Braess's Paradox Examples - Single Origin-Destination Pair

Jane Hagstrom
University of Illinois at Chicago

When citations are not given, please attribute these examples to this web site.

Braess's Original Example with Sensitivity to Demand Analysis

Braess's Original Network with Costs Demand from s to t is 6. This is Braess's original example. The system optimal distribution of flows makes some travelers better off while making none worse off. Appears in Über ein Paradox der Verkerhsplannung, Dietrich Braess, Unternehmenstorchung 12:258-268 (1968). Link to Excel file.

System Optimal Being Distinct from Equilibrium Does Not Indicate the Existence of a Braess Paradox

Braess Network with Different Costs Demand from s to t is 3.6. In this example, the system optimal distribution of flows makes some travelers worse off than in equilibrium. A generalized Braess Paradox does occur but requires a different distribution of flows to make some travelers better off while making none worse off. There is a neighborhood of the equilibrium distribution of flows in which no traveler can become better off without some traveler becoming worse off. Appears in Characterizing Braess's Paradox for Traffic Networks, J. N. Hagstrom and R. A. Abrams, Proceedings of IEEE 2001 Conference on Intelligent Transportation Systems 837-842. Link to Excel file.

A Generalized Braess Paradox in a Parallel Network with Strongly Monotone Costs

Parallel Network with Nonseparable Costs
 Demand from s to t is 2. In this example, the system optimal distribution of flows demonstrates the presence of a Braess Paradox. Appears in Two Alternative Definitions of Traffic Equilibrium, M. J. Smith, Transportation Research B, 18B: 63-65, 1984. Link to Excel file.

A Generalized Braess Paradox for Which No Arc Can Be Deleted

Braess Network with Different Costs Demand from s to t is 6. In this example, the system optimal distribution of flows demonstrates the presence of a generalized Braess Paradox. However, if any link is deleted some travelers are worse off than in equilibrium. Appears in Characterizing Braess's Paradox for Traffic Networks, J. N. Hagstrom and R. A. Abrams, Proceedings of IEEE 2001 Conference on Intelligent Transportation Systems 837-842. Link to Excel file.

It Is Possible to Have a Braess Paradox When Costs are Linear and No Braess Paradox When Costs Are Nonlinear

Demand from s to t is 6. In this example, the system optimal distribution of flows makes some travelers worse off than in equilibrium. A Braess Paradox does not occur with the hyperbolic cost functions used here. However, when costs are linearized about the equilibrium, a Braess Paradox occurs. Appears in Improving traffic flows at no cost, R. A. Abrams and J. N. Hagstrom, 2004. Link to Excel file.

A Generalized Braess Paradox Involving Two Cross Arcs

Six Node Network Demand from s to t is 12. In this example, the system optimal distribution of flows does not demonstrate the presence of a Braess Paradox. There is a generalized Braess Paradox illustrated by reducing flow on both the cross-arcs. Alternatively, one can delete one of the cross-arcs, but not both, to demonstrate the presence of a classic Braess Paradox. Link to Excel file.

Multiple Distributions Demonstrate the Existence of a Braess Paradox

Eight Node Network Demand from s to t is 8. This example was constructed to test the conjecture that the solution set for the Braess Optimization Problem is convex when the cost functions are convex and monotone. Based on this example, we conjecture that given two flow distributions which are feasible for the Braess Optimization Problem and have the same system cost, either every convex combination is feasible or we can construct a feasible solution with a lower system cost. The second alternative is true for this example. Link to Excel file.

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Revised, 15 May, 2004, by Jane Hagstrom.