Competition, Cooperation, and Game Theory

Game theory is a branch of mathematics that tries to model strategic interactions. One of the best-known examples is the Prisoner’s Dilemma, an imaginary scenario where two prisoners have committed a crime together but there’s not enough evidence to convict either without a confession. So the police take them to separate interrogation rooms and offer each of them the same deal: “If you confess and he doesn’t, we’ll give him 5 years and let you go free.”

In the unlikely event that you find yourself in that situation, your instinct might be to confess, even if it means selling out your accomplice. No honor among thieves, right? But not so fast: if you both confess, you both get three years. How things turn out isn’t entirely up to you. Can you trust your friend to stay quiet, faced with the same tempting offer you got?

So maybe you shouldn’t confess after all. Maybe you should just keep your mouth shut, and hope your friend does the same. That’s still not as good as ratting him out: if you both refuse to talk, the cops will get you both on some bogus charge that carries a one-year sentence. Which means that the overall situation is like this:
• If you turn your buddy in and he says nothing, you’re free to go.
• If you both stay quiet, you each get one year.
• If you both confess, it’s three years.
• And if he turns you in first, you get five.

What should you do?

Many game theorists would argue that your first instinct was the right one: confess immediately! After all, you have no control over the other guy, who’s off in another room doing whatever he’s going to do. Let’s assume he’s already made his decision. If he’s decided to stay quiet, your confession means you get off scot-free. If he’s decided to confess, your confession means you get three years, not five. Whatever he’s decided, your plan is the best one under the circumstances. (Sorry, friend!)

If both of you reason this way, you’ll end up in what game theorists call a “Nash equilibrium” (named for the economist memorably portrayed by Russell Crowe in the somewhat misleading film A Beautiful Mind). That’s a situation where nobody has an incentive to change their mind about what to do, even if they learn what everyone else plans to do.

And if both of you decide to keep quiet—maybe there is honor among thieves?—then that’s what game theorists call a “Pareto optimal” solution, where there’s no alternative that makes everyone better off.

All of this may sound pretty abstract, hypothetical, and math-y. And needless to say, we hope and trust you’ll never end up committing crimes! But game theory can actually be helpful in the real world. The Prisoner’s Dilemma was originally invented as a model of nuclear deterrence between countries. Game theory is also useful for thinking about business decisions, like price competition. And biologists have even claimed that it helps explain aspects of evolution, which makes sense, perhaps, given how Darwinian it can sound to talk about people rabidly defending their own interests.

But that raises a different kind of question: what kind of model is it for how we should actually behave? Fortunately we’ve got an actual game theorist to tell us: economist Jonathan Levin, who also happens to be the current president of Stanford University.