Testing for intransitivity of preferences predicted by a lexicographic semi-order

Abstract

A classic paper by Tversky (Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76,[31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] reported that some people systematically violated transitivity of preference when choosing between specially constructed risky gambles. This conclusion remained controversial because his statistical analysis did not allow each participant to have a different true preference order. Recently, however, it has been argued that an inherently intransitive process governs risky decision making. This paper uses a relatively new statistical technique for testing transitivity and analyzes two new experiments in which hundreds of participants made choices among the same gambles studied by Tversky. It was found that very few people repeated intransitive patterns. The incidence of violation of transitivity was slightly higher when probability was displayed graphically without numerical information, but even in this condition few participants were intransitive. Furthermore, even among those few who appeared to be intransitive, most showed dimension interaction, contrary to implications of the lexicographic semi-order. These results cast doubt on a lexicographic semi-order as a descriptive theory of risky decision making. Ó 2007 Elsevier Inc. All rights reserved.Keywords: Choice; Decision making; Errors; Lexicographic semi-order; Risk; Transitivity; Utility Transitivity of preference is the assumption that if a person prefers A to C and C to E, then that person prefers A to E. This principle is not only assumed by most theoreticians to be a rational principle to obey, but it is also assumed by most, but not all, descriptive theories of risky decision making. It was therefore surprising when May (1954), and later Tversky (1969) reported that some people are systematically intransitive when presented with specially constructed choices. They noted that if real, such findings ruled out many popular descriptive theories.The gambles used by Tversky (1969) are listed in Table 1. Each was a two-branch gamble with probability p to win cash prize x, otherwise receive nothing. These were designed so that if people used a lexicographic semi-order, their preferences would be intransitive. A semi-order (Luce, 1956) has the property that small differences are responded to as if there were no difference. A lexicographic order is illustrated by the task of alphabetizing a list of words. First one examines the first letters in each word, and if they differ, the ranking is based on this letter alone; only if the first letters are the same does one check the second letters, which if different determine the order, and so on (Fishburn, 1971;Hausner, 1954).According to the lexicographic semi-order (LS) interpretation of Tversky's (1969) results, people first compared probabilities to win in the two gambles. When this difference was large enough (greater than D), they chose the gamble with the higher probability to win. However, when this difference was small, they co...

Dimension integration: Testing models without trade-offs

“…semi-orders because they imply transitivity, so it seems inappropriate to assume them when evaluating such intransitive models (Birnbaum & Gutierrez, 2007). Nevertheless, if we were to apply these models to the present data, we would reach the same conclusions.…”

“…Tversky's selected data have been reanalyzed, with the result that not all analyses agree that anyone was significantly intransitive in his study (For contrasting analyses and arguments, see papers by Iverson & Falmagne, 1985;Iverson, Myung, & Karabatsos, submitted for publication;Regenwetter, Stober, Dana, & Kim, 2006). Birnbaum and Gutierrez (2007) conducted a study in which people were asked to choose between the same gambles used by Tversky, except using procedures similar to those used in most of the studies summarized by Brandstaetter et al (2006). Whereas Tversky (1969) used pie charts to represent probability and did not present probability information numerically, Birnbaum and Gutierrez presented both probabilities and prizes numerically.…”

Transitivity for height versus speed: To what extent do the under-7s really have a transitive capacity?

“…It is also an integral part of children's educational progression, such that "in history class, one has to determine which event or personality preceded another; in geography, altitudes are represented by different colors; in geometry, measurement operations are performed;..." (Artman & Cahan, 1993, pp.753). Transitivity even features in social and interpersonal contexts, such as in assessing possible health risks, gambles and friendships (Birenbaum & Gutierrez, 2007;Markovits & Dumas, 1999;Reyna & Farley, 2006). Initially, its importance to child cognition was very much a Piagetian pursuit (e.g., Piaget & Inhelder, 1967;Piaget, Inhelder & Szeminska, 1960; see also Muller et al, 1999 for a sound introduction to Piaget's framework for reasoning research).…”

Min- and Max-induced rankings: an experimental study

“…The true and error model can straightforwardly be extended to estimate the proportion of individuals with a given true preference pattern for the sequence of three choices (Birnbaum & Gutierrez, 2007;Birnbaum & Bahra, 2007;Birnbaum & Schmidt, 2008, 2010.…”