
Braess’ paradox - dedalus
https://en.wikipedia.org/wiki/Braess%27_paradox
======
tunesmith
Another example of the Nash equilibrium being worse than the group optimum is
some examples of Tragedy of the Commons, things like overfishing. It's
possible to create a model that shows that a more regulated environment leads
to more fish _per individual_ than the Nash equilibrium.

I don't think it's appreciated enough that some forms of regulation actually
do make things better for everyone, and that acting only in one's own self-
interest can lead to results that are provably worse for your own self-
interest.

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edanm
"I don't think it's appreciated enough that some forms of regulation actually
do make things better for everyone, and that acting only in one's own self-
interest can lead to results that are provably worse for your own self-
interest."

As someone who is libertarian-ish, I find that a funny thing to say which
doesn't jive with reality. Western countries usually have _huge_ governments
with _huge_ amounts of regulation, and the default assumption of almost
everyone around me seems to be that regulation is the way to deal with almost
anything we don't like.

So saying that it's not appreciated that some regulation makes things better
for everyone sounds to me like saying "it's not appreciated enough that
clothes make us warm".

(Unless, of course, you're referring only to specific circles, e.g.
libertarian economists).

~~~
alimw
This contributor to the talk page will appreciate no such thing:

this whole article is poorly veiled propaganda by the current communist
administration to justify government ownership of roads. After all, who but
the government could remove a link to benefit "everyone"? By generating this
absurd fake propaganda, it hopes to ply public sympathy toward state
involvement and ownership. Naturally the actual proposed "facts" are totally
rubbish: it is obvious that you could never decrease traffic congestion in an
area by removing already congested roads. The article is an obvious hoax from
the first sentence: "Braess's paradox, credited to the mathematician Dietrich
Braess" redlinks. I've added a deletion tag. 82.113.121.167 (talk) 01:27, 31
March 2010 (UTC)

~~~
coldtea
I am always amazed that people with such thought processes are not locked in
padded cells, or at least taken to some provincial farmhouse to rest for a
while before returning to society.

~~~
twic
The Wikipedia talk pages are a more effective prison than any mere set of
walls.

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whack
I had a hard time understanding how this could be true, until I read the
simple math example presented. The problem seems to be one of externalities.
Each individual driver is selfishly optimizing for his own travel time, but by
choosing to drive in a short-congested route, instead of a longer-sparse
route, he's negatively impacting the travel times of everyone else around him.

It's been well proven in economics that externalities, if unregulated, will
produce sub-optimal outcomes. That the way to restore optimal outcomes is to
impose a tax that's proportional to the negative externality imposed on
others. In an ideal high-tech world, this can be achieved by having
differential prices that each driver has to pay, to use each road/highway. In
a low tech world, narrowing/closing a road is akin to imposing a tax on
certain routes, which explains why it can counter-intuitively help us get to a
more optimal outcome.

~~~
Chris2048
So if I understand it, adding road signs before the new route adding estimated
travel times (that take congestion into account) will fix things? The paradox
arises from considering only route length, and not congestion?

~~~
whack
No, the problem comes from considering only your travel time, and not how your
choices impact other drivers. The example given in the wiki demonstrates that
even road signs with estimated travel times, will still not fix the above
problem.

~~~
chaosfox
However the example have a road that takes 45 minutes regardless of how many
drivers go through it, that is not very realistic is it. nowdays we have apps
that can tell us in real time how long each path will take according to
current traffic, and the time is never constant.

~~~
whack
It's actually pretty realistic for very lightly congested roads. At those
levels of congestion, the primary bottleneck is the distance and speed-limit.
Marginally increased traffic will have minimal impact on travel time.

In contrast, for heavily congested roads, the primary bottleneck is the
congestion itself. Hence, marginally increased traffic will have a
proportional impact on travel time.

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laszlokorte
The german wiki article has an paragraph about an analogue effect in mechanics
in which adding an additional spring to support a hanging weight can causes
the weight to hang lower than before. Which I found quite interesting.

~~~
mulmen
I don't speak German, can you provide a link to the paragraph and/or a
translation?

~~~
diziet
[https://de.wikipedia.org/wiki/Braess-
Paradoxon#Mechanisches_...](https://de.wikipedia.org/wiki/Braess-
Paradoxon#Mechanisches_Analogon)

~~~
schoen
The article describes a weight initially hanging from two strong springs
(blue) and two weak springs (yellow). The subsequent addition of the strong
red spring in the indicated position will, according to the article's
calculations, make the weight hang lower overall. I also find that pretty
counterintuitive!

Edit: the explanation given is that the red spring redistributes forces so
that more force is applied to the weak springs and less force to the strong
springs, causing an overall greater total extension.

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WhitneyLand
>if each driver is making the optimal self-interested decision as to which
route is quickest, a shortcut could be chosen too often for drivers to have
the shortest travel times

Does this just mean people are making bad decisions based on incomplete
information?

For example, would this still be a problem if everyone was using Google Maps
routing which changes routes based on congestion patterns?

~~~
argonaut
This applies even with perfect information, as long as drivers are completely
selfish.

~~~
lvh
How's that work? Is the idea that an individual driver would have to take a
penalty in order for everyone else to have better road use?

(Also, is this specified somewhere? It seems like the paradox relies on people
making their minds up ahead of time (hence imperfect information) and then not
changing their minds halfway.)

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saosebastiao
Exactly. In other words, in order to globally optimize for throughput, you
would have to deoptimize some people's routes.

~~~
snrplfth
Normally you would do this through prices, but most roads don't work that way.

~~~
trvlngsalesmn
Hwy 635 in Dallas has a variable-toll expressway lower level.

~~~
snrplfth
There are definitely many variable-tolled roads. The trouble is that
frequently they are embedded in a system of non-tolled (or, really, not-
allowed-to-be-tolled) roads, and that the tolling is not well-focused on
congestion. Basically, the Braess "paradox" emerges if you try to make a road
network that's optimal for travel patterns over an entire day, without using
pricing to ration scarce space.

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trvlngsalesmn
I learned about this in a course titled Dynamics of Complex Networks and
Systems taught by Mark Spong, a robotics control theorist active in dynamical
systems research. Ever since learning about Braess' Paradox in that course, I
have noticed examples of it in the highways of DFW. The simplest case to spot
is where an off-ramp feeds directly to the next on-ramp, creating a "shortcut"
for drivers who exit and re-enter the highway.

I've tried to make the case that by closing "key" exits on the highway, we
could reduce congestion. But who would be responsible for making that likely
unpopular decision?

~~~
obstinate
They do this on some exits. Speak with your county commissioners, the police,
etc.
[https://www.youtube.com/watch?v=9Ia4pzskae8&list=PLEEC4A6D0A...](https://www.youtube.com/watch?v=9Ia4pzskae8&list=PLEEC4A6D0A6CD0C5B&index=9)

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cousin_it
Yeah. Good road planning in the future should avoid situations where a driver
can save 5 minutes by delaying 10 others for 1 minute. Adding new roads can
certainly create such situations where they weren't before. You need serious
effort to figure it out though, because adding a new road can have lots of
other effects as well.

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basicplus2
This reminds me of the example of the placement of a barrier at an exit to
reduce the number of interactions to improve pedestrian flow

[http://www.gkstill.com/CV/PhD/Chapter3.html](http://www.gkstill.com/CV/PhD/Chapter3.html)

Scroll down to 3.5.3 Flow through a door.

I can't find the original article I read years ago but it showed a large round
barrier to be the most effective, and more than one barrier for larger numbers
of people

~~~
cr0sh
Dang you hit it before me - I was wondering the same thing, and posted about
it to another comment here...

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dx034
That's where variable toll expressways are a great idea. If you add a faster
way and equip it with variable tolls, the paradox shouldn't appear. The new
way will not be congested (if the toll has no upper limit) and drivers taking
the traditional way will experience less congestion.

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cs702
The Braess paradox applies to all networks, including the Internet, the
financial system, FaceBook, etc.

It seems crazy, but interconnecting more nodes to improve efficiency (as
measured by travel time, costs, etc.) can paradoxically make the whole network
less efficient!

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em3rgent0rdr
this wiki article seems to get posted every few months...
[https://hn.algolia.com/?query=%09Braess’%20paradox&type=stor...](https://hn.algolia.com/?query=%09Braess’%20paradox&type=story)

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beefman
See also, the Price of Anarchy [1] and How Bad is Selfish Routing [2].

1
[https://en.wikipedia.org/wiki/Price_of_anarchy](https://en.wikipedia.org/wiki/Price_of_anarchy)

2
[http://theory.stanford.edu/~tim/papers/routing.pdf](http://theory.stanford.edu/~tim/papers/routing.pdf)

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drewhk
Hm, can't this be circumvented by pushing towards a correlated equilibrium
([https://en.wikipedia.org/wiki/Correlated_equilibrium](https://en.wikipedia.org/wiki/Correlated_equilibrium))
instead of a Nash equilibrium?

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Pfhreak
Is this a function of turbulence? When I think about 'perfect' traffic, I
think about a pipe with water flowing through it without eddies. Branch and
merge a pipe along its length and you'd introduce all sorts of ripples and
turbulence.

~~~
ufo
IT doesn't have to

[https://en.wikipedia.org/wiki/Braess%27_paradox#Example](https://en.wikipedia.org/wiki/Braess%27_paradox#Example)

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zardeh
Someone who is perhaps better versed in machine learning:

Is there some sort of intuition here that would connect this paradox to the
effectiveness of dropout in deep neural networks, or am I drawing connections
where there aren't any?

~~~
argonaut
Having studied both topics (just coursework), I don't think so.

~~~
zardeh
That's what I assumed, it feels like there should be an intuitive connection
(both are kind of situations where a local optima is avoided by removing the
greedily optimal path on a graph structure), but I may be wrong, i dunno.

~~~
brianchu
I'm not sure dropout has anything to do with local optima or removing greedily
optimal paths, since it is random.

The original dropout paper's handwavy justification for dropout is that it
prevents co-adaptation. It prevents individual units (nodes/neurons) in the
network from relying on specific units in the previous layer firing as well.
This is a bad thing because it's fragile (if one unit is off). I say handwavy
because this is just intuition; there is not really any proof that this is
actually what is happening.

Another commonly cited motivation is that dropout is like learning an ensemble
of multiple networks.

The only paper I've seen that theoretically analyzes dropout is:
[https://arxiv.org/pdf/1506.02142v6.pdf](https://arxiv.org/pdf/1506.02142v6.pdf),
which proves it's equivalent to approximating gaussian processes (this is
beyond me).

~~~
zardeh
I think that you could consider the co-adaption as a form of greedy local
optima (I think this is a very handwavy explanation, though, and I'm having
trouble forming it better, so take with that what you will). Dropout prevents
that by not allowing the 'greedily optimal paths' to form, since sometimes
they don't exist, so you can't rely on them too much.

~~~
brianchu
Sure, but dropout actually increases training error (makes you _less_ likely
to find the globally optimal training error), with (sometimes) a decrease in
generalization error (test error). So any connection is very thin.

~~~
zardeh
Oh actually that's a really good point and I think makes the difference clear
(there isn't such a train/test divide in the driving example).

Thank you :)

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dksidana
It will be interesting to see a case where all drivers use real-time traffic
updates from services like Google Maps

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xyzzy4
HN might as well pin this article to the top since it gets reposted so much.

~~~
jayajay
I benefited from this repost.

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runeks
"Perfection is achieved, not when there is nothing more to add, but when there
is nothing left to take away."

\- Antoine de Saint-Exupery

