Valuing the Reliability of Arrival Time by Means of the Risk Premia

Valuing the Reliability of Arrival Time by Means of the Risk Premia


Richard Batley, ITS, University of Leeds, UK


This paper exploits the concept of the risk premium in comparing marginal valuations of travel time and scheduling under certainty with those under uncertainty. Where these valuations diverge, marginal valuations of reliability can be inferred.


Unreliability is endemic in many transport systems, and this stimulates interest in whether and how unreliability impacts upon the choices of travellers. The paper pursues specific interest in the effect of unreliability in arrival time on scheduling choice. Following the precedent of Noland & Small (1995) and Bates et al. (2001), this is developed through the marriage of Small?s (1982) scheduling function with von Neumann & Morgenstern?s (1947) theory of individual choice under uncertainty. Arising from the latter union is the proposition that unreliability imposes disutility on the traveller. Hence the potential for benefit should unreliability be reduced or indeed eliminated.

In contrast to the precedent works on reliability, the paper adopts a discrete representation of time, motivated in particular by a desire to promote implementation within the apparatus of the Random Utility Model and Stated Preference. The substantive contribution of the paper, however, is the scope of the theoretical exposition, which offers significant extensions beyond the extant literature. Drawing analogy with the theoretical literature on attitudes to risk (e.g. Pratt, 1964; Arrow, 1970), the paper considers the implications of Small?s scheduling function for travellers? attitudes to unreliability in arrival time, and in particular identifies circumstances under which travellers would be risk averse. In response to the latter observation, and drawing analogy with a proposition of Pratt?s (1964), the paper introduces the notion of the ?reliability premium?. This measures, for a given departure time, the delay in arrival time that the individual would be willing-to-pay in exchange for eliminating unreliability in arrival time.

The paper then reconciles this consideration of attitudes to risk with Bates et al.?s (2001) marginal valuations of travel time and scheduling under uncertainty. The latter arise from the proposition that an individual traveller would, in choosing between prospects, be willing to exchange travel cost for expected travel time, expected schedule delay early, expected schedule delay late, and the probability of late arrival. Exploiting the reliability premium, the paper establishes a basis for comparison between marginal valuations of travel time and scheduling under uncertainty, and analogous valuations derived under certainty. This comparison reveals the theoretical possibility that marginal valuations of travel time and scheduling might show divergence between conditions of certainty and uncertainty. Should this theoretical discrepancy manifest in empirical discrepancy - which remains to be seen - then it carries important interpretation as the marginal valuation of reliability.

Whilst marginal valuations of travel time and scheduling under uncertainty are adequate for demand forecasting, economic appraisal should mitigate the projected benefits of a scheme against the costs of risk bearing, and this is where the reliability premium becomes pertinent. It is crucial to acknowledge that reliability benefits arise only under the particular circumstances of risk aversion, and that risk aversion is, in the terms of the scheduling function, dictated by the relation of the possible arrival times to the preferred arrival time. Moreover, the prevalence of benefit will likely vary by departure time for given preferred arrival time. This assumes a single individual however; once the analysis is extended to a sample of individuals, the outcome will be complicated by heterogeneity in the preferred arrival time and, it follows, heterogeneity in the prevalence and magnitude of reliability benefits. Thus for any particular departure time, an improvement in the reliability of arrival time may yield benefit for some travellers but no benefit for others.


Association for European Transport