PRISONER’S DILEMMA THE MATHEMATICS OF BETRAYAL (AND DATING AND VARIOUS OTHER THINGS) Math Club 9/26/2011.

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PRISONER’S DILEMMA THE MATHEMATICS OF BETRAYAL (AND DATING AND VARIOUS OTHER THINGS) Math Club 9/26/2011

Prisoner’s Dilemma?  Two criminals are arrested, but the police do not have enough information to convict them.  The two men are put into separate rooms and are given a deal by the police: to give information (in other words, betray the other) in exchange for freedom.  If neither prisoner betrays the other, then the police cannot convict them and has to let them both off with a light, 1 month sentence.  If one prisoner betrays the other, then the ‘betrayer’ gets away free, and the ‘betrayee’ gets a heavy, 1 year sentence.  If both prisoners betray each other, then both get a medium, 3 month sentence.

Payoff Matrix CooperateDefect Cooperate(3,3)(0,5) Defect(5,0)(1,1)

Let’s try this!  We will play five rounds.  The goal is to get as many points as possible, not to beat your opponent.

(Naïve) Mathematical Analysis  If they cooperate, you’re better off defecting.  If they defect, you’re still better off defecting.  Result: you should always defect.  Yet cooperating yields the most overall net gain… CooperateDefect Cooperate(3,3)(0,5) Defect(5,0)(1,1)

Dating…  … can be extremely problematic.

Dating  You arrange a date with a friend at the Stampede.  But when you get there, you discover that the Stampede is huge and you have no idea where to look for her.  You prefer to go to the horseracing show, while your date prefers to go to the cooking show.  However, both of you prefer to be with each other and go to the same place, rather than be alone.  You know that your date has to make the same decision.  Should you go to the horseracing show or the cooking show?

Payoff Matrix Cooperate (Horseracing) Defect (Cooking) Cooperate (Cooking)(1,1)(4,5) Defect (Horseracing)(5,4)(2,2)

Let’s try this!  We will play five rounds.  The goal is to get as many points as possible, not to beat your opponent.

(Naïve) Mathematical Analysis  If they cooperate, you’re better off defecting.  If they defect, you’re better off cooperating.  But if you cooperate, they’re better off defecting and if you defect, they’re better off defecting.  Result: ????????? Cooperate (Horseracing) Defect (Cooking) Cooperate (Cooking)(1,1)(4,5) Defect (Horseracing)(5,4)(2,2)

Did math fail us?!  Well, sort of.  But if we are God…

Kant’s Categorical Imperative  In other words, assume that everyone in the world acts and makes decisions in the same way that you do.  Then how should you act? "Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.“ – Immanuel Kant

Prisoner’s Dilemma  Clearly it’s better if everyone cooperates all the time than if everyone defects all the time. CooperateDefect Cooperate(3,3)(0,5) Defect(5,0)(1,1)

Dating Cooperate (Horseracing) Defect (Cooking) Cooperate (Cooking)(1,1)(4,5) Defect (Horseracing)(5,4)(2,2)

Some algebra

Optional Exercise  What if you’re ‘meh’ about your date and you would rather watch horseracing alone rather than cooking with your date?  How often should you cooperate, assuming Kant’s categorical imperative? Cooperate (Horseracing) Defect (Cooking) Cooperate (Cooking)(0,0)(2,5) Defect (Horseracing)(5,2)(3,3)