So here is the famous prisoner’s dilemma, the basic problem which is used to kick off all introductions to game theory and which finds a surprisingly large number of applications.
We start with two people lets call them Alice and Bob. Alice and Bob collaborate to execute a robbery. The police are able to catch them, but find that they have insufficient evidence to convict them for robbery which would carry a jail term of 5 years. Instead, they figure that they can only convict the duo for a lesser crime (say stealing the getaway car) which carries a jail term of only 2 years.
So the police offer Alice a deal. If she collaborates and testifies against Bob, they will send Bob to prison for 5 years and let Alice go free without pressing any charges. They also tell Bob that if he testifies against Alice, he will be allowed to go free and Alice will be sent to prison for 5 years.But if they both testify, then they will get a jail term of 4 years each.
Alice and Bob are not allowed to communicate with each other. They must individually make up their mind whether to cooperate or betray their partner. So now we can have 4 possible choices.
1. Both Alice and Bob cooperate with each other and do not testify. In this case, they each get 3 years of jail. We can represent this by saying that the payoff for each of them is -2.
2. Alice agrees to testify by Bob doesn’t. Then Alice goes free but Bob gets 5 years in jail. So Alice’s payoff is 0 and Bob’s is -5.
3. Alice does not testify but Bob testifies. Then Alice gets 5 years in jail and Bob goes free. So Alice’s payoff is -5 and Bob’s is 0.
4. They both testify. Then both go to prison for 4 years. So both have a payoff of -4.
Now lets imagine both our criminals are rational and want to minimise the time they spend in jail themselves. So what should they do? This is the dilemma.
Alice thinks for 2 cases:
Case 1: Bob does not testify. Now either she can avoid testifying and go to jail for 2 years (option 1) or she could testify and escape (option 2). So in this case, she would testify.
Case 2: Bob testifies. Now, if Alice does no testify, then she will go to jail for 5 years (option 3). But if she testifies, she will only have to go to jail for 4 years (option 4). So she should testify.
So we see that irrespective of what Bob chooses to do, Alice will choose to testify against him. Bob will think similarly and always choose to testify against Alice. So we see that both of them end up testifying and going to prison for 4 years whereas if they had cooperated they could have only gone for 2 years.
This is a strange paradoxical situation where both of them are trying to make the best deal for themselves but end up worse than if they had both chosen their second best option.
This is related to another problem which is called the Tragedy of the Commons. When two people share a limited common resource. It is in each of their interests to exploit as much of it as possible before the other gets it. So both will move quickly and the limited supply is likely to get exhausted quickly giving less benefit. This problem is seen often in ecological issues.