Braess's paradox: adding roads can increase congestion
crowddynamics.co.uk
crowddynamics.co.uk
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.
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.
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. :-)
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
For example in probability the St. Petersburg paradox. http://en.wikipedia.org/wiki/St._Petersburg_paradox
There are generally 3 resolutions of a paradox 1) The premise(s) are wrong. 2) The logic is in correct. 3) The "answer" is not self contradictory.
If the resolution is 3 then there is no contradiction. Even if its 2, there isn't technically one (just a mistake). If its 1, there might be a contradiction.
I remember discussing this with someone and coming to the conclusion that the word "paradox" really should be "paradox for me" because its only a paradox to someone until they see how to resolve it.