Random Utility Representations of Finitem-ary Relations

Random Relations, Random Utilities, and Random Functions

“…This paper extends earlier results of Regenwetter (1996) and complements and reformulates closely related work of Niedere e (1989, 1992), Niedere e and Heyer (1997), and Suck (1995Suck ( , 1996 by discussing the relationship between three fundamental ways of mathematically representing judgments and preferences. To motivate these three approaches, suppose that each member of a group of people is presented with a finite set C of objects, persons, or questions.…”

“…More generally, R may be some m-ary relation (R C m ), where m is a positive integer. In this paper we present two key extensions of Regenwetter's (1996) random relation approach. First, we cover the cases where the individual's latent state is a relational structure (i.e., a k-tuple of relations) in the sense of representational measurement theory (Krantz, Luce, Suppes, 6 Tversky, 1971;Luce, Krantz, Suppes, 6 Tversky, 1990;Suppes, Krantz, Luce, 6 Tversky, 1989).…”

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We extend Regenwetter's (1996) results on the relationship between (1) random relations, i.e., a probability measure on m-ary relations, and (2) random utilities, i.e., families of random variables, to (3) random functions, i.e., a probability measure over a function space. In this third approach, we assume that each sampled respondent accesses an urn of (utility) functions over the choice alternatives and that hisÂher judgmentÂchoice is governed by the currently sampled (utility) function.

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A general concept of majority rule

“…The generalizations to strict weak orders and semiorders were pointed out by Regenwetter (1996), and the interval order case by Suck (1995) and Regenwetter (1997). Furthermore, Heyer andŃ iederee (1989Heyer andŃ iederee ( , 1992, Niederee and Heyer (1997), Regenwetter (1996), and Regenwetter and Marley (2001) provide general abstract results including binary relations as very special cases. We now add another pair of random utility representation results, which, to our knowledge, have not been previously stated in the literature.…”

“…In other words, it is equivalent, for instance, to assume that a randomly sampled observer has a strict linear order preference over all choice alternatives or to assume that s / he assigns utilities to objects according to a joint outcome of a family of random variables (satisfyinǵ certain properties). Recent developments in mathematical psychology (Niederee and Heyer, 1997;Regenwetter, 1996Regenwetter, , 1997Regenwetter and Marley, 2001;Suck, 1995) have extended this result to arbitrary relations, including the special cases where the individual preferences are transitive binary relations, strict weak orders (i.e. rankings with possible ties), semiorders (Luce, 1956(Luce, , 1959)-which capture thresholds of utility discrimination-and interval orders (Fishburn, 1970(Fishburn, , 1985-which are a generalization of semiorders to the case with variable thresholds.…”