
Braess's paradox: adding roads can increase congestion - RiderOfGiraffes
http://www.crowddynamics.co.uk/Thesis/Chapter%203.htm#Braess’s%20paradox
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btilly
I always prefer the string and spring version because people can physically
see it happening.

For an illustration suppose we have two springs A and B connected in the
middle. A is tied on the other end to the ceiling, and B is tied to a weight.
The springs each expand 1 cm/newton of force, and the weight exerts 100 N of
force pulling them apart. So each spring is 1 m long, and the whole
arrangement is 2 m. Let's attach two "safety strings" of 1 m in length. One is
attached from the top of B to the ceiling and the other is attached from the
bottom of A to the weight. Cut the tie in the middle and the weight will rise
up 50 cm!

People are shocked to see that cutting a string makes the whole arrangement
stronger, but it is really easy to replicate.

According to <http://www.maa.org/mathland/mathtrek_11_10.html> this visually
startling version of the paradox was developed by Joel E. Cohen.

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roundsquare
Neat visualization. Am I getting this right?

Arrangement 1:

Ceiling -- A -- B -- Weight

Arrangement 2 (Picture is a bit weird, but both are attached to the ceiling
and the weight and not to each other right?):

(Edit: Making picture clearer).

Ceiling -- A -- Weight

Ceiling -- B -- Weight

If thats right... it shouldn't be that surprising. In (1) B has to carry all
of the weight, and A has to carry the weight and B (even if B weighs 0 it
still has to carry it) so each spring is carrying the weight. In (2) they are
in effect sharing the weight.

Its probably easier to think of the action movie scene where William falls off
the cliff, Brad catches William but starts falling and Adam catches Brad and
holds on. Brad has to hold up William and Adam has to hold them both up. But,
if instead, Adam and Brad each took one of William's arms, it would be easier
for both of them.

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btilly
Change the picture for the second to

Ceiling -- A -- String -- Weight Ceiling -- String -- B -- Weight

and you've got it exactly. Also the reason why it works.

Once you understand it the result isn't that surprising. So the physical model
provides both more intense initial surprise and also a direct way to visualize
the explanation. Which is why I prefer that version. :-)

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adamo
Some links about the paradox itself:

Wikipedia page: <http://en.wikipedia.org/wiki/Braess%27s_paradox>

English translation of the original paper:
<http://homepage.rub.de/Dietrich.Braess/Paradox-BNW.pdf>

Preface to the translation:
<http://supernet.som.umass.edu/articles/preface_to_braess.pdf>

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asciilifeform
This is reminiscent of (in effect, if not in mechanism) Belady's Anomaly.
(<http://en.wikipedia.org/wiki/Belady%27s_anomaly>)

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jamesbritt
Side question: Is "paradox" commonly used to mean "what some people find
unexpected", rather than "a logical or self- contradiction"?

~~~
xel02
It is used in both cases.

For example in probability the St. Petersburg paradox.
<http://en.wikipedia.org/wiki/St._Petersburg_paradox>

