Question no. 1

Ten profit-maximizing firms play the following Cournot competition game. Each firm i simultaneously chooses a quantity xi . Its payoff is xi(12-X) – C(xi), where X is the    sum of all ten firms’ chosen quantities, and C is a fixed-cost function satisfying           C(0)=0 and C(xi)=4 for every xi>0.

1.   Show that there is no symmetric pure-strategy Nash equilibrium in the game.

2.   Find an asymmetric pure-strategy Nash equilibrium in the game.

3.   Find a symmetric mixed-strategy Nash equilibrium in the game, where each firm randomizes between only two quantity values.

Question no. 2

Two agents with quasi-linear utility decide whether to submit a request for a single, indivisible object of common value v. Agent 1 has priority: If she requests the object, she gets it for sure. When agent 2 requests the object, he can get it only if agent 1 does not request  it.  For  each  agent,  submitting  a request  entails  a  fixed  cost  c(0,1), independently of whether the request is granted.

Assume v takes two possible values, 0 and 1, with equal prior probability. The agents are asymmetrically informed about v. Agent 2 receives no information. As to agent 1, with probability 1-q he receives no information. With probability q, he receives a signal with accuracy p(½,1) - i.e., for every v, the signal is equal to v with probability p.

1.   Describe the interaction between the two agents as a Bayesian game.

2.   Define pure-strategy Nash equilibrium in this game.

3.   Characterize  the  game’s  pure-strategy Nash  equilibrium,  as  a  function  of the parameters c,q,p.

Question no. 3

Consider a three-period version of the Rubinstein bargaining model in which:

•   Player A (she) makes a proposal (x,1-x) in period 1 (where throughout the  problem the first number indicates the fraction of the pie that player A gets, before discounting).

•   If player B (he) rejects (x,1-x), he makes a proposal (y,1-y) in period 2.

•   If player A rejects (y,1-y), the players split the pie at (½, ½) in period 3.

Players discount time with a discount factor of (0,1) per period.

1.   Fully characterize the subgame perfect equilibrium of this game.

2.   For this part, suppose player A is “tough”: In period 2, she never accepts an offer that gives her less than player B, even if this is ultimately to her detriment. Take this posture as given, and perform backward induction to derive the players’       behavior in periods 1 and 2.

3.  Now suppose player A is the tough type from part 2 with probability p(0,1),        while with probability 1-p she is “normal” as in part 1. Player A’s type is observed by her but not by Player B. Does this game have a separating perfect Bayesian       equilibrium in which the offer is always accepted in period 1? If so, fully               characterize it; if not, explain why.

4.   Consider the game from part 3 and assume p>½ . Is there a perfect Bayesian       equilibrium in which both types of player A make the same offer in period 1? If so, fully characterize it; if not, explain why.

Question 4

Consider the following simultaneous-move game:

A

B

C

D

X           Y           Z           W