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The Prisoners' Dilemma: Using Game Theory to Achieve the Optimal Solution


I am a former maths teacher and owner of Doingmaths. I love writing about maths, its applications and fun mathematical facts.

John Von Neuman (1903–1957)

John Von Neuman (1903–1957)

What Is the Prisoners' Dilemma?

The Prisoners' Dilemma is an example of a problem analyzed in game theory where two rational people acting in their own best interests do not necessarily produce optimal results.

Game Theory

Before we jump into looking at the Prisoners' Dilemma, it will be useful to look into what game theory is.

Game theory is a fairly modern branch of mathematics, originally developed by the Hungarian-born American mathematician John Von Neumann (1903–1957) and the German-born American economist Oskar Morgenstern (1902–1977) to solve problems in economics. It is essentially the study of strategy—looking at how interacting choices by individuals produce outcomes based on the individuals’ preferences which may lead to outcomes not intended by any of the individuals.

The American mathematician John Nash (1928 – 2015) (of A Beautiful Mind fame) is best known for his work in expanding the field.

The Prisoners' Dilemma

The Prisoner’s Dilemma was developed by the Canadian mathematician Albert W. Tucker (1905–1995) in 1950, based on earlier work by Merrill Flood and Melvin Dresher, and goes like this:

Two criminals are arrested for a bank robbery, but unfortunately for the police, no hard evidence exists for the main crime; there is only enough evidence to convict on a lesser charge. The two prisoners are separated and questioned separately, with no means of communicating with each other. Each prisoner is given the option of betraying the other by testifying against them or cooperating with the other prisoner by remaining silent.

They are advised of the following outcomes:

1. If prisoner 1 testifies and prisoner 2 remains silent, then prisoner 1 will walk free while prisoner 2 will get 10 years in prison.

2. If prisoner 2 testifies and prisoner 1 remains silent, then prisoner 2 will walk free while prisoner 1 will get 10 years in prison.

3. If both prisoners betray each other and testify, they will each get 6 years in prison.

4. If both prisoners remain silent, they will each get 1 year in prison for the lesser charge.

These four outcomes can be displayed in a payoff matrix.

Prisoners' Dilemma Payoff Matrix


What Do the Prisoners Do?

You can see from the payoff matrix that individually the best option for either prisoner is to betray the other and testify. On average, this option gives the smaller prison sentences, 0 and 6 compared to 1 and 10. Also, whatever decision one prisoner makes, the other prisoner will be better off by betraying and testifying than they will be by remaining silent. Therefore it is highly likely that this is what each prisoner will do; betray the other prisoner and both get a six-year sentence.

The issue here is that despite this seemingly best of course of action, the prisoners would have been better off if they both trusted each other and remained silent. This is the dilemma; mutual cooperation gives the best result, but the rational choice individually is to betray.

The Prisoners' Dilemma on the DoingMaths YouTube channel

Implications of the Prisoners' Dilemma

We can see from the Prisoners’ Dilemma that often when individuals pursue their own self-interests, the outcome is worse than if they were to cooperate.

Real world examples of this include:

  • a business cutting prices in order to steal custom from a rival, but the rival then cutting their own prices too and both businesses losing out due to decreased profit margins;
  • an arms race between two countries where they both increase strength at the same time, hence have no net gain on each other on power, but the expenditure required to increase strength leaves them both worse off than if they had maintained the status quo;
  • any example of haggling over a shop purchase or negotiating a salary increase can also be modelled similarly.

Another example of game theory in action is attributed to the ancient Greek philosopher Socrates, who fought at the battle of Delium. He considered the problem facing an individual soldier on the front line, waiting for the enemy to attack. This soldier postulates that if the defence is going to be a success then his contribution towards that is unlikely to be essential, and there is a high chance that he will be injured or killed. He also realises that if the defence is going to fail, then his presence will have made no difference and he is also likely to be injured or killed. Either way, his chance of death or injury is high, while his individual contribution will be low. He therefore decides that the best option is to run away and avoid the battle.

The problem here is when each soldier reasons in much the same way. If they all run away, then the battle is definitely lost. The best decisions for each individual have led to the worst result for the group as a whole. It should be noted that one reason why the execution of deserters exists in so many armies is to reduce the ‘profit’ gained from running away. If running away definitely leads to death, then the incentive to stay and fight is increased.

An interesting extension of the Prisoners’ Dilemma is that the same basic rules apply to problems with any number of participants, not just the two in our problem.

Bibliography and Further Reading

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 David

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