Skip to main content

Zero-Sum Two Person Games

  • Reference work entry
Computational Complexity

Article Outline

Introduction

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

Epilogue

Acknowledgment

Bibliography

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 1,500.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,399.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. Alpern S, Gal S (2003) The theory of search games andrendezvous. Springer

    Google Scholar 

  2. Aumann RJ (1981) Survey of repeated games, Essays in Game Theory and Mathematical Economics, in Honor of Oscar Morgenstern.Bibliographsches Institut, Mannheim, pp 11–42

    Google Scholar 

  3. Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211:1390–1396

    Article  MathSciNet  MATH  Google Scholar 

  4. Bapat RB, Raghavan TES (1997) Nonnegative Matrices and Applications. In: Encyclopedia in Mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  5. Bellman R, Blackwell D (1949) Some two person games involving bluffing. Proc Natl Acad Sci USA 35:600–605

    Article  MathSciNet  MATH  Google Scholar 

  6. Berge C (1963) Topological Spaces. Oliver Boyd, Edinburgh

    MATH  Google Scholar 

  7. Berger U (2007) Brown's original fictitious play. J Econ Theory 135:572–578

    Article  MATH  Google Scholar 

  8. Berlekamp ER, Conway JH, Guy RK (1982) Winning Ways for your Mathematical Plays, vols 1, 2. Academic Press, NY

    Google Scholar 

  9. Binmore K (1992) Fun and Game Theory A text on Game Theory. DC Heath, Lexington

    MATH  Google Scholar 

  10. Blackwell D (1951) On a theorem of Lyapunov. Ann Math Stat 22:112–114

    Article  MathSciNet  MATH  Google Scholar 

  11. Blackwell D (1961) Minimax and irreducible matrices.Math J Anal Appl 3:37–39

    Article  MathSciNet  MATH  Google Scholar 

  12. Blackwell D, Girshick GA (1954) Theory of Games and Statistical Decisions. Wiley, New York

    MATH  Google Scholar 

  13. Bouton CL (1902) Nim-a game with a complete mathematical theory. Ann Math 3(2):35–39

    MathSciNet  Google Scholar 

  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 374–376

    Google Scholar 

  15. Bubelis V (1979) On equilibria in finite games. Int J Game Theory 8:65–79

    Article  MathSciNet  MATH  Google Scholar 

  16. Chin H, Parthasarathy T, Raghavan TES (1973) Structure of equilibria in N‑person noncooperative games. Int Game J Theory 3:1–19

    Article  MathSciNet  Google Scholar 

  17. Conway JH (1982) On numbers and Games, Monograph 16. London Mathematical Society, London

    Google Scholar 

  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 333–335

    Google Scholar 

  19. Dresher M (1962) Games of strategy. Prentice Hall, Englewood Cliffs

    Google Scholar 

  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 22:1–21

    Article  MathSciNet  MATH  Google Scholar 

  21. Fan K (1953) Minimax theorems. Proc Natl Acad Sci Wash 39:42–47

    Article  MATH  Google Scholar 

  22. Ferguson C, Ferguson TS (2003) On the Borel and von Neumann poker models. Game Theory Appl 9:17–32

    MathSciNet  Google Scholar 

  23. Ferguson TS (1967) Mathematical Stat, a Decision Theoretic Approach. Academic Press, New York

    Google Scholar 

  24. Filar JA, Vrieze OJ (1996) Competitive Markov Decision Processes. Springer, Berlin

    Book  Google Scholar 

  25. Fisher RA (1936) The Use of Multiple Measurements in Taxonomic Problems. Ann Eugen 7:179–188

    Article  Google Scholar 

  26. Fourier JB (1890) Second Extrait. In: Darboux GO (ed) Gauthiers Villars,Paris, pp 325–328; English Translation: Kohler DA (1973)

    Google Scholar 

  27. Gal S (1980) Search games. Academic Press, New York

    MATH  Google Scholar 

  28. Gale D (1979) The game of Hex and the Brouwer fixed-point theorem. Am Math Mon 86:818–827

    Article  MathSciNet  MATH  Google Scholar 

  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 317–329

    Google Scholar 

  30. Harsanyi JC (1967) Games with incomplete information played by Bayesian players, Parts I, II, and III. Sci Manag 14:159–182; 32–334; 486–502

    MathSciNet  Google Scholar 

  31. Isaacs R (1965) Differential Games: Mathematical A Theory with Applications toWarfare and Pursuit. Control and Optimization.Wiley, New York; Dover Paperback Edition, 1999

    Google Scholar 

  32. Jansen MJM (1981) Regularity and stability of equilibrium points of bimatrix games. Math Oper Res 6:530–550

    Article  MathSciNet  MATH  Google Scholar 

  33. Johnson RA, Wichern DW (2007) Applied Multivariate Statistical Analysis, 6th edn. Prentice Hall, New York

    MATH  Google Scholar 

  34. Kakutani S (1941) A generalization of Brouwer's fixed point theorem. Duke Math J 8:457–459

    Article  MathSciNet  Google Scholar 

  35. Kantorowich LV (1960) Mathematical Methods of Organizing and Planning Production. Manag Sci 7:366–422

    Article  Google Scholar 

  36. Kaplansky I (1945) A contribution to von Neumann's theory of games. Ann Math 46:474–479

    Article  MathSciNet  MATH  Google Scholar 

  37. Karlin S (1959) Mathematical Methods and Theory in Games, Programming and Econs, vols 1, 2. Addison Wesley, New York

    Google Scholar 

  38. Kohler DA (1973) Translation of a report by Fourier on his work on linear inequalities. Opsearch 10:38–42

    MathSciNet  Google Scholar 

  39. Kreps VL (1974) Bimatrix games with unique equilibrium points. Int Game J Theory 3:115–118

    Article  MathSciNet  MATH  Google Scholar 

  40. Krein MG, Rutmann MA (1950) Linear Operators Leaving invariant a cone in a Banach space. Amer Math Soc Transl 26:1–128

    Google Scholar 

  41. Krishna V, Sjostrom T (1998) On the convergence of fictitious play. Math Oper Res 23:479–511

    Article  MathSciNet  MATH  Google Scholar 

  42. Kuhn HW (1953) Extensive games and the problem of information. Contributions to the theory of games. Ann Math Stud 28:193–216

    Google Scholar 

  43. Lindenstrauss J (1966) A short proof of Liapounoff's convexity theorem. Math J Mech 15:971–972

    MathSciNet  MATH  Google Scholar 

  44. Loomis IH (1946) On a theorem of von Neumann. Proc Nat Acad Sci Wash 32:213–215

    Article  MathSciNet  MATH  Google Scholar 

  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, Princeton

    Google Scholar 

  46. Monderer D, Shapley LS (1996) Potential LS games. Games Econ Behav 14:124–143

    Article  MathSciNet  MATH  Google Scholar 

  47. Myerson R (1991) Game theory. Analysis of Conflict.Harvard University Press, Cambridge

    MATH  Google Scholar 

  48. Nash JF (1950) Equilibrium points in n‑person games. Proc Natl Acad Sci Wash 88:48–49

    Article  MathSciNet  Google Scholar 

  49. Owen G (1985) Game Theory, 2nd edn. Academic Press, New York

    Google Scholar 

  50. Parthasarathy T, Raghavan TES (1971) Some Topics in Two Person Games. Elsevier, New York

    MATH  Google Scholar 

  51. Radzik T (1988) Games of timing related to distribution of resources. Optim J Theory Appl 58:443–471, 473–500

    Google Scholar 

  52. Raghavan TES (1970) Completely Mixed Strategies in Bimatrix Games. Lond J Math Soc 2:709–712

    MathSciNet  MATH  Google Scholar 

  53. Raghavan TES (1973) Some geometric consequences of a game theoretic result. Math J Anal Appl 43:26–30

    Article  MathSciNet  MATH  Google Scholar 

  54. Rao CR (1952) Advanced Statistical Methods in Biometric Research. Wiley, New York

    MATH  Google Scholar 

  55. Reijnierse JH, Potters JAM (1993) Search Games with Immobile Hider. Int J Game Theory 21:385–94

    Article  MathSciNet  MATH  Google Scholar 

  56. Robinson J (1951) An iterative method of solving a game.Ann Math 54:296–301

    Article  MATH  Google Scholar 

  57. Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97–109

    Article  MathSciNet  MATH  Google Scholar 

  58. Schelling TC (1960) The Strategy of Conflict. Harvard University Press. Cambridge, Mass

    Google Scholar 

  59. Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int Game J Theory 4:25–55

    Article  MathSciNet  MATH  Google Scholar 

  60. Shapley LS (1953) Stochastic games. Proc Natl Acad of Sci USA 39:1095–1100

    Article  MathSciNet  MATH  Google Scholar 

  61. Sion M (1958) On general minimax theorem. Pac J 8:171–176

    Article  MathSciNet  MATH  Google Scholar 

  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 71–103

    Google Scholar 

  63. von Stengel B (1996) Efficient computation of behavior strategies. Games Econ Behav 14:220–246

    Article  MATH  Google Scholar 

  64. Thuijsman F, Raghavan TES (1997) Stochastic games with switching control or ARAT structure. Int Game J Theory 26:403–408

    Article  MathSciNet  MATH  Google Scholar 

  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 Borel

    Google Scholar 

  66. von Neumann J (1928) Zur Theorie der Gesellschaftspiele.Math Ann 100:295–320

    Article  MathSciNet  MATH  Google Scholar 

  67. von Neumann J, Morgenstern O (1947) Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, Princeton

    MATH  Google Scholar 

  68. Wald A (1950) Statistical Decision Functions. Wiley, New York

    MATH  Google Scholar 

  69. Weyl H (1950) Elementary proof of a minimax theorem due to von Neumann. Ann Math Stud 24:19–25

    MathSciNet  Google Scholar 

  70. Zermelo E (1913) Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proc Fifth Congress Mathematicians.Cambridge University Press, Cambridge, pp 501–504

    Google Scholar 

Download references

Acknowledgment

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.

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Raghavan, T. (2012). Zero-Sum Two Person Games. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_209

Download citation

Publish with us

Policies and ethics