Theory and Decision

, Volume 44, Issue 2, pp 149–171 | Cite as

Game Trees For Decision Analysis

  • Prakash P. Shenoy

Abstract

Game trees (or extensive-form games) were first defined by von Neumann and Morgenstern in 1944. In this paper we examine the use of game trees for representing Bayesian decision problems. We propose a method for solving game trees using local computation. This method is a special case of a method due to Wilson for computing equilibria in 2-person games. Game trees differ from decision trees in the representations of information constraints and uncertainty. We compare the game tree representation and solution technique with other techniques for decision analysis such as decision trees, influence diagrams, and valuation networks.

Game trees Decision trees Influence diagrams Valuation networks Roll-back method 

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REFERENCES

  1. Call, H. J. and Miller, W. A. (1990), A comparison of approaches and implementations for automating decision analysis, Reliability Engineering and System Safety 30, 115–162.Google Scholar
  2. Charnes J. M. and Shenoy, P. P. (1996), A forward Monte Carlo method for solving influence diagrams using local computation. Working Paper No. 273, School of Business, University of Kansas, Lawrence, KS.Google Scholar
  3. Covaliu, Z. and Oliver, R. M. (1995), Representation and solution of decision problems using sequential decision diagrams, Management Science 41(12), 1860–1881.Google Scholar
  4. Fung, R. M. and Shacter, R. D. (1990), Contingent influence diagram. Working Paper, Advanced Decision Systems, Mountain View, CA.Google Scholar
  5. Hart, S. (1992), Games in extensive and strategic form, in R. J. Aumann and S. Hart (eds.), Handbook of Game Theory with Economic Applications, Vol. 1, pp. 19–40. Amsterdam: North-Holland.Google Scholar
  6. Howard, R. A. and Matheson, J. E. (1981), Influence diagrams, reprinted in R. A. Howard and J. E.Matheson (eds.), The Principles and Applications of Decision Analysis (1984), Vol. 2, pp. 719–762. Menlo Park, CA: Strategic Decisions Group.Google Scholar
  7. Kirkwood, C. W. (1993), An algebraic approach to formulating and solving large models for sequential decisions under uncertainty, Management Science 39(7), 900–913.Google Scholar
  8. Kreps, D. and Wilson, R. (1982), Sequential equilibria, Econometrica 50, 863–894.Google Scholar
  9. Kuhn, H. W. (1953), Extensive games and the problem of information, in H. W. Kuhn and A. W. Tucker (eds.), Contributions to the Theory of Games, Vol. 2, pp. 193–216. Princeton, NJ: Princeton University Press.Google Scholar
  10. Olmsted, S. M. (1983), On representing and solving decision problems. Ph.D. thesis, Department of Engineering-Economic Systems, Stanford University, Stanford, CA.Google Scholar
  11. Raiffa H. and Schlaifer, R. O. (1961), Applied Statistical Decision Theory. Cambridge, MA: Harvard Business School.Google Scholar
  12. Raiffa, H. (1968), Decision Analysis: Introductory Lectures on Choices Under Uncertainty. Reading, MA: Addison-Wesley.Google Scholar
  13. Shachter, R. D. (1986), Evaluating influence diagram, Operations Research 34(6), 871–882.Google Scholar
  14. Shenoy, P. P. (1992), Valuation-based systems for Bayesian decision analysis, Operations Research 40(3), 463–484.Google Scholar
  15. Shenoy, P. P. (1993a), A new method for representing and solving Bayesian decision problems, in D. J. Hand (ed.), Artificial Intelligence Frontiers in Statistics: AI and Statistics III, pp. 119–138. London: Chapman & Hall.Google Scholar
  16. Shenoy, P. P. (1993b), Valuation network representation and solution of asymmetric decision problems. Working Paper No. 246, School of Business, University of Kansas, Lawrence, KS.Google Scholar
  17. Shenoy P. P. (1994), A comparison of graphical techniques for decision analysis, European Journal of Operational Research 78(1), 1–21.Google Scholar
  18. Shenoy, P. P. (1995), A new pruning method for solving decision trees and game trees, in P. Besnard and S. Hanks (eds.), Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference, pp. 482–490. San Francisco, CA: Morgan Kaufmann.Google Scholar
  19. Smith, J. E., Holtzman, S. and Matheson, J. E. (1993), Structuring conditional relationships in influence diagram, Operations Research 41(2), 280–297.Google Scholar
  20. von Neumann, J. and Morgenstern O. (1944), Theory of Games and Economic Behavior (2nd edn. 1947, 3rd edn. 1953). New York: John Wiley & Sons.Google Scholar
  21. Wilson, R. (1972), Computing equilibria of two-person games from the extensive form, Management Science 18(7), 448–460.Google Scholar
  22. Zermelo, E. (1913), Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiel, Proceedings of the Fifth International Congress of Mathematics, Vol. 2, pp. 501–504. Cambridge, UK: Cambridge University Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Prakash P. Shenoy
    • 1
  1. 1.School of BusinessUniversity of KansasLawrenceU.S.A

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