Question 1 Find the strategies that survive iterated elimination of strictly dom

Question 1
Find the strategies that survive iterated elimination of strictly dominated strategies and
all Nash Equilibria (in pure strategies and mixed strategies) in each of the following two
normal-form games:
(a) (5 points)
(b) (5 points)
Question 2
Four friends are thinking of playing a video-game tonight. They all are very busy during
the day and don’t have time to communicate whether they will play (strategy P) or not
(strategy N). The video-game can only be played with exactly two or four players. If the
game can be played, each of the four friends gets a benefit of 3 (even those who don’t
participate). Every friend who goes to the court bears a cost of 1 (regardless of whether the
match is played or not). A friend who doesn’t participate bears a cost of 2. Hence, the payoff
of a friend who chooses P is 2 if the match is played, and −1 if the match is not played,
while the payoff of a friend who chooses N is 3 if the match is played, and −2 otherwise.
(a) (1 points) How many pure strategy profiles does this game have?
(b) (2 points) What is the best response of a friend to every possible combination of
strategies for the other three friends?
(c) (4 points) Find all pure-strategy Nash equilibria of this game. How many purestrategy Nash equilibria are in this game?
(d) (3 points) Compute a symmetric mixed-strategy Nash equilibrium in this game in
which every friend goes to the court with the same probability p ∈ (0, 1). (There
could be more than one symmetric Nash equilibria)
Question 3
HEAD and Dunlop set prices simultaneously for their respective tennis rackets. Let p1 ≥ 0
denote the price set by HEAD and p2 ≥ 0 the price set by Dunlop. Consumers demand
4−2p1 +p2 billions of HEAD rackets and 4−2p2 +p1 billions of Dunlops’s. Assume that the
cost of producing a tennis rackets is 0, so the payoff of HEAD is v1(p1, p2) = p1(4−2p1 +p2),
and the payoff of Dunlop is v2(p1, p2) = p2(4 − 2p2 + p1).
(a) (3 points) What is the best-response p1 = BR1(p2) of HEAD to a price p2 chosen by
Dunlop? What is the best-response p2 = BR2(p1) of Dunlop to a price p1 chosen by
HEAD?
(b) (4 points) Find the Nash equilibrium of this game.
(c) (2 points) Which strategies are rationalizable (i.e., survive the process of iterated
elimination of strategies that are not best responses for any surviving play of the
opponent)?
(d) (1 points) Suppose both firms choose prices p = p2 = 2. Does this outcome Pareto
dominate the Nash equilibrium?