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Games with a Continuum of Strategies

  • Alan Washburn
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 201)

Abstract

It is not unusual to encounter games where the number of available pure strategies is infinite. Any game where the two players each select a time for action is an example, or a submarine can dive to any depth up to some maximum limit. Intervals of real numbers can of course be artificially subdivided to make the number of strategies finite, but that is merely an approximation technique. Sometimes it may even be enlightening to approximate a subdivided interval by a continuous one. The radio frequency spectrum, for example, contains only finitely many frequencies as far as modern digital receivers are concerned, but there are so many frequencies that for some purposes one might as well think of the spectrum as being continuous. In this chapter we consider games where the choice of strategy is not limited to a finite set.

References

  1. Blackett, D. (1954). Some Blotto games. Naval Research Logistics, 1(1), 55–60.CrossRefGoogle Scholar
  2. Burger, E. (1963). Introduction to the Theory of Games. Prentice-Hall.Google Scholar
  3. Charnes, A. (1953). Constrained games and linear programming. Proceedings of the National Academy of Sciences, 39(7), 639–41.CrossRefGoogle Scholar
  4. Dresher, M., Karlin, S., Shapley, L. (1950). Polynomial games. In H. Kuhn, & A. Tucker (Eds.), Contributions to the theory of games I (Annals of Mathematics Studies, Vol. 24, pp. 161–180), Princeton University Press.Google Scholar
  5. Dresher, M. (1961). Games of strategy theory and applications. Prentice-Hall, subsequently republished in 1980 as The Mathematics of Games of Strategy Theory and Applications, Dover.Google Scholar
  6. Karlin, S. (1959). Mathematical Methods and Theory in Games, Programming, and Economics (Vols. 1 and 2). Addison-Wesley.Google Scholar
  7. McKinsey, J. (1952). Introduction to the Theory of Games. McGraw Hill.Google Scholar
  8. Morse, P., & Kimball, G. (1950). Methods of Operations Research. Joint publishers Technology Press of MIT and Wiley, Section 5.4.5.Google Scholar
  9. Newman, D. (1959). A model for 'real' poker. Operations Research, 7(5), 557–560.CrossRefGoogle Scholar
  10. Thomas, C. (1964). Some past applications of game theory to problems of the United States Air Force. In A. Mensch (Ed.), Proceedings of a conference under the aegis of the NATO Scientific Affairs Committee, Toulon, France 250–267, American Elsevier.Google Scholar
  11. Washburn, A., & Ewing, L. (2011). Allocation of clearance assets in IED warfare. Naval Research Logistics, 58(3), 180–187.CrossRefGoogle Scholar
  12. von Neumann, J. (1937). Über ein Ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergebnisse eines Mathematischen Kolloquiums, 8, 73–83.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alan Washburn
    • 1
  1. 1.Operations Research DepartmentNaval Postgraduate SchoolMontereyUSA

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