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Braess paradox: Maximum penalty in a minimal critical network


  • Penchina, Claude M.


A 'simplest anti-symmetric' two-path network is described which exhibits the well-known Braess paradox: the user travel costs being higher after the paths are joined by a transversal link (bridge). This network, herein named a 'Minimal Critical Network', clearly demonstrates the essence of the paradox with the minimum number of independent parameters, a minimum of mathematical complexity and a maximum Braess penalty. Although the Braess paradox has been studied extensively in the past, this 'simplest' network has been overlooked. The critical ranges of flow, and of user travel cost, all agree with the theorems of Frank, thus extending the validity of those theorems to a wider range of networks. Only one result, showing the effects of bridge congestion, contrasts with Frank's conclusion. Examples are given of techniques, some old and some new, which modify or eliminate this paradoxical behavior. A discussion of the good effects (non-paradoxical) of a bridge (especially a two-way bridge) is also included for the first time. Our Minimal Critical Network and graphical solution technique give a clear understanding of the paradox for this network. They are also especially useful for analysis of sensitivity to such extensions as, e.g. changes in parameters, elastic demand, general non-linear (even non-continuous) cost functions, two-way bridges, tolls and other methods to control the paradox, and diverse populations of users. We show that the paradox occurs in a simpler network than previously noted, and with a larger Braess penalty than previously noted.

Suggested Citation

  • Penchina, Claude M., 1997. "Braess paradox: Maximum penalty in a minimal critical network," Transportation Research Part A: Policy and Practice, Elsevier, vol. 31(5), pages 379-388, September.
  • Handle: RePEc:eee:transa:v:31:y:1997:i:5:p:379-388

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    References listed on IDEAS

    1. Dafermos, Stella & Nagurney, Anna, 1984. "On some traffic equilibrium theory paradoxes," Transportation Research Part B: Methodological, Elsevier, vol. 18(2), pages 101-110, April.
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    Cited by:

    1. Sohn, Keemin, 2011. "Multi-objective optimization of a road diet network design," Transportation Research Part A: Policy and Practice, Elsevier, vol. 45(6), pages 499-511, July.
    2. Yang, Hai & Bell, Michael G. H., 1998. "A capacity paradox in network design and how to avoid it," Transportation Research Part A: Policy and Practice, Elsevier, vol. 32(7), pages 539-545, September.
    3. Morgan, John & Orzen, Henrik & Sefton, Martin, 2009. "Network architecture and traffic flows: Experiments on the Pigou-Knight-Downs and Braess Paradoxes," Games and Economic Behavior, Elsevier, vol. 66(1), pages 348-372, May.
    4. Rapoport, Amnon & Kugler, Tamar & Dugar, Subhasish & Gisches, Eyran J., 2009. "Choice of routes in congested traffic networks: Experimental tests of the Braess Paradox," Games and Economic Behavior, Elsevier, vol. 65(2), pages 538-571, March.
    5. Rapoport, Amnon & Mak, Vincent & Zwick, Rami, 2006. "Navigating congested networks with variable demand: Experimental evidence," Journal of Economic Psychology, Elsevier, vol. 27(5), pages 648-666, October.

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