Thought Toys · Games & choices · Exhibit 37

Braess's paradox

4,000 drivers, two routes, a perfectly balanced 65-minute commute. Add one free shortcut between the routes and — with nobody behaving irrationally — the commute gets longer. This is what happens when everyone optimizes selfishly: adding capacity to the network can make the network worse.

A four-node road network: Start on the left, End on the right, and two middle nodes C (top) and D (bottom). Two routes run Start-C-End and Start-D-End; a shortcut edge between C and D can be tuned. Line thickness shows how many of the 4,000 drivers use each road.

Drag it toward zero and watch every commute lengthen.

↳ your turn — drag the shortcut's cost up

What you're seeing

4,000 drivers need to get from Start to End. Two routes exist: Start→C→End and Start→D→End. The Start→C and D→End legs are congestion-sensitive — the more drivers pile on, the slower they get (time = drivers ÷ 100 minutes) — while C→End and Start→D are always exactly 45 minutes, jam or no jam. With no shortcut, drivers split evenly, 2,000 each way, and everyone's commute settles at a stable 65 minutes. Nobody can do better by switching — a Wardrop equilibrium: the traffic version of a Nash equilibrium, where every driver has independently picked their best response to everyone else.

Now open a shortcut straight from C to D. A third route appears — Start→C→D→End — that uses both congestion-sensitive legs and skips both of the fixed 45-minute ones. If the shortcut is cheap, every driver looking at their own commute prefers it — so all 4,000 pile onto it. But now both congestion-sensitive legs carry all 4,000 drivers instead of 2,000, and the shared commute rises to 80 minutes. Every single driver is individually behaving optimally; the network as a whole got worse anyway.

Drag the shortcut's own cost up and the trap loosens: past 5 minutes some drivers give up on it and split back toward the two original routes, and the shared commute actually falls as the shortcut gets pricier. Past 25 minutes, nobody bothers with it at all, and the network recovers its original 65-minute equilibrium exactly. The shortcut never once makes the commute better than 65 — at best it's ignored.

The rule, exactly. At a Wardrop equilibrium, every used route has equal, minimal travel time; the shared commute time T(L) as a function of the shortcut's cost L is T = 80+L for L<5,   90−L for 5≤L≤25,   65 for L>25. Verified in node (improve/verify/37-braess.js): this closed form matches an independent best-response-dynamics simulation started from a lopsided, non-equilibrium split; the equilibrium commute is ≥65 minutes for every tested L from 0 to 30 in steps of 0.5 — the shortcut can only hurt or do nothing, never help. Counter-example: rebuild the same network with all four legs at a fixed cost (no congestion sensitivity) and adding the identical shortcut never makes any driver's trip longer — ordinary shortest-path relaxation only ever helps. The paradox needs congestion-dependent cost, not merely "one more road."

Also in Games & choices: The secretary problem →

All 14 in Games & choices
  1. 10The evolution of trust
  2. 100The measure went up. The thing barely moved.
  3. 106They meet in the middle. The beach walks twice as far.
  4. 109Twice the soldiers is four times the army
  5. 116He told them nothing they did not already know
  6. 127The prize you only win when you have overpaid
  7. 129Everyone waits for somebody else
  8. 133A hundred prisoners and one sentence of agreement
  9. 27Nash equilibria
  10. 37Braess's paradox — you are here
  11. 47The secretary problem
  12. 67Better at both, and still better off trading
  13. 69Everyone was consistent. The vote wasn't.
  14. 97Same votes. Different winner.

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