A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories
type: Item
instance of: scholarly article
title: A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories
zbMATH Open document ID: 0839.90001
DOI: 10.1007/BF01079912
author: Donald B. Davis
author: M.-Elisabeth Paté-Cornell
published in: Theory and Decision
publication date: 1996-06-23T00:00:00Z
review text: Decisions among lotteries whose outcomes are themselves lotteries can be made by considering the collapsed lottery of the two stage lottery on the outcome space (view 1). Another approach is chosen by the authors: ``Especially when the underlying lottery uncertainties are resolved at different times or when they represent essentially different uncertainty concepts, a more complex model than that implied in view 1 is needed to fully capture the decision maker's primitive preference relationships.'' The authors prove a theorem which weakens the ``reduction of compound lotteries axiom''. Let \(Z\) be the outcome space and let \(D_2\) be a mixture set of Borel probability measures on the Borel \(\sigma\)-field of \(Z\) (second stage lotteries). Let \(D_1\) be a mixture space of Borel probability measures on the set \(X_2\) of all closed subsets of \(D_2\) (first stage lotteries). Let \(\succsim_i\) be a weak order on \(D_i\) which is continuous and fulfills strong independence \((i= 1,2)\). Furthermore, consistency is assumed, i.e.: For degenerate distributions \(d, d'\in D_1\) the preference \(d\succsim_1 d'\) implies consistent preference between the outcome probability distribution, i.e. if \(x\in X_2\), \(x'\in X_2\) are resultant states of \(d\) and \(d'\) respectively, then for every \(e'\in x'\) there exists an \(e\in x\) with \(e\succsim_2 e'\). Then there exist bounded functions \(u_2: Z\to \mathbb{R}\), \(u_1: X_2\to \mathbb{R}\) such that for \(d, d'\in D_i: d\succsim_i d'\Leftrightarrow E_d(u_i)\geq E_{d'}(u_i)\) \((i= 1,2)\). Moreover, there exists a unique continuous strictly increasing function \(\phi: \mathbb{R}\to \mathbb{R}\) s.t. \(u_1(d)= \phi(E_d(u_2))\) for all \(d\in D_2\); when the reduction of compound lotteries axiom is assumed \(\phi\) is linear but in general it is not. The paper contains a review of existing theories of two stage lottery. The results are compared to these theories.
reviewed by: Peter Kischka
Mathematics Subject Classification ID: 91B06
zbMATH DE Number: 710394
zbMATH Keywords: reduction of compound lotteries axiom
zbMATH Keywords: two stage lottery
Wikidata QID: Q114018479
MaRDI profile type: Publication
cites work: Q3922442
cites work: A Definition of Subjective Probability
cites work: Q5618987
cites work: Lottery Dependent Utility
cites work: Regret in Decision Making under Uncertainty
cites work: Disappointment in Decision Making Under Uncertainty
cites work: Some Probability Paradoxes in Choice from Among Random Alternatives
cites work: A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox
cites work: Q5832752
cites work: An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom
cites work: Q5556137
cites work: Risk, ambiguity and the Savage axioms
cites work: Slanted Subjective Probabilities and Randomization: Reply to Howard Raiffa and K. R. W. Brewer
cites work: A General Theory of Subjective Probabilities and Expected Utilities
cites work: Q5615833
cites work: Nontransitive measurable utility
cites work: Ellsberg revisited: A new look at comparative probability
cites work: Transitive measurable utility
cites work: SSB utility theory: An economic perspective
cites work: SSB utility theory and decision-making under uncertainty
cites work: Foundations of Decision Analysis: Along the Way
cites work: Unreliable probabilities, risk taking, and decision making
cites work: Reply to Levi
cites work: An Axiomatic Approach to Measurable Utility
cites work: Prospect Theory: An Analysis of Decision under Risk
cites work: Temporal Resolution of Uncertainty and Dynamic Choice Theory
cites work: Temporal von Neumann-Morgenstern and induced preferences
cites work: Ignorance, probability and rational choice
cites work: Note on von Neumann-Morgenstern's Strong Independence Axiom
cites work: Rational Behavior, Uncertain Prospects, and Measurable Utility
cites work: Cyclical preference logic
cites work: The Foundations of Decision Under Uncertainty: An Elementary Exposition
cites work: The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach
cites work: Non-additive probabilities in the work of Bernoulli and Lambert
cites work: The Framing of Decisions and the Psychology of Choice
cites work: Q5844986
description: scientific article; zbMATH DE number 710394