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.
An untolled equilibrium establishes the baseline before the same network is assigned a system-optimal split. A charge on the congestible link is chosen so that travelers, responding to generalized cost, select the system-optimal pattern on their own. The fee is also translated into travel-time terms using an assumed value of time. In the final act, the toll revenue is returned to travelers, illustrating that the charge can steer traffic without leaving a net monetary payment for the group. The central insight is that pricing changes route incentives; the revenue is a transfer that can be redistributed. The illustrative rebate comparison does not imply every individual automatically gains in every real network.
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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Route choice is learned through experience. If one road becomes slower than another, some travelers switch the next day, changing congestion on both roads. Repeating this adjustment can eventually bring the travel times of used routes close together.
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