## From Random Utility to Random Expected Utility: Theory and Application to Departure Time Choice

### Authors

J Polak and X Liu, Centre for Transport Studies, Imperial College London, UK

### Description

We propose a new approach to modelling choice under uncertainty combining key features of classical expected utility theory and classical random utility theory. The implications of this approach are discussed and empirical applications presented.

### Abstract

In recent years, the topic of travel decision making under risk and uncertainty has received increasing attention in the literature. The most widely used modelling approach is to assume that travellers? decision making, including their attitudes towards risk, can be characterised by Expected Utility Theory. Although EUT provides a framework in which to consider the effect of decision makers? uncertainty about the outcomes associated with their decisions, at least in its classical form, it does not explicitly accommodate modellers? uncertainty regarding decision makers? tastes and behaviour, which is classically the domain of Random Utility Theory. In practice, therefore, the existing transport literature has combined the EUT and RUT perspectives in a variety of entirely plausible but nevertheless essentially ad hoc manners to draw conclusions regarding attitudes towards risk, principally, but not exclusively, those associated with variability in travel time.

However, such ad hoc combination raises a number of important theoretical issues, many of which have so far not been satisfactorily resolved in the transport literature. This paper draws attention to these theoretical issues and argues that the incomplete integration of EUT and RUT has led to a number of unsatisfactory and potentially misleading interpretations in the literature affecting both the characterisation and valuation of attitudes towards risk. To address these shortcomings, we set out a new theoretical perspective, which we refer to as random expected utility theory (REUT) that achieves a more comprehensive integration between these two theoretical traditions, explore the implications of this framework and present a number of empirical applications that illustrate the benefits of this new approach.

Our theoretical perspective can be summarised briefly as follows. Classical expected utility theory (EUT) is concerned with situations in which the decision maker chooses between a number of risky prospects, in which each prospect consists of a set of possible outcomes, with each outcome being defined by the values of a vector of attributes and an associated probability of occurrence. The essence of the risky choice problem is that the choice task involves integrating information about the attributes characterising outcomes with information about the probability of each outcome. This integration is achieved by assigning each possible outcome a value (based on its attributes) and computing an evaluation measure for each prospect that is the expected value of a suitable non-linear transformation (termed the von Neumann and Morgenstern utility) of the outcome values. Attitudes to risk are characterised by the curvature of the V-M utility function and decision makers are assumed to choose the prospect that maximises the V-M utility. Random utility theory (RUM), on the other hand, is concerned with the choice between specific outcomes under conditions of risk free choice but taking account of the analyst?s uncertainty regarding the decision maker?s valuation of a particular outcome. This leads to the decomposition of outcome value into an observable component and a random component where the observable component is assumed to be a function of a vector of tastes acting on the attributes characterising individual outcomes. Our proposed fusion of EUT and RUM involves a RUM-style decomposition of outcome value into observable and unobservable components, coupled with an EUT-style non-linear composition rule.

The paper is divided into a number of sections. In the first section of the paper we briefly review existing approaches in the transport and related literature to the study of risky choice. In the second section we outline the proposed REUT approach and explore in detail the implications of this perspective. We identify a number of key issues, both for the interpretation of treatments of risky choice in the existing literature and the future development of theory in this area. Amongst the issues we consider are:

? The quite distinct nature of the concept of ?utility? expressed in RUT and EUT. In RUT, utility is a measure of individual value or benefit, akin to the welfare economic interpretation of the term. However, the (von Neumann and Morgenstern) concept of utility is quite different and represents no more and no less than a convenient characterisation of attitudes to risk.

? The treatment and implications of uncertainty affecting some but not all of the attributes of outcomes. This is relevant, for example, in circumstances where the attribute set may comprise a cost attribute which is known with certainty and a travel time attribute that is uncertain.

? The implications of non-linearities in the observable components of outcome value with respect to uncertain attributes. We argue that in this these circumstances (which are common for example, in the modelling of departure time choice where typical schedule delay formulations induce non-linearities in outcome value) there is the possibility of a confounding of these taste effects with non-linearities in the EUT composition rule, which represent attitudes towards risk.

? The implications of REUT for the valuation of changes in the uncertainty of travel attributes (e.g., changes in travel time variability).

In the third section presents our empirical work, in which we use data from two recent large-scale stated preference exercises, which examined risky decision making in the context of departure time choice. The first dataset focused on choice between risky rail travel while the second considered choice between risky car and rail alternatives. Preliminary results from the first dataset indicate that a simple implementation of REUT results in significantly improved model fit, compared to existing treatments in the literature and also shows evidence of significant inter-personal heterogeneity in attitudes towards risk. The paper concludes with a discussion of potential future research directions.

#### Publisher

Association for European Transport