Two players must choose among three alternatives a, b, and c. Player 1's preferences are given by a≻b≻c while player 2's preferences are given by c≻b≻a. The rules are that player 1 moves first and can veto one of the three alternatives. Then player 2 chooses one of the remaining two alternatives.

(a) Find all Nash equilibria of this game.

(b) Find the subgame-perfect NE of this game.

Consider the following two-player game:

First player 1 selects a number x, which must be greater than or equal to zero. Player 2 observes x. Then, simultaneously and independently, player 1 selects a number y1 and player 2 selects a number y2 at which point the game ends. Player 1's payoff is u1 = y1y2 + xy1 - y1^2 - (x^3/3), and player 2's payoff is u2 = -(y1 - y2)^2.

Find the subgame perfect Nash equilibrium of this game.

7.

If this game becomes a perfect information game and is played sequentially, two things can happen: (i) case 1: player 1 moves first and player 2 moves second, and (ii) case 2: player 2 moves first and player 1 moves later. In case 2, the perfect equilibrium strategies and corresponding payoffs are __________ and in case 1, the perfect equilibrium strategies and corresponding payoffs are ____________. This suggests that players will prefer to play ________ rather than __________.

Consider the following game. Player 1 has an infinite set of pure strategies, which is the interval [0, 1]. Player 2 has only two pure strategies, L and R. When Player 1 chooses his pure strategy x ∈ [0, 1], payoffs to Player 1 are u1(x, L) = 2 − 2x, u1(x, R) = x; and payoffs to Player 2 are u2(x, L) = x, u2(x, R) = 1 − x.

(i) Show that the game has no Nash equilibrium in pure strategies.

(ii) Find all Nash equilibria of the game when Player 2 is allowed to use a mixed strategy. Explain carefully why they are Nash equilibria. Give the payoffs to the two players.