Table of Contents
Game Theory
Volume 2013 (2013), Article ID 534875, 10 pages
http://dx.doi.org/10.1155/2013/534875
Research Article

Chess-Like Games May Have No Uniform Nash Equilibria Even in Mixed Strategies

RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854-8003, USA

Received 2 February 2013; Accepted 22 April 2013

Academic Editor: Walter Briec

Copyright © 2013 Endre Boros et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Boros, K. Elbassioni, V. Gurvich, and K. Makino, “On Nash equilibria and improvement cycles in pure positional strategies for Chess-like and Backgammon-like n-person games,” Discrete Mathematics, vol. 312, no. 4, pp. 772–788, 2012. View at Publisher · View at Google Scholar
  2. D. Andersson, V. Gurvich, and T. D. Hansen, “On acyclicity of games with cycles,” Discrete Applied Mathematics, vol. 158, no. 10, pp. 1049–1063, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. D. Andersson, V. Gurvich, and T. D. Hansen, “On acyclicity of games with cycles,” in Algorithmic Aspects in Information and Management, vol. 5564, pp. 15–28, 2009. View at Google Scholar
  4. E. Boros and V. Gurvich, “On Nash-solvability in pure stationary strategies of finite games with perfect information which may have cycles,” Mathematical Social Sciences, vol. 46, no. 2, pp. 207–241, 2003. View at Publisher · View at Google Scholar · View at Scopus
  5. E. Boros and V. Gurvich, “Why chess and backgammon can be solved in pure positional uniformly optimal strategies,” RUTCOR Research Report 21-2009, Rutgers University.
  6. E. Boros, V. Gurvich, K. Makino, and W. Shao, “Nash-solvable two-person symmetric cycle game forms,” Discrete Applied Mathematics, vol. 159, no. 15, pp. 1461–1487, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. E. Boros and R. Rand, “Terminal games with three terminals have proper Nash equilibria,” RUTCOR Research Report RRR-22-2009, Rutgers University.
  8. J. Flesch, J. Kuipers, G. Shoenmakers, and O. J. Vrieze, “Subgame-perfect equilibria in free transition games,” Research Memorandum RM/08/027, University of Maastricht, Maastricht, The Netherlands, 2008. View at Google Scholar
  9. J. Flesch, J. Kuipers, G. Shoenmakers, and O. J. Vrieze, “Subgame-perfection equilibria in stochastic games with perfect information and recursive payos,” Research Memorandum RM/08/041, University of Maastricht, Maastricht, The Netherlands, 2008. View at Google Scholar
  10. J. Kuipers, J. Flesch, G. Schoenmakers, and K. Vrieze, “Pure subgame-perfect equilibria in free transition games,” European Journal of Operational Research, vol. 199, no. 2, pp. 442–447, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. J. Flesch, J. Kuipers, G. Schoenmakers, and K. Vrieze, “Subgame perfection in positive recursive games with perfect information,” Mathematics of Operations Research, vol. 35, no. 1, pp. 193–207, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. D. Gale, “A theory of N-person games with perfect information,” Proceedings of the National Academy of Sciences, vol. 39, no. 6, pp. 496–501, 1953. View at Publisher · View at Google Scholar
  13. V. A. Gurvich, “On theory of multistep games,” USSR Computational Mathematics and Mathematical Physics, vol. 13, no. 6, pp. 143–161, 1973. View at Google Scholar · View at Scopus
  14. V. A. Gurvich, “The solvability of positional games in pure strategies,” USSR Computational Mathematics and Mathematical Physics, vol. 15, no. 2, pp. 74–87, 1975. View at Google Scholar · View at Scopus
  15. V. Gurvich, “Equilibrium in pure strategies,” Soviet Mathematics, vol. 38, no. 3, pp. 597–602, 1989. View at Google Scholar
  16. V. Gurvich, “A stochastic game with complete information and without equilibrium situations in pure stationary strategies,” Russian Mathematical Surveys, vol. 43, no. 2, pp. 171–172, 1988. View at Publisher · View at Google Scholar
  17. V. Gurvich, “A theorem on the existence of equilibrium situations in pure stationary strategies for ergodic extensions of (2×k) bimatrix games,” Russian Mathematical Surveys, vol. 45, no. 4, pp. 170–172, 1990. View at Google Scholar
  18. V. Gurvich, “Saddle point in pure strategies,” Russian Academy of Science Doklady Mathematics, vol. 42, no. 2, pp. 497–501, 1990. View at Google Scholar
  19. H. Kuhn, “Extensive games,” Proceedings of the National Academy of Sciences, vol. 36, pp. 286–295, 1950. View at Google Scholar
  20. H. Kuhn, “Extensive games and the problems of information,” Annals of Mathematics Studies, vol. 28, pp. 193–216, 1953. View at Google Scholar
  21. J. Nash, “Equilibrium points in n-person games,” Proceedings of the National Academy of Sciences, vol. 36, no. 1, pp. 48–49, 1950. View at Publisher · View at Google Scholar
  22. J. Nash, “Non-cooperative games,” Annals of Mathematics, vol. 54, no. 2, pp. 286–295, 1951. View at Publisher · View at Google Scholar
  23. A. Condon, “An algorithm for simple stochastic games,” in Advances in Computational Complexity Theory, vol. 13 of DIMACS series in discrete mathematics and theoretical computer science, 1993. View at Google Scholar
  24. I. V. Romanovsky, “On the solvability of Bellman's functional equation for a Markovian decision process,” Journal of Mathematical Analysis and Applications, vol. 42, no. 2, pp. 485–498, 1973. View at Google Scholar · View at Scopus
  25. R. A. Howard, Dynamic Programming and Markov Processes, The M.I.T. Press, 1960.
  26. H. Mine and S. Osaki, Markovian Decision Process, American Elsevier, New York, NY, USA, 1970.
  27. J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, 1960.