A beautiful idea?
from David Ruccio
What most of my students know about and game theory (at least before sitting through my two or three lectures on the topic or taking an entire course in game theory) they derived from the film, A Beautiful Mind, and the stupid example used to illustrate game theory in the film (of a blond woman walking into a bar).
As Kenneth Chang explains, the episode in the film is not an example of a Nash equilibrium.
A simpler example is what is known as the Prisoner’s Dilemma. Two conspirators in a crime are arrested and offered a deal: “If you confess and testify against your accomplice, we’ll let you off and throw the book at the other guy — 10 years in prison.”
If both stay quiet, the prosecutors cannot prove the more serious charges and both would spend just a year behind bars for lesser crimes. If both confess, the prosecutors would not need their testimony, and both would get eight-year prison sentences.
At first glance, keeping quiet might seem the best strategy. If both did so, both would get off fairly lightly.
But the calculation of the Nash equilibrium shows they would likely both confess.
This type of problem is called a noncooperative game, which means the two prisoners cannot convey intentions to each other. Without knowing what the other prisoner is doing, each is faced with this choice: If he confesses, he could end up with freedom or eight years in prison. If he stays quiet, he goes to prison for one year or 10 years.
In that light, confessing is the better option. And he knows that the other prisoner has the same incentive to confess, so it is less likely he would stay quiet.
Further, changing strategy to staying mum would be a bad move — longer prison term — unless the other prisoner somehow also decided to do that. Without any communication, that would be a highly risky guess, and thus, this strategy represents a Nash equilibrium.
When I teach game theory, I use a couple of other examples to illustrate the problems highlighted by game theory.
One, in the area of noncooperative games, has to do with the wages paid by capitalist employers: they want to pay their own workers low wages (to extract as much surplus as possible) but they want the workers of all other capitalists to be paid high wages (so they can purchase the commodities being produced). This is one of the basic contradictions of capitalism. These days, the noncooperative solution—even in the midst of declining unemployment—is to keep the wages of most workers low.
The other is the so-called ultimatum game, which, in the simplest form of the ultimatum game, involves a proposer who decides how much of a sum of money (say, $10) to give a responder, and the responder decides whether to accept or reject the offer. If the responder accepts, the players split the money in the way the proposer suggested. If the responder rejects, neither player gets any money. The neoclassical version of game theory predicts that, since both the proposer and the responder know that rejection of the offer results in neither receiving any money, the proposer will offer the smallest possible amount (anything greater than $0) and the responder will always accept. As it turns out, during actual experiments, participants choose outcomes that are much closer to a 50-50 split, which runs counter to the usual notion of self-interest—something we might call “fairness.”
But there’s an additional—to my mind, quite beautiful—outcome due to Robert H. Frank, Thomas Gilovich, and Dennis T. Regan: economists and students who have taken courses in neoclassical economics are much less likely to engage in cooperative behavior than noneconomists and students who have not been exposed to those ideas.