When each traveler chooses the cheapest available route, the result may cost society more than a coordinated assignment would. The price of anarchy measures how large that gap can become under a stated class of congestion rules.
An ordinary network comparison establishes user equilibrium and system optimum before the question shifts to the worst possible ratio across networks. A geometric argument for linear congestion functions shows why a general bound can be proved without inspecting every possible road map. The last act compares that bound with the bounds associated with steeper nonlinear congestion functions. The key insight is that selfish routing has a measurable efficiency cost, and its possible size depends strongly on how travel time grows with traffic. A bound is a guarantee across a model class, not a prediction that every real network will suffer the worst case.
Many commuters want to arrive at a similar time, but a bottleneck can admit only a limited number of vehicles per minute. If everyone departs near the preferred time, a long queue forms. If departures are spread out, the queue shrinks, but some people arrive early or late instead.
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Building an extra road sounds like an unambiguous improvement. Braess's paradox shows why that intuition can fail when drivers independently choose the route that seems fastest to them. A new link changes everyone's incentives, and the resulting route changes can overload roads that previously worked well.
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Drivers consider the time they personally spend on a road, but each additional driver may slow down many others. That difference between private and collective cost can make the route pattern chosen by individuals less efficient than the pattern that minimizes total travel time.
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