Mathematics and politics
Lecture notes, 2/20/03



Current assignments:

  • Today: Elements of Game Theory

    • Game Theory Introduction

    Prisoner's Dilemma
    • This is the most well-known example in game theory.
    • There are many variations.
    • This one is the original (I think):
      • There are two prisoners, isolated from each other.
      • The police have some evidence that they committed a crime together, but not much.
      • Each is given the choice between remaining silent and agreeing to testify against the other.
        • If both testify against each other, both get five years in prison.
        • If both remain silent, both get one year in prison.
        • If one testifies and the other remains silent, then the one who testifies goes free, and the one who remains silent gets ten years in prison.

    • Your assignment (part I):
      • Play the prisoner's dilemma ten times with ten different people.
      • Both of you choose simultaneously whether to cooperate (keep silent) or to not cooperate (testify).
      • Write down the name of the person you played against each time and the number of years you ended up with.
      • (If you both noncooperate, you both get 5 years; if you both cooperate, you both get 1 year; if you cooperate and opponent does not, you get 10 years; if you noncooperate and your opponent cooperates, you get 0 years.)
      • Remember the strategy you use; how do you decide which to do?
      • Lowest score wins, and gets a prize.
      • Rule: You can discuss whatever you want with your opponent but you can't see your opponent's score so far.

    • Discussion:
      • Who won?
      • What strategy did s/he use?
      • Who lost?
      • What strategy did s/he use?
      • Does this game reward cooperation or noncooperation?

    • Your assignment (part II):
      • Play the prisoner's dilemma ten times with ten different people.
      • Both of you choose simultaneously whether to cooperate (keep silent) or to not cooperate (testify).
      • Write down the name of the person you played against each time and the number of years you ended up with.
      • (If you both noncooperate, you both get 5 years; if you both cooperate, you both get 1 year; if you cooperate and opponent does not, you get 10 years; if you noncooperate and your opponent cooperates, you get 0 years.)
      • Remember the strategy you use; how do you decide which to do?
      • Lowest score wins, and gets a prize.
      • Rule: Each player must show the other his/her entire history.

    • Discussion:
      • Who won?
      • What strategy did s/he use?
      • Who lost?
      • What strategy did s/he use?
      • Does this game reward cooperation or noncooperation?

    • In the prisoner's dilemma, it's always better to testify. Why?
      • If the other player testifies, you can either testify or not.
        • If you also testify, you get 5 years in jail.
        • If you don't testify, you get 10 years in jail.
      • If the other player keeps silent, you can either testify or not.
        • If you testify, you get 0 years.
        • If you also keep silent, you get 1 year.
      • Either way, you get a better outcome by testifying.

    • But here's the paradox:
      • If both testify, both get 5 years in prison.
      • If both keep silent, both get 1 year in prison.
      • If both follow their optimal individual strategies, their joint outcome is worse for both.
    • We'll come back to this (and its political implications) on Tuesday.

    Two-by-two Ordinal Games in General

    • Definition: A two-by-two ordinal game is a game where two players each have two options.
    • The actual payoffs (e.g. years in jail) don't matter.
    • All that matters is the preferences of each player.
    • ("Ordinal" describes ranking, while "cardinal" describes actual values.)

    • Any two-by-two ordinal game can be represented as a matrix.
    • Example: the prisoner's dilemma as a two-by-two game.
      Two is
      silent      		 Two
      testifies
      One is
      silent
      One
      testifies
      One gets 1,
      Two gets 1      		 One gets 10,
      Two gets 0
      One gets 0,
      One gets 10      		 One gets 5,
      Two gets 5

    • To represent this as an ordinal game, we just look at the preferences.
    • Prisoner One clearly prefers 0, then 1, then 5, then 10. So does Prisoner Two.
    • Following Mathematics and Politics, we rank these in order from 1 to 4, 4 being the best.

    • Each prisoner's preferences:
      Prisoner One's preferences Prisoner Two's preferences
      Two is
      silent      		 Two
      testifies
      One is
      silent
      One
      testifies
      3 1
      4 2
      Two is
      silent      		 Two
      testifies
      One is
      silent
      One
      testifies
      3 4
      1 2

    • Both players' preferences together (the full two-by-two ordinal game):
      Two is
      silent      		 Two
      testifies
      One is
      silent
      One
      testifies
      (3,3) (1,4)
      (4,1) (2,2)

    Dominant strategies