AMS 335/ECO 355: GAME THEORY

Instructions
(1) Answer all three questions.
(2) The exam lasts from 12:00pm EST on Friday, Jan 21st to 12:00pm EST Saturday,
Jan 22nd.
(3) This is an open-book exam. You may use any of the materials posted on blackboard,
including the problem set video recordings.
(4) This is an individual exam. Working in groups is not permitted. Any suspicion
of cheating/copying will be directly reported to the Academic Judiciary Office for
possible dishonesty.
Question 1
Find the strategies that survive iterated elimination of strictly dominated strategies and
at least one Nash Equilibria (in pure strategies or mixed strategies) in each of the following
two normal-form games:
(a) (3 points)
Player 1
Player 2
L C R
T 0, −3 3, 2 −7, −4
M 1, 5 4, 8 6, 7
B 0, 8 5, 5 4, 9
(b) (3 points)
Player 1
Player 2
D E F
A 0, 4 2, 3 2, 3
B 5, 1 3, 2 4, 4
C 5, 5 1, 6 1, 2
(c) (4 points) Is it possible that the action profile (C,D) is played in every round (t =
1, 2, . . .) in a subgame perfect equilibrium on the infinitely repeated game (T = ∞) in which
two players play the game from part (b) in each round (observing past play) and discount
payoffs with a common discount factor δ ∈ (0, 1) per period? If yes, find the lowest δ for
which such an equilibrium exists. If no, explain why not.
2
Question 2
Consider the following version of the simultaneous-move game.
Player 1
Player 2
a b
A 2, 2 3, 4
B 3, 4 2, 2
(a) (3 points) Find all Nash equilibria (both in pure and mixed strategies) of the game.
Suppose now that players move in sequence: player 1 moves first, and player 2 chooses
her action after observing player 1’s action.
(b) (1 points) Draw the extensive game form of this game.
(c) (4 points) Find all subgame perfect equilibria of the dynamic game.
(d) (2 points) Does the dynamic game have a Nash equilibrium which gives payoffs (3,4)
and that is not a subgame perfect equilibrium of the game? If yes, specify strategies
in such an equilibrium. If not, argue why not.
Question 3
The North Face and Salomon set prices simultaneously for their respective hiking boots.
Let p1 ≥ 0 denote the price set by The North Face and p2 ≥ 0 the price set by Salomon.
Consumers demand 3 − 2p1 + p2 billions of The North Face boots and 3 − 2p2 + p1 billions of
Salomon’s. Assume that the cost of producing a hiking boot is 0, so the payoff of The North
Face is v1(p1, p2) = p1(3−2p1+p2), and the payoff of Salomon is v2(p1, p2) = p2(3−2p2+p1).
(a) (2 points) What is the best-response p1 = BR1(p2) of The North Face to a price p2
chosen by Salomon?
(b) (3 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) (3 points) Suppose the two companies play the stage game an infinite number of
times (T = ∞) and have a common discount factor δ ∈ (0, 1). Consider the following
strategy profile in the infinitely repeated game: the two companies start playing
(p
c
1
, pc
2
) = (2, 2), and keep playing (p
c
1
, pc
2
) as long as neither company deviated from
a prices of 2; as soon as one of the companies deviates, they switch to playing the
Nash equilibrium of the stage game. For which values of δ does this strategy profile
constitute a subgame perfect equilibrium

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