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 script c sign and a family of jointly distributed random variables indexed by script c sign. 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 overthe family's index set. Conversely, conditions are presented for a probability distribution on a collection of m-ary relations over a finite set script c sign to induce (on a given sample space) a family of jointly distributed random variables indexed by script c sign. 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 Niederée (1989, in E. E. Roskam (Ed.), Mathematical psychology in progress (pp. 99-112). Berlin: Springer-Verlag; 1992, Mathematical Social Sciences, 23, 31-44).