A portal · the mechanism seam · a combine across three layers
The Price of Everyone Being Right
Open a faster road and every drive gets slower. Add one more herder and the shared field dies. Follow the car ahead sensibly and a traffic jam crystallizes out of nothing. Three of this archive's layers, told apart, are one shape told three times — and the shape has a name.
In each of them, every agent does the locally right thing. The commuter takes the faster route. The herder adds the profitable animal. The driver brakes when the car ahead brakes. Nobody is wrong, nobody is greedy, nobody is confused — and the crowd still settles somewhere worse than it could have reached together, at a place no one can leave alone. Economists call the gap between what selfish choice reaches and what coordination could reach the price of anarchy. Below, one switch adds a single rule to all three worlds at once. Watch what it costs to be free, and what it costs to be right.
1 · The road that helps by closing
Braess's paradox — 4,000 drivers, two routes, one tempting shortcut. full instrument →
The free shortcut (A→B) is open, so every driver rationally uses it — and every driver pays 80. Deviating costs 85, so no one moves. It is a stable trap.
2 · The field that dies by sharing
The tragedy of the commons — 8 herders on one pasture. full instrument →
Each herder adds animals until his own next one barely pays — but he ignores the crowding he lays on everyone else. The more hands, the less the field is worth to anyone.
3 · The jam that comes from nothing
The phantom traffic jam — 22 identical cars, no bottleneck, one small nudge. full instrument →
Flip the switch. In the first world it closes the shortcut; in the second it caps the herd; in the third it asks every driver to keep a little more space. Three different rules — and yet the same kind of move: each one takes a free choice away, and every one makes the whole better off. The cure is never "try harder." It is always a constraint.
One shape, two languages
It would be a tidy lie to call these "the same theorem." They are not. Two of them are equilibria — stable resting points of a game, where each player is already doing their best given everyone else, so no one has any reason to move. The third has no resting point at all.
Equilibrium · a stable trap
Braess and the commons settle. The 80-minute commute and the dying pasture are Nash equilibria: change your own choice and you personally lose, so the bad outcome holds itself in place. Here "local optimization" means strategic best-response, and the price of anarchy is a number you can measure.
Instability · harm from nothing
The phantom jam never rests. There is no scarce field, no added road, no strategy — just identical drivers reacting to the car ahead. Above a density, that reasonable reaction amplifies instead of settling. It is the reductio: local reasonableness, with absolutely nothing to fight over, still manufactures collective harm.
So the honest unifying claim is not about equilibrium at all. It is this: wherever agents follow only their own local gradient, the aggregate can drift to a place worse than an achievable coordinated one, and it will not correct itself — because each agent, checked individually, is behaving perfectly. That is true of a game with a fixed point and true of a flow with none. The line between them is real, and it is drawn here rather than smoothed over.
Some anarchies are cheap. Some are bottomless.
When there is an equilibrium, the price of anarchy is a ratio: the selfish cost divided by the coordinated cost. And here the two equilibria diverge sharply — a fact none of the three pages states, because it only appears when you set them side by side.
The price of anarchy, as the crowd grows
For a road network with straight-line congestion costs, selfish routing is never more than 4/3 as bad as optimal — a hard ceiling, proven by Roughgarden and Tardos in 2002, and hit exactly by the smallest Braess network. The commons has no such mercy: its price of anarchy is (n+1)²/4n, which climbs without limit — at ten herders the field is worth a third of its potential, at a hundred barely a twenty-fifth. The same sentence — "everyone rational, everyone worse off" — describes a bounded inefficiency in one world and a ruin with no floor in the other. Knowing which kind you are standing in is the whole game.
The check
Every number on this page is recomputed offline in
research/price-of-anarchy/verify.mjs — 33/33 —
which imports the exact engines the three source pages were verified with
(braess/engine.mjs, a Frank–Wolfe routing solver; phantom-traffic-jam/idm.mjs,
the Intelligent Driver Model). It pins Braess's 80→65 and its 256/207 ≈ 1.237 price of
anarchy under the proven 4/3 affine ceiling; the commons' 4n/(n+1)² efficiency and its
unbounded (n+1)²/4n price; and — for the jam, which is an instability with no price
of anarchy — that a 20-car ring decays a nudge while a 45-car ring blows up, that the jam
wave travels backward, and that a larger safe headway (T: 1.5 → 2.8 s) restabilizes the
identical road. The animation above runs the same model in your browser.
The three layers this walks
- The Road That Made Everyone Late Braess's paradox, fully operable: sweep the drivers and the highway cost and watch the same road help, hurt, or wash. Source of the 80, the 65, and the 4/3 bound.
- The Tragedy of the Commons The pasture as an exactly-solved Cournot game, with a stint toggle that restores the optimum — and Ostrom's Nobel counter-story that a commons need not end in tragedy.
- The Jam That Isn't There A stop-and-go wave born on a road with no bottleneck at all, from the instability of dense car-following — reproduced with 22 real drivers by Sugiyama in 2008.