TY - JOUR

T1 - Random relations, random utilities, and random functions

AU - Regenwetter, M.

AU - Marley, A. A.J.

N1 - Funding Information:
This work was supported by NSERC Collaborative Research Grant CGP0164211, NSERC Research Grant 282-82, the Faculty of Science at McGill University, the Fuqua School of Business at Duke University, and NSF Grants SBR 97-30578 (to Grofman and Marley) and SBR 97-30076 (to Regenwetter). This research was carried out while the first author was at McGill and Duke Universities. We owe special thanks to Reinhard Niederee, from whom we received tremendous encouragement as well as extensive and constructive critical evaluation. We are also grateful to the action editor, Peter Fishburn, for handling our submission, and to Martin Chabot, Harry Joe, Duncan Luce, and Reinhard Suck for many helpful suggestions.

PY - 2001

Y1 - 2001

N2 - We extend Regenwetter's (1996) results on the relationship between (1) random relations, i.e., a probability measure on m-ary relations, and (2) random utilities, i.e., families of random variables, to (3) random functions, i.e., a probability measure over a function space. In this third approach, we assume that each sampled respondent accesses an urn of (utility) functions over the choice alternatives and that his/her judgment/choice is governed by the currently sampled (utility) function. Although the three approaches usually involve completely different sample spaces, we show, under reasonable conditions, that if any one of the representations holds then so do each of the others. We also develop the results for valued m-ary relations and relational structures. Our theoretical findings are illustrated with probabilistic models of magnitude estimation, probabilistic extensive measurement, probabilistic metric spaces, and (binary) subjective expected utility. The theoretical results complement and reformulate closely related research, e.g., that of Heyer and Niederée (1989, 1992), Niederée and Heyer (1997), and Suck (1995, 1996).

AB - We extend Regenwetter's (1996) results on the relationship between (1) random relations, i.e., a probability measure on m-ary relations, and (2) random utilities, i.e., families of random variables, to (3) random functions, i.e., a probability measure over a function space. In this third approach, we assume that each sampled respondent accesses an urn of (utility) functions over the choice alternatives and that his/her judgment/choice is governed by the currently sampled (utility) function. Although the three approaches usually involve completely different sample spaces, we show, under reasonable conditions, that if any one of the representations holds then so do each of the others. We also develop the results for valued m-ary relations and relational structures. Our theoretical findings are illustrated with probabilistic models of magnitude estimation, probabilistic extensive measurement, probabilistic metric spaces, and (binary) subjective expected utility. The theoretical results complement and reformulate closely related research, e.g., that of Heyer and Niederée (1989, 1992), Niederée and Heyer (1997), and Suck (1995, 1996).

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U2 - 10.1006/jmps.2000.1357

DO - 10.1006/jmps.2000.1357

M3 - Article

AN - SCOPUS:0035709921

VL - 45

SP - 864

EP - 912

JO - Journal of Mathematical Psychology

JF - Journal of Mathematical Psychology

SN - 0022-2496

IS - 6

ER -