# 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.