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More than you ever Wanted to Know about the Prisoner's Dilemma

Ad1e78459902691e92658a3ba3c58778?s=47 aaboyles
July 04, 2012

More than you ever Wanted to Know about the Prisoner's Dilemma

A basic to mid-level overview of the Prisoner's Dilemma (and game theory) tailored to my Georgia Tech cohort's knowledge base.

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aaboyles

July 04, 2012
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Transcript

  1. More than you ever wanted to know about The Prisoner’s

    Dilemma Tony Boyles Georgia Institute of Technology
  2. What we’re doing here The Three Models of the Prisoner’s

    Dilemma • Normal Form Game • Game of Mixed Strategy • Repeated Game …As Applied to Nuclear Strategy, And what’s really goes on.
  3. The Prisoner’s Dilemma

  4. The Prisoner’s Dilemma Cooperate Defect Cooperate 3, 3 1, 4

    Defect 4, 1 2, 2 Player 1 Player 2
  5. The Prisoner’s Dilemma Cooperate Defect Cooperate 3, 3 1, 4*

    Defect *4, 1 *2, 2* Player 1 Player 2 Nash Equilibrium
  6. The Prisoner’s Dilemma, Generalized Cooperate Defect Cooperate Good Worst, Best

    Defect Best, Worst Bad Player 1 Player 2
  7. The Prisoner’s Dilemma, Generalized Cooperate Defect Cooperate C, C A,

    D Defect D, A B, B Player 1 Player 2 If we know that (Defect, Defect) is the Nash Equilibrium, for what values of A, B, C, and D is this a Prisoner’s Dilemma?
  8. The Prisoner’s Dilemma, Generalized Cooperate Defect Cooperate C, C A,

    D Defect D, A B, B Player 1 Player 2 D > C C > B B > A So, D > C > B > A
  9. Is This A Prisoner’s Dilemma? Cooperate Defect Cooperate 4, 4

    2, 13 Defect 13, 2 1, 1 Player 1 Player 2
  10. Is This A Prisoner’s Dilemma? Cooperate Defect Cooperate 4, 4

    *2, 13* Defect *13, 2* 1, 1 Player 1 Player 2 Nope!
  11. Is Nuclear War a Prisoner’s Dilemma? Cooperate Defect Cooperate 0,

    0 -100, 10 Defect 10, -100 -90, -90
  12. Is Nuclear War a Prisoner’s Dilemma? Cooperate Defect Cooperate 0,

    0 -100, 10 Defect 10, -100 -90, -90
  13. Is Nuclear War a Prisoner’s Dilemma? Cooperate Defect Cooperate 0,

    0 -100, 10 Defect 10, -100 -90, -90 Substantive Assumptions: • Threat is a nontrivial factor • Decision makers view themselves as better off if everyone suffers
  14. Could a Nuclear War have Happened? p (1-p) Cooperate Defect

    q Cooperate 0, 0 -100, 10 (1-q) Defect 10, -100 -90, -90
  15. Could a Nuclear War have Happened? p (1-p) Cooperate Defect

    q Cooperate 0, 0 -100, 10 (1-q) Defect 10, -100 -90, -90 E(Cooperate) = 0(p) + -100(1-p)
  16. Could a Nuclear War have Happened? p (1-p) Cooperate Defect

    q Cooperate 0, 0 -100, 10 (1-q) Defect 10, -100 -90, -90 E(Cooperate) = 0(p) + -100(1-p) E(Defect) = 10(p) + -90(1-p)
  17. Could a Nuclear War have Happened? E(Cooperate) = 0(p) +

    -100(1-p) E(Defect) = 10(p) + -90(1-p)
  18. Could a Nuclear War have Happened? E(Cooperate) = 0(p) +

    -100(1-p) E(Defect) = 10(p) + -90(1-p) 0(p) + -100(1-p) = 10(p) + -90(1-p)
  19. Could a Nuclear War have Happened? E(Cooperate) = 0(p) +

    -100(1-p) E(Defect) = 10(p) + -90(1-p) 0(p) + -100(1-p) = 10(p) + -90(1-p) -100 + 100p = 10p – 90 + 90p
  20. Could a Nuclear War have Happened? E(Cooperate) = 0(p) +

    -100(1-p) E(Defect) = 10(p) + -90(1-p) 0(p) + -100(1-p) = 10(p) + -90(1-p) -100 + 100p = 10p – 90 + 90p -10 = 0
  21. Could a Nuclear War have Happened? E(Cooperate) = 0(p) +

    -100(1-p) E(Defect) = 10(p) + -90(1-p) 0(p) + -100(1-p) = 10(p) + -90(1-p) -100 + 100p = 10p – 90 + 90p -10 = 0 There does not exist any possible probability value which satisfies this relationship!
  22. What if we could assign a probability?

  23. What if we could assign a probability? Pr(No Attack today)

    = p = .999
  24. What if we could assign a probability? Pr(No Attack today)

    = p = .999 Pr(No Attack today AND tomorrow) = .9992
  25. What if we could assign a probability? Pr(No Attack today)

    = p = .999 Pr(No Attack today AND tomorrow) = .9992 Pr(No Attack over n days) = (.999)n
  26. What if we could assign a probability? Pr(No Attack today)

    = p = .999 Pr(No Attack today AND tomorrow) = .9992 Pr(No Attack over n days) = (.999)n Pr(Attack over n days) = 1 – (.999)n
  27. Probability of a Nuclear Attack 0 0.1 0.2 0.3 0.4

    0.5 0.6 0.7 0.8 0.9 1 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 Pr(Attack) Year
  28. If Probability Theory doesn’t help… What if it’s a repeated

    game? Cooperate Defect Cooperate 0, 0 -100, 10 Defect 10, -100 -90, -90
  29. If Probability Theory doesn’t help… What if it’s a repeated

    game? Cooperate Defect Cooperate 0, 0 -100, 10 Defect 10, -100 -90, -90 Cooperate Defect Cooperate 0, 0 -100, 10 Defect 10, -100 -90, -90
  30. If Probability Theory doesn’t help… What if it’s a repeated

    game? Cooperate Defect Cooperate 0, 0 -100, 10 Defect 10, -100 -90, -90 Cooperate Defect Cooperate 0, 0 -100, 10 Defect 10, -100 -90, -90
  31. If Probability Theory doesn’t help… What if it’s a repeated

    game? Grim Trigger?
  32. If Probability Theory doesn’t help… What if it’s a repeated

    game? Grim Trigger? Tit-for-tat!
  33. If Probability Theory doesn’t help… What if it’s a repeated

    game? Grim Trigger? Tit-for-tat! Tit-for-two-tats?
  34. The Prisoner’s Dilemma, Generalized Cooperate Defect Cooperate Good Worst, Best

    Defect Best, Worst Bad Player 1 Player 2
  35. The Prisoner’s Dilemma, Generalized Cooperate Defect Cooperate Good Worst, Best

    Defect Best, Worst Bad Player 1 Player 2 Substantive Assumptions: • Threat is a nontrivial factor • Decision makers view themselves as better off if everyone suffers
  36. The Nuclear Prisoner’s Dilemma? Cooperate Defect Cooperate Defect Player 1

    Player 2
  37. The Nuclear Prisoner’s Dilemma? Cooperate Defect Cooperate 0, 0 -100,

    0 Defect 0, -100 -100, -100 Player 1 Player 2
  38. The Nuclear Game Cooperate Defect Cooperate *0, 0* *-100, 0*

    Defect *0, -100* *-100, -100* Player 1 Player 2
  39. The Nuclear Game Cooperate Defect Cooperate *0, 0* *-100, 0*

    Defect *0, -100* *-100, -100* Player 1 Player 2
  40. And Deterrence is Born! Cooperate Defect Cooperate *0, 0* *-100,

    0* Defect *0, -100* *-100, -100* Player 1 Player 2
  41. Thank you very much. Tony Boyles Georgia Institute of Technology