The Prisoner's Dilemma: When Everyone Acts Sensibly and Both Lose

Game Theory, Part 4 of 9

Santa Monica, California, 1950. Merrill Flood and Melvin Dresher, two mathematicians at the RAND Corporation, came up with a simple two-player game. That same day they asked two colleagues to play it a hundred times in a row: the economist Armen Alchian and the mathematician John Williams.

The theory Flood and Dresher knew well predicted the result: both players would choose to "defect" every time. The players didn't comply. According to William Poundstone's account of the story, both players cooperated in 60 of the 100 games, and mutual defection happened only 14 times. In the very first game, Williams wrote in the margin: "Hope he's bright."

The game was later given the name the whole world knows it by: the prisoner's dilemma.

A story about two prisoners

The name came from Albert Tucker, a mathematician at Princeton. In May 1950 he had to explain Flood and Dresher's idea to psychologists at Stanford. A table of payoffs wouldn't have grabbed anyone, so he told a story.

Here is a modern version, with numbers we have made up for illustration. Two accomplices are arrested and put in separate rooms. Each is offered a deal: confess and testify against the other. They can't talk to each other.

Take a minute to decide what you would do before reading on.

Why betraying is "always better"

Let's go through the table calmly, one case at a time.

Suppose your accomplice stays silent. If you stay silent too, you get a year. If you confess, you walk free. Confessing is better.

Now suppose your accomplice confesses. If you stay silent, you get ten years. If you confess, five. Again, confessing is better.

Whatever the other person does, confessing gets you a shorter sentence. Game theorists call a move like this a dominant strategy: it is better against every possible reply. But your accomplice reasons the same way. Both confess and get five years each — when they could have had one.

That is the dilemma. Each acts perfectly sensibly for themselves, and together they arrive at an outcome that is worse for both. Individual logic and shared interest pull apart.

Any game has the same skeleton if four conditions hold. Betraying someone who trusts you is the best outcome (the temptation). Mutual cooperation comes second (the reward).

Mutual betrayal comes third (the punishment). And trusting a betrayer is the worst (the sucker's payoff). The numbers can be anything; only the order matters.

Why chess logic doesn't work here

In the previous part we saw that at the chessboard it makes sense to assume your opponent's worst move for you. There, your gain is their loss, so they "wish you the worst" simply by playing for themselves.

The prisoner's dilemma is different. It isn't zero-sum: both can win together (a year instead of five) or lose together. Your accomplice isn't your enemy.

They, too, are better off if both stay silent. That is exactly why the outcome hurts: betrayal doesn't make anyone stronger, it just destroys what both could have had.

There is one more subtlety. "Both confess" is a very stable position: neither gains by changing their choice alone. Points of balance like this have a name. It belongs to the mathematician the next part of the series is about.

How to spot a dilemma without a prison

The prison story is only packaging. The same game is easy to miss when it is dressed in something ordinary. Three signs are worth checking:

1. Everyone decides on their own, and no promise is binding.
2. Each person, individually, does better by "defecting", whatever the others do.
3. If everyone defects, each gets less than if everyone had cooperated.

Here is an example from sport. Two cyclists break away from the main group. If they take turns at the front, sheltering each other from the wind, the breakaway has a chance of reaching the finish.

But each would rather sit on the other's wheel and save energy for the final sprint. If both do that, the group catches them, and neither of the two wins.

Or an everyday one: a shared office kitchen. It is easier for each person not to wash their mug straight away — it takes five minutes, and a clean kitchen belongs to everyone. When everyone thinks that way, everyone ends up with a dirty kitchen. Here there are many players rather than two, but the logic is the same.

The opposite matters too: not every conflict is a prisoner's dilemma. A game of chess isn't one, because there is no shared gain to lose. A situation where cooperating is better whatever the other person does isn't one either, because there is no temptation. A dilemma appears exactly where temptation and shared interest pull in opposite directions.

What real people do

If people behaved as the table suggests, everyone would defect. Experiments show something else.

In 1995, David Sally analysed 35 years of laboratory studies of the prisoner's dilemma, from 1958 to 1992. His conclusion: a model of pure self-interest usually doesn't explain what participants actually do. Results are furthest from the theory when participants are allowed to talk to each other.

But laboratory games are usually played for small sums. What happens when the stakes are high?

In the British TV game show Golden Balls (ITV, 2007–2009), the final always came down to one decision. Two contestants secretly chose a ball marked "Split" or "Steal":

• both split — each takes half the jackpot;
• one splits, the other steals — the stealer takes everything, and the splitter gets nothing;
• both steal — both go home empty-handed.

The economists Martijn van den Assem, Dennie van Dolder and Richard Thaler analysed 287 episodes — 574 finalists. The average final jackpot was more than £13,000. Even so, 52.8% of contestants chose "Split".

So roughly half of people cooperate even when betrayal is worth real thousands. A purely "rational" player from the table doesn't make choices like that. Real people do.

What changes the game

Why do people so often depart from the "correct" strategy? One answer is that they aren't really playing the game drawn in the table. They have a reputation, a conscience, a wish to be someone who doesn't let others down. All of these are "payoffs" too, and the table can't see them.

A second answer, according to Poundstone, came from John Nash himself. Flood showed him the results of the experiment, and Nash objected: a hundred games in a row aren't a hundred separate games but one big game with many moves. The games aren't independent: your opponent will remember today's betrayal tomorrow. What happens when the dilemma repeats is the subject of Part 6.

This leads to a practical lesson that works far beyond the board. When you see a dilemma, the useful question isn't "how do I outsmart the other person?" but "how do I change the game?"

Make it repeated. Let the sides talk. Change the payoffs so that betrayal stops paying. The best move in a prisoner's dilemma is often made before the game begins.

Exercise

1. Change one number in the table: now the person who confesses while the other stays silent gets 2 years instead of 0. Is there still a dominant strategy? Is it still a prisoner's dilemma?
2. Is the Golden Balls final a true prisoner's dilemma? Look carefully at what the stealer gets when the other player steals too.

Answers are at the end.

Questions for discussion

• Think of a time when "defecting" would have paid off for you, but you didn't do it. What stopped you?
• If the Golden Balls contestants hadn't been on television, would they have split just as often?
• Where in your life could you "change the game" so that cooperation pays for both sides?

Next in the series: "Nash Equilibrium", on a state that nobody wants to be the first to leave — even when it is bad for everyone.

Answers. 1) There is no dominant strategy any more. If your accomplice stays silent, you are better off staying silent too (one year instead of two). If they confess, you are better off confessing (five instead of ten). Now the best move is to do whatever the other person does, so everything depends on trust. This is no longer a prisoner's dilemma but another classic game, the stag hunt. 2) Almost, but not quite. If the other player steals, you get nothing whether you chose "Split" or "Steal". So stealing is never worse, but it isn't always better. In a strict prisoner's dilemma, defecting is better against every reply.