Lexicographic probabilities and choice under uncertainty

Lawrence Blume, Adam Brandenburger, Eddie Dekel

Research output: Chapter in Book/Report/Conference proceedingChapter

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.

Original languageEnglish (US)
Title of host publicationWorld Scientific Series in Economic Theory
EditorsAdam Brandenburger
PublisherWorld Scientific
Pages137-160
Number of pages24
DOIs
StatePublished - Sep 1 2023

Publication series

NameWorld Scientific Series in Economic Theory
Volume5
ISSN (Print)2251-2071

Keywords

  • admissibility
  • conditional probabilities
  • lexicographic probabilities
  • non-Archimedean preferences
  • subjective expected utility
  • weak dominance

ASJC Scopus subject areas

  • Economics and Econometrics
  • Statistics, Probability and Uncertainty
  • Applied Mathematics
  • Statistics and Probability

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