Introduction to the Theory of Games

  • Tamer BaşarEmail author
Reference work entry


This chapter provides a general introduction to the theory of games, as a prelude to other chapters in this Handbook of Dynamic Game Theory which discuss in depth various aspects of dynamic and differential games. The present chapter describes in general terms what game theory is, its historical origins, general formulation (concentrating primarily on static games), various solution concepts, and some key results (again primarily for static games). The conceptual framework laid out here sets the stage for dynamic games covered by other chapters in the Handbook.


Game theory Dynamic games Historical evolution of game theory Zero-sum games Nonzero-sum games Strategic equivalence Saddle-point equilibrium Nash equilibrium Correlated equilibrium Stackelberg equilibrium Computational methods Linear-quadratic games 


  1. Alpcan T, Başar T (2011) Network security: a decision and game theoretic approach. Cambridge University Press, CambridgeGoogle Scholar
  2. Aumann RJ (1974) Subjectivity and correlation in randomized strategy. J Math Econ 1(1):67–96Google Scholar
  3. Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55(1):1–18Google Scholar
  4. Başar T (1974) A counter example in linear-quadratic games: existence of non-linear Nash solutions. J Optim Theory Appl 14(4):425–430Google Scholar
  5. Başar T (1976) On the uniqueness of the Nash solution in linear-quadratic differential games. Int J Game Theory 5:65–90Google Scholar
  6. Başar T (1977) Informationally nonunique equilibrium solutions in differential games. SIAM J Control 15(4):636–660Google Scholar
  7. Başar T (1978) Decentralized multicriteria optimization of linear stochastic systems. IEEE Trans Autom Control AC-23(2):233–243Google Scholar
  8. Başar T (1985) An equilibrium theory for multi-person decision making with multiple probabilistic models. IEEE Trans Autom Control AC-30(2):118–132Google Scholar
  9. Başar T (1987) Relaxation techniques and on-line asynchronous algorithms for computation of noncooperative equilibria. J Econ Dyn Control 71:531–549Google Scholar
  10. Başar T, Bernhard P (1995) H Optimal control and related minimax design problems: a dynamic game approach. Birkhäuser, BostonGoogle Scholar
  11. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. Classics in applied mathematics, 2nd edn. SIAM, PhiladelphiaGoogle Scholar
  12. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, BostonGoogle Scholar
  13. Bertsekas DP (2007) Dynamic programming and optimal control. Athena Scientific, BostonGoogle Scholar
  14. Brown GW (1951) Iterative solutions of games by fictitious play. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 374–376Google Scholar
  15. Chazan D, Miranker W (1969) Chaotic relaxation. Linear Algebra Appl 2:199–222Google Scholar
  16. Fudenberg D, Tirole J (1991) Game theory. MIT Press, CambridgeGoogle Scholar
  17. Han Z, Niyato D, Saad W, Başar T, Hjorungnes A (2011) Game theory in wireless and communication networks: theory, models, and applications. Cambridge University Press, Cambridge/ New YorkGoogle Scholar
  18. Harsanyi J (1967) Games with incomplete information played by “Bayesian” players, Parts I–III. Manage Sci 14(3):159–182, Nov 1967; 14(5):320–334, Jan 1968; 14(7):486–502, Mar 1968Google Scholar
  19. Ho Y-C (1965) Review of ‘Differential Games’ by R. Isaacs. IEEE Trans Autom Control AC-10(4):501–503Google Scholar
  20. Isaacs R (1975) Differential games, 2nd edn. Kruger, New York; First edition: Wiley, New York (1965)Google Scholar
  21. Kohlberg E, Mertens JF (1986) On the strategic stability of equilibria. Econometrica 54:1003–1037MathSciNetCrossRefGoogle Scholar
  22. Kreps M, Wilson R (1982) Sequential equilibria. Econometrics 50:863–894MathSciNetCrossRefGoogle Scholar
  23. Miyasawa K (1961) On the convergence of learning processes in a 2 × 2 non-zero-sum two person game. Technical report 33, Economic Research Program, Princeton University, PrincetonGoogle Scholar
  24. Monderer D, Shapley LS (1996) Fictitious play property for games with identical interests. J Econ Theory 68:258–265MathSciNetCrossRefGoogle Scholar
  25. Myerson RB, Selten R (1978) Refinement of the Nash equilibrium concept. Int J Game Theory 7:73–80MathSciNetCrossRefGoogle Scholar
  26. Nash JF Jr (1950) Equilibrium points in n-person games. Proc Natl Acad Sci 36(1):48–49MathSciNetCrossRefGoogle Scholar
  27. Nash JF Jr (1951) Non-cooperative games. Ann Math 54(2):286–295MathSciNetCrossRefGoogle Scholar
  28. Nguyen KC, Alpcan T, Başar T (2010a) Security games with decision and observation errors. In: Proceedings 2010 American control conference, Baltimore, pp 510–515Google Scholar
  29. Nguyen KC, Alpcan T, Başar T (2010b) Fictitious play with time-invariant frequency update for network security. In: Proceedings of the IEEE multi-conference on systems and control (MSC) and IEEE CCA, Yokohama, pp 8–10Google Scholar
  30. Owen G (1974) Existence of equilibrium pairs in continuous games. Int J Game Theory 5:97–105MathSciNetCrossRefGoogle Scholar
  31. Owen G (1995) Game theory, 3rd edn. Academic Press, San DiegozbMATHGoogle Scholar
  32. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Interscience Publishers, New YorkGoogle Scholar
  33. Robinson J (1951) An iterative method for solving a game. Ann Math 54(2):296–301MathSciNetCrossRefGoogle Scholar
  34. Rosen JB (1965) Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33(3):520–534MathSciNetCrossRefGoogle Scholar
  35. Saad W, Han Z, Debbah M, Hjorungnes A, Başar T (2009) Coalitional game theory for communication networks [A tutorial]. IEEE Signal Process Mag Spec Issue Game Theory 26(5):77–97CrossRefGoogle Scholar
  36. Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55MathSciNetCrossRefGoogle Scholar
  37. Shamma JS, Arslan G (2004) Unified convergence proofs of continuous-time fictitious play. IEEE Trans Autom Control 49(7):1137–1142MathSciNetCrossRefGoogle Scholar
  38. Shamma JS, Arslan G (2005) Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria. IEEE Trans Autom Control 50(3):312–327MathSciNetCrossRefGoogle Scholar
  39. Shapley LS (1964) Some topics in two-person games. In: Shapley LS, Dresher M, Tucker AW (eds) Advances in game theory. Princeton University Press, Princeton, pp 1–29Google Scholar
  40. Simaan M, Cruz JB Jr (1973) On the Stackelberg strategy in nonzero sum games. J Optim Theory Appl 11:533–555MathSciNetCrossRefGoogle Scholar
  41. Smith JM (1974) The theory of games and the evolution of animal conflicts. J Theor Biol 47: 209–221MathSciNetCrossRefGoogle Scholar
  42. Smith JM (1982) Evolution and the theory of games. Cambridge University Press, Cambridge/New YorkCrossRefGoogle Scholar
  43. Smith JM, Price GR (1973) The logic of animal conflict. Nature 246:15–18CrossRefGoogle Scholar
  44. Starr AW, Ho YC (1969) Nonzero-sum differential games. J Optim Theory Appl 3:184–206MathSciNetCrossRefGoogle Scholar
  45. van Damme E (1984) A relationship between perfect equilibria in extensive games and proper equilibria in normal form. Int J Game Theory 13:1–13CrossRefGoogle Scholar
  46. van Damme E (1987) Stability and perfection of equilibria. Springer, New YorkCrossRefGoogle Scholar
  47. von Neumann J (1928) Zur theorie der Gesellschaftspiele. Mathematische Annalen 100:295–320MathSciNetCrossRefGoogle Scholar
  48. von Neumann J, Morgenstern O (1947) Theory of games and economic behavior, 2nd edn. Princeton University Press, Princeton; first edition: 1944Google Scholar
  49. von Stackelberg H (1934) Marktform und gleichgewicht. Springer Verlag, Vienna (An English translation appeared in 1952 entitled “The Theory of the Market Economy,” published by Oxford University Press, Oxford)Google Scholar
  50. Vorob’ev NH (1977) Game theory. Springer, BerlinCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Coordinated Science Laboratory and Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations