Braess's Paradox
In 1968, mathematician Dietrich Braess discovered a counter-intuitive result: adding a new road to a traffic network can make everyone's commute worse. This happens because individually rational route choices (Nash equilibrium) can diverge from the socially optimal allocation. Below, a diamond network routes drivers from S to T. Toggle the shortcut to see the paradox in action.
Nash equilibrium cost ≥ Social optimum cost — adding capacity can increase equilibrium cost
Avg Travel Time—
Social Optimum—
Route S-A-T—
Route S-B-T—
Route S-A-B-T—
How it works: The network has four nodes. Edges S→A and B→T have latency proportional
to the number of drivers using them (flow/N). Edges S→B and A→T have fixed latency of 1.
Without the shortcut, drivers split evenly between the two paths and each experiences a cost of
0.5 + 1 = 1.5. When you add a zero-cost shortcut A→B, every driver prefers S→A→B→T
because the flow-dependent edges are always cheaper than the fixed edges when not fully loaded. At Nash
equilibrium, all drivers take the shortcut and each experiences a cost of
1 + 0 + 1 = 2 — worse for everyone.
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