Braess's paradox

Setup

Traffic runs from a source to a destination across a small network. Some links are congestible: travel time on them rises with the number of vehicles. Others take a fixed time whatever the load. Every driver picks the route that minimises their own travel time and keeps switching until no unilateral switch helps.

That state is Wardrop’s first principle: at equilibrium every route in use costs the same, and no unused route costs less. It is the standard equilibrium concept in transport modelling, and it is entirely reasonable behaviour. The trouble starts when you add a link.

The model

The paradox is not that the new route is bad. It is that taking it is a best response for every individual driver, so the equilibrium cannot avoid it, and the resulting equilibrium is worse than the one available without the link.

Results

Koutsoupias and Papadimitriou (1999) formalised how bad this can get as the ratio between the equilibrium outcome and the socially optimal one, later named the price of anarchy. Roughgarden and Tardos (2002) proved a clean bound for the common case: when every link’s latency is an affine function of flow, selfish routing costs at most 4/3 of the optimum, no matter how large or tangled the network. They also proved a result that is more useful to a planner, that the equilibrium cost of selfish routing is no worse than the optimal cost of routing twice the traffic.

The bound is reassuring and the mechanism is not. A ceiling of one third on the damage still means that adding capacity can be worse than doing nothing, and there is no local signal that tells you when. From inside any single trip the paradox is invisible: every driver is on the fastest route available to them.

Evidence

Youn, Gastner and Jeong (2008) took the theory to measured road networks. Working with single origin-to-destination routes inside real cities, Harvard Square to Boston Common, Borough to Farringdon in London, Washington Market Park to the Queens Midtown Tunnel in New York, they computed the price of anarchy across a range of traffic volumes and identified links whose closure would reduce equilibrium travel time for everyone.

The price of anarchy was not monotonic in load. It showed three regimes: a value of one while traffic is light and the network has slack, a band of fluctuation at intermediate volumes, and a decay back toward one once traffic is heavy enough that there is no slack left to misallocate. Boston’s peaked near 1.30, meaning drivers gave up close to 30 per cent of their travel time to the lack of coordination, with London and New York peaking somewhat lower. Note that these are route-level results inside those cities rather than a figure for the network as a whole.

That finding is what turns the paradox from a constructed counterexample into a property of networks people drive on. Braess’s original 1968 paper was a planning result published in German in an operations research journal, and it went largely unnoticed in English-language transport engineering until the 2005 translation.

Widely cited cases include the removal of a section of 42nd Street in New York and the demolition of the Cheonggyecheon elevated freeway in Seoul, both followed by reported improvements in flow. These are observational, confounded by simultaneous changes in demand and land use, and are best read as consistent with the model rather than as tests of it.

Limitations

Wardrop equilibrium assumes drivers have full information, identical preferences and no scheduling flexibility, and that latency depends only on the flow on that link. Real networks have signals, junction interactions, heterogeneous vehicles and drivers who choose departure times as well as routes. Adding any of these changes the numbers, though not usually the direction.

The deeper caveat is that demand is not fixed. Induced demand means a new road attracts trips that were not being made at all, which is a separate and often larger effect than the routing paradox. Braess tells you that a link can hurt at constant demand. It does not tell you what happens when the link changes how many people drive, and in most real projects that second effect dominates.

Finally, identifying a Braess link in a real network requires knowing the latency functions, which are estimated rather than measured. The Youn et al. results are as good as their calibration.

References

1. Braess, D. (1968). Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung, 12(1), 258–268. https://doi.org/10.1007/BF01918335
2. Braess, D., Nagurney, A., & Wakolbinger, T. (2005). On a paradox of traffic planning. Transportation Science, 39(4), 446–450. https://doi.org/10.1287/trsc.1050.0127
3. Koutsoupias, E., & Papadimitriou, C. (1999). Worst-case equilibria. In C. Meinel & S. Tison (Eds.), STACS 99 (Lecture Notes in Computer Science, Vol. 1563, pp. 404–413). Springer. https://doi.org/10.1007/3-540-49116-3_38
4. Roughgarden, T., & Tardos, É. (2002). How bad is selfish routing? Journal of the ACM, 49(2), 236–259. https://doi.org/10.1145/506147.506153
5. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(3), 325–362. https://doi.org/10.1680/ipeds.1952.11259
6. Youn, H., Gastner, M. T., & Jeong, H. (2008). Price of anarchy in transportation networks: Efficiency and optimality control. Physical Review Letters, 101(12), 128701. https://doi.org/10.1103/PhysRevLett.101.128701
7. Youn, H., Gastner, M. T., & Jeong, H. (2009). Erratum: Price of anarchy in transportation networks: Efficiency and optimality control [Phys. Rev. Lett. 101, 128701 (2008)]. Physical Review Letters, 102(4), 049905. https://doi.org/10.1103/PhysRevLett.102.049905

Verification Every reference on this page has been checked against Crossref. The claims this page makes about what those sources report have also been checked against the sources, and any corrections that check produced are recorded in the project’s claims log.