Bounded Rationality in Traffic Assignment

As a relaxation of the otherwise restrictive Wardropian assumption, the notion of bounded rationality was proposed by Simon (1957, 1990, 1991) and introduced to traffic modelling by Mahmassani and Chang (1987). In prose, the common notion of bounded rationality postulates a range of acceptable travel costs that, when achieved, do not incentivize travelers to change their departure times or route choices. Such a range is phrased by Mahmassani and Chang (1987) as 'indifference band'. The width of such band, usually denoted by is either derived through a behavioural study of road users (for example, by surveys) or calibrated from empirical observation through inverse modelling techniques. It could be defined differently for each traveler (e.g. Mahmassani & Chang, 1987), on the traffic flow level, and on the origin-destination pair (e.g. Ge & Zhou, 2012).

Bounded rationality has gradually become a major field of inquiry especially in static traffic assignment, with an incomplete list of papers including Karakostas, Kim, Viglas, and Xia (2011), Zhang (2011), Di, Liu, Pang, and Ban (2013), Di, He, Guo, and Liu (2014) and Zhao and Huang (2014a, 2014b). Some studies further considered boundedly rational user equilibrium in road toll applications (e.g. Lou, Yin, & Lawphongpanich, 2010) and planning and policy applications (e.g. Arslan & Khisty, 2005; Gifford & Checherita, 2007; Marsden, Frick, May, & Deakin, 2012).

Bounded rationality has been examined and used in both within-day and day-to- day DTA frameworks. It was investigated via simulation-based approaches in the venue of within-day dynamic modelling (Gao, Frejinger, & Ben-Akiva, 2011; Hu & Mahmassani, 1997; Jayakrishnan, Mahmassani, & Hu, 1994; Jou, Lam, Liu, & Chen, 2005; Mahmassani & Jayakrishnan, 1991; Mahmassani & Liu, 1999; Mahmassani, Zhou, & Lu, 2005; Nakayama, Kitamura, & Fujii, 2001; Srinivasan & Mahmassani, 1999). It was also used in within-day DTA models (e.g. Ge, Sun, Zhang, Szeto, & Zhou, 2014; Ge & Zhou, 2012; Han et al., 2014; Szeto, 2003; Szeto & Lo, 2006). These models do not consider en-route adjustment. Ridwan (2004) applied the theory of fuzzy systems to the study of bounded rationality with the consideration of en-route adjustment. Fonzone and Bell (2010) proposed an en-route adjustment model for passenger traffic assignment. Others considered boundedly rational user equilibrium in day-to-day dynamic problems (e.g. Di, Liu, Ban, & Yu, 2014; Guo & Liu, 2011; Han & Timmermans, 2006; Wu et al., 2013; Yang & Jayakrishnan, 2013) and their toll applications (e.g. Guo, 2013).

The notion of bounded rationality was used rather imprecisely (in terms mathematics) during the early days of within-day DTA research and defined by various ways. In particular, bounded rationality was studied in a so-called laboratory setting by Mahmassani and Chang (1987), but they did not provide a meaningful mathematical articulation of bounded rationality for within-day DTA. They defined differently for each traveler and named the resultant equilibrium 'boundedly rational user equilibrium'. Bounded rationality was used in a similarly ad hoc fashion for simulations by Mahmassani and Jayakrishnan (1991), Peeta and Mahmassani (1995), Mahmassani and Liu (1999), and Chiu and Mahmassani (2002), in which efforts were again limited by the lack of a complete mathematical model of bounded rationality for within-day DTA. Recognising the lack of a theory of traffic assignment that directly incorporates bounded rationality, Ridwan (2004) applied the theory of fuzzy systems to the study of bounded rationality. Bogers, Viti, and Hoogendoorn (2005), again driven by the lack of a suitable theory, conducted more laboratory studies about bounded rationality on route choice.

For analytical within-day DTA models, Szeto (2003) and Szeto and Lo (2006) proposed a mathematical model for the route-choice boundedly rational dynamic user equilibrium (BR-DUE) traffic assignment problem, although the authors did not take into account drivers' departure time choices. εmax was defined on the traffic flow level, irrespective of OD pairs, and the resultant equilibrium principle was referred to as tolerance-based DUO principle. The route-choice BR-DUE traffic assignment problem was formulated as a discrete-time nonlinear complementarity problem in the study of Szeto and Lo (2006), where a heuristic routeswapping algorithm adapted from Huang and Lam (2002) was proposed to solve the problem. Ge and Zhou (2012) considered the route-choice BR-DUE traffic assignment problem with signals and endogenously determined tolerances by allowing the width of the indifference band εmax to depend on OD pair, departure time and the actual path Hows. Ge et al. (2014) extended the concept of DUO with variable tolerances to more scenarios, including the discontinuity of path travel times or costs, too high demand levels, capacity shortage or sharp change, etc. Therefore, the applications of this concept are not limited to signalised road networks any more. Ge et al. (2014) focused on a comparison of the concept of DUO with variable tolerances to DUE and tolerance-based DUO, including the differences between the three alternative definitions of DUO, their relationships and the existence conditions of these DUO states. However, no solution method was proposed in these two papers. Contributions by Szeto (2003), Szeto and Lo (2006), Ge and Zhou (2012) and Ge et al. (2014) achieved enhanced (yet partial) integration of bounded rationality and DUE but did not establish and analyse a complete theory, where by 'complete' we mean a mathematical formulation consistent with known empirical results and surmised behaviours, qualitative properties and a computational approach that is demonstrably effective.

To the best of our knowledge, there has not been any analytical treatment of the simultaneous route and departure time choice BR-DUE traffic assignment problem with exogenous tolerances or with variable (endogenous) tolerance in the literature, in terms of formulation, qualitative properties and computation until Han et al. (2014) which successfully bridged such a gap in the literature by providing the first complete analytical framework capable of formulating BR-DUE problems into canonical mathematical forms and continuous time, analysing their qualitative properties and computing solutions with convergent algorithms.

As a final remark in this section, although Mahmassani and Chang (1987) defined εтaх at the individual level and Szeto and Lo (2006) defined it at the flow level, their terminologies 'boundedly rational user equilibrium' and 'tolerance-based user equilibrium' were used interchangeably in the traffic assignment literature.

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