Similarity and preferences in the space of simple lotteries

A. Rubinstein's similarity-relation approach to decision making under uncertainty is extended by introducing a new concept of correlated similarity relations. It is shown that every expected utility preference is consistent with the -difference similarity relation on the prize space and some correlated similarity relations on the probability space, and that a similarity relation on the prize space and correlated similarity relations on the probability space overdetermine the preference (in A. Rubinstein's sense).Earlier versions were presented at the Edwin Smart Symposium on Games and Economic Behavior, Ohio State University, 1989, the ESEM 1989 Meeting, the EEA 1989 Meeting, the Workshop on Rational and Boundedly Rational Principles of Strategic Behavior, Universität Bielefeld, 1989, and the ASSET 1991 Meeting. We wish to thank the comments received from the participants in those meetings and the stimulating suggestions of S. Ch. Kolm, R. D. Luce, and an anonymous referee. Financial support from the Universidad del Pais Vasco-Euskal Herriko Unibertsitatea, DGCYT (PS 87-0039) and Gobierno de Navarra are gratefully acknowledged.     

Choice procedure consistent with similarity relations

We deal with the approach, initiated by Rubinstein, which assumes that people, when evaluating pairs of lotteries, use similarity relations. We interpret these relations as a way of modelling the imperfect powers of discrimination of the human mind and study the relationship between preferences and similarities. The class of both preferences and similarities that we deal with is larger than that considered by Rubinstein. The extension is made because we do not want to restrict ourselves to lottery spaces. Thus, under the above interpretation of a similarity, we find that some of the axioms imposed by Rubinstein are not justified if we want to consider other fields of choice theory. We show that any preference consistent with a pair of similarities is monotone on a subset of the choice space. We establish the implication upon the similarities of the requirement of making indifferent alternatives with a component which is zero. Furthermore, we show that Rubinstein's general results can also be obtained in this larger class of both preferences and similarity relations.The nontransitiveness of indifference must be recognized and explained on any theory of choice and the only explanation that seems to work is based on the imperfect powers of discrimination of the human mind whereby inequality becomes recognizable only when of sufficient magnitude.