Game Theory + Rational Women = Dead Husbands

By Robert P. contributor

OK I admit it, the title of this post is probably more interesting than the content will be. But two posts dealing with the Angry wifebattle of the sexes and game theory recently caught my attention, and economists writing for the Internet need to use marketing tricks. In this post I’ll explain the virtues and vices involved in using hyper-formalistic models of human behavior in what’s called “game theory.”

First, the fun thought puzzle, which I’m adapting from a brain teaser apparently used in job interviews by Microsoft. Here’s the story:

There’s a village consisting solely of 100 married couples. Because of strict religious customs and the odd architecture of the village, there are some unusual rules we can state about these villagers: First, whenever a wife uses the restroom, her husband has the option of cheating on her, in full view of everyone else in the village. Further, nobody ever gossips in this village, so that the observance of a husband’s cheating is not shared with anybody else.

Second, if a particular wife ever becomes certain that her OWN husband has cheated, then she must kill him that day.

Finally, it just so happens that in this particular village, every husband has in fact cheated on his wife.

Notice that in this condition, no wife will kill her husband. Even though each woman in the village has observed the other 99 husbands cheating on their respective wives, no wife can be sure that HER OWN husband has been unfaithful, and no wife will ever inform another woman that she saw betrayal.

One day, the Queen visits the village. She assembles the entire village and announces that at least one husband has cheated on his wife.

What happens?

Now it’s tempting to conclude that the Queen’s announcement surely has no effect. After all, every woman in the village already knew that at least one husband had committed adultery; the Queen’s announcement seems to offer no new information.

However, that conclusion is incorrect. After the Queen’s announcement, each woman knows that at least one husband has been unfaithful (as before), but now she also knows that everyone else knows this fact, and those are different things. (This is what Microsoft apparently wanted its job applicants to see.)

What effect will this have? The official answer is that 100 days will go by, after which every wife will kill her no-good philandering hubby. Let me quickly walk through the inductive logic that the “official” answer wants someone to grasp:

==> The Queen’s announcement means that it is “common knowledge” that at least 1 husband has cheated. However, suppose only 1 husband had cheated. In that scenario, 99 women would have  observed one instance of adultery, while 1 woman in the village would have observed 0 instances. Then that sole woman, when she heard the Queen say, “At least one husband has been unfaithful,” would realize her own husband must be the guy. She would therefore kill him that first day.

==> If nothing happens on Day 1, then it means there does not exist a woman in the village who has witnessed 0 instances of adultery. (To repeat, this is because if there were such a woman, then–coupled with the Queen’s announcement–she would realize her husband had cheated on her, and would have killed him on Day 1.) Therefore, as Day 2 begins, the women know that at least 2 husbands have cheated. Now suppose there were a woman who had only observed one instance of adultery. She would realize that her own husband must have been the other guy to bring the total count up to two, and hence she would kill her husband during Day 2.

==> If nothing happens on Day 2, then it means there does not exist a woman in the village who has witnessed only 1 instance of adultery, and therefore we conclude that at least 3 husbands have committed adultery. Etc.

==> If nothing happens on Day 99, then we conclude that at least 100 husbands have committed adultery. All the wives then kill their husbands on Day 100.

[EDITED: In the original version of this post, I had expressed doubts–which I’ve now deleted from this section–about the above solution, since the women already started out knowing that the other women all knew that several adulteries had occurred. However, I didn’t take the logic far enough. In formal game theory, something is “common knowledge” when every player in the game knows Fact X, every player in the game knows that every player knows Fact X, every player in the game knows that every player in the game knows that every player in the game knows Fact X, and so on for an infinite number of rounds. The Queen’s announcement makes the fact of at least one cheating husband “common knowledge” in this strong sense, which does indeed change the beliefs of the women in the village. In order for the lack of action by other women on Days 1 through 99 to lead to decisive action on Day 100, the women need a very deep structure of knowledge, consisting of many layers of “she knows that she knows that she knows that…” If you’re having trouble seeing why this is so important, read this comment and then this comment at my personal blog, where two of my readers spelled it out.]

Besides fun thought puzzles, this distinction has practical relevance. For example, companies that provide “network goods”–such as cell phone providers–apparently are more likely to advertise on highly publicized TV events, other things equal. The idea is that Apple will want to advertise during the Superbowl not just because many millions of people will see the commercial, but because many millions of people will know that many millions of people are seeing that same commercial. When you’re trying to decide between a MacBook and a PC, it helps to know if other people are doing something similar.

This type of thinking also explains the behavior of dictators. It’s fine if most people hate the dictator and wish he would die, so long as these opponents of the regime feel relatively isolated. However, if it becomes common knowledge that people hate the dictator, then he could be overthrown overnight. That’s why dictatorial regimes are so intent on erasing subversive graffiti and controlling media outlets, even though one might think that with all the guns and prisons at his disposal, a dictator wouldn’t care what some smart-aleck teenager spraypaints on an overpass.

* * *

Let me switch examples to Tyler Cowen’s recent discussion of experimental results in game theory, broken down according to sex:

In one set of these experiments, called the dictator game, women were found to be more generous than men. Players were given $10 and allowed but not required to hand out some of it to a hidden and anonymous partner. Women, on average, gave away $1.61 of the $10, whereas men gave away only 82 cents.

In another test, called the ultimatum game, one player received $10 and then decided how much of it to offer to a partner. (Let’s say the first player suggests, “$8 for me, $2 for you.” If the respondent accepts the offer, that’s what each gets. If the respondent is offended by the unequal division or dislikes it for any other reason, he or she may refuse, and then no one gets anything.)

The depressing news was this: Both men and women made lower offers, on average, when the responder was female. Male proposers offered an average of $4.73 to male respondents, but only $4.43 to women. More painful yet was the behavior of female proposers, who, on average, offered $5.13 to men but only $4.31 to women. It seems that women were seen as softies who were willing to settle for less — and the discrimination was worse coming from the women themselves.

But why is this “depressing,” and why does Cowen interpret it as women being seen as “softies”? After all, the “correct,” “rational” play–according to standard economic reasoning–is for a player to accept any offer in the Ultimatum Game. Indeed, I would be on much stronger ground, according to standard game theory, if I said that these results showed women are more rational than men, and that women themselves recognized their superior rationality more than men did. In other words, not only were men more petulant and self-destructive, but they apparently were ignorant of these weaknesses.

I am largely being facetious with my criticism of Tyler Cowen, but it really is the case that the standard way economists analyze the Ultimatum Game stands in sharp contrast to what “normal people” think. (If you want to see a more detailed treatment of typical results in game theory, see my articles here and here.) The fact that Cowen so casually throws out the standard analysis to focus on apparent discrimination against women highlights just how susceptible economics is to our preconceptions and values. As I’ve often said, with formal economic analysis, you can get just about any conclusion you want, and you won’t be “lying.”

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