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The last time this colloquium was offered, I assigned myself as homework a model of high school students' decisions which universities to apply to and attend, but I didn't get my homework in. Fortunately, I didn't have to give myself a grade. Here is my late homework.
In this game there are 100,000 players, seniors in high school. There are 100 universities. The universities are not players in the game -- just mindless, predictable automata. The students are ranked from the most to the least "promising," with the most "promising" getting 100,000 points, the second most "promising" getting 99,999 points, and so on down to the least "promising" student, who gets one point. Each university will admit 1000 students, and they will be the 1000 highest-ranking students who apply. The payoff to each student is the average "promise" ranking of the students who enroll in the same university she or he does.
Thus, suppose the "best" 1000 students enroll in Old Ivy University. Their average "promise" ranking is 99,500, so that is the payoff to every student at Old Ivy. (Well, actually, 99500.5, but we will round off to integers). Suppose the next ranked 1000 enroll in Pixel University. Their average ranking is 98,500, so that is the payoff to each student at Pixel. And so it goes.
The student's strategies are to apply to one and -- for simplicity -- only one university. We will assume that each student knows where she or he is in the "promise" ranking. Thus the student knows the best university that will accept her or him. We may assume each student will apply to and attend the university that will give her or him the best payoff, that is, the university with the highest average "promise" ranking, provided that the student is confident of being admitted. (We are ignoring tuition and also parents' preferences for a college nearer home).
This game has 100! distinct Nash equilibria, but, happily, they are all very similar to one another. Suppose, for example, that (as we have said before)
and so on, with each group of 1000 students ranked together applying to the same university. Then each university will admit the 1000 students that apply, and the payoffs will be highest to students enrolled in Old Ivy, second highest to students enrolled in Pixel, third highest to those enrolled in Pinhead, and so on. Every student knows what university to apply to and is enrolled in the university she or he applies to.
This is a Nash equilibrium. To see why, suppose a single student in the top thousand were to switch his or her application from Old Ivy to Pixel. The student who switches will be accepted, but that student's payoff drops from 99,500 to 98,500. Conversely, suppose a student in the second 1000 switches her or his application from Pixel to Old Ivy. She or he will not be accepted, so cannot improve the payoff by switching to a more highly ranked university.
Thus, the ranking of universities with Old Ivy at the top, Pixel second, and so on is a Nash Equilibrium; but it is not the only one. As we have said, there are 100! equilibria in this game. For example, there are equilibria in which Old Ivy is ranked last, instead of first. If Old Ivy were ranked last in terms of the average promise of its students, then only the 1000 worst students would bother applying to Old Ivy. No student who could get admitted to Pixel would bother applying to Old Ivy, since that would just reduce their payoff to 500, the minimum.
In other words, this game is a coordination game. So long as each group of students in the same thousand all apply to the same university, we have an equilibrium -- and it doesn't matter which university that is. If the best 1000 students happened to apply to Podunk State, Podunk State would be the best university in the country, and Harvard and MIT would be so much chopped liver. (Notice that it also doesn't depend on the quality of the faculty, the facilities, or the food in the lunchroom. All that matters is agreement among the students).
But it gets worse. Once all of the students have sorted themselves out into groups of 1000, each group with next-door promise rankings and attending separate universities, the payoffs to the students will range from a low of 500 to a high of 99,500. The average payoff will be 50,000. What would happen if the students were deprived of their decision to apply to one school or another, and instead were assigned to universities at random? Each university would then have an average promise ranking of -- average, that is, about 50,000. So that the average payoff to students would be 50,000. So all this struggle among the students hasn't changed the average student payoff at all. It has just taken from those who have less (promise) and given to those who have more (promise), like Robin Hood in reverse. If that seems discouraging, look at it this way: it's your decision. Harvard may be the best or Harvard may be the worst. It's the students who decide. The faculty and the trustees don't have any say at all. Just those high school seniors.
Roger A. McCain 