Randomness in outcomes, utilities and tastes; an application to travel time risk

The Random Utility Model (RUM), as originally proposed by Block and Marschak (1960), specifies a probabilistic relation between choice and utility, within a paradigm of individual choice under certainty. According to this specification, utility is comprised of deterministic and random elements, with the latter reflecting variability in preferences over certain outcomes within and across individuals. Our paper exploits Marschak et al.’s (1963) extension to RUM, which admits the possibility of variability in the outcomes themselves, and specifies a probabilistic relation between choice and the expectation of utility across these outcomes. Emanating from Marschak et al.’s analysis is the notion of a Random Expected Utility Model (REUM), which might be seen as an analogue to RUM within a paradigm of individual choice under risk or uncertainty. Although previous researchers have implemented various formulations of REUM, these have seemingly been rather ad hoc, with little acknowledgement of the distinction between RUM and REUM, and the implications of this distinction for model specification. A number of recent works (e.g. Batley & Daly, 2004; Michea & Polak, 2006; Polak, Hess & Liu, 2007; Liu & Polak, 2008) have been more instructive in this regard, and our paper begins by distilling these works and presenting a typology of methods. We build upon previous such typologies by reconciling specific practical model specifications (e.g. logit, probit, mixed logit) with different dimensions of randomness; in marginal utility (i.e. the notion of a ‘taste distribution’), in outcomes (i.e. the notion of ‘expected utility’), and in preferences (i.e. the notion of ‘random utility’). The second stage of our paper distinguishes between different ‘currencies’ of risk. Whereas the vast literature on economic decision-making under risk or uncertainty is devoted almost entirely to variability in monetary outcomes, of more immediate concern to transport users is variability in travel time. The paper discusses various implementations of REUM in the context of travel time variability, including the so-called ‘mean-variance’, ‘scheduling’ and ‘mean lateness’ approaches. Drawing analogy with the economic literature on attitudes to risk (e.g. Pratt, 1964), we explicate the properties of the utility functions supporting these implementations. In particular, we observe that some implementations imply rather restrictive properties concerning travellers’ attitudes towards travel time risk. Moreover, we identify substantive discrepancies between the methods adopted as ‘standard’ on different transport modes. Exploiting data from a recent Stated Preference study of rail reliability, we implement various REUM models, experimenting with different specifications of the underlying utility function (e.g. mean-variance) and different dimensions of risk (e.g. in terms of total travel time, in-vehicle time, or lateness). We report the ‘risk premia’ emanating from these models, thereby isolating the economic cost of unreliability. Developing the analysis further, we show that Marschak et al. ’s notion of REUM implies somewhat restrictive properties on error specifications, such that only a subset of model specifications can be entirely REUM-compliant. Our most general model involves a mixed logit specification, and offers the means by which we can parameterise each of the aforementioned dimensions of randomness; across outcomes, utilities and tastes. An important outcome from this model is insight into the distribution of attitudes to travel time risk across our population.