Order indifference and rank-dependent probabilities
The paper presents a new axiomatization of the anticipated utility model in a two-stage lotteries framework. It is a common assumption in decision theory that a multi-stage lottery is equally attractive as the simple lottery yielding the same prizes with the compound probabilities (reduction of compound lotteries). This paper adopts a different point of view. Only a weakening of the reduction axiom which is called order indifference is assumed. Let \(L_ 1\) be the set of all finite lotteries with non-negative outcomes, and \(L_ 2\) the set of all two-stage lotteries over \(L_ 1\), i.e., the set of all lotteries whose outcomes are tickets for lotteries in \(L_ 1\). For lotteries in \(L_ 1\) the following notation is used: \((x_ 1,p_ 1;\dots;x_ n,p_ n)\) denotes the lottery giving \(x_ i\) with probability \(p_ i\), \(i= 1,\dots,n\). Analogously, lotteries in \(L_ 2\) are denoted by \((X_ 1,p_ 1;\dots;X_ n, p_ n)\), where \(X_ 1,\dots,X_ n\) are lotteries in \(L_ 1\). Lotteries in \(L_ 1\) can be embedded in \(L_ 2\) in two different ways. The lottery \((x_ 1,p_ 1;\dots; x_ n,p_ n)\) can either be represented by the two-stage lottery \(((x_ 1,p_ 1;\dots; x_ n,p_ n),1)\) or by the lottery \((x_ 1,1),p_ 1;\dots; (x_ n, 1),p_ n)\). The time neutrality axiom states that the decision maker is indifferent between these two lotteries. The key axiom in the axiomatization is called order indifference. It assumes that for every one-stage lottery \((x_ 1,q_ 1;\dots; x_ n,q_ n)\) and every \(p\in [0,1]\) the decision maker is indifferent between the two-stage lotteries \(((0,1),1- p;(x_ 1,q_ 1;\dots;x_ n,q_ n),p)\) and \(((0,1-p; x_ 1,p),q_ 1);\dots; (0,1-p;x_ n,p),q_ n)\). This means that the decision maker does not care on which stage he is informed that he will gain nothing. Under time neutrality, order indifference, and four other technical axioms (continuity, first- order stochastic dominance, irrelevance, Archimedean) the preference relation over one-stage lotteries can be represented by the anticipated utility functional. For the representation of the preference relation over two-stage lotteries the author assumes compound independence instead of the reduction of compound lotteries as stated above. Compound independence implies that the decision maker is always indifferent between the lotteries \((X_ 1,q_ 1;\dots; X_ n,q_ n)\) and \(((CE(X_ 1),1),q_ 1;\dots; (CE(X_ n),1),q_ n)\), where \(CE(X)\) is the certainty equivalent of \(X_ i\), \(i= 1,\dots,n\). Thus every two-stage lottery can be reduced to an equally attractive one-stage lottery. The author further discusses relations to other versions of order indifference. If the common outcome 0 in the order indifference axiom is replaced by an arbitrary outcome \(y\), the preferences can be represented by the anticipated utility functional with a symmetric distortion function. If order indifference is replaced by a more general axiom which is called general order indifference the preference relation can be represented by expected utility. Finally, relations to different axiomatizations in the literature are discussed.
scientific article; zbMATH DE number 429102