Description

1.Two players engage in the following interaction regarding the battle of the sexes. Player
1 first decides whether she wants to “play” (Y ) or not (N). If player 1 chooses N, then
the game is over, and players receive payoffs (2, 3). If player 1 chooses Y , then player 2
(observes player 1’s choice and) decides whether he wants to “play” (y) or not (n). If player
2 chooses n, then the game is over, and players receive payoffs (3, 2). If player 2 chooses
y, then the players play the following version of the battle of the sexes game in which they
choose actions simultaneously. (a) Draw the extensive-form of the game.
(b) How many subgames does this game have (including the game itself)? Draw each
subgame as a separate tree.
(c) Find all pure-strategy subgame perfect equilibria of the game.
(d) Does the game have any Nash equilibrium in which both players decide to “play”
(i.e., player 1 chooses Y and player 2 chooses y)?

2.Consider the following version of the ultimatum game in which two players have the
opportunity to split $10. First, player 1 proposes a number z, where z can take one of (only)
five values 0, 1, 3, 5 or 8. Then, player 2 decides whether to accept or reject the proposal. If
player 2 accepts the proposal, then player 2 receives z dollars, and player 1 is left with 10−z
dollars (e.g., if player 2 accepts z = 3, then player 2 gets $3, and player 1 gets $7). If player
2 rejects it, then both players get $0.(a) Draw the extensive form of this game.
(b) How many pure strategies does each player have in the game?
(c) Find all pure-strategy subgame perfect equilibria of the game.
(d) Does the game have a Nash equilibrium in which player 1 gets only $2?

3.Two players play the following stage-game two times in a row (T = 2) and observe each
other’s first-stage action before choosing actions in the second stage. Both players discount
second-stage payoffs using a discount factor δ ∈ [0, 1].(a) How many pure strategies does each player have in the two-stage game?
(b) Which action profiles can be sustained in the first stage of a pure-strategy subgame
perfect equilibrium for δ = 1?
(c) What is the smallest δ for which the profile (F, m) can be played in the first stage of
a subgame perfect equilibrium?