Breas’s Paradox and You! and how nothing makes any sense-
Ever wonder If getting rid of one of your lanes of traffic would make your commute quicker? Ever wonder how you could add another path to for current flow in a circuit and increase voltage drop? Ever wonder if adding salt to a pineapple will make it taste sweeter instead of more salty? No? It does really. But Braes’s Paradox!
Braes’s Paradox is essentially a situation in which a network adds some general improvement and suffers from a lower optimization as a result. Make sense? No? good, this makes my job way easier. Imagine two roads road A and road B, the first half of road has two lanes. A takes 5 minutes and an extra minute for every 2 cars that drive on it. the second half of A has to go around a town so due to distance it takes an hour but opens up to 6 lanes and the number of cars on that road matter less so it only adds 1 minute to the travel time for every 10 cars. Now imagine road B is exactly like road A but reversed: the 6 lane is at the beginning of the route and the two lane is on the second half.
If 100 cars want to start at the beginning of A and B then 50 of the cars can go down A and 50 of the cars can go down B and everyone has a travel time of 95 minutes.
Now if we were to add an exit between route A and route B at the midpoint of the two routes so that halfway in anyone from route A and anyone from route B may switch routes there’s a problem.
Now everyone’s best travel time changes, because now going down route B for 20 minutes and then switching to A so 100 cars have to take the same route or reach their time later.
before routes A and B’s time looked like this
5 + 25 + 60 + 5 = 95
now with this new road the fastest route for anyone becomes
5 + 50 + 5 + 50 = 110
Everyone’s travel time is now 15 minutes longer thanks to the new road that theoretically kept everyone from taking the longest route, but caused so much traffic congestion that it increased everyone’s travel time by 15 minutes. This is Braes’s paradox.
In fact over in New York City it turns out that closing 42nd street actually sped up travel times for the general populace, due to Braes’s paradox.
http://www.nytimes.com/1990/12/25/health/what-if-they-closed-42d-street-and-nobody-noticed.html
The paradox actually shows up in several places, for instance in electric grids, This article by Joel E. Cohen and Paul Horowitz explains how mechanical and electrical equilibriums are subject to Braes’s Paradox most notably how adding a current path can actually cause less current to flow in the circuit. http://lab.rockefeller.edu/cohenje/PDFs/185CohenHorowitzNature1991.pdf