## The Applicability of Prospect Theory to the Analysis of Transport Networks

### Authors

R Batley, R Connors, A Sumalee, ITS, University of Leeds, UK

### Description

We extend conventional RUM models of trip scheduling to accommodate travel time uncertainty - with particular reference to Prospect Theory - before implementing this extension in a transport network model.

### Abstract

As the demands placed on transport systems have increased relative to extensions in supply, problems of network unreliability have become ever more prevalent. The response of some transport users has been to accommodate expectations of unreliability in their decision-making, particularly through their trip scheduling.

In the analysis of trip scheduling, Small?s (1982) approach has received considerable support. Small extends the microeconomic theory of time allocation (e.g. Becker, 1965; De Serpa, 1971), accounting for scheduling constraints in the specification of both utility and its associated constraints. Small makes operational the theory by means of the random utility model (RUM). This involves a process of converting the continuous departure time variable into discrete departure time segments, specifying the utility of each departure time segment as a function of several components (specifically journey time, schedule delay and the penalty of late arrival), and adopting particular distributional assumptions concerning the random error terms of contiguous departure time segments (whilst his 1982 paper assumes IID, Small?s 1987 paper considers a more complex pattern of covariance).

A fundamental limitation of Small?s approach is that individuals make choices under certainty, an assumption that is clearly unrealistic in the context of urban travel choice. The response of microeconomic theory to such challenge is to reformulate the objective problem from the maximisation of utility, to one of maximising expected utility, with particular reference to the works of von Neumann & Morgenstern (1947) and Savage (1954). Bates et al. (2001) apply this extension to departure time choice, but specify choice as being over continuous time; the latter carries the advantage of simplifying some of the calculations of optimal departure time. Moreover Bates et al. offer account of departure time choice under uncertainty, but retain a deterministic representation. Batley & Daly (2004) develop ideas further by reconciling the analyses of Small (1982) and Bates et al. (2001). Drawing on early contributions to the RUM literature by Marschak et al. (1963), Batley and Daly propose a probabilistic model of departure time choice under uncertainty, based on an objective function of random expected utility maximisation.

Developing ideas further, we acknowledge the fundamental role that RUM conventionally plays as a component of transport network models. The concept of a transport network acknowledges that travellers may make choices with reference to the condition of the transport system and, it follows, to their interactions with other users. Furthermore, travellers? experiences of travel conditions may influence their future travel choices. When faced with the task of reconciling these phenomena, network analysts seek to identify the conditions under which the network reaches some sense of ?equilibrium?; a popular such criteria is Wardrop?s equilibrium (1952). This notion of equilibrium may, however, be challenged on several counts; of particular relevance to the current discussion is the proposition that travellers have perfect information. Moreover, it becomes apparent that the challenges to RUM that arise from trip scheduling under uncertainty - as discussed above - also manifest themselves in the operation of transport network models. In response to such challenges, transport network modellers have experimented with RUMs of increasingly generality. Thus, although the conventional formulation has been logit or probit (Sheffi, 1985), more recent research has seen the application of cross-nested logit (Prashker & Bekhor, 1999), error components (Nielsen et al., 2002) and gamma link component distributions (Cantarella & Binetti, 2002).

Despite this progression in the generality and sophistication of methods, significant challenges to the normative validity of RUM and transport network models remain. Of increasing prominence in transport research, is the conjecture that expected utility maximisation may represent an inappropriate objective of choice under uncertainty. Significant evidence for this conjecture exists, and a variety of alternative objectives proposed instead; Kahneman & Tversky (2000) offer a useful compendium of such papers. With regards to these alternatives, Kahneman & Tversky?s (1979) own Prospect Theory commands considerable support as a theoretical panacea for choice under uncertainty. This theory distinguishes between two phases in the choice process - editing and evaluation. Editing may involve several stages, so-called ?coding?, ?combination?, ?cancellation?, ?simplification? and ?rejection of dominated alternatives?. Evaluation involves a value function that is defined on deviations from some reference point, and is characterised by concavity for gains and convexity for losses, with the function being steeper for gains than for losses.

Hence to the contribution of our own paper. First, we reconcile Prospect Theory with the extant theory of trip scheduling; a novel aspect of the investigation is our consideration of the overlap between Prospect Theory and a generalised representation of the random expected utility maximisation model. This draws on earlier work by Batley & Daly (2003) on the equivalence between RUM and elimination-by-aspects (Tversky, 1972); the latter representing one example of a possible ?editing? model within Prospect Theory. We then extend our analysis to consider the interface with transport network models, with the aim of developing a consistent framework for representing the travel time uncertainty arising from uncertainty in demand. To this end, we consider the existence and uniqueness of an equilibrium solution (fixed point condition) to the network problem, given our representation of Prospect Theory. Finally, the paper proposes a mathematical algorithm for computing this equilibrium state.

#### Publisher

Association for European Transport