also cooperates (top-left cell, second number) versus $35 if he defects ( top-right cell, second number).

Similarly if Cain believes Abel will defect (i.e., choose the bottom row), then Cain gets $15 by cooperating versus $20 by defecting. Thus no matter what Abel does, Cain is better off defecting! Similarly, Abel prefers to defect no matter what Cain does, and so rational players both defect. This is what’s known as a Nash equilibrium: it’s a pair of strategies, one for each player, such that neither player can improve his or her payoff by doing something else. So for our game, it is an equilibrium for Cain and Abel to both defect, and hence that’s the behavior we expect to see. (In fact, it’s also a dominant strategy equilibrium, i.e., each player’s strategy is a best-response to all strategies of the opponent, not just the Nash equilibrium strategy of the opponent.)

If you doubt this outcome, try to get two friends to agree to the following experiment: have the first friend (the one with deeper pockets; we’ll call him/her the banker) give you and the second friend two envelopes and two $20 dollar bills. You and the second friend will each take an envelope and a $20 dollar bill and go to separate rooms. There you will each either put the $20 bill in the envelope or keep it for yourself. At night, the banker should collect the envelopes, take out the money, increase the total by 50 percent, and then redistribute the money evenly among the two envelopes. Thus if both you and your friend put $20 in the envelope (this corresponds to “cooperating” in the game discussed above), the banker will collect $40 at night, increase this amount to $60, and then you’ll each have $30 in your envelopes in the morning.

On the other hand, if neither of you puts money in the envelope (which corresponds to ‘defecting’), you’ll each end up with just $20 in the morning. If only one of you puts money in the envelope, say your friend, then the banker will collect $20 at night, increase it to $30, and put $15 in each envelope. Thus your friend wakes up with only $15 whereas you get $35! Similar to the game here, no matter

“By casting the
world into a simple
mathematical
framework, we are
able to illuminate key
aspects of society. We
learn that evolving
friendships are
important—provably
so—for a smoothly
functioning society.”

what your friend does, you get the most money by keeping it all to begin with. If you’re rational, you will defect. (This game works well when you play it with someone you don’t like all that much, or better, if you don’t know him/her at all and expect never to see him/her again; otherwise you may rationally cooperate expecting him/her to spend his/her money buying you coffee.)

We have admittedly abstracted our setting quite far from the original story of bank robbery, but, and this is the art of modeling, the essence of the issue is still there: it is very tempting to rob a bank or, more generally, cheat in life. And this issue is very important. It appears throughout our lives, from personal dilemmas to political quandaries. In fact, a general form of this game, called the Prisoner’s Dilemma, was introduced during the Cold War to study arms races between the United States and the Soviet Union, and has been extensively studied ever since by economists, social scientists, and politicians. And the results are by-and-large upheld: small perturbations of the model do not significantly affect our dire prediction. Rational people should rob banks.

 

CONFLICTS BET WEEN THE MODEL AND REALITY Rational people don’t rob banks. This conflict between theory and practice has puzzled researchers for decades.

If people don’t rob banks, there must be some essential element of interaction that our simple model fails to capture. What is it? Is there some model in which rational people don’t rob banks, or must we abandon our assumption of rationality? If you want to get a feel for what it’s like to be a researcher in this area, stop for a moment and compile a list of things that bother you about our simple model. Contemplate whether addressing these concerns would intuitively advert our dire prediction, and how these concerns might be incorporated mathematically. Question whether the components you’ve now incorporated are themselves realistic, or whether the assumptions you’ve made are so strong that they essentially assume what you want to derive. Now play with your new mathematical model and try to see what it predicts. This is computer science and economics research, and this is what my colleagues and I do on a daily basis.

I now want to tell you about some answers for this paradox in the literature. One clear flaw in our story with Cain and Abel is that these games aren’t played once. They are played many times throughout our lives, and often multiple times with the same partner. In other words, the game is repeated and this repetition causes people to cooperate today so that their partners will continue to cooperate in the future. Furthermore, these games aren’t played with just one other partner, but with an evolving set of partners, and again this causes people to cooperate so that others will want to play with them. I will explain mathematically how each of these intuitions solves the paradox in our game.

 

REPEATED GAMES

Let’s consider the idea that games are repeated. To model this, suppose that every day Cain and Abel toss a fair coin. If the coin comes up heads, they play their bank-robber game. If it comes up tails, they stop playing. This complicates matters significantly. In particular, Cain and Abel can tailor their actions on the history of the game. If Cain’s been nice in the past and cooperated, perhaps Abel is more likely to cooperate. Consider a strategy in which a player cooperates on day zero