Game Theory 1.0 — Prisoner’s dilemma(Part I)

Let’s start with a brief introduction about what Game theory is all about. So,Game theory is a branch of mathematics which deals with analysing the strategies in a competitive situation where the move of one player depends on the actions of the other player. In other words it is the study of strategically interdependent behaviour.

Source

Game theory is used widely in deciding government policies, decision making during war, economic policies, pricing strategies etc.

Interesting fact — Game theory is used by Uber for its pricing strategy.

What we will cover?

In these series of lectures we will talk about strict dominance, pure strategy Nash equilibrium, mixed strategy Nash equilibrium, iterated elimination of strictly dominated strategies, the mixed strategy algorithm, weak dominance, backward induction, subgame perfect equilibrium, comparative statics, and much more. So please make this an interactive blog series where we dive deep into Game theory and make it a win-win situation for both of us!

Why we need Game theory?

Game theory helps us to understand and analyse the actions of other competitors and act accordingly. It doesn’t necessarily have to be a win-loss situation. It can also be a win-win situation. For example — if we talk about the two giants Wal-Mart and Unilever, both are in equilibrium which means if Unilever makes the best decision possible, taking into account Wal-Mart’s decision and Wal-Mart will also make the best decision taking into account Unilever’s decision, which will in turn create a win-win situation for both the players.

With Game theory we will be able to make good government policies by understanding people’s need and Government’s budget. It also helps us in building strategies during war by analysing the opposition. It is also used in corporate for pricing strategies based on several parameters.

Now let’s start with the first topic in Game theory which is very interesting. It’s called the prisoner's dilemma.

Prisoner’s dilemma

This is a little bit tricky and you will have a lot of questions about the solution of the problem. So before you have those questions in mind, I want you to remember that it’s a “real world scenario”. Don’t worry if you didn’t get my point, so lets dive deep.

The Situation

  • Two suspects are arrested by the cops.
  • Police thinks that the suspects were trying to rob a store but they only have proof of trespassing.
  • The police wants one criminal to rat out the other.

The Deal

  • If none of them confesses to the crime, then the police can charge them only for trespassing which is- 1 month.
  • If one of them confesses to the crime and the other doesn’t, then the police will be lenient on the rat and will not punish him while the police will punish the person who didn’t confess for the crime for 12 months.
  • Another scenario where both of them confess to the crime, they will be sentenced to 8 months in prison.

So, I just gave you a whole lot of information regarding the problem which is a bit difficult to visualise, so what we will do is to create a matrix for easy understanding of the problem.

Source — Game Theory 101

So, as you can see we have mapped the results into this simple matrix. If both player 1 and player 2 keep quiet, then they both get 1 month jail for trespassing, while if player 1 keep quiet and player 2 confesses then player 1 gets 12 months jail while player 2 is set free. Coming to the third scenario where player 1 confesses but player 2 keep quiet, we can see player 1 is set free and player 2 gets 12 months of jail. The last scenario is when both of them confesses, they both serve 8 months in jail.

Now the question lies “If the suspect wants to minimise his jail term, what should they do? Keep quiet or confess.

Before you go forward and read the solution, I want you to write in the comment section about what could be the possible solution?

I hope you guys have written your solution in the comment section. So, lets dive deep without wasting more time.

Scenario I

Let’s assume a scenario when Player 1 knows that Player 2 will keep quiet.

So, in this case if player 1 confesses then he will be set free and if player 1 keeps quiet, then he serves one month in jail. Here, it’s clear that player 1 is better of confessing.

Scenario II

Let’s assume a scenario when Player 1 knows that Player 2 will confess.

Here, if player 1 confesses, he gets 8 months of jail and if player 1 keeps quiet, he gets 12 months of jail. In this case, Player 1 is better of confessing.

So, in both the scenarios we can see a pattern that whatever Player 2 does, Player 1 is always better of confessing.

Scenario III

Let’s assume a scenario when Player 2 knows that Player 1 will confess.

Same as Scenario II, player 2 is better of confessing, as he serves less months in jail when confessing in the case Player 1 confesses.

Scenario IV

Let’s assume a scenario when Player 2 knows that Player 1 will keep queit.

Same as Scenario I, player 2 is better of confessing as he serves less months in jail when player 1 keep quiet.

So, in both the scenarios we can see a pattern that whatever Player 1 does, Player 2 is always better of confessing.

So, in all the 4 situations we discussed, the players are better of confessing than keeping quiet. So, it is the best solution for them to confess and serve 8 months jail each.

Now, many people will ask me Why not keeping quiet a better solution for both of them as they only serve 1 month of jail term each?

If you see the matrix once again carefully, (-1, -1) is not the optimal solution because they can always do better individually by confessing.

I know it sounds confusing. So, if you have any doubts regarding Prisoner’s dilemma or the article, please feel free to reach out in the comment section.

You can also contact me through my twitter handle or email.

Hope you guys enjoy it, pleased stay tuned for Part II of this series where I will cover — Iterated Elimination of Strictly Dominated Strategies and Nash Equilibrium.

AI Engineer. Deep learning enthusiast and an avid tech follower.