Stochastic transitivity and quadratic representation functions
type: Item
instance of: scholarly article
title: Stochastic transitivity and quadratic representation functions
zbMATH Open document ID: 0812.90003
author: Uzi Segal
published in: Journal of Mathematical Psychology
publication date: 1994-04-19T00:00:00Z
review text: Let \(L\) be the set of finite lotteries with outcomes in \([0,M]\). Let \(\gtrsim\) be a complete, transitive, continuous, monotone (first order stochastic dominance) and strict quasi-concave relation on \(L\). It is assumed that comparing two lotteries \(X\) and \(Y\) the real choice set is the set of compound lotteries \(\{(X,\alpha; Y,1-\alpha)| \alpha\in [0,t]\}\) which can be identified -- assuming the reduction of compound lotteries axiom -- with the set \(\{\alpha X+ (1-\alpha)Y| \alpha\in [0,1]\}\subset L\). If \(V\) is the functional representing \(\gtrsim\) the decision maker will choose a compound lottery \((X,\alpha^*; Y,1- \alpha^*)\), where \(\alpha^*\) is determined by \(\max_ \alpha V(\alpha X+(1- \alpha)Y)\). Let \(C: L\times L\to \mathbb{R}\) be a mapping with \(C(X,Y)= \alpha^*\) if \(X\neq Y\), \(C(X,X)= {1\over 2}\). Then the relation \(\gtrsim\) implies a new relation \(\gtrsim^*\) on \(L\) by defining: \(X\gtrsim^* Y\Leftrightarrow C(X,Y)\gtrsim{1\over 2}\), \(X\sim^* Y\Leftrightarrow C(X,Y)= {1\over 2}\). It is shown: Either the relation \(\gtrsim^*\) is intransitive or the relation \(\gtrsim\) is quadratic (\(\gtrsim\) is quadratic if \(V(x_ 1,p_ 1;\dots; x_ n, p_ n)=\sum^ n_{i=1} \sum^ n_{j=i} p_ i p_ i\varphi(x_ i, x_ j)\) with a symmetric function \(\varphi\)). Since it is optimal for decision makers with a strict quasi-concave preference relation to randomize, the result shows that intransitivity must be observed if the preference relation \(\gtrsim\) is not quadratic. The proofs make strong use of geometrical considerations.
reviewed by: Peter Kischka
Mathematics Subject Classification ID: 91B06
Mathematics Subject Classification ID: 91B08
zbMATH DE Number: 554106
zbMATH Keywords: quadratic preference relations
zbMATH Keywords: set of finite lotteries
zbMATH Keywords: strict quasi- concave preference relation
zbMATH Keywords: intransitivity
MaRDI profile type: Publication
full work available at URL: jmps.1994.1006
OpenAlex ID: W2072148581
DOI: 10.1006/JMPS.1994.1006
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description: scientific article; zbMATH DE number 554106