Random Utility Representations of Finite m-ary Relations

Block and Marschak (1960, in Olkin et al. (Eds.), Contributions to probability and statistics (pp. 97-132). Stanford, CA: Stanford Univ. Press) discussed the relationship between a probability distribution over the strict linear rankings on a finite set C and a family of jointly distributed random variables indexed by C. The present paper generalizes the concept of random variable (random utility) representations to m-ary relations. It specifies conditions on a finite family of random variables that are sufficient to construct a probability distribution on a given collection of m-ary relations over the family's index set. Conversely, conditions are presented for a probability distribution on a collection of m-ary relations over a finite set C to induce (on a given sample space) a family of jointly distributed random variables indexed by C. Four random variable representations are discussed as illustrations of the general method. These are a semiorder model of approval voting, a probabilistic model for betweenness in magnitude judgments, a probabilistic model for political ranking data, and a probabilistic concatenation describing certainty equivalents for the joint receipt of gambles. The main theorems are compared to related results of Heyer and Niederee (1989, in E. E. Roskam (Ed.), Mathematical psychology in progress (pp. 99-112). Berlin: Springer-Verlag; 1992, Mathematical Social Sciences, 23, 31-44).