Flexibility of Tolls for Optimal Flows in Networks with Fixed and Elastic Demands

Claude M. Penchina*
Department of Physics, University of Massachusetts at Amherst, Amherst, MA 01003 USA.

Additional Affiliations: Department of Physics, King's College, Strand London WC2R-2LS, UK, ECE Department of UCSD, La Jolla CA 92093 USA, and Gilora Associates, Flemington, NJ, USA

Article Metrics

CrossRef Citations:
Total Statistics:

Full-Text HTML Views: 330
Abstract HTML Views: 751
PDF Downloads: 258
Total Views/Downloads: 1339
Unique Statistics:

Full-Text HTML Views: 232
Abstract HTML Views: 527
PDF Downloads: 158
Total Views/Downloads: 917

Creative Commons License
© 2009 Claude M. Penchina;

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Department of Physics, University of Massachusetts at Amherst, Amherst, MA 01003 USA; E-mails: Penchina@Physics.UMass.Edu,


Tolls are used to influence users to "voluntarily" choose paths in the system which optimize flows (VSO) to minimize social costs. When the toll solutions are not unique, administrators gain the flexibility to vary tolls in order to meet other goals while still maintaining optimal flows. For Fixed-Demand networks, it was known that path-tolls can be adjusted. However, the link-toll solution was thought to be unique if the number of paths exceeds the number of links. We show, however, that link-toll solutions are non-unique even for many very large networks in which the number of paths greatly exceeds the number of links. Uniqueness of tolls under Elastic Demand is studied here for the first time in the literature. We examine the uniqueness (or lack thereof) of tolls needed to optimize flows with Elastic User Demand. We find that elastic demand eliminates the adjustability of path-toll solutions, and partially restricts the newly-found flexibility of link-tolls. These results should provide guidance for traffic network administrators in planning second-best toll policies for situations where marginal cost pricing may be politically or otherwise unpopular.

Keywords: Braess's paradox, elastic demand, fixed demand, link-tolls, path-tolls, uniqueness of tolls, wheatstone bridge.