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Context-Dependent Choice with Nonlinear and Nontransitive Preferences

Peter C. Fishburn
Irving H. LaValle

Abstract

The main purpose of this paper is to explore implications for one-stage and two-stage decision processes of a theory of choice that accommodates intransitivities in preferences. To motivate our analysis, we argue that intransitivities and preference cycles deserve serious consideration in normative theories of preference and choice. Examples from decision making under certainty, risk, and uncertainty are used to suggest that there are simply too many settings in which intelligent people and groups can have good reasons for cyclic preferences to exclude intransitivities from normative theory. Our analysis of decision processes enriches the basic alternative set by probabilistic convexification, as in the formation of mixed strategies in game theory. It is noted (and has been known for 30 years) for one-stage processes that if the convex set is finitely generated then it can have an enriched alternative that is maximally preferred regardless of intransitivities among basic alternatives. The core of the paper focuses on two-stage processes, typified by the Bayesian formulation of a choice of experiment followed by selection of a terminal act once the experiment's outcome is observed. We suppress consideration of Bayes's theorem and other facets of specialized formulations of two-stage processes to concentrate on a general treatment in which the "holistic," single-stage choice procedure is compared with two sequential procedures that first choose among different subsets and then choose something from the selected subset. The sequential procedures, referred to as "naive" and "sophisticated," base the choice of subset on a natural extension of holistic preferences. Many combinations of these three procedures are available in settings involving at least three stages of choice. A main result of the comparative analysis is that, when convex sets are finitely generated, the three choice procedures all have unambiguous solutions and these solutions can differ radically when preferences are intransitive. On the other hand, they coincide when preferences are fully transitive, and this can be taken as an argument in favor of transitivity. Moreover, since the imposition of transitivity on our model still allows nonlinearities that violate the von Neumann-Morgenstern utility model, the procedural equivalence obtains for a theory that is more general than their expected utility theory.