Economic and Game Theory
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1. Consider a game with two players, player 1 and player 2, in which each player i can choose an action from a finite action set Mi that contains mi actions. Player is payoff if the actions choices are (m1, m2) is Ui (m1, m2).
a) Suppose, first, the two players move simultaneously. How many strategies does each player have?
b) Now suppose that player 1 moves first and that player 2 observes player 1s move before choosing her move. How many strategies does each player have?
c) Suppose that the game in (b) has multiple subgame perfect equilibria. Show that if this is the case, then there exist two pairs of moves (m1, m2) and (m1, m2) (where either m1 does not equal m1or m2 does not equal m2) such that either
U1 (m1, m2) = U1 (m1, m2)
U2 (m1, m2) = U2 (m1, m2)
d) Suppose that for all (m1, m2) and (m1, m2) such that m1 does not equal m1 or m2 does not equal m2, U2 (m1, m2) = U2 (m1, m2). Suppose also that there exist a pure strategy Nash equilibrium for the game in (a) in which p1 is player 1s payoff. Show that in any subgame perfect equilibrium of the game in (b). Player 1s payoff cannot be smaller than p1. [Manage messages]