1.Two players engage in the following interaction regarding

1.Two players engage in the following interaction regarding the battle of the sexes. Player1 first decides whether she wants to “play” (Y ) or not (N). If player 1 chooses N, thenthe 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 player2 chooses n, then the game is over, and players receive payoffs (3, 2). If player 2 choosesy, then the players play the following version of the battle of the sexes game in which theychoose actions simultaneously. (a) Draw the extensive-form of the game.(b) How many subgames does this game have (including the game itself)? Draw eachsubgame 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 theopportunity 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. Ifplayer 2 accepts the proposal, then player 2 receives z dollars, and player 1 is left with 10−zdollars (e.g., if player 2 accepts z = 3, then player 2 gets $3, and player 1 gets $7). If player2 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 eachother’s first-stage action before choosing actions in the second stage. Both players discountsecond-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 subgameperfect equilibrium for δ = 1?(c) What is the smallest δ for which the profile (F, m) can be played in the first stage ofa subgame perfect equilibrium?