quantum Prisoner’s Dilemma? Quant. Inf. Process. 5, 43–67 (2006)MathSciNetzbMATHGoogle Scholar
  121. 121.
    Chen, K., Hogg, T.: Experiments with probabilistic quantum auctions. Quant. Inf. Process. 7, 139–152 (2008)zbMATHGoogle Scholar
  122. 122.
    Schelling, T.C.: Arms and Influence. Yale University Press, New Haven (1966)Google Scholar
  123. 123.
    Zabaleta, O.G., Barrangú, J.P., Arizmendi, C.M.: Quantum game application to spectrum scarcity problems. Phys. A Stat. Mech. Appl. 466, 455–461 (2017)MathSciNetGoogle Scholar
  124. 124.
    Challet, D., Zhang, Y.-C.: Emergence of cooperation and organization in an evolutionary game. Phys. A Stat. Mech. Appl. 246(3–4), 407–418 (1997)Google Scholar
  125. 125.
    Flitney, A.P., Hollenberg, L.C.L.: Multiplayer quantum minority game with decoherence. In: Fluctuations and Noise in Photonics and Quantum Optics III. International Society for Optics and Photonics 5842, 175–183 (2005)Google Scholar
  126. 126.
    Solmeyer, N., Dixon, R., Balu, R.: Quantum routing games, arXiv preprint arXiv:1709.10500 (2017)
  127. 127.
    Roughgarden, T.: On the severity of Braess’s paradox: designing networks for selfish users is hard. J. Comput. Syst. Sci. 72(5), 922–953 (2006)MathSciNetzbMATHGoogle Scholar
  128. 128.
    Hanauske, M., Bernius, S., Dugall, B.: Quantum game theory and open access publishing. Phys. A Stat. Mech. Appl. 382(2), 650–664 (2007)Google Scholar
  129. 129.
    de Sousa, P., Ramos, R.: Multiplayer quantum games and its application as access controller in architecture of quantum computers. Quant. Inf. Process. 7, 125–135 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Faisal Shah Khan
    • 1
    Email author
  • Neal Solmeyer
    • 2
  • Radhakrishnan Balu
    • 2
  • Travis S. Humble
    • 3
  1. 1.Center on Cyber Physical Systems and Department of MathematicsKhalifa UniversityAbu DhabiUAE
  2. 2.Army Research LabAdelphiUSA
  3. 3.Quantum Computing InstituteOak Ridge National LabOak RidgeUSA

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