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Respond to the question: Prove it?

 01/15/2001 03:19 PM by Rodrigo; About the game you suggested
Regarding part (a) of your question, player 1 has m1 strategies and player 2 has m2 strategies. Regarding part (b), player 1 has m1 strategies, but now player 2 can observe player 1's move before taking her action, so player 2 has

 12/23/2000 03:07 AM by Pavel; Prove it !!
I've an interesting (though an easy) problem set for you to try to solve ... e-mail your answers to me and you will have a chance to win J45 ...

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 is 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 1s 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 m1or m2 does not equal m2) such that either

U1 (m1, m2) = U1 (m1, m2)

or

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 1s payoff. Show that in any subgame perfect equilibrium of the game in (b). Player 1s payoff cannot be smaller than p1. [Manage messages]