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.
Many of us are familiar with the Fifth Commandment, given by the eponymous title of this post. But what does this mean in practice? In the Jewish tradition, the Rabbis interpreted one’s obligations under this commandment as the requirement to feed and clothe one’s parents, along with other similar duties. In other words, basically, one must take care of one’s parents in their old age.
We would like it if the kids had reason to actually fulfill these duties. For, as Immanuel Kant put it, “Nevertheless, in the practical problem of pure reason, i.e., the necessary pursuit of the summum bonum, such a connection is postulated as necessary: we ought to endeavour to promote the summum bonum, which, therefore, must be possible.” In other words (abusing Kant a little bit), we would like the kids to actually take care of their parents in their old age in a Nash equilibrium solution.
Here’s the problem. The kids may well be rotten, and tell their parents, “So long, and thanks for all the fish!” Why should they waste their precious time and resources taking care of them when there’s nothing in it for them?
The standard reply is that, since you were raised by your parents from infancy, you ought to take care of them. Just as, when you were incapable of maintaining yourself, they clothed and fed you, you ought to do the same when they are incapable of maintaining themselves in their old age. If they would anticipate that you’d be so ungrateful, they wouldn’t have raised you in the first place! But, while the kid is happy that his parents raised him, this is still utterly absurd from a game theory perspective. Here’s the game tree:
As per the story in the following paragraph, , and .
We see from the game tree that, if we reach the second stage (where the kid has to make his decision, and the parents have already decided to raise the kid), then the optimal thing for the kid to do is to betray his parents. Since there’s nothing they can do about it, they are sort of stuck if that happens. Knowing this, the parents should expect that, if they raise the kid, they will receive a payoff of . So, they should stick with the dog.
This solution seems paradoxical. After all, we don’t see parents choosing not to have children because of this. One might say that really, they’d still rather have the kid anyway, i.e. ; but this is not universally true, and was certainly more frequently not the case before the modern era. It used to be that children were considered a financial asset, as they could work the fields, bring in extra cash from factory work (in the 19th century), and/or support them in their old age. Without these incentives, the children would not have been thought worthwhile in the first place.
But in any case, we don’t see that all kids abandoning their parents in their old age. The stigma of not taking care of them no longer exists to a large extent, since we hear all the time of children estranged from their parents, sending them to nursing homes, etc. Even if , game theory would seem to predict that they would do so, or else they are not acting rationally; as per the quote from Kant before, we’d like it to be in one’s rational interest to help one’s parents.
Kant would likely say that the game tree does not appear as it morally ought. That is, morally, one ought to have preferences where , so that parents would choose to have children and children would choose to take care of their parents, as this leads to the socially optimal outcome. This falls in line with Kant’s categorical imperative: one ought to act in the way that one wishes were universal law. Since one would wish that it were universal law that children took care of their parents, their preferences and actions should fall into line.
But what if the preferences remain as above? Is there any way to save Kant, along with (more importantly) the incentive to honor one’s parents?
One (naïve) way to try to resolve this is by making the above game tree into a repeated game. If the kids are rotten, then this triggers the “dog strategy,” in which the parents expect that the kids will always be rotten going forward, and so always get the dog instead. Wary of this, the kids will be sure to toe the line. With discount factors sufficiently close to 1, this will be a subgame-perfect Nash equilibrium.
This, however, doesn’t quite work, since the kids are only kids once. By the time they have the ability to act rottenly, they have no concern for future stages of the game – the parents have no means by which to punish them for their deviance. So, the above proposed solution fails.
A more sophisticated method of enforcing taking care of one’s parents involves repeating this game with overlapping generations. After all, nobody lives forever: the kids will one day grow old and feeble themselves. So, we can have THEIR kids punish them for not taking care of their parents.
Here’s the idea: each generation lasts for two periods, each consisting of the two stages in the game tree above. In the first, they are the kids; in the second, they are the parents. If the kids (Generation B) decide not to take care of their parents (generation A), then their kids (Generation C) will not take care of them, either. Thus generation B will get a payoff of , since they will know that their kids will not bother to take care of them to punish them for their own malfeasance.
Now, we might be worried that Generation C will not have incentive to punish their parents, out of fear of being punished themselves by their kids in turn (Generation D). We can resolve this by making an exception to the above rule, so that a generation (C) is not punished for not taking care of its parents (Generation B) if its parents (Generation B again), in turn, did not take care of their parents (here, Generation A). Thus Generation C would get the best of both worlds: free-riding from their parents (if they, Generation B, is dumb enough to have kids anyway), and support from their own children (Generation D). Moreover, the entire process resets once we get to generation D, so even if someone screws up and does the wrong thing, it doesn’t screw doom everyone forever.
Thus this proposed strategy profile is a subgame-perfect Nash equilibrium as long as , so that all generations prefer to take care of their parents and be taken care of, over backstabbing their parents and being content with a dog. I think this is likely the case for many individuals, and so we can rest easy that our kids will likely not be so inconsiderate as to send us to a nursing home if they wouldn’t want that for themselves.
That being said, as per a fact known as the folk theorem, multiple equilibria will exist in the repeated game framework, so everyone not taking care of their parents will also be an equilibrium. This can explain why some people fail to do their moral duty. Kant would not approve.
It seems likely, doesn’t it? After all, as White, you get to develop your pieces first, putting them in position to better attack Black, or at least defend against his/her attacks. And the statistics seems to bear this out (at least, according to Wikipedia, my source for all things true). Though it could turn out that Black has an advantage – it might be that any move by White fatally weakens his/her position, so that they are at a comparative disadvantage to Black.
Whatever the outcome might be, it turns out that if either side has an advantage, that advantage is necessarily complete. Put more formally: either (a) there exists a strategy profile for White that guarantees victory, or (b) a strategy profile for Black that guarantees victory, or (c) strategy profiles for both White and Black that guarantee a draw. It sounds somewhat trivial, but it’s not: for example, if (a) is true, this means that no matter what Black does, the outcome of the game is not in doubt – White will win.
Results such as these are common among many games without exogenous uncertainty, and many games have been solved, so we know which of the analogues to the above possibilities are true. For example, checkers was recently shown to have strategy profiles for both players which guarantee a draw. So that this holds true for chess as well should not come as a surprise.
To show that one of these three possibilities must hold, we can draw a game tree which contains all the possible move sequences in chess. This is because chess ends in a finite number of moves: a draw is automatically declared if the same position is reached three times, or if fifty moves have gone by without a pawn move or a piece capture. Since there are only a finite number of possible pawn moves (given the size of the board) and piece captures (since there are only 32 pieces), the game is finite.
Next, we can use backward induction (as in my post on tic-tac-toe) from each possible ending of a game to determine the outcome from the beginning. At each node, the player involved (White or Black) deterministically selects the branch that leads to the best final outcome for him/her (using tie-breakers if necessary if several outcomes are equally good). We proceed in this manner all the way up to the initial node, corresponding to the starting position of the game. We can then go back down the tree, and since we have already determined the best response to any position, we can deterministically get to the best outcome for Black or White. This automatically yields a win for one of them, or a draw.
Unfortunately, while this works well in theory, in practice it is virtually impossible. Given the combinatorial explosion of positions in chess, the computing necessary to determine which possibility is correct is infeasible. I guess we’ll be stuck with just a good game of chess.
 That is, theoretically; the actual game tree is WAY too big to actually depict
This is something I’ve asked a few of people about. It seems odd that in our modern, post-feminist age, it is almost always men who do the asking out. This is not so good for both men and women. For men, it puts a lot of pressure on them to make all of the moves. For women, I cite Roth and Sotomayor’s classic textbook on matching, which shows that, though the outcome from men always choosing partners is stable, it is the worst possible stable outcome for women. That is, women could get better guys to date if they made the moves.
I have a few hypotheses, but none of them seem particularly appealing:
1) Women aren’t as liberated as we think.
Pro: There doesn’t seem to be any point in history where this was any different, so this social practice may indeed be a holdover from the Stone Age (i.e. before 1960).
Con: If this is true, then it is a very bad social practice, and we should buck it! This is not a good reason to maintain it!
2) If a woman asks a man out, it reveals information about her. This could be a case of multiple equilibria. Suppose that a small percentage of “crazy types” of both men and women exists, and under no circumstances do you ever want to date one of them. The equilibrium in which we are is fully separating for women, where the “normal types” always wait for men to ask them out, while the “crazy types” ask men out. Since this is a perfect Bayesian equilibrium, men know that if they get asked out, the woman must be crazy, and so they reject. Knowing this, the “normal” women would never want to ask a man out, since it would involve the cost of effort/rejection with no chance of success.
Suppose the chance that someone is crazy is some very small . Consider the game tree:
Notice that the crazy women always want to ask the guy out, no matter what the beliefs of the guy are.
There are a few perfect Bayesian equilibria of this game, but I will highlight two. The first is that the normal women never ask guys out, and guys never accept. As , this gives expected payoff to people of . No one wants to deviate, because only crazy women ask guys out, and so a guy would never accept an offer, as that would give payoff instead of ; knowing this, normal women will never ask men out, because that gives them payoff instead of .
Another equilibrium is that all women ask men out, and men always accept. As , the expected payoff vector is . Thus the former is a “bad” equilibrium, while the latter is a “good” one. In other words, we may be stuck in a bad equilibrium.
Pro: I think that there definitely some guys out there who think that women who would ask them out are “aggressive” or “desparate,” and so they wouldn’t go out with them.
Con: I don’t think the above sentiment is true in general, at least for guys worth dating! If a guy has that attitude, he’s probably an @$$#0!3 who’s not worth your time.
There may also be some elements of the problem with (1), but these would be harder to overcome, as the scenario here is an equilibrium.
Finally, while this might have some plausibility for people who don’t really know each other yet, I definitely don’t think this is true for people who know each other somewhat better, and therefore would already know whether the woman in question was crazy. That being said, I would expect it to be more likely that a woman who has known the man in question for longer to be proportionally more likely to ask him out (relative to the man), even if it is still less likely.
3) Women just aren’t as interested. If he’s willing to ask her out, then fine, she’ll go, but otherwise the cost outweighs the benefit.
Pro: It doesn’t have any glaring theoretical problems.
Con: I want you to look me in the eyes and tell me you think this is actually true.
4) They already do. At least, implicitly, that is. Women can signal interest by trying to spend significant amounts of time with men in whom they have interest, and eventually the guys will realize and ask them out.
Pro: This definitely happens.
Con: I’m not sure it’s sufficient to even out the scorecard. Also, this seems to beg the question: if they do that, why can’t they be explicit?
When I originally showed this to some friends, they liked most of these possibilities (especially (1) and (2)), but they had some additional suggestions:
5) Being asked out is self-validating. To quote my (female) friend who suggested this,
…many girls are insecure and being asked out is validation that you are pretty/interesting/generally awesome enough that someone is willing to go out on a limb and ask you out because they want you that badly. If, on the other hand, the girl makes the first move and the guy says yes it is much less clear to her how much the guy really likes her as opposed to is ambivalent or even pitying her.
Pro: This is true of some women.
Con: Again to quote my friend, “There are lots of very secure, confident girls out there, so why aren’t they asking guys out?”
6) Utility from a relationship is correlated with interest, and women have a shorter window. This one is actually suggested by Marli:
If asking someone out is a signal of interest level , and higher interest level is correlated with higher longterm/serious relationship probability, then women might be interested in only dating people with high interest level because they have less time in which to date.
Pro: It is true, women are often conceived to have a shorter “window,” in that they are of child-bearing age (for those for whom that matters) for a shorter period.
Con: This doesn’t seem very plausible. Going on a date doesn’t take very long, at least in terms of opportunity cost relative to the length of the “window.” As a friend put it in response,
Obviously one date doesn’t take up much time; the point of screening for interest is to prevent wasting a year or two with someone who wasn’t that into you after all. But then it would seem rational for (e.g.) her to ask him on one date, and then gauge his seriousness from how he acts after that. Other people’s liking of us is endogenous to our liking of them, it really seems silly to assume that “interest” is pre-determined and immutable.
So overall, it seems like there are reasons which explain how it happens, but no good reason why it should happen. I hope other people have better reasons in mind, with which they can enlighten me!
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.
Relax, I’m not referring to actual Sicilians. I’m referring, of course, to Vizzini from the movie “The Princess Bride.” The hero, Westley, is trying to rescue his true love, Buttercup, from the clutches of Vizzini and his henchmen, Inigo Montoya and Fezzik. After outdueling Inigo and knocking out Fezzik, he overtakes Vizzini, who threatens to kill Buttercup if Westley comes any closer. This leads to an impasse: Vizzini cannot escape, but Westley cannot free Buttercup. So, Westley challenges Vizzini to a “battle of wits”:
The structure of the game is simple: there are two glasses of wine. Westley has placed poison (in the form of the odorless, tasteless, yet deadly iocaine powder) somewhere among the two cups, and allows Vizzini to choose which to take. Afterwards, they drink, and they see “who is right, and who is dead.”
Presumably, when Vizzini encounters the game, he is supposed to think that Westley has restricted himself to poisoning one of the glasses. In this case, we have a standard extensive form game of incomplete information, which is equivalent to a normal-form game:
|Vizzini\Westley||Poison Westley’s cup||Poison Vizzini’s cup|
|Drink Westley’s cup||(Dead, Right)||(Right, Dead)|
|Drink Vizzini’s cup||(Right, Dead)||(Dead, Right)|
|Fig. 1: Battle of Wits (outcomes)
Immediately we see that this game is symmetric (or, more precisely, anti-symmetric), in that whatever doesn’t happen to one player happens to the other. In this way, this game is strategically equivalent to the game of matching pennies. This lets us know right away that the equilibrium outcome is for Westley to randomize 50-50 between the choices: do anything else, and Vizzini has a better chance of winning if he plays optimally, as he could just choose the cup that is less likely to have the poison. Similarly, if Vizzini was a priori less likely to choose a given cup, then that is where Westley should have put the poison.
Yet Vizzini does not reason this way. Instead, he attempts to make vacuous arguments about the psyche of Westley, namely, where Westley would have put the poison. He may be reasoning as if Westley is a behavioral type, but clearly, that’s not the best thing to do in a “battle of wits,” where presumably everyone is rational. Instead of making the game-theoretic choice based on mixed strategies, he tries to find an optimal pure strategy.
In the end, Vizzini takes his own cup, which indeed contains the poison. As it turns out, both cups contained poison: Westley has built up tolerance to iocaine, and so it didn’t make any difference which was chosen. So in a way, Westley did make Vizzini indifferent between the two outcomes; it’s just that Vizzini was mistaken in which game was being played. In reality, no matter what, Vizzini would be dead, and Westley would win. This makes one wonder that perhaps Vizzini should have thought something was afoul when Westley proposed the game in the first place, and even more so when he falls for such an obvious trick of misdirection which tries to get Westley to look the other way (see 3:04 in the video). But no matter – while Vizzini may have been smarter than Plato, Aristotle, and Socrates, he could have used some of the 20th century wisdom of John Nash.
The classic question for high school seniors: “So, what are you doing next year? What colleges are you applying to?” Please, give them a break. They get this question way too much, and it only makes them more nervous about their futures. After all, they seem to be under the impression that where they go is a make-or-break issue, and bringing up the subject as if it is important just reinforces that fear.
But while we’re on the topic, where should they apply?
For simplicity, let’s suppose that each college (indexed by a number ) has a particular quality level, , at which every potential student values that college. This can be through the quality of academics, the alumni network, the cost, the location, you name it. One might think that it would be best to apply to as many colleges as possible, since that maximizes your chances of getting in somewhere good. But, like everything in life, there is a cost to doing so. This can be the actual application fee, the time involved in putting together the materials, getting ETS to send your SAT scores, whatever. Let’s fix this cost for each college at . We can relax these assumptions, but the qualitative result will still be the same.
Suppose there is a large number of people, and we restrict people to applying to one school. Yes, I know, this is an unreasonable assumption, but the qualitative results will again be the same even if we allow for applying to multiple schools; it just makes the math hairier. The probability of getting in is , where is the number of slots that the school has, and is the number of people who apply.
Let’s consider the Nash equilibrium. Since everyone values each school equally, we will expect that everyone will be indifferent between applying to the various colleges. Thus, we will have, for every college ,
This can give us the relative admission rates of each school:
This equation in and of itself is informative. It shows that the admission rate to a school will be increasing in the cost of applying, all other things being equal. This makes sense: if you make it harder to apply, less people will do so, and this will drive up the admission rate needed by the school to fill all of its slots. Similarly, an increase in the quality of the school drives down the admission rate, since more people will then want to go there, making it more competitive.
So, in summary, what should you do? You should apply to the school which maximizes , which is your expected benefit from applying there. Assuming that everyone else is being rational and doing the same thing, though, then it won’t make much difference where you apply. That being said, this last result will no longer hold if not all people value different schools the same (though the trends for the relative admission rates will still hold), but that makes the analysis too complicated for a mere blog post.
Edit: For a more sophisticated theoretical and empirical model whose basic idea is the same, click here.