RANDOM UTILITY REPRESENTATIONS OF FINITE M-ARY RELATIONS

Block and Marschak (1960, in Olkin et al. (Eds.), Contributions to pro bability and statistics (pp. 97-132), Stanford, CA: Stanford Univ. Pre ss) 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 genera li zes 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 o n 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 i ndexed by C. Four random variable representations are discussed as ill ustrations of the general method. These are a semiorder model of appro val voting, a probabilistic model for betweenness in magnitude judgmen ts, a probabilistic model for political ranking data, and a probabilis tic concatenation describing certainty equivalents for the joint recei pt of gambles. The main theorems are compared to related results of He yer 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). (C) 1996 Academic Press, Inc.