TY - JOUR

T1 - Random utility representations of finite m-ary relations

AU - Regenwetter, Michel

N1 - Funding Information:
This paper is based on a chapter of my Ph.D. dissertation in Mathematical Behavioral Sciences at the University of California at Irvine. I am grateful to my Ph.D. advisor J.-Cl. Falmagne and to X. Dai, J.-P. Doignon, B. Grofman, J. Heller, J. Liter, R. D. Luce, Y.-F. Hsu, and J. Rouder for their feedback on previous drafts of this material. Thanks go to M. Koppen, L. Narens, R. Suck, and a referee for pointing out the work by Heyer and Niederee (1989, 1992) after the first submission. D. Heyer and R. Niederee have been very supportive and patient in explaining to me the subtleties of their papers and in commenting upon mine. A. A. J. Marley has been very helpful through several lengthy discussions about this and related problems. I gratefully acknowledge the fellowship support from Mr. H. L., the Ministere de l’Education Nationale in Luxembourg, and the School of Social Sciences at UCI. The final work on this paper was supported through a Postdoctoral Fellowship in the framework of NSERC Collaborative Research Grant CGP 0164211 to A. A. J. Marley and others.

PY - 1996/9

Y1 - 1996/9

N2 - 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).

AB - 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).

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U2 - 10.1006/jmps.1996.0022

DO - 10.1006/jmps.1996.0022

M3 - Article

AN - SCOPUS:0030240427

SN - 0022-2496

VL - 40

SP - 219

EP - 234

JO - Journal of Mathematical Psychology

JF - Journal of Mathematical Psychology

IS - 3

ER -