Nontransitive preferences in decision theory

Intransitive preferences have been a topic of curiosity, study, and debate over the past 40 years. Many economists and decision theorists insist on transitivity as the cornerstone of rational choice, and even in behavioral decision theory intransitivities are often attributed to faulty experiments, random or sloppy choices, poor judgment, or unexamined biases. But others see intransitive preferences as potential truths of reasoned comparisons and propose representations of preferences that accommodate intransitivities. This article offers a partial survey of models for intransitive preferences in a variety of decisional contexts. These include economic consumer theory, multiattribute utility theory, game theory, preference between time streams, and decision making under risk and uncertainty. The survey is preceded by a discussion of issues that bear on the relevance and reasonableness of intransitivity. The later sections of this article discuss quantitative representations of preferences that accommodate preference cycles and milder forms of intransitivity. Many also feature additivity, as in nontransitive additive conjoint measurement, additive differences, skew-symmetric bilinear utility, and skew-symmetric additive utility for decision under uncertainty. The effects on the representations of adding transitivity or weak order to their axioms will be noted. Prior to that discussion, the next two sections offer a point-and-counterpoint medley on the theme of transitivity. Section 1 uses the simple preference cyclex > y > z > x to argue for transitivity by showing what is wrong or unreasonable about this cycle. Section 2 counters by exposing weak points or fallacies of those arguments. The rest of this introduction defines terms and mentions three topics not covered later: partially ordered preferences, choice probabilities, and the distinction between normative and descriptive decision theory. Our starting point is a binary relation of preference on a setX. It is customary to adopt one relation as primitive and then to define other relations from it. Some writers begin with an tk preferred to relation > and define is indifferent to ( N ) and/s preferred or indifferent to ( > ) by x y if neitherx > y n o r y > x , x > . y i f x > y o r x y . *I thank Irving LaValle and Duncan Luce for valuable comments on a previous version of this article. 114 PETER C. FISHBURN Others begin with >~ and define >, , and [[ (incomparability) by x > y ifx > y and not(y ~> x), x y ifx >_ y andy > x, x ]ly if neitherx _> y nory _> x. This article uses the former approach unless noted otherwise. In a few cases we will include the cannot be compared to relation [[ explicitly, and in those cases ~> will be the primitive. The latter approach reduces to the former if the primitive >_ is complete (i.e., x > y ory >_ x always obtains). We will assume that > is asymmetric, so x > y ~ not(y > x). If > is transitive also (i.e.,x > y andy > z ~ x > z), then > is a partial order. If both > and are transitive, then > is a weak order, and in this case ~> is transitive and complete. Weak orders are widely assumed in preference theory and apply whenever there is a utility function u : X ~such that, for allx, y ~ X, x > y ~> u(x) > u(y). Examples include basic ordinal utility theory, additive conjoint measurement in which U(Xl . . . . . Xn) = ul(xl) + . + Un(Xn), as axiomatized by Debreu (1960) and Krantz et al. (1971), the expected utility theories ofvon Neumann and Morgenstern (1944) and Savage (1954), and the nonlinear theories of Allais (1979), Machina (1982), and Schmeidler (1989). Partial orders admit the representationx > y => u(x) > u{y), but notx ~ y ~ u(x) = u(y) when indifference is intransitive. Wiener (1914), Luce (1956), and Aumann (1962) illustrate seminal contributions, and further information is available in Fishburn (1970a, 1970b, 1985) and Fishburn and Monjardet (1991). Although intransitive indifference fits the theme of the article, the focus here shall be on the more interesting and more controversial case of intransitive strict preference. Another topic not detailed later is notions of stochastic transitivity in probabilistic choice theory. Attention here focuses on the probabil i typ(a,A) that alternative a will be chosen from setA whenA is the available set of alternatives, and on the binary specializationp(ab) = p(a,{a, b}). Thep function is usually the primitive, and > and >~, when considered at all, are defined by a > b ifp(ab) > 1/2, and a >. b ifp(ab) >_ 1/2, or in some related way. An example of stochastic transitivity is {p(ab) > 1/2,p(bc) > 1/2} ~p (ac ) > 1/2. Luce (1959), Luce and Suppes (1965), Fishbum (1973a), and Suppes et al. (1989) provide broad coverage of thep approach. Decision theorists often take care to distinguish between normative theory and descriptive theory when discussing axioms, examples, and empirical findings: see, for example, Savage (1954), Tversky and Kahneman (1986), Fishburn (1988), Brown (1989), and Rapoport (1989). Normative theory deals with ideals and principles of"good" or "rational" decision making. It is heavily influenced by Platonism and Scottish common-sense philosophy, and asserts that knowledge of the correct principles of decision making can be arrived at by intuition and introspection. One might go further and suggest that all reasonable and clear-thinking people will agree on the principles or axioms of rational