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Lexicographic Probabilities and Choice Under Uncertainty

In: The Language of Game Theory Putting Epistemics into the Mathematics of Games

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  • Lawrence Blume
  • Adam Brandenburger
  • Eddie Dekel

Abstract

Two properties of preferences and representations for choice under uncertainty which play an important role in decision theory are: (i) admissibility, the requirement that weakly dominated actions should not be chosen; and (ii) the existence of well defined conditional probabilities, that is, given any event a conditional probability which is concentrated on that event and which corresponds to the individual's preferences. The conventional Bayesian theory of choice under uncertainty, subjective expected utility (SEU) theory, fails to satisfy these properties — weakly dominated acts may be chosen, and the usual definition of conditional probabilities applies only to nonnull events. This chapter develops a non-Archimedean variant of SEU where decision makers have lexicographic beliefs; that is, there are (first-order) likely events as well as (higher-order) events which are infinitely less likely but not necessarily impossible. This generalization of preferences, from those having an SEU representation to those having a representation with lexicographic beliefs, can be made to satisfy admissibility and yield well defined conditional probabilities and at the same time to allow for “null” events. The need for a synthesis of expected utility with admissibility, and to provide a ranking of null events, has often been stressed in the decision theory literature. Furthermore, lexicographic beliefs are appropriate for characterizing refinements of Nash equilibrium. In this chapter we discuss: axioms on, and behavioral properties of, individual preferences which characterize lexicographic beliefs; probability-theoretic properties of the representations; and the relationships with other recent extensions of Bayesian SEU theory.

Suggested Citation

  • Lawrence Blume & Adam Brandenburger & Eddie Dekel, 2014. "Lexicographic Probabilities and Choice Under Uncertainty," World Scientific Book Chapters,in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 6, pages 137-160 World Scientific Publishing Co. Pte. Ltd..
  • Handle: RePEc:wsi:wschap:9789814513449_0006
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    Keywords

    Game Theory; Epistemic Game Theory; Foundations; Applied Mathematics; Social Neuroscience; Rationalizability; Nash Equilibrium; Probability; Uncertainty;

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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