Exercise 1: Find the Nash equilibrium in mixed strategies of the following static game of complete information.
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Player 2
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L
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R
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Player 1
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T
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2, -2
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6, -6
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M
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5, -5
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5, -5
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B
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7, -7
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4, -4
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Exercise 2: For the following base game determine whether or not (2, 1) is an equilibrium payoff of the corresponding infinitely repeated game. If it is an equilibrium payoff, describe an equilibrium leading to that payoff. If not, justify your answer.
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Player 2
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L
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R
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Player 1
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T
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0, 0
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2, 2
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B
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1, 1
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0, 0
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Exercise 3: Find a Bayesian equilibrium in the following game with incomplete information:
- N = [I, II}
- T1 = [I1, I2} and TII = {II1}: Player I has two types, and Player II has one type.
- p(I1, II1) = 1/3
- p(I2, II1) = 2/3
- Every player has two possible actions, and state games are given by the following matrices:
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State game for t = (I1, II1)
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Player 2
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L
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R
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Player 1
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T
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2, 0
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0, 3
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B
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0, 4
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1, 0
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State game for t = (I2, II1)
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Player 2
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L
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R
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Player 1
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T
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0, 0
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2, 2
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B
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1, 1
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0, 0
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Exercise 4: Suppose a parent and a child play the following game, first analyzed by Becker (1974). First, the child takes an action, A, that produces income for the child, IC(A), and income for the parent, IP(A). Second the parent observes the income IC and IP and then chooses a bequest, B, to leave to the child. The child's payoff is U(IC + B): the parent's is V(IP - B)+ kU(IC + B), where k reflects the parent's concern for the child's well-being. Assume that: the action is a nonnegative number, A ≥ 0; the income functions IC(A) and IP(A) are strictly concave and are maximized at AC > 0 and AP > 0 , respectively; the bequest B can be positive or negative; and the utility functions U and V are increasing and strictly concave. Prove the "Rotten kid Theorem": in the backwards-induction outcome, the child chooses the action that maximizes the family's aggregate income, IC(A)+ IP(A), even though only the parent's payoff exhibits altruism.
References:
Gibbons, R., 1992. A Primer in Game Theory, Prentice Hall.
Maschler, M., Solan, E. & Zamir, S., 2013. Game Theory, Cambridge, UK: Cambridge University Press.