# When in New York, do as the New Yorkers do.

I was recently walking through a busy section of midtown Manhattan with Jeffrey and another friend, and, as almost all foot-travelers on busy sidewalks do, we walked on the right. Now and again there would be the odd tourist, camera around neck, standing blissfully oblivious in the sidewalk, taking in the sights of Times Square as the traffic flows around him [1].

Now if you live in North America, you most likely drive on the right side of the road. It’s the law. Likewise, in the UK, Japan, India, Australia, and a handful of other countries, you would drive on the left. There are no such laws for pedestrians, and indeed there is no need — for the most part, we follow the convention.

The convention is an equilibrium — a focal point in a coordination game. Americans could just as well all walk (or drive) on the left with no ill consequences. Imagine that you are on a sidewalk with a number of other pedestrians, who all walk on the right. You could walk on the left, but even if the oncoming traffic were not shooting you glares of death, you would waste precious time dodging them. Your optimal strategy is then to walk on the right, the path of no resistance. The same is true for each other pedestrian on that sidewalk, whose situations are (with some abuse of terminology) symmetric.

Tourists, who happen to be abundant in the theatre district, add a random element to the game. Pedestrian conventions are nowhere as well-established in less-trafficked locales, since really, walking on either side of the street is equally good when encounters with other pedestrians are infrequent. Even when we introduce this random factor, traffic tends toward an equilibrium. Imagine a block on which roughly equal numbers of pedestrians choose to travel in each direction on both sides of the sidewalk: If there are even slightly more people following the right-hand traffic rule or the left-hand traffic rule, there will be an advantage for every pedestrian to follow that rule.

In other parts of the city, which comprise almost exclusively of tourists, there may exist a third equilibrium, an unstable knife edge equilibrium where evey pedestrian is randomly choosing between walking on the left or on the right (0.5, 0.5). Since every other pedestrian is doing the same, it is not advantageous for any one pedestrian to change her strategy (they are equally bad). As soon as a large enough group of convention-following natives joins the sidewalk that the change in proportion is locally perceivable, the traffic is tipped toward one of the pure strategy equilibria.

In fact, it was the case that we found ourselves on a block along Broadway on which more people seemed to be walking on the left than the right, and, since we wanted to escape an impending storm as soon as possible, we walked on the left. But no sooner than a long gap appeared on the right did the two people immediately ahead in turn switch to the walking on the right, and, perceiving this and the oncoming right-walkers approaching from the next block, I and my companions followed suit. Convention won out again.

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[1] If you do this downtown during rush hours, you will be stampeded.