Jeff’s post last Sunday on Jewesses in Skirts got me thinking, how is it that we have Orthodox Jewish communities that are tolerant of pants-wearing by women, and communities that are not tolerant of pants-wearing, but rarely a community with large factions of each type? The question, of course, applies to more than just skirts worn by Jewish women – we can talk about many aspects of our culture, such as our changing views on LGBT people, or miscegeny, or a gold standard vs. silver standard using the same language.
We’ve established that, assuming that the types of women in the previous post are static (that is, Nature or Circumstance assigned you to one group and you can’t change allegiances), that it is optimal for the more conservative group to adhere to skirt-wearing, and the other group not to bother. Those static proportions of types in the population affect how the “others” in Jeff’s game form their prior beliefs about which type a woman is based on her choice of clothing. But, what if the women could choose or change their ideology, and what if we consider the effects of judgement and peer-pressure?
In the following model, we can look at a scenario where the proportion of each type of player in the population is endogeneous. Suppose that a new community forms, consisting of some random number of “conservative,” skirts-only types (Jeff puts them in class 1) and some number of “progressive” types who sometimes wear pants (Jeff puts them in class 2). This represents what we would expect to happen if everyone formed their own opinions and ideologies totally independently of everyone else. Each person will randomly encounter other members of the community on a one-on-one basis, and receive social payoffs from the encounter. If she encounters a likeminded person, they both feel validated in their choices, and if not, they feel judged. As before, we assume some disutility for a restriction on wardrobe.
Now, say that the population starts out with a percentage of progressive types and of conservative types. Then, assuming that the population is large, if you are one of the members, then of the people you meet, will be progressive and will be conservative. Therefore, if you are a woman who chooses to be conservative and wear only skirts, your expected utility is and if you choose to wear pants, then your expected utility in any one encounter is . You would be indifferent if — that is, if (the fraction of progressive types) is 1/3.
Maintaining is incredibly difficult, because it is so sensitive to shocks. If for any reason becomes a little more or a little less — say, a contingent of pants-wearers suddenly move in — the balance would tilt and one of the ideologies would start providing the better payoff, the whole population would start snowballing in that direction, and it would become the predominant convention (or evolutionarily stable state) . That may be why these larger faction groups tend not to exist in real life: they are a lot like a flipped coin that lands on its edge.
So what kinds of things affect what is? The payoff matrix, obviously, and how I’ve assigned the payoffs. No one said that I must assign those particular numbers (and indeed, I don’t. The numbers that go into those matrices don’t really matter. What matters is their order of size and relative distance to each other. Try it: multiply all of the numbers by a constant, or add a constant to all of them. The solution should be the same.) What if the inconvenience of wearing only skirts is very large? (Imagine replacing the 2s with, say, 5s.) Then, (the tipping point) could be much smaller, and it would take a much smaller group of rebels to send the equilibrium going the other way. Issues like women’s suffrage are like this — they are so significant that a grassroots movement picks up momentum very quickly. If the inconvenience is less, then p would be greater, and if the existing equilibrium is skirts-only/conservative, it would be harder to change. Equivalently, we can think about the effects of mutually judgemental behavior (making the 0s in the matrix more negative). If people are less tolerant when they meet the other type, conventions are harder to change. If they are more tolerant, change is easier.
If you enjoyed the ideas in this post, you may also enjoy When in New York, do as the New Yorkers do, which describes a special, symmetrical case of the kind of game we’ve discussed here.
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 .
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.
 If you do this downtown during rush hours, you will be stampeded.