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ECON7070
Tutorial 8
Q1.
has to turn in an
important assignment at Colin Clark Building but needs to rush into an examination at
the other end of the St Lucia Campus. She has two options: either turn in the assignment
after the exam (option L(ate)); or ask an unacquainted student, player 2, to deliver the
assignment for her (option S(tudent)). If player 1 chooses S, player 2 can then either
deliver the assignment on time (option D(eliver)) or throw the assignment in the nearest
trash can (option T (rash)). To player 1, the payoffs is 0 if the assignment is turned in
late, 3 if it is turned in on time, and −3 (negative three) if it is thrown away. Player 2 gets
a payoff of 0 if he is not asked to do anything, a payoff of x if he delivers the assignment
and y if he throws it away.
(a) Draw the game tree that represents this game.
(b) What is the condition on x and y so that player 1 will turn in the assignment late
in a subgame perfect equilibrium of this game?
Q2.
There is an incumbent firm (player 1) and a potential entrant (player 2) in a market.
The incumbent firm first chooses whether to invest in its production capacity or not. Upon
observing the incumbent firm’s choice, the potential entrant chooses between entering the
market or staying out. If the potential entrant stays out, the game ends. If the potential
entrant enters, the incumbent can choose between fighting a price war or accommodating
the entrant.
The payoffs are as follows: The investment costs C > 0. If the entrant stays out, the
incumbent gets a profit (before deducting investment cost, if any) of 2 while the entrant’s
profit is 0, regardless of whether the investment is made. If the entrant enters and the
incumbent accommodates, the incumbent gets a profit (before deducting investment cost,
if any) of 1 while the entrant’s profit is 1, regardless of whether the investment is made.
The investment makes a difference if the incumbent decides to fight. If no investment is
made and the incumbent fights the entrant, the incumbent gets 0 while the entrant gets
−1. If an investment is made and the incumbent fights the entrant, the incumbent gets
1 (after deducting the investment cost) and the entrant gets −2.
(a) Draw the game tree for this game.
(b) Suppose C = 3. Solve for the subgame perfect equilibrium of this game. (Dis-
playing the subgame perfect equilibrium in the extensive form is sufficient.)
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(c) For what range of values of C may the incumbent invest in a subgame perfect
equilibrium?
Q3.
A single firm requires the services of exactly one worker to produce a gross profit of
one unit. Assume that only one worker is available, with a reservation wage of zero. The
firm bargains with the worker in the following manner. At time zero, the firm makes
an offer w to the worker. If he accepts, then the game ends, the payoffs received by the
worker, and by the firm, being w and 1− w respectively. If the worker rejects the offer,
he makes a counter-offer w′ at time 1. If the firm accepts, the game ends. The payoffs to
the worker, and the firm, are now equal to δw′ and δ(1−w′), where δ ∈ (0, 1) represents
their common discount factor. If the firm rejects, both the worker and the firm receive a
payoff of zero.
(a) Define Nash equilibrium for this game.
(b) Define subgame perfect equilibrium for this game.
(c) Find the subgame perfect equilibria of this game.
(d) Given an example of a Nash equilibrium that is not subgame perfect.