1. There are three players, who are about to walk into a room one after the other. There is a switch in the room, which can be in one of three positions: Up, Middle, or Down. The players know the switch is initially in the Down position (they know this in parts a,b, and c below). When a player walks into the room he can change the switch position or leave it unchanged. A player who changes the switch position can only change it by one position. That is, if the switch is in D, he can move it to M, if the switch is in M, he can move it to D or to U, and if the switch is in U he can move it to M. The game ends after the third player walks into the room and makes his decision.

(a) (10 points) Suppose that each player sees what all the players before him did. Draw the game in extensive form, ignoring payoffs. How many, information sets and strategies does each player have? Explain.

(b) (10 points) Suppose that when each player enters the room he only sees the position of the switch. Draw the game in extensive form, ignoring payoffs. How ma, information sets and strategies does each player have? Explain.

Now suppose that the Up position is blocked (and the players know this), so the switch can only be toggled between D and M. Nothing else changes: when a player walks into the room he can still leave the switch position unchanged or change its position.

(c) (10 points) Repeat parts a and b.

2. Two firms engage in Cournot competition. Each firms i chooses a quantity qi 0 and the resulting price is p = 12 qi-q2 if qi + q2 12 and otherwise p = 0,.The firms have a marginal production cost of 0.

(a) (10 pts) Suppose that firm 1 thinks that firm 2 is going to choose quantity q2=4. What is firm's 1's profit as a function of its quantity choice qi

(b) (15pts) Suppose that firm 1 thinks that firm 2 is going to...quantity q2 = 0 with probability 1/2 and quantity q2 = 8 with probability 1/2. What is firm's 1's profit as a function of its quantity choice q1

c. (15 pts) For which quantities qi would firm 1 strictly prefer that firm 2 produce q2 =4 rather than q2 = 0 with probability 1/2 and q2=8 with probability 1/2 and for which quantities q2 would firm 1 have the reverse preference?

3. Consider the following two-player game. in which x and y are numbers in [0, 10] known to the players.

LCR

U1,62,103,0

M2,x3,51,10

D3,31,y2,0

(a)(10 pts) For which values of x and y is R dominated by a pure strategy strategy?

For which values of x and y is R dominated by a mixed strategy, Explain.

(b)(10 pts) For which values of and y is C dominated by a pure strategy strategy,

(c)For which values of x and y is C dominated. by a mixed strategy, Explain.

(c) (20 pts) For which values of x and y isL dominated by a pure strategy strategy,

For which values of x and y is L dominated by a mixed strategy, Explain.