Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Zero-Sum Two Person Games

  • T.E.S. Raghavan
Reference work entry

Article Outline


Games with Perfect Information

Mixed Strategy and Minimax Theorem

Behavior Strategies in Games with Perfect Recall

Efficient Computation of Behavior Strategies

General Minimax Theorems

Applications of Infinite Games





Mixed Strategy Pure Strategy Payoff Matrix Terminal Vertex Minimax Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.



The author wishes to acknowledge the unknown referee's detailed comments in the revision of the first draft. More importantly he drew theauthor's attention to the topic of search games and other combinatorial games. The author would like to thank Ms. Patricia Collins for her assistance inher detailed editing of the first draft of this manuscript. The author owes special thanks to Mr. Ramanujan Raghavan and Dr. A.V. Lakshmi Narayanan fortheir help in incorporating the the graphics drawings into the LaTeX file.


  1. 1.
    Alpern S, Gal S (2003) The theory of search games andrendezvous. SpringerGoogle Scholar
  2. 2.
    Aumann RJ (1981) Survey of repeated games, Essays in Game Theory and Mathematical Economics, in Honor of Oscar Morgenstern.Bibliographsches Institut, Mannheim, pp 11–42Google Scholar
  3. 3.
    Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211:1390–1396MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bapat RB, Raghavan TES (1997) Nonnegative Matrices and Applications. In: Encyclopedia in Mathematics. Cambridge University Press, CambridgeGoogle Scholar
  5. 5.
    Bellman R, Blackwell D (1949) Some two person games involving bluffing. Proc Natl Acad Sci USA 35:600–605MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Berge C (1963) Topological Spaces. Oliver Boyd, EdinburghzbMATHGoogle Scholar
  7. 7.
    Berger U (2007) Brown's original fictitious play. J Econ Theory 135:572–578zbMATHCrossRefGoogle Scholar
  8. 8.
    Berlekamp ER, Conway JH, Guy RK (1982) Winning Ways for your Mathematical Plays, vols 1, 2. Academic Press, NYGoogle Scholar
  9. 9.
    Binmore K (1992) Fun and Game Theory A text on Game Theory. DC Heath, LexingtonzbMATHGoogle Scholar
  10. 10.
    Blackwell D (1951) On a theorem of Lyapunov. Ann Math Stat 22:112–114MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Blackwell D (1961) Minimax and irreducible matrices.Math J Anal Appl 3:37–39MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Blackwell D, Girshick GA (1954) Theory of Games and Statistical Decisions. Wiley, New YorkzbMATHGoogle Scholar
  13. 13.
    Bouton CL (1902) Nim-a game with a complete mathematical theory. Ann Math 3(2):35–39MathSciNetGoogle Scholar
  14. 14.
    Brown GW (1951) Iterative solution of games by fictitious play. In: Koopmans TC (ed) Activity Analysis of Production and Allocation. Wiley, New York, pp 374–376Google Scholar
  15. 15.
    Bubelis V (1979) On equilibria in finite games. Int J Game Theory 8:65–79MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chin H, Parthasarathy T, Raghavan TES (1973) Structure of equilibria in N‑person noncooperative games. Int Game J Theory 3:1–19MathSciNetCrossRefGoogle Scholar
  17. 17.
    Conway JH (1982) On numbers and Games, Monograph 16. London Mathematical Society, LondonGoogle Scholar
  18. 18.
    Dantzig GB (1951) A proof of the equivalence of the programming problem and the game problem. In: Koopman's Actvity analysis of production and allocationation, Cowles Conumesion Monograph 13. Wiley, New York, pp 333–335Google Scholar
  19. 19.
    Dresher M (1962) Games of strategy. Prentice Hall, Englewood CliffsGoogle Scholar
  20. 20.
    Dvoretzky A, Wald A, Wolfowitz J (1951) Elimination of randomization in certain statistical decision problems and zero-sum two‐person games. Ann Math Stat 22:1–21MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fan K (1953) Minimax theorems. Proc Natl Acad Sci Wash 39:42–47zbMATHCrossRefGoogle Scholar
  22. 22.
    Ferguson C, Ferguson TS (2003) On the Borel and von Neumann poker models. Game Theory Appl 9:17–32MathSciNetGoogle Scholar
  23. 23.
    Ferguson TS (1967) Mathematical Stat, a Decision Theoretic Approach. Academic Press, New YorkGoogle Scholar
  24. 24.
    Filar JA, Vrieze OJ (1996) Competitive Markov Decision Processes. Springer, BerlinCrossRefGoogle Scholar
  25. 25.
    Fisher RA (1936) The Use of Multiple Measurements in Taxonomic Problems. Ann Eugen 7:179–188CrossRefGoogle Scholar
  26. 26.
    Fourier JB (1890) Second Extrait. In: Darboux GO (ed) Gauthiers Villars,Paris, pp 325–328; English Translation: Kohler DA (1973)Google Scholar
  27. 27.
    Gal S (1980) Search games. Academic Press, New YorkzbMATHGoogle Scholar
  28. 28.
    Gale D (1979) The game of Hex and the Brouwer fixed-point theorem. Am Math Mon 86:818–827MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Gale D, Kuhn HW, Tucker AW (1951) Linear Programming and the theory of games. In: Activity Analysis of Production and Allocation. Wiley, New York, pp 317–329Google Scholar
  30. 30.
    Harsanyi JC (1967) Games with incomplete information played by Bayesian players, Parts I, II, and III. Sci Manag 14:159–182; 32–334; 486–502MathSciNetGoogle Scholar
  31. 31.
    Isaacs R (1965) Differential Games: Mathematical A Theory with Applications toWarfare and Pursuit. Control and Optimization.Wiley, New York; Dover Paperback Edition, 1999Google Scholar
  32. 32.
    Jansen MJM (1981) Regularity and stability of equilibrium points of bimatrix games. Math Oper Res 6:530–550MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Johnson RA, Wichern DW (2007) Applied Multivariate Statistical Analysis, 6th edn. Prentice Hall, New YorkzbMATHGoogle Scholar
  34. 34.
    Kakutani S (1941) A generalization of Brouwer's fixed point theorem. Duke Math J 8:457–459MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kantorowich LV (1960) Mathematical Methods of Organizing and Planning Production. Manag Sci 7:366–422CrossRefGoogle Scholar
  36. 36.
    Kaplansky I (1945) A contribution to von Neumann's theory of games. Ann Math 46:474–479MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Karlin S (1959) Mathematical Methods and Theory in Games, Programming and Econs, vols 1, 2. Addison Wesley, New YorkGoogle Scholar
  38. 38.
    Kohler DA (1973) Translation of a report by Fourier on his work on linear inequalities. Opsearch 10:38–42MathSciNetGoogle Scholar
  39. 39.
    Kreps VL (1974) Bimatrix games with unique equilibrium points. Int Game J Theory 3:115–118MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Krein MG, Rutmann MA (1950) Linear Operators Leaving invariant a cone in a Banach space. Amer Math Soc Transl 26:1–128Google Scholar
  41. 41.
    Krishna V, Sjostrom T (1998) On the convergence of fictitious play. Math Oper Res 23:479–511MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kuhn HW (1953) Extensive games and the problem of information. Contributions to the theory of games. Ann Math Stud 28:193–216Google Scholar
  43. 43.
    Lindenstrauss J (1966) A short proof of Liapounoff's convexity theorem. Math J Mech 15:971–972MathSciNetzbMATHGoogle Scholar
  44. 44.
    Loomis IH (1946) On a theorem of von Neumann. Proc Nat Acad Sci Wash 32:213–215MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Miyazawa K (1961) On the convergence of the learning process in a \( { 2\times 2 } \) non-zero-sum two‐person game, Econometric Research Program, Research Memorandum No. 33. Princeton University, PrincetonGoogle Scholar
  46. 46.
    Monderer D, Shapley LS (1996) Potential LS games. Games Econ Behav 14:124–143MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Myerson R (1991) Game theory. Analysis of Conflict.Harvard University Press, CambridgezbMATHGoogle Scholar
  48. 48.
    Nash JF (1950) Equilibrium points in n‑person games. Proc Natl Acad Sci Wash 88:48–49MathSciNetCrossRefGoogle Scholar
  49. 49.
    Owen G (1985) Game Theory, 2nd edn. Academic Press, New YorkGoogle Scholar
  50. 50.
    Parthasarathy T, Raghavan TES (1971) Some Topics in Two Person Games. Elsevier, New YorkzbMATHGoogle Scholar
  51. 51.
    Radzik T (1988) Games of timing related to distribution of resources. Optim J Theory Appl 58:443–471, 473–500Google Scholar
  52. 52.
    Raghavan TES (1970) Completely Mixed Strategies in Bimatrix Games. Lond J Math Soc 2:709–712MathSciNetzbMATHGoogle Scholar
  53. 53.
    Raghavan TES (1973) Some geometric consequences of a game theoretic result. Math J Anal Appl 43:26–30MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Rao CR (1952) Advanced Statistical Methods in Biometric Research. Wiley, New YorkzbMATHGoogle Scholar
  55. 55.
    Reijnierse JH, Potters JAM (1993) Search Games with Immobile Hider. Int J Game Theory 21:385–94MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Robinson J (1951) An iterative method of solving a game.Ann Math 54:296–301zbMATHCrossRefGoogle Scholar
  57. 57.
    Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97–109MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Schelling TC (1960) The Strategy of Conflict. Harvard University Press. Cambridge, MassGoogle Scholar
  59. 59.
    Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int Game J Theory 4:25–55MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Shapley LS (1953) Stochastic games. Proc Natl Acad of Sci USA 39:1095–1100MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Sion M (1958) On general minimax theorem. Pac J 8:171–176MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Sorin S (1992) Repeated games with complete information, In: Aumann RJ, Hart S (eds) Handbook of Game Theory, vol 1, chapter 4. North Holland, Amsterdam, pp 71–103Google Scholar
  63. 63.
    von Stengel B (1996) Efficient computation of behavior strategies. Games Econ Behav 14:220–246zbMATHCrossRefGoogle Scholar
  64. 64.
    Thuijsman F, Raghavan TES (1997) Stochastic games with switching control or ARAT structure. Int Game J Theory 26:403–408MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Ville J (1938) Note sur la theorie generale des jeux ou intervient l'habilitedes joueurs. In: Borel E, Ville J (eds) Applications aux jeux de hasard, tome IV, fascicule II of the Traite du calcul des probabilites et de sesapplications, by Emile BorelGoogle Scholar
  66. 66.
    von Neumann J (1928) Zur Theorie der Gesellschaftspiele.Math Ann 100:295–320MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    von Neumann J, Morgenstern O (1947) Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, PrincetonzbMATHGoogle Scholar
  68. 68.
    Wald A (1950) Statistical Decision Functions. Wiley, New YorkzbMATHGoogle Scholar
  69. 69.
    Weyl H (1950) Elementary proof of a minimax theorem due to von Neumann. Ann Math Stud 24:19–25MathSciNetGoogle Scholar
  70. 70.
    Zermelo E (1913) Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proc Fifth Congress Mathematicians.Cambridge University Press, Cambridge, pp 501–504Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • T.E.S. Raghavan
    • 1
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of IllinoisChicagoUSA