# Failing public schools

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?

 Smiths\Joneses Public Private Public (A,A) (L,H) Private (H,L) (B,B)

For the aforementioned equilibrium to hold, we require that $B>L$. Note that I don’t set $H=B$ (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 $L. This means that everything comes down to $H$ – 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.

# Honor Thy Father and Thy Mother

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.”[1] 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,  $T>D>N$, and $K>C>0$.

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 $N$. 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. $N>D$; 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 $N>D$, 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 $D>N$, 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 $K+D$, 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 $K+D<(K-C)+T$, 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,[2] 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.

[1] Critique of Practical Reason, Book II, Chapter II, Section V. No, I’m not quite a Kantian, at least in this regard, but I like the quote.

[2] So called because it was among the “folklore” of game theory, sort-of known by everyone before it was rigorously proven.

# If you give your future self a carrot

I spend a lot of time trying to figure out how to be productive. How can I fit more work, sleep, exercise, and leisure into my day? How can I overcome procrastination? How can I focus on a task for longer periods of time? For a long time, my strategy was something like the one illustrated in the following (brilliant) comic:

Commitment devices like StickK received a lot of press when it was mentioned in Freakonomics – the premise was basically that people would commit to achieving a goal, like exercising, and pay a penalty when they didn’t stick to the task (the money could go to a person of their choice, a random charity, or an anti-charity – one whose cause the user opposes.) Users lost more weight when there was money on the line.

StickK gets its name from the carrot-or-stick analogy. The idea is that people might respond better to sticks (punishments) than to carrots (rewards), because they are loss averse: assuming that the income effect is negligible, losing \$5 hurts a lot more than winning \$5 is pleasurable.  So, when I create a StickK account and set a goal, I’m playing a game with my future self.  I commit to, for instance, working out every day, and if I don’t succeed that day, I have to give my roommate a \$5. The commitment is self-executing – say my roommate wants those \$5, so she’s definitely going to come and get the money from me if I deserve to lose it. Then, when my future self is debating whether to go to dance class, I’ll have to think, “Would I rather go to the class, or would I rather lose \$5?” Of course, I’d rather not have to make the commitment at all, but Future Me won’t stick to the task if I don’t.

One problem is that I might value the time I would get back from being lazy far above \$5. I might have to set the penalty at a price > the most I would be willing to give up to get that chunk of time back at the time that Future Me is making the decision (and that might be a pretty high number). Another issue is that for many people, succeeding in something like “not procrastinating” might feel like an even bigger loss than the procrastination itself – what if you don’t do as well as you would like at the task? What if you fail at it? Maybe you’d rather not find out – procrastinate instead. (In that particular case, the solution isn’t to penalize yourself with five pushups, punching yourself in the nose, and giving \$1 to the NRA. You should probably figure out how to change how you evaluate your payoffs so that failing doesn’t hurt so much.)

# Does White have an advantage in chess?

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[1]. 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.

[1] That is, theoretically; the actual game tree is WAY too big to actually depict