The Braess Paradox: Choice as Famine

A friend of mine who chiseled his own undergraduate curriculum at UCLA is invaluable when it comes to getting an opinion on city planning and transportation outside of the very large echo chamber that is the urbanist blogosphere. He sent me an interesting thread from Quora, the crowd sourced question-and-answer forum like Yahoo! Answers if it was made up of mostly smart people and a few PhDs, where the topic was on transportation efficiency: If you replaced 2 train tracks with a dedicated 2 lane road and ran passenger buses would it be more efficient? It’s a question that isn’t all that rare anymore with Bus Rapid Transit emerging as a low cost alternative to light rail in cities through South America and Asia; shelling out hundreds of millions of dollars to dig out new tunnels and install what amounts to be one of the more complex feats of human engineering shouldn’t be a priority for emerging economies.

The top answer on this Quora thread (again, it sounds like Yahoo! Answers but I promise it isn’t) comes from a guy named Amar Prabhu who posted a detailed economic model of transportation modal choice based on efficiency ratings (passenger load, CO2 emissions, average speed, etc.) complete with caveats and assumptions. I mean he pared down the analysis so it didn’t turn into a dissertation but, Christ, this is on a glorified knowledge forum so apparently there’s more than three dozen people actually interested in transportation that don’t hold professorships after all. I won’t go into Prabhu’s analysis (you should read it yourself) but his conclusion is that in the very short term BRT is more efficient but after three or four years a metro system becomes the better choice. Prabhu openly admits to flaws in his analysis especially on the human and policy side which are inherently tougher to model from an empirical perspective, but his number crunching is right on even if I choose to disagree with it.

As a follow up to this Quora thread, my friend sent me a Wikipedia entry on something called Braess’ Paradox (a large part of his independent curriculum was advanced economics) and, because I regret never dipping more than a toe into the intersection of traffic engineering and consumer economics, I got giddy just reading through the first paragraph. The problem is essentially this (and I encourage you to read the Wikipedia entry because I am likely to butcher this delivery and the functional notation): If commuters (let’s say exactly 4000 of them) are given two paths to work, A and B, where half of each route is a function of the number of commuters on the route (the 1st half of A and the 2nd of B; (Fc) = C/100 where C is the number of commuters) and the opposing halves have constant travel times (45 minutes in this example) then the drivers will logically split into two evenly weighted groups reducing the total travel time to an efficient equilibrium. However, if you provide the commuters with a transfer point at the halfway mark of their journeys and the transfer time is effectively 0, commuters will always choose the selfish option: take the 1st half of Route A (40 minutes with 4000 commuters) and the 2nd half of Route B (ditto) which gives us an 80 minute commute.

Easy enough right? Unfortunately these commuters have become lobotomized by choice. If you evenly split the groups into groups of 2000 like we did before the path between A and B was built then the travel times for each group would be 65 minutes; 20 minutes on the function-based halves and 45 minutes on the constant ones. Instead the commuters are choosing to reduce their travel times by five minutes by all traveling on the same route on either side of the halfway point but increasing their overall travel times by 15 minutes because of the existence of that transfer point.

500px Braess paradox road example The Braess Paradox: Choice as Famine

I’ll give everyone a chance to consider their arguments against this sort of choice paradox (here’s some starting points: “this only applies in very limited circumstances,” this is true because you have to have a population of commuters that makes both ends of the equation work which is why this is an elegant theory and not an easily applicable policy; “this applies to road choice rather than modal choice,” again, true, but on routes where two modes of transit overlap you could apply the same theoretical rigor.) but please remember that this is a vein of economics with a lot of history in transportation planning and that most of us are idiotic creatures of habit when it comes to route choice so the likelihood that we would take an 80 minute commuter over a 65 minute one because we “saved five minutes” each leg is not small.

 The Braess Paradox: Choice as Famine

It’s also something not uncommon in real world planning: commuters (especially drivers) are constantly choosing roads and highways that may have an apparent rather than real impact which is why you may see some deserted stretches of arterials that could potentially redistribute travelers efficiently. Other issues come into play when you’re discussing real world transportation issues (latent/induced demand associated with new lane construction is probably the biggest) but the Braess paradox is a novel way to look at commuter psychology.

It’s also a potential argument for more robust state-sponsored transportation systems (I’ll have to give a citation to my friend for raising that theory) because in a typical economic system choice leads to a higher performing equilibrium. If each set of commuters has two extremely well funded and maintained routes to work then there’s a higher likelihood there would be no funneling of actors from A to B and you’d reach the more stable inflection point where everyone’s travel times are reduced to a minimum. Don’t you love reading about economic paradoxes on Fridays?


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