Thursday, February 14, 2008

Game Theory and Human Behavior, Part I

Game theory is a subfield in applied mathematics that is used by social scientists to quantify and predict behavior. The main concept is that of the game, consisting of players, a set of moves available to the players, and the benefits or costs associated with each move.

One of the most famous games in game theory is known as the Prisoner's Dilemma. Here is the classic setup: Suppose that you and your accomplice have been arrested for a crime. The police place you in separate cells, and you cannot communicate with one another. They then tell you the following: they can convict you both on a lesser charge, but they're lacking the evidence to convict you for the big crime you committed. But, if you cooperate with them, they may be able to work something out with you. If you rat out your accomplice, and assuming he doesn't rat you out, you'll go free and he'll be sentenced to ten years in prison. If you don't rat him out but he rats on you, he'll go free and you'll be facing ten years in prison. If you both rat each other out, you'll each get five years in prison. If neither of you cooperates, you'll each get six months in prison on that lesser charge.

So, in this game, the players are you and your accomplice. The possible moves are to cooperate or not cooperate. The outcomes can be summarized in a table:

You Cooperate You Don't Cooperate
Accomplice Cooperates 5 yrs / 5 yrs 10 yrs / free
Accomplice Doesn't Cooperate free / 10 yrs 6 mos / 6 mos

The best outcome for both of you together is if neither one cooperates; however, by not cooperating, you risk facing the worst personal outcome. The best personal outcome occurs if you cooperate, so, no matter what your accomplice chooses, it is best for you personally to cooperate. If he doesn't cooperate, you get out free; if he does cooperate, at least you got only five years rather than ten.

Another problem in game theory has to do with signaling. Generally, we have one player who is sending messages and another who is receiving messages. Each player has a knowledge space (the set of all facts that the player knows), and these knowledge spaces may or may not intersect (overlap). Each player can assess the veracity of a statement, within the limits of his or her knowledge space. We could express the likelihood of a statement being true as a number between zero and one, with 0 = completely untrue, and 1 = completely true.

So, for example, if I'm the sender and you're the receiver, if I tell you that I have a tattoo on the bottom of my left foot, I know the veracity of that statement, since it's my foot and I can take a look at it. You can assess the veracity of the statement based on the last time you examined the sole of my foot; or your knowledge of my personality, sense of humor, and threshold of pain; or, if you've never met me, you might consider the likelihood of someone like me having a tattoo of any type. In the end, the better you knew me, the more you would be persuaded that the likelihood of me having a tattoo on the bottom of my left foot is very low (indeed, it is zero).

A classic signaling game is the job-market signaling game. Let's suppose that there are two types of employees: good employees and bad employees. Employers want to hire good employees, and are willing to pay them more than they would pay bad employees. The problem is, it's not possible to distinguish between good and bad employees without hiring them. This is fine for people who are bad employees, because they can get hired and coast on the productivity of others. It's not fine for the good employees, however. So they have to somehow signal to the employers that they are the employees that should be hired.

The way they can do this is through education. In this game, we assume that education does not actually enhance the skills of the employee; we assume that an employee would have to possess a certain level of natural ability (or other positive traits) that would allow the employee to make it through the educational experience. If a prospective employee makes it through a prestigious program of study in a timely manner, then he or she is a good employee. A bad employee wouldn't bother to pay the cost (in time, money, and effort) to produce this signal.

Of course, this game is imperfect, as in real life there is not a true one-to-one correlation between degrees from prestigious institutions of higher learning and intelligent, hard-working, successful employees (George W. Bush, anyone?), but in the game theory world, where the only way to make it through school is to work hard, this game makes sense.

A practical application of signaling is in the game of contract bridge. I love bridge, and I used to play duplicates in bridge tournaments. For those unfamiliar with the game, it is played with a standard deck of 52 cards, by two pairs of partners. Let's go with the convention of the bridge advice column in the newspaper and refer to them as North/South and East/West, respectively. Each player is dealt a hand of 13 cards, and then there is a round of bidding. After the bidding, the cards are played.

It's not really that important to know much about the bidding, except to say that the winning bid is what determines which card suit will be trump. But during the bidding, players often make bids that signal something completely different than what they sound like on the surface. For example, practitioners of the Blackwood convention recognize that when they bid "Four no-trump," the suit their partner responds with tells them how many aces are in their partner's hand. So if North bids "Four no-trump" and South responds with "Five Diamonds," South could have no diamonds in her hand; she's signaling that she has one ace.

Knowing these signals is very useful, even if you don't use the Blackwood convention yourself. You might perk up every time you hear a "Four no-trump" bid, and assess the likelihood that the bid responding to it is a Blackwood response, based on your evaluation of the experience and expertise of your opponents. In bridge tournaments, each partnership fills out a form explaining which conventions they use and the frequency of usage (always, often, sometimes, rarely, never). So you can check your opponents' form to get an idea of what signals they're using.

It's a shame that at the time I was playing in bridge tournaments, I didn't know game theory. I might have done a little better!

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