The Prisoner’s Dilemma

The prisoner’s dilemma is a very popular situation in crime fiction, and can be found in many areas of real life. It originated from Merrill Flood and Melvin Dresher, however Albert Tucker is credited for formalizing it into its current form. So let us take a look.

Two criminal partners, A and B, are arrested and detained. They cannot communicate with each other by any means. Though law enforcement could make the arrest on a lesser charge, the prosecutors lack the evidence to charge them with the principal crime. So, the prosecutor offers both prisoners a bargain.

If A and B both cooperate, they will both serve 5 years in prison.

If A cooperates and B does not, A will be set free while B serves 10 years. (This works the other way as well)

If both A and B remain silent, they both serve 2 years.

The question then becomes, “what is the best strategy to win?” This question turns out to not have an exact answer. Let us take a look at a table,

Prisoner's Dilemma

This table just takes the rules of the game and puts them in matrix form. For example, if both cooperate then go to row C and column C to find that both serve 5 years. Though there is no clear cut way to win this game, we can find the strategy that is in the best interest of prisoner A by finding the Nash Equilibrium. So in this situation, prisoner A must base his choice on his belief in the choice of prisoner B. If B cooperates, A will either serve 5 years by cooperating or 10 years by not cooperating. However if A believes that B will not cooperate, then A has the option of 0 years by cooperating, or 1 year by not cooperating. Clearly then, cooperating is the best choice for A. In the other room, B is doing the same calculation. Therefore, the Nash equilibrium of this game is for both A and B to cooperate. The prisoner’s dilemma is very similar to a philosophical concept known as Pascal’s Wager, which goes like this.

  • “I can believe that God exists or I cannot.”
  • “Either God exists or God does not exist.”
  • If I believe and God exists then I will spend an eternity in happiness.
  • If I believe and God does not exist, then I have neither lost nor gained anything .
  • If I do not believe and God does not exist then I have neither lost nor gained anything.
  • If I do not believe and God does exist, then I will spend eternity in despair.

If one can put emotions aside and analyze this, it is clear that in this game, believing is the dominant strategy. However, to step outside the mathematics, this wager makes countless number of assumptions and lacks proper precision. For example, believing is all it takes to spend eternity in happiness. Another example would be that it does not specify exactly which god to believe in for this game. I encourage the reader to make the table for this game and explore other assumption this wager makes. I also encourage the reader to identify the prisoner’s dilemma in real life!

 

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