Congestion derivatives for a traffic bottleneck with heterogeneous commuters Tao Yao a,1 , Mike Mingcheng Wei b,2 , Bo Zhang c,3 , Terry Friesz d,⇑ a Department of Industrial & Manufacturing Engineering, Pennsylvania State University, 349 Leonhard Building, University Park, PA 16802, United States b Olin Business School, Washington University in St. Louis, One Brookings Drive, Campus Box 1133, St. Louis, MO 63130, United States c Department of Industrial & Manufacturing Engineering, Pennsylvania State University, 240 Leonhard Building, University Park, PA 16802, United States d Department of Industrial & Manufacturing Engineering, Pennsylvania State University, 305 Leonhard Building, University Park, PA 16802, United States article info Article history: Received 23 February 2012 Received in revised form 17 July 2012 Accepted 18 July 2012 Keywords: Congestion derivative Bottleneck model Heterogeneous commuters Market-based mechanism abstract Deterministic congestion pricing has attracted most attentions in the literature. But little attention has been given to pricing under uncertainty, especially for heterogeneous com- muters. In this paper, we investigate congestion externalities by considering commuters’ risk preferences and heterogeneity. In particular, when price involves exogenous uncer- tainty which is independent of both central authority and individual commuters, we are able to express commuters’ departure equilibria and the total social cost in closed-form in terms of the departure time and uncertainty. Moreover, we find that uncertainty will lead heterogeneous risk-averse commuters not only to avoid traveling at the time when uncertainty level is high, but also to deviate from their optimal departure sequence. Hence, we are able to show that uncertainty can tremendously increase the total social cost. Fur- thermore, we also prove that both the central planner and the market-base mechanism have the potential to reduce the total social cost and alter commuters’ departure behavior. Specifically, we find out that the central planner can always find a class of financial deriv- atives to induce the socially optimal departure behavior, while the market-based mecha- nism may do so at specific cases. Finally, numerical formulation and experiments are given to assess the robustness of our results for more general forms of uncertainties and derivatives. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Congestion pricing is widely accepted as an effective method to reduce congestion, and has the potential to reduce the total social cost (Walters, 1961). However, with exception of a few (e.g. Yao et al., 2010, etc., please check the Literature sec- tion for details), most congestion pricing schemes and results are derived under two strict, sometimes unrealistic, assump- tions: there is no underlying uncertainty and commuters are risk neutral. Hence, whether those dynamic tolls derived from deterministic case are still optimal, or at least suboptimal, in terms of mitigating the total social cost is questionable. More importantly, congestion pricing is not a market-based mechanism: a mechanism facilitates a negotiation process between sellers and buyers to determine prices of services according to supply–demand relationship. As the congestion pricing is 0191-2615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trb.2012.07.003 ⇑ Corresponding author. Tel.: +1 814 863 2445; fax: +1 814 863 4745. E-mail addresses: taoyao@psu.edu (T. Yao), weim@wustl.edu (M.M. Wei), bzz104@psu.edu (B. Zhang), tfriesz@psu.edu (T. Friesz). 1 Tel.: +1 814 865 8040; fax: +1 814 863 4745. 2 Tel.: +1 314 632 6606; fax: +1 314 935 8840. 3 Tel.: +1 814 865 9984; fax: +1 814 863 4745. Transportation Research Part B 46 (2012) 1454–1473 Contents lists available at SciVerse ScienceDirect Transportation Research Part B journal homepage: www.elsevier.com/locate/trb