Advertisement

Quantization and experimental realization of the Colonel Blotto game

  • A. C. Maioli
  • M. H. M. Passos
  • W. F. Balthazar
  • C. E. R. Souza
  • J. A. O. Huguenin
  • A. G. M. SchmidtEmail author
Article
  • 87 Downloads

Abstract

We present a quantum mechanical version of the Colonel Blotto game, where two players, Blotto and Enemy, collocate their soldiers (resources) sequentially in a finite number of territories. We analyse the representative classical cases of this game as well as the trivial case—which on its turn has no interest at all in the point of view of classical game theory—where, surprisingly, a player that could control a single parameter can win the game even if he/she is greatly outnumbered by his/her opponent. Besides the theoretical study we present an experimental realization of classical game by using linear optics circuits as well as a proposal of an experimental investigation of the quantized game. Finally, in order to check our quantization scheme we also present computer simulation results.

Keywords

Quantum games Optical circuits Transverse modes Polarization modes 

Notes

Acknowledgements

The authors gratefully acknowledge the financial support of Brazilian’s agencies CAPES, FAPERJ, CNPq and INCT—Quantum Information. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

References

  1. 1.
    Press, W.H., Dyson, F.J.: Iterated prisoner’s dilemma contains strategies that dominate any evolutionary opponent. Proc. Natl. Acad. Sci. 109(26), 10409 (2012)ADSCrossRefGoogle Scholar
  2. 2.
    von Neumann, J., Morgenstern, O.: The Theory of Games and Economic Behavior, 60th edn. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  3. 3.
    Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052 (1999)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077 (1999)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Fra̧ckiewicz, P.: The ultimate solution to the quantum battle of the Sexes game, J. Phys. A 42(36) (2009)Google Scholar
  6. 6.
    Guo, H., Zhang, J., Koehler, G.J.: A survey of quantum games. Decis. Support Syst. 46(1), 318 (2008)CrossRefGoogle Scholar
  7. 7.
    Challet, D., Zhang, Y.C.: Emergence of cooperation and organization in an evolutionary game. Physica A 246(3), 407 (1997)ADSCrossRefGoogle Scholar
  8. 8.
    de Ponte, M.A., Santos, A.C.: Adiabatic quantum games and phase-transition-like behavior between optimal strategies. Quantum Inf. Process. 17(6), 149 (2018)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Flitney, A.P., Abbott, D.: Quantum version of the Monty Hall problem. Phys. Rev. A 65, 062318 (2002)ADSCrossRefGoogle Scholar
  10. 10.
    Hogg, T., Harsha, P., Chen, K.Y.: Quantum auctions. Int. J. Quantum Inf. 5(05), 751 (2007)CrossRefGoogle Scholar
  11. 11.
    Zeng, Q., Davis, B.R., Abbott, D.: Reverse auction: the lowest unique positive integer game. Fluctuation Noise Lett. 7(04), L439 (2007)CrossRefGoogle Scholar
  12. 12.
    Makowski, M.: Transitivity versus intransitivity in decision making process—an example in quantum game theory. Phys. Lett. A 373, 2125 (2009)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Makowski, M., Piotrowski, E.W.: Transitivity of an entangled choice. J. Phys. A 44, 075301 (2011)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Flitney, A.P., Abbott, D.: Quantum two-and three-person duels. J. Opt. B 6(8), S860 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    Schmidt, A.G.M., Paiva, M.M.: Quantum duel revisited. J. Phys. A 45(12), 125304 (2012)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Balthazar, W.F., Huguenin, J.A.O., Schmidt, A.G.M.: Simultaneous quantum duel. J. Phys. Soc. Jpn. 84(12), 124002 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    Balthazar, W.F., Passos, M.H.M., Schmidt, A.G.M., Caetano, D.P., Huguenin, J.A.O.: Experimental realization of the quantum duel game using linear optical circuits. J. Phys. B 48(16), 165505 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    Amengual, P., Toral, R.: Truels, or survival of the weakest. Comput. Sci. Eng. 8(5), 88 (2006)CrossRefGoogle Scholar
  19. 19.
    Chowdhury, S.M., Kovenock, D., Sheremeta, R.M., Roman, M.: An experimental investigation of Colonel Blotto games. Econ. Theory 52(3), 833 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gross, O., Wagner, R.: A continuous colonel blotto game. Tech. rep, RAND PROJECT AIR FORCE SANTA MONICA CA (1950)Google Scholar
  21. 21.
    Roberson, B.: The colonel blotto game. Econ. Theory 29(1), 1 (2006)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Roberson, B., Kvasov, D.: The non-constant-sum Colonel Blotto game. Econ. Theory 51(2), 397 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hendricks, K., Weiss, A., Wilson, C.: The war of attrition in continuous time with complete information. Int. Econ. Rev. 663–680 (1988)Google Scholar
  24. 24.
    Hodler, R., Yektaş, H.: All-pay war. Games Econ. Behav. 74(2), 526 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Myerson, R.B.: Incentives to cultivate favored minorities under alternative electoral systems. Am. Polit. Sci. Rev. 87(4), 856 (1993)CrossRefGoogle Scholar
  26. 26.
    Szentes, B., Rosenthal, R.W.: Three-object two-bidder simultaneous auctions: chopsticks and tetrahedra. Games Econ. Behav. 44(1), 114 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Golman, R., Page, S.E.: General Blotto: games of allocative strategic mismatch. Public Choice 138(3–4), 279 (2009)CrossRefGoogle Scholar
  28. 28.
    Lu, J., Zhou, L., Kuang, L.M.: Linear optics implementation for quantum game with two players. Phys. Lett. A 330(1–2), 48 (2004)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kolenderski, P., Sinha, U., Youning, L., Zhao, T., Volpini, M., Cabello, A., Laflamme, R., Jennewein, T.: Aharonov–Vaidman quantum game with a Young-type photonic qutrit. Phys. Rev. A 86(1), 012321 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    Pinheiro, A.R.C., Souza, C.E.R., Caetano, D.P., Huguenin, J.A.O., Schmidt, A.G.M., Khoury, A.Z.: Vector Vortex implementation of a quantum game. J. Opt. Soc. Am. B 30(12), 3210 (2013)ADSCrossRefGoogle Scholar
  31. 31.
    Borges, C.V.S., Hor-Meyll, M., Huguenin, J.A.O., Khoury, A.Z.: Bell-like inequality for the spin-orbit separability of a laser beam. Phys. Rev. A 82(3), 033833 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    Kagalwala, K.H., Di Giuseppe, G., Abouraddy, A.F., Saleh, B.E.: Bell’s measure in classical optical coherence. Nat Photonics 7(1), 72 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Balthazar, W.F., Souza, C.E.R., Caetano, D.P., Galvão, E.F., Huguenin, J.A.O., Khoury, A.Z.: Tripartite nonseparability in classical optics. Opt. Lett. 41(24), 5797 (2016)ADSCrossRefGoogle Scholar
  34. 34.
    Milione, G., Nguyen, T.A., Leach, J., Nolan, D.A., Alfano, R.R.: Using the nonseparability of vector beams to encode information for optical communication. Opt. Lett. 40(21), 4887 (2015)ADSCrossRefGoogle Scholar
  35. 35.
    Souza, C.E.R., Borges, C.V.S., Khoury, A.Z., Huguenin, J.A.O., Aolita, L., Walborn, S.: Quantum key distribution without a shared reference frame. Phys. Rev. A 77(3), 032345 (2008)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Balthazar, W.F., Caetano, D.P., Souza, C.E.R., Huguenin, J.A.O.: Using polarization to control the phase of spatial modes for application in quantum information. Braz. J. Phys. 44(6), 658 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    Balthazar, W.F., Huguenin, J.A.O.: Conditional operation using three degrees of freedom of a laser beam for application in quantum information. J. Opt. Soc. Am. B 33(8), 1649 (2016)ADSCrossRefGoogle Scholar
  38. 38.
    da Silva, B.P., Leal, M.A., Souza, C.E.R., Galvão, E.F., Khoury, A.Z.: Spin-orbit laser mode transfer via a classical analogue of quantum teleportation. J. Phys. B 49(5), 055501 (2016)ADSCrossRefGoogle Scholar
  39. 39.
    Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286 (1951)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Gerrard, A., Burch, J.M.: Introduction to Matrix Methods in Optics. Courier Corporation (1994)Google Scholar

Copyright information

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

Authors and Affiliations

  • A. C. Maioli
    • 1
    • 3
  • M. H. M. Passos
    • 1
    • 3
  • W. F. Balthazar
    • 2
  • C. E. R. Souza
    • 3
    • 4
  • J. A. O. Huguenin
    • 1
    • 3
  • A. G. M. Schmidt
    • 1
    • 3
    Email author
  1. 1.Instituto de Ciências ExatasUniversidade Federal FluminenseVolta Redonda - RJBrazil
  2. 2.Instituto Federal do Rio de JaneiroVolta Redonda - RJBrazil
  3. 3.Programa de Pós Graduação em Física Instituto de FísicaUniversidade Federal FluminenseNiterói - RJBrazil
  4. 4.Instituto de FísicaUniversidade Federal FluminenseNiterói - RJBrazil

Personalised recommendations