Nice to see you in the "Let's play a game" series! In this series of posts I'm gonna show you a number of applications of game theory. You may not have realized that game theory can be applied in a wide range of areas, including but not limited to economics, biology, mathematics and computer science. I hope every reader, no matter what your background is, can have a good moment in reading my posts and so I will try making each post precise without complex maths. If you like my posts, please upvote, resteem and follow me @manfredcml : )

Prisoner's Dilemma ----- Betray or not?

(Picture source: https://pixabay.com/en/fence-freedom-prison-hands-fingers-2163951/)

You are Andy. Your friend Brian and you are arrested for a theft and a robbery. Unfortunately the prosecutor does not have sufficient evidence to charge you for the robbery. You are delighted at first but soon you struggle to make a decision, when the prosecutor says the following to you:

1. If you and Brian both betray by claiming the other for having committed the crime, both of you will be sentenced to 5 years in prison.
2. If you betray Brian by claiming that Brian committed the crime while Brian remains silent, you will be freed and Brian will be sentenced to 10 years in prison.
3. If you remain silent while Brian betrays you by claiming that you committed the crime, you will be sentenced to 10 years in prison and Brian will be freed.
4. If both of you remain silent, both of you will be sentenced to 1 year in prison only.

Of course, the prosecutor says the same words to Brian. You and Brian are in 2 separate rooms so you have no way to communicate with Brian. At first, you think that if both of you don't say a word, you both are better off. But what if Brian betrays you? You will be in jail for 10 years! Now, what will you do? Betray or not?

An optimal way?

Let us draw a simple table as follow to see how Andy should act:

There are only 2 actions available, namely "betray" and "be silent". If Brian betrays you, the best response is that you betray him as well. Why? If Brian betrays and you remain silent, you will be in prison for 10 years. But if you respond by betraying him as well, you are in prison for 5 years only. On the other hand, if Brian remains silent, you will be freed if you betray him while you will be in jail for 1 year if you don't say a word. Thus, the best response to Brian remaining silent is to betray him.

Let's wrap up!

We arrive at a conclusion: no matter what Brian chooses, you should betray him. But Brian will think in the same way. So what will happen at last? Both of you betray each other and end up being sentenced to jail for 5 years. You and Brian both act the best response and none of you have the incentive to change the strategy. In game theory, such a state is known as a Nash equilibrium --- all players act the best responses and no one can be better off by unilaterally changing strategy without making others worse off.

Is this result optimal? Clearly NOT. We know that if both remain silent, both can be better off. But this is the magic of Prisoner's Dilemma --- if you and other players each acts the best response to others' strategies, all players will end up in a Nash equilibrium state which is not socially optimal.

A short film

Recently I have seen a great short film on Youtube about Prisoner's Dilemma and I would like to share with you : )

Coming soon...

In the next post, I will share a famous game called "Matching Pennies" with you and we will look at how each player should act. See you!

References

The original version of Prisoner's Dilemma is due to Merrill Flood and Melvin Dresher in 1950, according to Wikipedia.

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I write articles on machine learning, applied statistics and economics to the best of my knowledge : ) If you like my posts, please upvote, resteem and follow me @manfredcml.

Other articles:
Be a smart gambler! #1 - Gambler's fallacy
Paradox is fun! #1 - Boy or Girl?