Problem Set #8

PlSc 349
Due: Tuesday, 2 November 2010

  1. [5 points] Problem U11.2
  2. [9] Problem U11.4
  3. [9] Problem U11.6. You may calculate δ instead.
  4. [9] [from Watson 2008] Consider the following game:

    Payoffs: Player 1, Player 2
    Player 2
    x y z
    Player 1 x 3, 3 0, 0 0, 0
    y 0, 0 5, 5 9, 0
    z 0, 0 0, 9 8, 8

    a. What are the (pure-strategy) Nash equilibria of this game?
    b. If the players could meet and make a self-enforced agreement regarding how to play this game, which of the Nash equilibria would they jointly select?
    c. Suppose the preceding matrix describes each period in a two-period repeated game. Show that there is a subgame perfect equilibrium in which (z, z) is played in the first period.
    d. One can interpret the equilibrium from part c. as a self-enforced, dynamic contract. Suppose that, after they play the first round but before they play the second round, the players have an opportunity to renegotiate their self-enforced contract. Can the equilibrium from part c. be sustained?
  5. [9] [from Rasmusen 2001] There is a long sequence of players. One player is born in each period t, and he lives for periods t and t + 1. Thus, two players are alive in any one period, a youngster and an oldster. Each player is born with one unit of chocolate, which cannot be stored. Utility is increasing in chocolate consumption, and a player is very unhappy if he consumes less than 0.3 units of chocolate in a period: the per-period utility functions are U(C) = -1 for C < 0.3 and U(C) = C for C ≥ 0.3 , where C is consumption. Players can give away their chocolate, but, since chocolate is the only good, they cannot sell it. A player’s action is to consume X units of chocolate as a youngster and give away 1 - X to some oldster. Every person’s actions in the previous period are common knowledge, and so can be used to condition strategies upon.
    a. If there is finite number of generations, what is the unique Nash equilibrium?
    b. If there are an infinite number of generations, what are two perfect equilibria? Which one (if any) is Pareto-superior?
    c. If there is a probability θ at the end of each period (after consumption takes place) that barbarians will invade and steal all the chocolate (leaving the civilized people with payoffs of -1 for any X ), what is the highest value of θ that still allows for an equilibrium with X = 0.5?
  6. [9] [loosely from McCarty and Meirowitz 2007] Consider the following Generalized Prisoner's Dilemma:

    Payoffs: Player 1, Player 2
    Player 2
    Cooperate Defect
    Player 1 Coooperate a, a d, c
    Defect c, d b, b

    a. For what values of δ can a cooperative equilibrium be sustained using a Grim Trigger strategy? Interpret the resulting equation.
    b. For what values of δ can a cooperative equilibrium be sustained using a Tit-for-Tat strategy? Interpret the resulting equation, and compare to part a.
  7. [for future planning] Which problem was most useful in learning the concepts? Which problem was least useful?

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