
Braess' paradox: adding a new road to a city can slow down traffic - Thorondor
https://en.wikipedia.org/wiki/Braess%27_paradox
======
Asbostos
Taken to the extreme, if you made such a dense network of roads that it was
effectively just one giant paved surface, then I think it's obvious that it'd
be inefficient since everyone is barging through trying to go in their own
straight line and getting in each other's way. In that case, adding barriers
and one-way lanes would intuitively speed up people's journeys. So perhaps
Braess's paradox is only unintuitive for simple cases that are very close to
our existing road networks.

~~~
Certhas
But that's not all there is to it. It really is a network phenomenon.
Otherwise you wouldn't expect it to also appear in power networks:
[http://iopscience.iop.org/article/10.1088/1367-2630/14/8/083...](http://iopscience.iop.org/article/10.1088/1367-2630/14/8/083036/pdf)

~~~
adekok
A standard physics question is to calculate the electrical flow between two
points on an infinite square grid of resistors.

Not surprisingly, the flow isn't large. The electricity takes many path
through the circuit.

Here, too, having fewer paths would result in large flow.

~~~
mikexstudios
But the equivalent resistance always decreases as the grid of resistors
increase. I don't see how having "fewer paths would result in a large flow".

See: [http://physics.stackexchange.com/questions/2072/on-this-
infi...](http://physics.stackexchange.com/questions/2072/on-this-infinite-
grid-of-resistors-whats-the-equivalent-resistance)

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Leszek
I once saw a particularly interesting physical manifestation of this paradox
(performed I believe by Chris Bishop), where a weight was suspended from two
partially elastic ropes, both attached to the ceiling. These two suspending
ropes were connected in the middle by another rope. The weight was analogous
to the destination, and the ropes were the roads, with the total duration of
the route being analogous to the distance of the weight from the ceiling. When
we removed the central road (by cutting the connecting rope), the weight
paradoxically went up instead of down.

I wish I had a video of this demonstration, it really hammered in the point
(for me) that this is a real phenomenon and not just mathematical trickery or
electrical weirdness.

~~~
Leszek
This is close enough, it's not quite as clear as the demonstration I saw but
it shows the same effect:
[https://www.youtube.com/watch?v=xiOEYNGV5P8](https://www.youtube.com/watch?v=xiOEYNGV5P8)

~~~
eru
I found
[https://www.youtube.com/watch?v=nMrYlspifuo](https://www.youtube.com/watch?v=nMrYlspifuo)
linked from your video. It has much more explanation.

------
Agustus
The paradox does not take into account the rigidity of a roadway user's route.
If your current route to work takes x minutes amount of time to drive and a
new roadway is about to be opened that will reduce the drive time by ten
minutes (x-10). News announces the new route, ribbons are cut, and signage is
put in place announcing the new roadway.

* The user is aware of the new roadway and utilizes the roadway.

* The user knows that the old route was 10 minutes more and was the current ideal.

* Other users utilize the route under the same 10 minute saving condition, driving up the amount of traffic over time, even when the new route ends up adding travel time to the original time.

* Users do not consider going back to the old route, even though it may be better now as the system builders had declared this route to be the best. There is also an internal feeling that if the new route was like this, what will the old route look like.

* When talking to people who drive, once a known route has been established, it takes a lot to get them to change. That is why accepting Waze provides such a great opportunity for balancing traffic and utilizing roadway capacity.

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micwawa
From my (mathematician's) perspective, when the solution to the optimal
transportation problem corresponds to a Nash equilibrium, this is called a
Cournot-Nash equilibrium. This does not happen generically.

In other words, it is very unlikely to simultaneously minimize both the
expected commute cost for the group and for each individual.

However, in the continuous case, you can fix this using taxes, tolls or
incentives (in theory - in practice I don't know. )

Blanchet and Carlier have some nice mathematical articles including

[https://www.ceremade.dauphine.fr/~carlier/blanchetcarlierfin...](https://www.ceremade.dauphine.fr/~carlier/blanchetcarlierfinal.pdf)

~~~
alimw
Are you saying that an equilibrium is Cournot-Nash if it _does_ manage to
"simultaneously minimize both the expected commute cost for the group and for
each individual"? Not sure that's right...

~~~
micwawa
In the literature I've encountered a Cournot-Nash Equilibrium is a solution to
an optimal transportation problem. There could be some discrepancies in
definitions as to the parameters one is allowed to vary. This is also assumed
to be a global minimum - not just local.

~~~
alimw
In the paper you reference, an 'optimal transport' problem does indeed arise
in connection with Cournot-Nash equilibrium. However the naming is a
coincidence, it is unrelated to the problem of finding the most efficient
routing of traffic. Think rather 'earth-moving'.
[https://en.wikipedia.org/wiki/Earth_mover%27s_distance](https://en.wikipedia.org/wiki/Earth_mover%27s_distance)

The paper does however note the traffic problem in passing: "the variational
approach we develop presents some similarities with the variational approach
to Wardrop equilibria on congested networks and in both cases equilibria are
socially inefficient."

~~~
micwawa
The 'optimal transportation' is misleading in this case : People are not
interchangeable, so you are actually trying to find a map between the space of
{ people who have to go places } and {routes they might take } . Once
everybody has chosen a route, there is a measure on the space of routes. The
cost of each route to each person depends on the measure on the route space.
Once you have this pairing of measures, you can ask if it's optimal or not.

~~~
alimw
I think we agree :) Thanks for the reference btw, I'm supposed to be writing
my thesis on such things.

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matthj
Networked traffic routing apps like Waze ought to nullify the Braess' paradox;
if everyone used Waze, new roads would always have a positive marginal impact.

~~~
pash
The situation is complicated by induced demand [0], the phenomenon in which
providing more roadway capacity tends to raise demand for it by reducing the
cost of travel.

What typically happens when a new roadway opens or a congested route is
widened is that travel times first fall, as expected; but soon the reduced
travel times attract drivers who previously had used other routes, or who had
commuted at off-peak times, or who had used public transit. Thus the improved
roadway ends up with significantly more traffic than the old roadway, and the
price (in travel time) of the increased capacity is bid back up near its
previous level.

Added roadway capacity can lead to a welfare improvements, but it mostly comes
in the form of fewer people shifting their schedules to beat rush-hour traffic
or deigning to take public transit, and not through reductions in travel
times. In actual fact, as we've seen throughout the era of American
suburbanization, vast new roadways have probably substantially increased
average travel times [1] by markedly changing patterns of land use (i.e., by
giving rise to the modern low-density, car-dependent suburb), albeit while
also engendering some welfare gains (e.g., increased suburban living space).

If you want to prevent induced demand from gobbling up expected improvements
in travel times when you build new roads, you pretty much either have to (a)
decrease demand by deliberately increasing the financial cost of travel (e.g.,
through tolls or congestion charges), or (b) over-build roadway capacity in
the extreme, to the extent that just about everyone in the region can fit on
the roads at once, alone in his or her own car (as is the situation in cities
like mine).

0\.
[https://en.m.wikipedia.org/wiki/Induced_demand](https://en.m.wikipedia.org/wiki/Induced_demand)

1\. I mean in comparison to a counterfactual alternative history in which the
urban-suburban commuter freeway systems typical of American cities went
unbuilt. Commute times in European metropolises, which mostly lack the
extensive commuter roadways and concomitant suburban sprawl of their American
counterparts, are broadly lower than in American metros of similar sizes, for
example.

~~~
cLeEOGPw
You kind of imply that building a new road is useless if travel time stays the
same. But increased traffic bandwidth IS improvement, even if individual
drivers don't feel it.

Also same thing applies to this paradox. Even if individual travel time is
reduced for some, I doubt that overall traffic throughput is reduced. At worst
case scenario it should stay the same.

~~~
chipsy
That only appears to be the best option if you specify all trips to be made
via private automobile. Road bandwidth is more efficient on bike or bus, but
people will default to a car if the road is well suited and there is parking
at the destination. It's not that the road is useless, it's that it
perpetuates demand for driving and so can't solve congestion except by brute
force saturation.

It's one of the big shifts in planning thought to get away from a simple
engineering problem of moving more cars faster and try to integrate different
modes and give land use a more careful impact assessment.

~~~
cLeEOGPw
That assumes that people will change from a car to other options. At least
where I live, that will not happen. They tried to do exactly what you said -
instead of increasing size of roads, they broadened roads but made the new
lanes only for public transport. The expected happened - everyone who used
cars before, were using them after, and only thing that improved is people who
ride bus reach destination a little faster. Same would be with bikes and other
things - nothing will have impact, car throughput is basically only thing that
matters in traffic congestion reduction.

~~~
chipsy
If you expand capacity in one mode but hold the other ones steady, you're
still running into Braess' paradox. It doesn't matter that there is more bus
capacity if people were tolerating the existing car congestion before. People
will not switch until the roads are jam-packed.

The paradox still applies when we consider the Tokyo rail network. There's
lots of rail in Tokyo - it transports most of the commuters. Major lines are
quite literally packed during rush hour. [0] Adding more rail will make the
system even more popular, so the problem won't get better.

OTOH consider congestion charges. When interviewed[1], individuals do not
believe they were significantly affected by the introduction of a charge, even
though overall traffic levels go down substantially, by one fifth in the
example used of Stockholm.

What we are changing when we change capacities is not speed or comfort, but a
preference for what kind of congestion we get and how cheaply it will be
filled. If our only goal is congestion reduction we should not be looking at
the roads at all.

The San Francisco area has plenty of car traffic, but more recently, within
the past decade or so, the public transit has become extremely popular. This
is not because the transit has gotten substantially better by quality or speed
metrics. It is because the population is making more trips and longer ones,
and the public transit systems were the last resort for capacity. Demand for
new trips is in turn caused by available housing being located distantly from
workplaces. If newcomers were able to live where they worked, congestion would
drop significantly.

[0]
[https://www.youtube.com/watch?v=pRBLnth4oSg](https://www.youtube.com/watch?v=pRBLnth4oSg)
[1] [https://www.youtube.com/watch?v=wC33HAq--
x8](https://www.youtube.com/watch?v=wC33HAq--x8)

~~~
cLeEOGPw
My point was that in most places none of these things you mentioned actually
helps reduce congestion except brute force capacity increase by building more
or wider roads somewhere.

Sure, there always are some cases where fiddling with lanes, transport types
and other relatively cheap means of changes makes traffic better, but in the
end simply more roads are needed.

------
milkers
That reminds me another notion from The Mythical Man-Month; Brooks's law:
Adding manpower to a late software project makes it later.

------
Tinyyy
My understanding of this is that you have a highway and a road (with a low
speed limit). The highway gets congested if many people use it so they’ll all
move slower. If everyone takes the highway, it’ll still be faster than the
road, so everyone chooses to take the highway. On the other hand, if everyone
spends half their time on the highway and the other half on the road, all of
them will be able to travel faster (But there is no incentive to do so).

At Nash equilibrium, the highway is over-utilised while the road is under-
utilised. So stripped to the core, this is an example of tragedy of the
commons where if every single individual works for their sole interest, they
will all lose out and yet nobody has an incentive to make a change.

------
ocfnash
Brian Hayes wrote a nice article about this a few months ago in American
Scientist [1] and also wrote a little JS demo to tinker with linked from here
[2].

1\.
[http://www.americanscientist.org/libraries/documents/2015617...](http://www.americanscientist.org/libraries/documents/201561716294611219-2015-07CompSciHayesRev.pdf)

2\. [http://bit-player.org/2015/traffic-jams-in-javascript](http://bit-
player.org/2015/traffic-jams-in-javascript)

------
personjerry
Anyone who wants to try this might consider playing or watching a video of the
Cities: Skylines game. It's said to be a pretty good "traffic planner"
simulation.

~~~
mjevans
That game has really taught me how poorly most drivers algorithms probably
are. A lot of that is probably a lack of visibility about the actual
conditions ahead of the drivers.

I wonder how that would change if the actual cost (in time to that driver) of
each potential route were known; even if that is in current time costs? (IE
how would they operate if they had real time traffic enabled GPS?)

Also, their refusal to bypass turning sims is inane.

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somebodyother
Ah, I love this stuff. Tim Roughgarten's work on selfish routing was my bible
through an undergrad research project, it's succinct and packed with excellent
proofs. [https://mitpress.mit.edu/books/selfish-routing-and-price-
ana...](https://mitpress.mit.edu/books/selfish-routing-and-price-anarchy)

------
jasonjei
I remember reading a piece by WIRED that discussed this phenomenon in Southern
California: [http://www.wired.com/2014/06/wuwt-traffic-induced-
demand/](http://www.wired.com/2014/06/wuwt-traffic-induced-demand/) ("Building
Bigger Roads Actually Makes Traffic Worse")

~~~
WildUtah
When the 880 connector between 980 and the Bay Bridge was rebuilt in 1998,
traffic engineers were predicting that it would increase traffic congestion
and lengthen almost all trips that used it or passed near it. And it did
indeed slightly worsen the traffic conditions in the area compared to the
interregnum after it fell down in the Loma Prieta quake in 1989. Traffic is
still worse that it would be if CDOT just closed it and turned it into an
urban garden or walking path with views or something.

That's not why it took so long to rebuild. The nine year process was the
result of the usual corruption, insider jockeying, incompetence, bureaucracy,
and lack of urgency from Bay Area government. The objections of CDOT and local
residents that ardently opposed the freeway in their neighborhood were ignored
and steamrollered as usual. Of course, the local officials, contractors, and
CDOT shared and enjoyed the lucre from the $1,200,000,000 we all paid for the
three mile connector.

Meanwhile the 1994 earthquake in LA triggered a special exception to the usual
legal process where Gov Wilson could take personal responsibility for
selecting a design, a contractor, a price, a schedule, and contract terms for
rebuilding several segments of LA highway. They were all delivered on time and
well under expected budget. Some were rebuilt in a couple months. No one had
time to figure out if those could be beneficially abandoned.

And today Oakland and CDOT still cannot make the simple decision to admit
error and close the 880 connector.

------
cbr
What fraction of the time does adding a new road slow down traffic? Or,
equivalently, what fraction of the time does removing a road speed up traffic?
If this fraction is high we should be experimenting with closing roads: that's
a very cheap way to improve infrastructure.

------
abhgh
I learnt this as a part of a course I was doing on social networks. This was
under the game theory module. Pretty interesting stuff.

------
Theodores
Why does everything have to have a new 'paradox' when perfectly good physics
exists already:

[https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws](https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws)

It is not as if cars are equipped with some superior intelligence when
compared to electrons in a circuit, the behaviour is identical and the normal
laws of physics apply.

~~~
teraflop
I think you might have misunderstood. This example is notable precisely
because cars _don 't_ behave like electrons in a circuit.

Or do you have an example of a situation in which Kirchhoff's laws predict
that adding a wire to a circuit decreases the total current?

~~~
madez
The weaker question "do you have an example of a situation in which adding a
wire to a circuit decreases the total current?" is easy to answer positively.
Adding a wire sometimes decreases total current.

~~~
teraflop
Like I said, an example would be helpful, because I don't see how that's
possible. (Unless we're talking about nonlinear devices like transistors,
which doesn't seem like what the parent poster was talking about.)

~~~
madez
I was thinking about electric circuits in general, so very much including
transistors. I mean, we were also talking about street networks in general, or
were we excluding intersections?

The simple example I was thinking about is a "digital potentiometer". Think
about the audio output of your motherboard, which's amplification you can
control digitally.

Kirkhoff's circuit laws are extremely simplifying.

An interesting exercise would be to prove the statement we were talking about
in the fictional world of "Kirkhoff circuits".

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rumcajz
Are there any examples of this happening when a new internet link is added?

~~~
shitloadofbooks
I don't think it really applies to a packet switched network: > adding extra
capacity to a network when the moving entities selfishly choose their route
can in some cases reduce overall performance.

Network frames don't "selfishly choose their route" and are entirely at the
whims of the Routers they pass through.

Because of complex routing and suboptimal peering/routing, a "shortcut" could
be added that makes it quicker to get from A to X1 but makes it take much
longer to get from A to X2 Additional routes (as in "directing packets the
right way", not physical routes) would fix this and get the best of both
worlds.

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biggestbob
Why is paradox? Seem obvious to me. If person can explain, thank you.

~~~
eru
Check out
[https://www.youtube.com/watch?v=nMrYlspifuo](https://www.youtube.com/watch?v=nMrYlspifuo)
Does it still seem obvious in the physical context?

