Sixty-five minutes, and eighty once the shortcut opened

Two ways round, and a road between them

The two long stretches — S→A and B→T — are the narrow ones: they take one minute for every hundred cars on them, so their cost depends on who else chose them. The other two, A→T and S→B, are wide motorways that take 45 minutes whoever is on them. With no road between A and B the two routes are mirror images, the traffic splits evenly by symmetry, and the crowded halves carry two thousand cars each.

Open A→B and there is a third route, S→A→B→T, which is made entirely of the two crowd-dependent stretches and skips both motorways. Take the shortcut and you never sit in a 45-minute stretch — you sit in two stretches whose length is decided by how many other people had the same thought.

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Nobody is being stupid

The eighty-minute outcome is not a traffic jam and it is not a mistake. It is the only arrangement no single driver can improve on alone, which is the whole difficulty: to fix it, four thousand people have to change their minds at once, and each of them is worse off for going first.

Read the last two rows first. If all four thousand agreed to ignore the new road and keep to the old split, they would each take 65 minutes — and any one of them who broke the agreement would take 40 minutes and a fraction. The agreement is worth twenty-five minutes to break and nothing to keep. Read the top rows second: once everybody has broken it, leaving the shortcut costs you five minutes, so nobody does.

The stable outcome is the bad one, and the good one is not an outcome at all. Closing the road makes it one, which is the only reason a page about a road that helps nobody is worth drawing: the fix is not persuasion, and there is nothing to enforce. There is a bollard.

The harm has a shape, and it ends

The road is not a bad road. Below three thousand drivers it is a good one: at a thousand drivers the shortcut takes them from fifty minutes to twenty, because the crowd-dependent stretches are nearly empty and skipping two motorways is simply faster. The harm begins where the crowd on it has grown until its two stretches together — 2N/100, since a driver taking it sits in both — come to more than the even split would have cost, motorway and all. At three thousand drivers those are 60 minutes and 60 minutes.

And it ends. Past nine thousand drivers the shortcut is so busy that using it is no better than a motorway, the traffic drifts back to the old split, and the extra road quietly stops mattering. Everything interesting happens in one window, and the worst point in it is exactly 4,500 drivers, where the road costs each of them 22.5 minutes — a third of the journey they would have had.

The curve is drawn from the arithmetic, not sampled: the equilibrium at every driver count is the place where no route in use costs more than any route not in use, and for this network that condition has a closed form. The page does not take it on trust — one of the checks at the bottom finds the same flows a second way, by minimising the quantity whose least value a selfish flow always sits at, and they agree at every driver count it tries.

Four thirds

Now the general fact, which is older than the paradox and larger than it. Take one road that takes an hour whoever is on it and one road that takes an hour if everybody uses it and proportionately less if fewer do. Send an hour's traffic from one end to the other. Every driver reasons that the second road is never slower than the first — and they are right, each of them, individually — so everybody takes it and it takes an hour. Had half of them taken the slow road, the average journey would have been three quarters of an hour.

Four thirds is not an accident of that example. For any road network at all, however tangled, in which each road's cost is a straight line in the traffic on it, drivers choosing for themselves never take more than 4/3 of the time an all-seeing planner could have arranged — and the little two-road network above is what makes the bound exact rather than generous. Selfishness in traffic has a price, and the price has a number.

The network at the top of this page reaches the same number, from the other direction. At 4,500 drivers — its worst point, the one already marked on the curve above — the equilibrium is 90 minutes and the best a planner could have done is 67.5. That is four thirds exactly, in a network that looks nothing like Pigou’s, arrived at by a different quantity. The two are the same theorem twice.

And the planner is not doing anything clever. Put one more car on a stretch that takes a minute per hundred cars and it costs that car its own journey — and it also costs everybody else already on that stretch a hundredth of a minute each, which comes to exactly as much again. A crowded road costs twice what the driver on it pays. A planner reckoning in the true figure would send traffic exactly where drivers would send themselves if the crowded stretches charged double, and one of the checks below confirms that at every driver count from 1 to 12,000. That factor of two is where 4,500 comes from: it is 9,000, halved.

They have actually done this

This page cannot check the rows below. They are history, and the honest claim about them is that the arithmetic above says such a thing is possible, not that it is what happened in any of the three. Traffic is a system with a hundred other things in it — new routes, changed habits, a different year.

The Stuttgart case is the one Braess's own circle cited, and it is the cleanest shape: a street was opened, traffic worsened, the street was shut again, and traffic recovered. That is the only one of the three that runs the experiment in both directions, which is what makes it worth more than the other two put together.

What this page checks about itself

Every number on this page is a fraction of whole numbers held in BigInt and divided once, at the very end, when it is printed. Nothing is simulated and nothing is sampled, so 65, 80, 22.5 and 4/3 are exact rather than rounded — the last of these is checked as a fraction, not as 1.3333.