The Prisoners’ Dilemma

The reward received by a player in a game, such as the profit earned by an oligopolist, is that player’s payoff.

Game theory deals with any situation in which the reward to any one player—the payoff—depends not only on his or her own actions but also on those of other players in the game. In the case of oligopolistic firms, the payoff is simply the firm’s profit.

A payoff matrix shows how the payoff to each of the participants in a two-player game depends on the actions of both. Such a matrix helps us analyze situations of interdependence.

When there are only two players, as in a duopoly, the interdependence between the players can be represented with a payoff matrix like that shown in Figure 14-1. Each row corresponds to an action by one player (in this case, ADM); each column corresponds to an action by the other (in this case, Ajinomoto). For simplicity, let’s assume that ADM can pick only one of two alternatives: produce 30 million pounds of lysine or produce 40 million pounds. Ajinomoto has the same pair of choices.

A Payoff Matrix Two firms, ADM and Ajinomoto, must decide how much lysine to produce. The profits of the two firms are interdependent: each firm’s profit depends not only on its own decision but also on the other’s decision. Each row represents an action by ADM, each column, one by Ajinomoto. Both firms will be better off if they both choose the lower output, but it is in each firm’s individual interest to choose the higher output.

The matrix contains four boxes, each divided by a diagonal line. Each box shows the payoff to the two firms that results from a pair of choices; the number below the diagonal shows ADM’s profits, the number above the diagonal shows Ajinomoto’s profits.

These payoffs show what we concluded from our earlier analysis: the combined profit of the two firms is maximized if they each produce 30 million pounds. Either firm can, however, increase its own profits by producing 40 million pounds while the other produces only 30 million pounds. But if both produce the larger quantity, both will have lower profits than if they had both held their output down.

Prisoners’ dilemma is a game based on two premises: (1) Each player has an incentive to choose an action that benefits itself at the other player’s expense (2) When both players act in this way, both are worse off than if they had acted cooperatively.

The particular situation shown here is a version of a famous—and seemingly paradoxical—case of interdependence that appears in many contexts. Known as the prisoners’ dilemma, it is a type of game in which the payoff matrix implies the following:

The original illustration of the prisoners’ dilemma occurred in a fictional story about two accomplices in crime—let’s call them Thelma and Louise—who have been caught by the police. The police have enough evidence to put them behind bars for 5 years. They also know that the pair have committed a more serious crime, one that carries a 20-year sentence; unfortunately, they don’t have enough evidence to convict the women on that charge. To do so, they would need each of the prisoners to implicate the other in the second crime.

So the police put the miscreants in separate cells and say the following to each: “Here’s the deal: if neither of you confesses, you know that we’ll send you to jail for 5 years. If you confess and implicate your partner, and she doesn’t do the same, we’ll reduce your sentence from 5 years to 2. But if your partner confesses and you don’t, you’ll get the maximum 20 years. And if both of you confess, we’ll give you both 15 years.”

Figure 14-2 shows the payoffs that face the prisoners, depending on the decision of each to remain silent or to confess. (Usually the payoff matrix reflects the players’ payoffs, and higher payoffs are better than lower payoffs. This case is an exception: a higher number of years in prison is bad, not good!) Let’s assume that the prisoners have no way to communicate and that they have not sworn an oath not to harm each other or anything of that sort. So each acts in her own self-interest. What will they do?

The Prisoners’ Dilemma Each of two prisoners, held in separate cells, is offered a deal by the police—a light sentence if she confesses and implicates her accomplice but her accomplice does not do the same, a heavy sentence if she does not confess but her accomplice does, and so on. It is in the joint interest of both prisoners not to confess; it is in each one’s individual interest to confess.

An action is a dominant strategy when it is a player’s best action regardless of the action taken by the other player.

The answer is clear: both will confess. Look at it first from Thelma’s point of view: she is better off confessing, regardless of what Louise does. If Louise doesn’t confess, Thelma’s confession reduces her own sentence from 5 years to 2. If Louise does confess, Thelma’s confession reduces her sentence from 20 to 15 years. Either way, it’s clearly in Thelma’s interest to confess. And because she faces the same incentives, it’s clearly in Louise’s interest to confess, too. To confess in this situation is a type of action that economists call a dominant strategy. An action is a dominant strategy when it is the player’s best action regardless of the action taken by the other player.

It’s important to note that not all games have a dominant strategy—it depends on the structure of payoffs in the game. But in the case of Thelma and Louise, it is clearly in the interest of the police to structure the payoffs so that confessing is a dominant strategy for each person. So as long as the two prisoners have no way to make an enforceable agreement that neither will confess (something they can’t do if they can’t communicate, and the police certainly won’t allow them to do so because the police want to compel each one to confess), Thelma and Louise will each act in a way that hurts the other.

PITFALLS: PLAYING FAIR IN THE PRISONERS’ DILEMMA

PITFALLS

PLAYING FAIR IN THE PRISONERS’ DILEMMA
One common reaction to the prisoners’ dilemma is to assert that it isn’t really rational for either prisoner to confess. Thelma wouldn’t confess because she’d be afraid Louise would beat her up, or Thelma would feel guilty because Louise wouldn’t do that to her.
But this kind of answer is, well, cheating—it amounts to changing the payoffs in the payoff matrix. To understand the dilemma, you have to play fair and imagine prisoners who care only about the length of their sentences.
Luckily, when it comes to oligopoly, it’s a lot easier to believe that the firms care only about their profits. There is no indication that anyone at ADM felt either fear of or affection for Ajinomoto, or vice versa; it was strictly about business.

So if each prisoner acts rationally in her own interest, both will confess. Yet if neither of them had confessed, both would have received a much lighter sentence! In a prisoners’ dilemma, each player has a clear incentive to act in a way that hurts the other player—but when both make that choice, it leaves both of them worse off. When Thelma and Louise both confess, they reach an equilibrium of the game. We have used the concept of equilibrium many times; it is an outcome in which no individual or firm has any incentive to change his or her action.

A Nash equilibrium, also known as a noncooperative equilibrium, results when each player in a game chooses the action that maximizes his or her payoff given the actions of other players, ignoring the effects of his or her action on the payoffs received by those other players.

In game theory, this kind of equilibrium, in which each player takes the action that is best for her given the actions taken by other players, and vice versa, is known as a Nash equilibrium, after the mathematician and Nobel laureate John Nash. (Nash’s life was chronicled in the best-selling biography A Beautiful Mind, which was made into a movie.) Because the players in a Nash equilibrium do not take into account the effect of their actions on others, this is also known as a noncooperative equilibrium.

Now look back at Figure 14-1: ADM and Ajinomoto are in the same situation as Thelma and Louise. Each firm is better off producing the higher output, regardless of what the other firm does. Yet if both produce 40 million pounds, both are worse off than if they had followed their agreement and produced only 30 million pounds. In both cases, then, the pursuit of individual self-interest—the effort to maximize profits or to minimize jail time—has the perverse effect of hurting both players.

Prisoners’ dilemmas appear in many situations. The upcoming For Inquiring Minds describes an example from the days of the Cold War. Clearly, the players in any prisoners’ dilemma would be better off if they had some way of enforcing cooperative behavior—if Thelma and Louise had both sworn to a code of silence or if ADM and Ajinomoto had signed an enforceable agreement not to produce more than 30 million pounds of lysine.

!worldview! FOR INQUIRING MINDS: Prisoners of the Arms Race

Between World War II and the late 1980s, the United States and the Soviet Union were locked in a seemingly endless struggle that never broke out into open war. During this Cold War, both countries spent huge sums on arms, sums that were a significant drain on the U.S. economy and eventually proved a crippling burden for the Soviet Union, whose underlying economic base was much weaker. Yet neither country was ever able to achieve a decisive military advantage.

As many have pointed out, both nations would have been better off if they had both spent less on arms. Yet the arms race continued for 40 years.

Why? As political scientists were quick to notice, one way to explain the arms race was to suppose that the two countries were locked in a classic prisoners’ dilemma. Each government would have liked to achieve decisive military superiority, and each feared military inferiority. But both would have preferred a stalemate with low military spending to one with high spending.

Caught in the prisoners’ dilemma: heavy military spending hastened the collapse of the Soviet Union.
©ITAR-TASS/Sovfoto

However, each government rationally chose to engage in high spending. If its rival did not spend heavily, its own high spending would lead to military superiority; not spending heavily would lead to inferiority if the other government continued its arms buildup. So the countries were trapped.

The answer to this trap could have been an agreement not to spend as much; indeed, the two sides tried repeatedly to negotiate limits on certain weapons. But these agreements weren’t very effective. In the end the issue was resolved as heavy military spending hastened the collapse of the Soviet Union in 1991.

Unfortunately, the logic of an arms race did not disappear. A nuclear arms race developed between Pakistan and India, two neighboring countries with a history of mutual antagonism. In 1998 both countries confirmed the unrelenting logic of the prisoners’ dilemma by publicly testing nuclear weapons in a tit-for-tat sequence.

However, by 2013 a glimmer of hope emerged, as the prime ministers of these South Asian nuclear rivals began a series of meetings aimed at making “a new beginning.”

But in the United States an agreement setting the output levels of two oligopolists isn’t just unenforceable, it’s illegal. So it seems that a noncooperative equilibrium is the only possible outcome. Or is it?