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Game theory: Prisoner’s Dilemma

                   Game theory: Prisoner’s Dilemma  

1 Here comes the problem 

Have you heard of Prisoner’s dilemma? Well, it’s something that has come to have groundbreaking influence in modern mathematics, economics, biology and politics. There are various versions of Prisoner’s dilemma. In this article, I’d like to talk to you about just one variant of it: the non-iterated Prisoner’s dilemma. Two people have been caught by the police for a particular crime. They have been kept in two different rooms for interrogation. Now, they are put forward with a situation. They should both either confess the crime and thus get a reduced crime sentence as Reward for Cooperation, or both accuse each other and thus both get a major prison sentence as Punishment for Defection. However the main issue of dilemma isn’t still there. There is an another condition that makes the whole problem very interesting, at least for us observers. If one of them chooses to accuse the other person but the the other person chooses to confess, the one accusing the other person will get to be free of shackles of the prison while the one who confessed would be given the maximum punishment. Now what would be the most logical thing to do in this case? 

Following table shows the whole dilemma in a beautiful way:



Now, look at the table (the points assigned are random points just so as to convey the severity of punishment or reward. A point of 0 means a very bad position while a point of 5 means the best possible position) considering that you are the person taking the decisions in the first column and that the other person is taking the decisions given in the first row. It may be tempting to see how you can choose to defect because it gives the highest amount of reward. But that depends on what the other person does. Because, if the other person also thinks the way that you are thinking(and presumably they think the same thing considering they are assessing the situation rationally), they are likely to think that defecting would the best possible strategy as well. That means you are sure to get a possible punishment for defection(point = 1). Can you somehow manipulate the other person to get to cooperate? Because, if both the persons cooperate it will lead to a better condition for both(point = 3). The problem lies in the fact that you cannot communicate and that you cannot persuade each other. And therefore, the whole decision taking process depends just on the speculation of what the other person might do.


 2 Solution

 Now, what’s the best possible choice then? Do you defect(accuse the other) or cooperate(confess the guilt)? While both can be benefited equally if both cooperate, this however isn’t the equilibrium of the case. Let’s look at this logically. That is because you are more likely to get a harsher punishment if you choose to cooperate on average. If you choose to cooperate and the other person chooses to defect, you get 0 points. If you choose to cooperate and the other person chooses to cooperate as well, you get 3 points. On average, cooperating gives you 1.5 points. 

(0+3)/2 = 1.5

 In contrast, if you choose to defect and the other person chooses to defect as well, you get 1 point. If you choose to defect and the other person chooses to cooperate you get 5 whooping points. 

On average, (1+5)/ 2 = 3 

On average, you get 3 point for defecting. Thus, defecting seems to be a better choice on average. Either you get a very good result or you get a fairly bad result. That’s in contrast to the very bad or a fairly good result for cooperating. 

3 Conclusion 

So, it seems like being ”bad” is what gets you most benefit. Does this always give you an edge?, you might ask. There is an iterated version of Prisoner’s Dilemma. It is where the decisions taken by your opponent on earlier similar encounters are known to you. What is the most rational decision then? Does ”niceness” take an edge on the ”badness” when iterations are possible and a same person can meet the other person twice and past reactions of opponents are available?(Spoiler: It does!) More on that later. For now, pause and ponder! Enjoy maths! 


-Manoj Dhakal

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