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Game theory

  1. Jan 6, 2014 #1
    1. The problem statement, all variables and given/known data

    A and B put an amount of money in a bank for a period of two years. If both players take the money during the first year, each one gets 75$. If A (B) takes it y B (A) keeps it, A (B) gets 100$ and B (A) 50$. If none of them takes it, they move to the second year. Then, if both players take the money, each one gets 150$, which is the same amount of money that they get if they keep it. If A (B) takes it y B (A) keeps it, A (B) gets 200$ and B (A) 100$. Players act independently from one another.

    Questions: a) pure strategy equilibria, b) probability of both players taking the money in the first year.

    2. Relevant equations

    Expected value for the second question.

    3. The attempt at a solution

    Since I do not know how to draw the tree diagram in the present format, I can specify the pay-offs differently.

    First year

    A takes, so B can take (75,75) or can keep (100, 50).
    A keeps, so B can take (50, 100).

    Second year

    If B keeps
    • then A can take, so B can take (150, 150) or keep (200, 100)
    • then A can keep, so B can take (100, 200) or keep (150, 150).

    Regarding the questions:

    a) the game reminds me the prisoner's dilemma. If they collaborate, though inadvertently, and wait till the second year, they can earn 150$ instead of 75$ or at least 100$. But I am not sure whether their collaboration is a logical expectation. Does A take it in his first move because he knows that B will take it if he keeps it (50, 100)?.

    b) The pay-offs.
    T, T = 75, 75
    T, K = 100, 50
    K, T = 50, 100
    K, K = 150, 150 --> is this correct?

    Be p and q the probabilities of taking the money for A and B:
    EV=75p+100(1-p)
    EV=50p+150(1-p)
    p=50/75=0.66
    Given that they are symmetrical q=50/75
    So p(T, T)=(50/75)*(50/75)= 0,4356, so they would keep the money.

    If this probability is right, which depends on the (K, K) pay-off, would elucidate a).
     
  2. jcsd
  3. Jan 6, 2014 #2

    Office_Shredder

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    For your payoffs, you are listing only two values of take or keep, presumably one for each player. But each player has two choices: whether to take or keep the first year, and whether to take or keep the second year.
     
  4. Jan 6, 2014 #3

    haruspex

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    Solve for the second year first.
     
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