Abstract
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.
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References
Wiesner, S.: Conjugate coding. ACM SIGACT News Spec. Issue Cryptogr. 15, 77–78 (1983)
Ingarden, R.S.: Quantum information theory. Rep. Math. Phys. 10(1), 43–72 (1976)
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press, Cambridge (2000)
Feynman, R.: Simulating physics with computers. Int. J. Theoret. Phys. 21, 467–488 (1982)
Deutsch, D.: Quantum theory, the Chruch–Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97–117 (1985)
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)
Bullock, S.S., O’Leary, D.P., Brennen, G.K.: Asymptotically optimal quantum circuits for \(d\)-level systems. Phys. Rev. Lett. 94, 230502 (2005)
Charles Bennett, G.B.: Quantum cryptography: public key distributions and coin tossing. Theor. Comput. Sci. 560, 7–11 (2014)
Barnett, S.: Quantum Information. Oxford Master Series in Physics. Oxford University Press, Oxford (2009)
Wilde, M.: Quantum Information Theory. Quantum Information Theory. Cambridge University Press, Cambridge (2013)
Miyake, A., Wadati, M.: Geometric strategy for the optimal quantum search. Phys. Rev. A 64, 042317 (2001)
Tzu, S.: The Art of War: The Denma Translation. Shambhala Library, Shambhala (2002)
Nash, J.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36, 48–49 (1950)
Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8(3), 457–459 (1941)
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)
Binmore, K.: Playing for Real: A Text on Game Theory. Oxford University Press, Oxford (2007)
Myerson, R.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)
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)
Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052–1055 (1999)
Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077–3080 (1999)
Bleiler, S.: Quantized poker, preprint: arXiv:0902.2196
Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272(5), 291–303 (2000)
Khan, F.S., Phoenix, S.: Mini-maximizing two qubit quantum computations. Quant. Inf. Process. 12, 3807–3819 (2013)
Sutton, B.: Computing the complete cs decomposition. Numer. Algorithm. 50, 33–65 (2009)
Khan, F.S., Humble, T.S.: Nash embedding and equilibrium in pure quantum states, arXiv:1801.02053 [quant-ph] (2018)
Frackiewicz, P.: A new model for quantum games based on the marinattoweber approach. J. Phys. A Math. Theoret. 46(27), 275301 (2013)
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)
Samadi, A.H., Montakhab, A., Marzban, H., Owjimehr, S.: Quantum barrogordon game in monetary economics. Phys. A Stat. Mech. Appl. 489, 94–101 (2018)
Khan, F.S., Phoenix, S.: Gaming the quantum. Quant. Inf. Comput. 13, 231–244 (2013)
van Enk, S.J., Pike, R.: Classical rules in quantum games. Phys. Rev. A 66, 024306 (2002)
Aumann, R.J.: Subjectivity and correlation in randomized strategies. J. Math. Econom. 1(1), 67–96 (1974)
Benjamin, S.C., Hayden, P.M.: Comment on quantum games and quantum strategies. Phys. Rev. Lett. 87, 069801 (2001)
Eisert, J., Wilkens, M.: Quantum games. J. Mod. Opt. 47(14–15), 2543–2556 (2000)
Benjamin, S.C., Hayden, P.M.: Multiplayer quantum games. Phys. Rev. A 64, 030301 (2001)
Johnson, N.F.: Playing a quantum game with a corrupted source. Phys. Rev. A 63, 020302 (2001)
Iqbal, A., Toor, A.: Evolutionarily stable strategies in quantum games. Phys. Lett. A 280(5–6), 249–256 (2001)
Flitney, A.P., Abbott, D.: Quantum version of the Monty Hall problem. Phys. Rev. A 65, 062318 (2002)
Iqbal, A., Toor, A.H.: Quantum mechanics gives stability to a Nash equilibrium. Phys. Rev. A 65, 022306 (2002)
Du, J., Li, H., Xu, X., Zhou, X., Han, R.: Entanglement enhanced multiplayer quantum games. Phys. Lett. A 302(5), 229–233 (2002)
Flitney, A.P., Abbott, D.: Quantum games with decoherence. J. Phys. A Math. Gen. 38(2), 449 (2005)
Iqbal, A., Weigert, S.: Quantum correlation games. J. Phys. A Math. Gen. 37(22), 5873 (2004)
Chen, J.-L., Kwek, L.C., Oh, C.H.: Noisy quantum game. Phys. Rev. A 65, 052320 (2002)
Nayak, A., Shor, P.: Bit-commitment-based quantum coin flipping. Phys. Rev. A 67, 012304 (2003)
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)
Fitzi, M., Gisin, N., Maurer, U.: Quantum solution to the Byzantine agreement problem. Phys. Rev. Lett. 87, 217901 (2001)
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)
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)
Marriott, C., Watrous, J.: Quantum Arthur-Merlin games. Comput. Complex. 14, 122–152 (2005)
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)
Aumann, R.: Game Theory. Palgrave MacMillan, Basingstoke (1989)
Brandenburger, A.: Cooperative Game Theory, Lecture Notes
Piotrowski, E., Sadkowski, J.: Quantum market games. Phys. A Stat. Mech. Appl. 312(1), 208–216 (2002)
Piotrowski, E.W., Sadkowski, J., Syska, J.: Interference of quantum market strategies. Phys. A Stat. Mech. Appl. 318(3), 516–528 (2003)
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)
Quantum game theory. https://scholar.google.com.au/citations?user=wkfPcaQAAAAJ&hl=en. Accessed 12 March 2018
Guo, H., Zhang, J., Koehler, G.J.: A survey of quantum games. Decis. Support Syst. 46(1), 318–332 (2008)
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)
Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38, 886–893 (1952)
Partha Sarathi Dasgupta, E.M.: Commentary—physical sciences—mathematics: Debreus social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 112, 15769–16770 (2015)
Piotrowski, E.W., Sładkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003)
Zhang, P.E.A.: Quantum gambling based on Nash-equilibrium. NPJ Quant. Inf. 3, 24 (2017)
Bouyer, P., Brenguier, R., Markey, N., Ummels, M.: Pure Nash equilibria in concurrent deterministic games. Log. Methods Comput. Sci. 11(2) (2015)
de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. Theor. Comput. Sci. 386, 188–217 (2007)
Zabaleta, O., Arizmendi, C.: Quantum games based communication protocols. J. Adv. Appl. Comput. Math. 4, 35–39 (2017)
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)
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)
Anand, N., Benjamin, C.: Do quantum strategies always win. Quant. Inf. Process. 14, 4027–4038 (2015)
Mishima, H.: Non-abelian strategies in quantum penny flip game. Progr. Theor. Exp. Phys. 2018(1), 013A04 (2018)
Bao, N., Yunger Halpern, N.: Quantum voting and violation of arrow’s impossibility theorem. Phys. Rev. A 95, 062306 (2017)
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)
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)
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. 36
Khan, F., Elsokkary, N., Humble, T.: arXiv:1808.06926v2 [cs.GT], 2018
Rai, A., Paul, G.: Strong quantum solutions in conflicting-interest bayesian games. Phys. Rev. A 96, 042340 (2017)
Brunner, N., Linden, N.: Connection between Bell nonlocality and Bayesian game theory. Nat. Commun. 4, 2057 (2013)
Harsanyi, J.C.: Games with incomplete information played by Bayesian players. Mgt. Sci. 14, 159–182 (1967)
Cheon, T., Iqbal, A.: Bayesian Nash equilibria and Bell inequalities. J. Phys. Soc. Jpn. 77, 024801 (2008)
Fine, A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23, 1306–1310 (1982)
Silman, J., Machnes, S., Aharon, N.: On the relation between Bell’s inequalites and nonlocal games. Phys. Lett. A 372, 3796–3800 (2008)
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)
Iqbal, A., Abbott, D.: Equivalence between Bell inequalities and quantum minority games. Phys. Lett. A 374, 3155–3163 (2010)
LA Mura, P.: Correlated equilibria of classical strategic games with quantum signals. Int. J. Quant. Inf. 03(01), 183–188 (2005)
Brandenburger, A., Mura, P.L.: Team decision problems with classical and quantum signals. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 374, 2058 (2016)
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)
Auletta, V., Ferraioli, D., Rai, A., Scarpa, G., Winter, A.: Belief-invariant equilibria in games with incomplete information, CoRR, arXiv:1605.07896 (2016)
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)
Melo-Luna, C., Susa, C., Ducuara, A., Barreiro, A., Reina, J.: Quantum locality in game strategy. Sci. Rep. 7, 44730 (2016)
Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24, 379–385 (1991)
Guney, V., Hiller, M.: Bell inequalities from group actions: three parties and non-abelian groups. Phys. Rev. A 91, 052110 (2015)
Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Birkhauser, Basel (1992)
Chang, M.-H.: Quantum Stochastics. Cambridge University Press, Cambridge (2015)
Barry, Jennifer, Barry, Daniel T., Aaronson, S.: Quantum partially observable Markov decision processes. Phys. Rev. A 90, 032311 (2014)
Shapley, L.S.: Stochastic games. Proc. Natl. Acad. Sci. 39(10), 1095–1100 (1953)
Solana, E., Vieille, N.: Stochastic games—perspective. In: Proceedings of the National Academy of Sciences, vol. 112 (2015)
Johari, R.: Lecture Notes in Game Theory. preprint. Stanford University
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)
Blackwell, D.: Discrete dynamic programming. Ann. Math. Stat. 33(2), 719–726 (1962)
Hora, A., Obata, N.: Quantum Probability and Spectral Analysis of Graphs. Springer, Berlin (2007)
Gleason, A.: Measures on the closed subspace of a Hilbert space. J. Math. Mech. 6, 885–893 (1957)
Bouten, V.B.L.M., Edward, S.: Bellman equations for optimal control of qubits. J. Phys. B At. Mol. Opt. Phys. 38(3) (2005)
Kurt Jacobs, H.W., Wang, X.: Coherent feedback that beats all measurement-based feedback protocols. New J. Phys. 16, 073036 (2014)
DiVincenzio, D.P.: The physical implementation of quantum computation. Fortschr. Phys 48, 771–783 (2000)
Pfaff, W., et al.: Unconditional quantum teleportation between distant solid-state quantum bits. Science 345, 532 (2014)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Buluta, I.M., Fujiwara, S.: Quantum games in ion traps. Phys. Lett. A 358, 100–104 (2006)
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)
Debnath, S., et al.: Demonstration of a small programmable quantum computer with atomic qubits. Nature 536, 63 (2016)
Hucul, D.: Modular entanglement of atomic qubits using photons and phonons. Nat. Phys. 11, 37–42 (2015)
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)
Chen, K., Hogg, T.: How well do people play a quantum Prisoner’s Dilemma? Quant. Inf. Process. 5, 43–67 (2006)
Chen, K., Hogg, T.: Experiments with probabilistic quantum auctions. Quant. Inf. Process. 7, 139–152 (2008)
Schelling, T.C.: Arms and Influence. Yale University Press, New Haven (1966)
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)
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)
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)
Solmeyer, N., Dixon, R., Balu, R.: Quantum routing games, arXiv preprint arXiv:1709.10500 (2017)
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)
Hanauske, M., Bernius, S., Dugall, B.: Quantum game theory and open access publishing. Phys. A Stat. Mech. Appl. 382(2), 650–664 (2007)
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)
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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.
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Khan, F.S., Solmeyer, N., Balu, R. et al. Quantum games: a review of the history, current state, and interpretation. Quantum Inf Process 17, 309 (2018). https://doi.org/10.1007/s11128-018-2082-8
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DOI: https://doi.org/10.1007/s11128-018-2082-8