Instructor: Akio Yamazaki
? Due date: August 10th, 2016
? Work alone or in groups of TWO (maximum). Turn in the assignments individually.
? Please write down how you arrived to your conclusions. If reasoning is rigorous and correct,
you will get partial credits even if the answer is not.
? If you have problems, you can ask me during my office hours or ask TA during her office
hours, but please do not expect me or her to solve the problem for you or tell you the
solution. Please do not email me or TA about the assignment.
Question 1: NE vs. BIE (15 points)
Consider the following game: Player 1 makes a choice of either U or D. After observing player 1?s
choice, player 2 chooses either L or R. If player 1 chooses U, player 2?s choice of L yeilds a payoff
of (0, 0), where the first number is the payoff to player 1 and the second number is the playoff to
player 2. Alternatively, if player 2 chooses R, the players receives a payoff of (5, 7). If player 1
chooses D, player 2 generates payoff (4, 4) from choosing L and (6, 4) from choosing R.
(a) Specify the strategy set for each player
(b) Write down the normal form of representation of this game, and find the Nash equilibria
(c) Write down the extensive form of representation of this game, and fine the Backward Induc-
Question 2: NE vs. SPNE (15 points)
Consider the following game:
(a) Is this a perfect information game or imperfect information game?
(a) How many sub-games are there?
(b) Find the Nash equilibria
(c) Find Sub-game Perfect Nash equilibria
Question 3: Envy ? a capital sin (15 points)
Consider the following ultimatum bargaining game. There is 1 unit of a good and player 1 offers
a split (x, 1 ? x), where x 2 [0, 1] is chosen by player 1. Player 2 accepts the offer (Y ) or refuse it
(N). If player 2 accepts the offer, then player 1 gets x and player 2 gets 1 ? x. If player 2 refuses
the offer, then both players get 0. We assume that when player 2 is indifferent between accepting
and refusing, then he accepts.
(a) Suppose that each player maximizes his payoff. Find the Backward Induction Equilibrium.
Suppose now players are envious. More precisely, the utility of player 1 is equal to his payoff minus
times the payoff of player 2 and the utility of player 2 is equal to his payoff minus times the
payoff of player 1, with > 0. The parameter can therefore be interpreted as a measure of
(b) Find the BIE as a function of .
Question 4: Strategic Investment (15 points)
Recall that in the class, we have analyzed ?Strategic Investment? game where two firms, firm 1
and firm 2, are competing in the Cournot model fashion. But we added an additional decision
making for firm 1 prior to the competition. That is firm 1 can choose to rent a new machine that
will lower his costs by 50%. It will cost $d (million) to rent this new machine. Both firms face a
linear demand curve, p = 4 ? Q where Q = q1 + q2. We assume that each firm?s marginal cost is
1, i.e., it costs $1 per unit of output.
(a) Find Nash equilibrium quantities (bq1, bq2) and equilibrium profits (b1, b2) in the subgame
where firm 1 has decided to rent the new machine.
(b) Find sub-game perfect equilibria of this game. [Hint: You will have to find a condition for
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