A preference relation on a set X is nontransitive if there are for which and not , and is cyclic if there are with such that for all i < m and . The representations described thus far assume that is transitive, or at least acyclic. I believe that this reflects a strong attraction of decision theorists and perhaps others to transitivity as an intuitively obvious basis for rational thought and action, an apparently natural desire for order in practical affairs, and the supposed efficiency of optimization based on maximization. Although I find some merit in these contentions, I argue elsewhere (Fishburn 1991a) against transitivity as an undeniable tenet of rational preferences because I believe that reasonable people sometimes hold nontransitive or cyclic preference patterns that account for their true feelings.

My purpose here is not to recount the arguments for or against transitivity set forth in Fishburn (1991a). Instead, I will illustrate a few situations that might give rise to cyclic patterns and then describe four representations that can account for cyclic preferences by straightforward and elegant generalizations of representations that presume transitivity.

The genesis of cyclic patterns in decision theory may be Condorcet's (1785) phenomenon of cyclic majority. The simplest example uses candidates x,y and z and three voters with preference orders , and . Then, with the strict simple majority relation, . Arrow's (1950) famous extension shows that every reasonable rule for aggregating voters' preference orders that makes binary social comparisons without regard to other candidates' positions in voters' orders, and which allows a variety of voter preference profiles, must have profiles for which the social comparison relation is nontransitive. Subsequent contributions on this theme are reviewed in Fishburn (1987).

Multiattribute comparisons provide a source of cyclic preferences for an individual. May (1954) asked 62 college students to make binary comparisons between hypothetical marriage partners x,y and z characterized by three attributes, intelligence, looks and wealth:



x: very intelligence, plain, well off y: intelligent, very good looking, poor z: fairly intelligent, good looking, rich.

Let (a,p) denote the lottery that pays $a with probability p and nothing otherwise. Tversky (1969) observed that a significant number of people have the cyclic pattern

In a four-state example with subjective probability of 1/4 for each state, consider four acts with monetary prizes:

states 1 2 3 4 $10000 $9000 $8000 $7000 acts $9000 $8000 $7000 $10000 $8000 $7000 $10000 $9000 $7000 $10000 $9000 $8000

We now describe four representations that accommodate cyclic preferences. The first two apply to multiattribute situations, the third to lottery comparisons in decision under risk, and the fourth to act comparisons in decision under uncertainty.

Let and denote items described by n attributes. Assume that the attribute levels within a given attribute are unambiguously ordered by a weak order . The additive difference representation is

where preserves as in (1) and is a strictly increasing functional on its domain with . Many possibilities for the allow and by way of positive sums on the right side of (15) for the three comparisons. Discussions and axioms for (15) and related representations are in Tversky (1969), Croon (1984), Chapter 17 in Suppes et al. (1989), and Fishburn (1992a).

An alternative to (15) is the nontransitive additive utility representation

in which is a real-valued function on ordered pairs of levels of attribute i with and . Axioms for (16) are in Vind (1991) and Fishburn (1990, 1991b). The latter axiomatizations imply that each is skew symmetric, i.e.,

and all three imply that the are unique up to proportionality transformations with a common scale multiplier.

Skew symmetry is also used in our other two representations. The representation for lotteries x and y on a set C of consequences is , where is a real-valued, skew-symmetric and bilinear function on ordered pairs of lotteries. Bilinearity means that and . We refer to the representation as the SSB representation, short for skew-symmetric and bilinear. When all individual consequences are in X and it is convex, we have the bilinear expected utility expression

The SSB representation was first described in Kreweras (1961) and is axiomatized in Fishburn (1988), with unique up to a proportionality transformation. A constant-threshold SSB representation that has is axiomatized in Fishburn and Nakamura (1991).

Our final representation applies to a Savage act set for decision under uncertainty. The representation is

where is a skew-symmetric functional on and is a finitely additive probability measure on , with unique and unique up to a proportionality transformation. When decomposes as , (17) reduces to Savage's subjective expected utility representation. Axioms that imply (17) for all acts that use only finite numbers of consequences are in Chapter 9 in Fishburn (1988). They are like Savage's axioms in most respects with weak order replaced by an asymmetry condition. Extension to all acts is also discussed in Chapter 9. Other representations that are closely related to (17) appear in Loomes and Sugden (1987), Fishburn and LaValle (1987) and Fishburn (1988).

Although cyclic preference patterns have been studied in depth for aggregate relations in voting and social choice theory, they have received very little attention in individual decision theory. Unlike most decision theorists, I think the aversion to cyclic preferences for individuals is unjustified, and I hope that more will be done on the subject in the years ahead.

Peter.Fishburn