Game Theory is Grrreat
After learning a little more about it last year and applying that knowledge over the summer, I decided I wanted to take a class in game theory this fall. The class I found was a graduate-level economics class, which seemed like it would be a nice change of pace from business classes. I realized that the pace would be a little quicker and the professor wouldn’t be conditioned to be as nice, but that seemed like fun.
It turns out that I was in for quite a surprise on the first day of class, when the professor informed us that his travel schedule would only allow him to teach during five weeks of the semester, so we would have to increase our meeting time from two-and-a-half hours to nine during those weeks. (This is accomplished by adding three-hour evening sessions twice a week.) As excited as I was for the class, this seemed unworkable in my schedule so I was ready to give it up. However, I tried the first three-hour class and it convinced me to make the time. The professor was superb; we covered most of the game theory I knew and quite a bit more on the first day. We got to use a fixed-point theorem, which I haven’t done in eight years or so. Even though I had already been to four hours of class when it started, I was energized and transfixed for almost the entire lecture (which was only interrupted when the class pizzas were delivered).
The professor told us to read the first several chapters of Myerson’s Game Theory, which I am working through now. In the middle of class, our professor made a comment to the effect of “make sure to read the section on common knowledge, which is important and I cannot explain any better than the author does”. It is indeed a fine section, and features a great fable that is worth thinking through. So I am going to shamelessly rip it off (my apologies for the sexist overtones):
This story concerns a village of 100 married couples, who were all perfect logicians but had somewhat peculiar social customs. Every evening the men of the village would have a meeting, in a great circle around a a campfire, and each would talk about his wife. If when the meeting began a man had any reason to hope that his wife had always been faithful to him, then he would praise her virtue to all the assembled men. On the other hand, if at any time before the current meeting he had ever gotten proof that his wife had been unfaithful, then he would moan and wail and invoke the terrible curse of the (male) gods on her. Furthermore, if a wife was ever unfaithful, then she and her lover would immediately inform all the other men in the village except her husband. All of these traditions were common knowledge among the people of this village.
In fact, every wife had been unfaithful to her husband. Thus every husband knew of every infidelity except for that of his own wife, whom he praised every evening.
This situation endured for many years, until a traveling holy man visited the village. After sitting through a session around the campfire and hearing every man praise his wife, the holy man stood up in the center of the circle of the husbands and said in a loud voice “A wife in this village has been unfaithful.” For ninety-nine evenings thereafter, the husbands continued to meet and praise their wives, but on the hundredth evening they all moaned and wailed and invoked the terrible curse.
To understand what happened in this fable, notice first that, if there had been only one unfaithful wife, her husband would have moaned and wailed on the first evening after the holy man’s visit, because (knowing of no other infidelities and knowing that he would have known of them if they existed) he would have known immediately that the unfaithful wife was his. Furthermore, one can show by induction that, for any integer k between 1 and 100, if there were exactly k unfaithful wives, then all husbands would praise their wives for k-1 evenings after the holy man’s visit and then, on the kth evening, the k husbands of unfaithful wives would moan and wail. Thus, on the hundredth evening, after 99 more evenings of praise, every husband knew that there must be 100 unfaithful wives, including his own.
Now let us ask, What did this holy man tell the husbands that they did not already know? Every husband already knew of 99 unfaithful wives, sot hat was not news to anyone. But the holy man’s statement also made it common knowledge among the men that there was an unfaithful wife, since it was common knowledge that he announced this to all the men…Thus the lesson to be drawn from this fable is that the consequences that follow if a a fact is common knowledge can be very different from the consequences that would follow if (for example) it were merely known that everyone knew that everyone knew it.
Even if it takes up all my time, who can pass that up.
(I should note that most of the book is written in set theory rather than anecdotes. It is a fabulous introduction to game theory, but for the most part a technical one.)