and Game Theory
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An inquiry from Gale Holland, a reporter writing an article on the school magnetic program in LA City, triggered the following game theoretic thoughts. For those who do not live in LA, here is a brief summary of the "rules of the game" set by the LA City Schools. In addition to regular city schools there are "magnet schools" which are presumably supposed to be superior to ordinary schools in some way. These schools are difficult to get into, and there is an elaborate point system based on such factors as race and ethnicity that determines which students are admitted. One part of the point system, and the part I analyze here, gives points to students who are wait listed for a particular magnet, but do not actually get in. The problem with this is that people apply to schools with no intention of getting in, but in hopes of getting points for being on the wait list so they can go to some different magnet later. Here is the game-theoretic analysis:
Analysis of the LA City School magnet admission system is addressed in a branch of game theory called "mechanism design." Mechanism design theory asks what the consequences of different systems of rules might be. In this case: what are the consequences of giving points for being on the wait list in contrast to other alternatives, for example, not giving points for being on the wait list.
Mechanism design focuses particularly on the problem of "private information." In this case the private information is the true intention of the family - whether they really want to go to a particular school or not. This is private information because they know and the school system doesn't. A key problem in mechanism design is that it is easy to design rules that lead people to misrepresent their private information. In this context, let's call the problem one of "strategic application," that is, applying to a school with no intention of going there, but in hopes of getting points for being on the wait list. Because you get points for being on the wait list, the rules create an incentive to make strategic applications. As a game-theorist I can predict with some confidence that people will do exactly this. This is an example of the law of unintended consequences: the rules were designed to reward families that wanted to get in but didn't; by doing so they create an incentive for families that do not want to get in to try to collect the same reward.
That's pretty obvious, mostly just translating common sense into jargon. The next step is to observe that both the school and the family have a strong incentive to make sure that a student who "strategically applies" does not actually get accepted. The school doesn't want to admit students who won't come because it is costly to find someone else; and they don't want to have students who don't want to be there, for obvious reasons. Similarly the family, if they are admitted, must either lose their points or send their child to a school they don't want. This creates an incentive for an "informal" system to run alongside the "formal" system. Since families don't get points for calling and complaining about not getting in, families that are strategically applying won't call. The school understands this, and so has an incentive to given priority to children whose parents call and complain (or make waves other ways either before or after the application process). This informal system would be referred to in technical terms as "incentive compatible" meaning that there is no reason for strategic misrepresentation in the informal system. So as a game theorist, I would predict that this is exactly what will happen: parents who really want to get in will call, and the school will admit kids whose parents call over those whose parents do not. Of course, not all parents who want to get in will know about this informal system, and so the kids of parents who are more knowledgeable about the system will be admitted over those whose parents are not. This is another unintended consequence of the rules.
The next step is to notice that there is a problem game theorists refer to as "time consistency." While the school would like to admit the students who really want to get in, by admitting children whose parents call, they encourage strategic applications in the future. That is - the reason for not strategically applying is that you may accidentally be accepted. By admitting only students whose parents call, the cost of accidental acceptance is reduced, and so there is more incentive to strategically apply. The time consistency comes about because something that is good for the school now - admitting the right kids - is costly for it next time around - getting more strategic applications. However: the second problem - getting more strategic applications - will only happen if parents learn how the informal system works - that is they know that their kids won't get in if they don't call and complain. So this creates an incentive for the school system to deny that the informal system exists. So the prediction here is that there will be an informal system in which parents who make waves will be admitted over those who do not AND that the school system will deny that such a system exists.
There is quite a bit of evicence that this is what really happens; from a game-theoretic perspective it is a predictable consequence of the rules. Whether it is good or bad is a more difficult question; in fact the relevant question is - is there a better system? One obvious alternative is not to give points for being on the wait list. If magnets have the capacity to admit all students who want to attend, this would clearly be a better system. No one would apply unless they wanted to get in; everyone who wanted to get in would get in, so there would be no need to compensate families that wanted to get in but didn't. Notice that we can't tell right now how the capacity of magnets compare to the number of students that want to get in. Of course many more applications are received than there are slots; but many of those applications are strategic, and we don't know how many are of this sort.
Game theory can't answer the question of how many applications are strategic. It does, however, tell us how we might find out. The school system could contact a small random sample of parents with children on the waiting list, and offer them the opportunity to get in, with no loss of points for refusing. Only those families that really wanted to go would accept, and so we could get a pretty accurate idea of what proportion of the applications are strategic.
If the magnets don't have enough capacity, the current rules may be the best. It depends on how many strategic applications there are; how costly it is to have strategic applications; and how many families who apply non-strategically and wind up on the wait list. These questions have empirical answers, but not theoretical answers.
Finally, the existence of strategic applications, if not well understood by the school district, can lead to foolish policy decisions. Generally speaking in LA parents seem reasonably satisfied with local elementary schools, but are unhappy with local middle schools. As a result, most strategic application are probably to elementary schools in order to get points that can be used for a middle school magnet. If in fact there are many more applications to elementary school magnets than there are slots, the school board might reasonably decide to create more elementary magnets. But if most of the magnet applications are strategic - no one actually wants additional elementary school magnets. So in order to understand how many magnets to create, the board would need to first find out how many applications are strategic.