Prisoner’s Dilemma: What Game Are you Playing?

In this classic game theory experiment, you must decide: rat out another for personal benefit, or cooperate? The answer may be more complicated than you think.

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What does it take to make people cooperate with each other when the incentives to act primarily out of self-interest are often so strong?

The Prisoner’s Dilemma is a thought experiment originating from game theory. Designed to analyze the ways in which we cooperate, it strips away the variations between specific situations where people are called to overcome the urge to be selfish. Political scientist Robert Axelrod lays down its foundations in The Evolution of Cooperation:

Under what conditions will cooperation emerge in a world of egoists without a central authority? This question has intrigued people for a long time. And for good reason. We all know that people are not angels and that they tend to look after themselves and their own first. Yet we also know that cooperation does occur and that our civilization is based on it. But in situations where each individual has an incentive to be selfish, how can cooperation ever develop?

…To make headway in understanding the vast array of specific situations which have this property, a way is needed to represent what is common to these situations without becoming bogged down in the details unique to each…the famous Prisoner’s Dilemma game.

The thought experiment goes as such: two criminals are in separate cells, unable to communicate, accused of a crime they both participated in. The police do not have enough evidence to sentence both without further evidence, though they are certain enough to wish to ensure they both spend time in prison. So they offer the prisoners a deal. They can accuse each other of the crime, with the following conditions:

  • If both prisoners say the other did it, each will serve two years in prison.
  • If one prisoner says the other did it and the other stays silent, the accused will serve three years and the accuser zero.
  • If both prisoners stay silent, each will serve one year in prison.

In game theory, the altruistic behavior (staying silent) is called “cooperating,” while accusing the other is called “defecting.”

What should they do?

If they were able to communicate and they trusted each other, the rational choice is to stay silent; that way each serves less time in prison than they would otherwise. But how can each know the other won’t accuse them? After all, people tend to act out of self-interest. The cost of being the one to stay silent is too high. The expected outcome when the game is played is that both accuse the other and serve two years. (In the real world, we doubt it would. After they served their time, it’s not hard to imagine each of them still being upset. Two years is a lot of time for a spring to coil in a negative way. Perhaps they spend the rest of their lives sabatoging each other.)

The Iterated Prisoner’s Dilemma

A more complex form of the thought experiment is the iterated Prisoner’s Dilemma, in which we imagine the same two prisoners being in the same situation multiple times. In this version of the experiment, they are able to adjust their strategy based on the previous outcome.

If we repeat the scenario, it may seem as if the prisoners will begin to cooperate. But this doesn’t make sense in game theory terms. When they know how many times the game will repeat, both have an incentive to accuse on the final round, seeing as there can be no retaliation. Knowing the other will surely accuse on the final round, both have an incentive to accuse on the penultimate round—and so on, back to the start.

Gregory Mankiw summarizes how difficult it is to model cooperation in Business Economics as follows:

To see how difficult it is to maintain cooperation, imagine that, before the police captured . . . the two criminals, [they] had made a pact not to confess. Clearly, this agreement would make them both better off if they both live up to it, because they would each spend only one year in jail. But would the two criminals in fact remain silent, simply because they had agreed to? Once they are being questioned separately, the logic of self-interest takes over and leads them to confess. Cooperation between the two prisoners is difficult to maintain because cooperation is individually irrational.

However, cooperative strategies can evolve if we model the game as having random or infinite iterations. If each prisoner knows they will likely interact with each other in the future, with no knowledge or expectation their relationship will have a definite end, the cooperation becomes significantly more likely. If we imagine that the prisoners will go to the same jail or will run in the same circles once released, we can understand how the incentive for cooperation might increase. If you’re a defector, running into the person you defected on is awkward at best, and leaves you sleeping with the fishes at worst.

Real-world Prisoner’s Dilemmas

We can use the Prisoner’s Dilemma as a means of understanding many real-world situations based on cooperation and trust. As individuals, being selfish tends to benefit us, at least in the short term. But when everyone is selfish, everyone suffers.

In The Prisoner’s Dilemma, Martin Peterson asks readers to imagine two car manufacturers, Row Cars and Col Motors. As the only two actors in their market, the price each sells cars at has a direct connection to the price the other sells cars at. If one opts to sell at a higher price than the other, they will sell fewer cars as customers transfer. If one sells at a lower price, they will sell more cars at a lower profit margin, gaining customers from the other. In Peterson’s example, if both set their prices high, both will make $100 million per year. Should one decide to set their prices lower, they will make $150 million while the other makes nothing. If both set low prices, both make $20 million. Peterson writes:

Imagine that you serve on the board of Row Cars. In a board meeting, you point out that irrespective of what Col Motors decides to do, it will be better for your company to opt for low prices. This is because if Col Motors sets its price low, then a profit of $20 million is better than $0, and if Col Motors sets its price high, then a profit of $150 million is better than $100 million.

Gregory Mankiw gives another real-world example in Microeconomics, detailed here:

Consider an oligopoly with two members, called Iran and Saudi Arabia. Both countries sell crude oil. After prolonged negotiation, the countries agree to keep oil production low in order to keep the world price of oil high. After they agree on production levels, each country must decide whether to cooperate and live up to this agreement or to ignore it and produce at a higher level. The following image shows how the profits of the two countries depend on the strategies they choose.

Suppose you are the leader of Saudi Arabia. You might reason as follows:

I could keep production low as we agreed, or I could raise my production and sell more oil on world markets. If Iran lives up to the agreement and keeps its production low, then my country ears profit of $60 billion with high production and $50 billion with low production. In this case, Saudi Arabia is better off with high production. If Iran fails to live up to the agreement and produces at a high level, then my country earns $40 billion with high production and $30 billion with low production. Once again, Saudi Arabia is better off with high production. So, regardless of what Iran chooses to do, my country is better off reneging on our agreement and producing at a high level.

Producing at a high level is a dominant strategy for Saudi Arabia. Of course, Iran reasons in exactly the same way, and so both countries produce at a high level. The result is the inferior outcome (from both Iran and Saudi Arabia’s standpoint) with low profits in each country. This example illustrates why oligopolies have trouble maintaining monopoly profits. The monopoly outcome is jointly rational for the oligopoly, but each oligopolist has an incentive to cheat. Just as self-interest drives the prisoners in the prisoners’ dilemma to confess, self-interest makes it difficult for the oligopoly to maintain the cooperative outcome with low production, high prices and monopoly prices.

Other examples of prisoners’ dilemmas include arms races, advertising, and common resources (see The Tragedy of the Commons). Understanding the Prisoner’s Dilemma is an important component of the dynamics of cooperation, an extremely useful mental model.

Thinking of life as an iterative game changes how you play. Positioning yourself for the future carries more weight than “winning” in the moment.