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?

 

4 Comments

  • 1
    Charlie
    September 28, 2012 - 2:36 pm | Permalink

    I believe the first question about BRT vs rail has already been studied in detail. BRT can carry 9,000 to 30,000 per hour and LRT (light rail) can carry 12,200 to 26,900. However, the highest potential line capacity is of MRT (metro/subway), 67,200 to 72,000.

    http://journalistsresource.org/studies/environment/transportation/bus-versus-rail/?wpmp_switcher=mobile

    One important thing to note also is that studies have shown that rail in general attracts more transit riders of choice (people who could drive but choose to use transit instead) than BRT. So even if you replaced rail with BRT with the same capacity as the rail it replaced, you would likely lose a chuck of riders who would prefer the comfort of rail but would rather drive than ride a bus.

    • 2
      TM Brown
      September 28, 2012 - 5:32 pm | Permalink

      Charlie —

      Spot on with the second paragraph. Just heuristically speaking I have dozens of friends who wouldn’t be caught dead on the bus because it just seems “icky”. I don’t think the aesthetic argument is marginal because, like you said, you would definitely lose a large share of riders to other modes and the typical shift would be from bus to single driver vehicles. It’s unfortunate but the revealed preferences in transit are pretty clear.

      In terms of the passenger loads for the three modes discussed I don’t think there’s much argument. My personal preference is to argue for the inclusion of robust and improved bus service over the construction of new metro lines simply because of the flexibility in planning. Even with BRT systems which are inherently less dynamic that regular bus systems you have the chance to change the placement and paths of lanes instead of being stuck with an entrenched infrastructure. Transit systems have to adjust to a more dynamic modeling system and subways are really, really bad at that especially in terms of economics.

  • 3
    TM Brown
    September 28, 2012 - 5:20 pm | Permalink

    This is a comment from the Friend referenced throughout the post:

    This sort of thing actually probably happens much more than we think. Urban landscapes and transportation topography are changing so quickly it’s difficult for anyone to claim empirical cause and effect. Traffic systems, almost by definition, are dynamic so changing one element (adding or removing a choice) can have systemic changes that make policy makers accustomed uncomfortable.

    I bet there’s a dissertation being written right now on explaining why the mess the Big Dig caused was obvious from a game theory perspective even before they started digging (the snarky byline will be: guess what? writing shit on a whiteboard is much cheaper than building it and waiting to find out later). These kind of infrastructure decisions might benefit from looking more at agent based models that can simulate non-equilibrium outcomes or at least show that there are a lot of local maximums / minimums that the system can settle on that result in drastically worse performance than the original plan. You can also run these kind of simulations on existing traffic systems. If you’re looking at incremental return on infrastructure investment there is a long tail of small tweaks that would probably yield multiples higher ROIs than planning new mega projects.

    Most of the people capable of this kind of analysis sadly will probably never go into transportation. There’re thick envelopes and stock option packages waiting for recent PhD grads willing to work in finance and tech. High frequency trading and big data processing are traffic optimization problems too. They just pay a whole lot better.

    A few random links that might be interesting:

    http://www.preservenet.com/freeways/FreewaysEmbarcadero.html

    http://blogs.cornell.edu/info2040/2011/09/25/real-life-example-of-braesss-paradox-bostons-big-dig/

    Some good food for thought from the best mind under 30 I am personally acquainted with.

  • 4
    Charlie
    October 1, 2012 - 8:26 am | Permalink

    Interesting stuff TM. The dynamic nature of buses is indeed a plus in many ways since routes can be tweaked as demand and need changes. On the other hand, that lack of permanence can be bad for other goals, particularly for spurring new development along a corridor. Fixed rail has been shown to be much more appealing for developers who are considering building new residences, offices, or retail along a corridor, particularly when they are purposefully included less off-street parking as compared to in a corridor without rapid transit.

    This also affects people looking to live a car-free or car-lite lifestyle and are considering where to live. I would much rather buy a house near a subway or light rail line than a BRT or standard bus line. With rail nearby, the chances of the transit service being severely cut or removed altogether is very low. Whereas, when transit agencies are looking to cut service due to budget issues, bus routes are usually hit first and hit the hardest.

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