You might think that I’m going to be talking about the (somewhat) recent cheating scandals plaguing schools around the country. Sorry to disappoint. What I’m more interested in is discussing why there is such a large achievement gap between public schools and private schools. We often read about how the public school system in this country is a mess. We thus have various national initiatives to try to help improve public schools, whether it be No Child Left Behind, or Race to the Top, or whatever. In any case, a lot of work has to be done on the public school system. For example, in New York City, less than half of students met the English standards. Compare this with New York City private schools, many of which are so successful that the state doesn’t even bother to test their students, as their curricula significantly surpass those of the state tests.
I think that a large factor in determining the quality of the public schools is how much the general public (including those with more resources to invest in education) actually are willing to contribute. If the wealthier students are all opting out and going to private school, then the money that these students could have been spending on the public school system is lost. Moreover, since students from a wealthier background will probably have a home environment more conducive to studying (after all, if the parents want to send them to private school, it shows a commitment to their children’s education), there will likely be, on average, more of an atmosphere at school as well of academic seriousness. In short, having the students from more affluent backgrounds attend the public schools would probably improve their quality by quite a bit.
Obviously, things are not so simple to fix – the families whose kids go to private school prefer this option, and are free to choose to pursue it. To illustrate the dynamic of sending children to private schools, we have to consider the incentives of the wealthier families. To simplify things, we reduce the number of agents to two (it makes it easier to see the rationale). We’ll call them the Smiths and the Joneses. As it stands now, their kids are mostly going to private schools, so the equilibrium we are in has both the Smiths and the Joneses choosing this option. The question is, is this the only equilibrium?
For the aforementioned equilibrium to hold, we require that . Note that I don’t set (though it might be), since it’s possible that private schools aren’t quite as good if there are not as many wealthy people attending; for example, they have the same fixed costs to pay for on less income from tuition, which may make it more difficult to maintain a higher quality.
We can say some more things about this. Given that schools are worse off when the wealthier students leave, we would expect this trend to hold even when only some of them leave. Thus we can stipulate that . This means that everything comes down to – if it is higher than $A$, then the families have a dominant strategy to send their kids to private school. But if not, then (Public, Public) is also an equilibrium. Thus it might be possible to “shock” the system to get the parents to send their kids to public school instead. I’m no education policy expert, so I don’t have any suggestions as to how to execute this, or whether it’s even desirable: perhaps we just want to pour our efforts into our students with the best chance of succeeding. But it’s a thought worth considering.
Note: This post was originally written in 2011, which is why the references are a little bit out of date, but the point is still valid.
You know when you open the door, and you find someone else is trying to get in at the same time, and so you both end up right in each other’s faces? And then you each try to get out of the other’s way, only to go in the same direction and still be in each other’s face? And then you do a sort of weird dance?
I’ve been trying to find some good Youtube clips of this, and while I know they’re out there, I can’t find them in a brief search. But I think you know what I’m talking about.
Well, surprise surprise, we can model this interaction as a game! Each person has two strategies: move left (L), or move right (R).(1) If they both move in the same direction, then they are still stuck doing the awkward dance, and get payoff -1. Otherwise, they move out of each other’s way, and so they happily go along their way, getting payoff 1.
|Fig. 1: Awkward dance game
There are a couple of pure-strategy Nash equilibria: one player goes left, and the other goes right. But which equilibrium is going to be chosen? A priori, there is no way to tell. Here, social conventions can be useful, such as always moving forward on the right side (for a similar post, see Marli’s post, “When in New York, do as the New Yorkers do“). The problem is when some people didn’t get the memo (*sigh*).
There is a third Nash equilibrium in mixed strategies, where each person chooses to go in one direction with a 50-50 chance. This means that they will have a 50-50 chance each time they play that they will bump into each other, but eventually, after perhaps dancing for a while, they will get it right.
(1) This will all be from the perspective of player 1.
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.