Braess's Paradox

A worked example

Suppose 4,000 cars travel from S to T by one of two routes, S→A→T or S→B→T. Links S→A and B→T take (number of cars)/100 minutes, while A→T and S→B take 45 minutes regardless of traffic. If cars split evenly, each route takes 20 + 45 = 65 minutes. Now add a zero-minute shortcut from A to B. For each driver, S→A→B→T is always at least as fast, so everyone switches and travel time rises to 40 + 40 = 80 minutes. Yet this is a Nash equilibrium: no driver can gain by changing routes alone. Braess (1968) published the result in German, and Braess, Nagurney and Wakolbinger (2005) translated it into English.

Theory: equilibrium and the price of anarchy

Wardrop (1952) distinguished user equilibrium, in which all used routes have equal travel times, from system optimum, which minimizes total travel time. They diverge because drivers ignore the congestion costs they impose on others, an externality already discussed for congested roads by Pigou (1920) and Knight (1924). Murchland (1970) introduced the paradox to transportation research. Koutsoupias and Papadimitriou (1999) defined the ratio of equilibrium to optimal cost as the price of anarchy, and Roughgarden and Tardos (2002) proved that with linear delay functions it is at most 4/3. Roughgarden (2005) synthesized this line of research.

Real cases and experiments

Cohen and Horowitz (1991) showed that the same paradox appears in mechanical and electrical networks of springs, strings, resistors and diodes, where adding a connection can increase sag. Youn, Gastner and Jeong (2008) analyzed road networks in Boston, New York and London and showed that closing certain roads could reduce total travel time. The removal of Seoul's Cheonggyecheon elevated highway, after which traffic disruption was smaller than expected, is often cited. Rapoport et al. (2009) reported that participants in lab route-choice experiments do converge to the paradoxical equilibrium. The fundamental law of road congestion reported by Duranton and Turner (2011), that traffic grows in proportion to added road capacity, is often discussed alongside it as evidence of the limits of road expansion.

FAQ

What is Braess's paradox in simple terms?

When a new shortcut opens, everyone piles onto it, and the result can be that everyone takes longer than before the road existed.

Does it happen in real life?

Yes. It has been found in road network analyses and lab experiments, and the same structure appears in mechanical and electrical circuits. Not every new road triggers it, though; it depends on network structure and traffic volume.

Sources reviewed: 2026-10-26

References

1. Braess, D. (1968). Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung, 12(1), 258–268.
2. Braess, D., Nagurney, A., & Wakolbinger, T. (2005). On a paradox of traffic planning (Translation of Braess 1968). Transportation Science, 39(4), 446–450.
3. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(3), 325–362.
4. Pigou, A. C. (1920). The Economics of Welfare. London: Macmillan.
5. Knight, F. H. (1924). Some fallacies in the interpretation of social cost. Quarterly Journal of Economics, 38(4), 582–606.
6. Murchland, J. D. (1970). Braess's paradox of traffic flow. Transportation Research, 4(4), 391–394.
7. Cohen, J. E., & Horowitz, P. (1991). Paradoxical behaviour of mechanical and electrical networks. Nature, 352(6337), 699–701.
8. Roughgarden, T., & Tardos, É. (2002). How bad is selfish routing? Journal of the ACM, 49(2), 236–259.
9. Roughgarden, T. (2005). Selfish Routing and the Price of Anarchy. Cambridge, MA: MIT Press.
10. Koutsoupias, E., & Papadimitriou, C. (1999). Worst-case equilibria. In Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (pp. 404–413). Berlin: Springer.
11. Youn, H., Gastner, M. T., & Jeong, H. (2008). Price of anarchy in transportation networks: Efficiency and optimality control. Physical Review Letters, 101(12), 128701.
12. Duranton, G., & Turner, M. A. (2011). The fundamental law of road congestion: Evidence from US cities. American Economic Review, 101(6), 2616–2652.
13. Rapoport, A., Kugler, T., Dugar, S., & Gisches, E. J. (2009). Choice of routes in congested traffic networks: Experimental tests of the Braess Paradox. Games and Economic Behavior, 65(2), 538–571.