Q1. Consider the following game describing the agency problem between supervisor and a laborer. Each player has two strategies. The laborer can either work or shirk, and the supervisor can either monitor the laborer or not monitor. If the laborer works, he produces output valued by the supervisor at v, but if he shirks, the value to the supervisor of the laborer's output is 0. The laborer earns a wage w unless he is caught shirking, in which case the laborer earns 0. Of course, this wage is a cost minus(-w) to the supervisor. The laborer experiences disutility from working which reduces his wage by g. Whenever, the supervisor monitors the laborer, she incurs a cost h.

A. Write a payoff matrix which summarizes the strategies and payoffs for each player in this game.

B. Find each player's best response strategy, given the strategy played by his/her opponent. Does either player have a dominant strategy?

C. Is there a NE (in pure strategies) in this game?

Q2. Draw a graph to show the core of a three-person game which has the following characteristic function: ({})=0; ({1})=0; ({2})=1; ({3})=2 ({1, 2})=4; ({1, 3})=3; ({2, 3})=5 {(1, 2, 3)}=7

Q3. Assume there are N firms in a market producing a homogenous product. Assume = for all i=1, 2, ...N. If the firms are competing for market share in terms of supplying output in the market, what is the Cournot-Nash equilibrium quantity supplied by each firm?

Q4. Let the demand curve for water-bottles be P = 22- (Qa + Qb). Two firms A and B are present in the market to supply these bottles. Both firms face the same marginal cost of $10 and fixed cost of $15. Output is in millions of bottles. Suppose that CEOs of both firms talk to each other at a party and agree to reduce their output by 1 million each so that price can rise by $2 and profits become triple. The decision to reduce output is called Cooperation and decision to maintain Cournot output is called Defection.

a) Write the payoff matrix for the game between the two firms with two strategies to Cooperate or Defect.

b) What is the Nash equilibrium if it is a one-shot game?

c) What is the Nash equilibrium if the game is repeated twice since these shirts will go out of fashion in two years (assume zero discount rate)?

Q5. Two firms decide whether to build a production facility. If both do not build, then they make profits of $5 million each. If they both decide to build, then they make profits of $4 million each. If one builds and the other does not build, then the former makes $10 million and the latter gets nothing. Firm 1 chooses in the first stage and then firm 2 acts in the second stage.

a) Represent the game in extensive form.

b) Is there is a first mover's advantage?

c) Show the extensive-form game in the normal form.