Random relations, random utilities, and random functions

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 his/her judgment/choice is governed by the currently sampled (utility) function. Although the three approaches usually involve completely different sample spaces, we show, under reasonable conditions, that if any one of the representations holds then so do each of the others. We also develop the results for valued m-ary relations and relational structures. Our theoretical findings are illustrated with probabilistic models of magnitude estimation, probabilistic extensive measurement, probabilistic metric spaces, and (binary) subjective expected utility. The theoretical results complement and reformulate closely related research, e.g., that of Heyer and Niederée (1989, 1992), Niederée and Heyer (1997), and Suck (1995, 1996).