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 Niedere e (1989, in E. E. Roskam (Ed.), Mathematical psychology in progress (pp. 99 112).