Alternative Presentations of the Random Utility Model

Alternative Presentations of the Random Utility Model


N Ibanez, R Batley, ITS, University of Leeds, UK



The Random Utility Model (RUM) was conceived by Marschak (1960) and Block & Marschak (1960) as a probabilistic representation of the Neo-Classical microeconomic theory of individual choice. The probabilistic content of RUM arises from the propensity for an individual, when faced with the repetition of the same choice task, to exhibit variability in his or her preference ordering. Motivated by an interest in its practical applicability, subsequent researchers - in particular the parallel teams of Daly & Zachary (1976), Williams (1977,) and McFadden (1978) - proposed an alternative interpretation of RUM, and equipped the model with a formal set of necessary and sufficient conditions. Within the alternative interpretation, the probabilistic content of RUM derives from the propensity for variability in behaviour across a population of individuals, as distinct from the intra-individual variability of a single individual in the original presentation. The alternative presentation opened the floodgates to practical RUM analysis, the subsequent years witnessing a plethora of applications across a diverse range of economic contexts, including transport research. Notwithstanding this experience gained in application, there persists some misunderstanding among practitioners concerning fundamental aspects of RUM.

The purpose of our paper is to compare and contrast the respective conditions of Daly & Zachary and McFadden for RUM consistency (noting that Williams? approach is similar to the former). McFadden?s conditions - referred to as the Generalised Extreme Value (GEV) model - have acquired particular prominence. These yield a whole family of specific models, permitting considerable flexibility in their properties, whilst ensuring closed form for their probability statements. Popular examples of such models include logit and nested logit. Subsequent to the initial works of the late 1970s, the conditions for RUM consistency have received some attention in the literature. Our analysis relates to three such works, in particular. First, to Börsch-Supan (1990), who sought to demonstrate that the conditions applied commonly to two-level nested logit are excessively restrictive. Second, to Herriges & Kling (1996), who corrected an oversight in Börsch-Supan, and offered proof of the definitive conditions relevant to two-level nested logit. Third, to Gil-Moltó & Risa (2004), who extended Herriges & Kling?s analysis to three-level nested logit.

The particular contributions of our paper are as follows. First, we assist deeper understanding of the conditions leading to RUM consistency, by demonstrating their necessity from first principles, and rationalising their motivation. Second, we illuminate the differences between the Daly & Zachary and GEV presentations; in particular, we demonstrate the greater generality of Daly & Zachary, relative to GEV. Third, we illustrate the previous point by means of application to logit, two-level nested logit, and three-level nested logit. Fourth, we reveal that a restrictive assumption is common to the analyses of Herriges & Kling and Gil-Moltó & Risa; we therefore generalise their presentations.


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Daly, A.J. and Zachary, S. (1976) Improved multiple choice models. In Proceedings of the Fourth PTRC Summer Annual Meeting. PTRC, London.

Gil-Moltó, M.J. & Risa, A.R. (2004) Tests for the consistency of three-level nested logit models with utility maximization. Economics Letters, 85, pp133-137.

Herriges, J.A. & Kling, C.L. (1996) Testing the consistency of nested logit with utility maximization. Economics Letters, 50, pp33-39.

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Marschak, J. (1960) Binary choice constraints and random utility indicators. In Marschak, J. (1974) Economic Information, Decision and Prediction: Selected Essays (Volume 1). D. Reidel, Dordrecht.

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