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Game Theory to Determine Form Representation

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Practice Exam
Faculty of Continuing Education


August 18, 2008

1. This game involves two roommates. Each of them has 2 hours that can be devoted either to preparing a common meal or to studying. The payoff to person i, (where i = 1; 2), to hours spent studying, Si, is given in the table on the left below. The payoff to person i from the joint cooking venture depends on the total number of hours spent by the two roommates preparing the meal, T = T1 + T2, (where Ti is the number of hours spent cooking by player i) and is given in the table on the right below.

a) Construct the normal form representation of this game in which the strategies are time spent preparing the meal (10 points).
b) Find all pure strategy Nash equilibria. Are they Pareto-optimal? (10 points).

2. Consider the following extensive game.

a) Construct the normal form representation of this game and find all pure strategy Nash equilibria (10 points).
b) Which of the pure Nash equilibria is sequentially rational (Perfect Bayesian Equilibrium)? Completely define the strategy profile and the belief system for the information set associated to player 2 such that we can generate a Perfect Bayesian Equilibrium in this game (15 points).

3. There are 2 firms in a market. Firms simultaneously choose prices; a price can be low (pL) or high (pH). If they both set pH, each receives profits of $320. If one sets a pL while the other sets pH, the low-price firm earns profits of $360 while the high-price firm earns $100. If they both set pL, each receives profits of $285.
a) Construct the normal form representation of this game and find all pure and mixed strategy Nash equilibria, if the game is played once (Bonus, 5 points).
b) If these two firms decide to play this game for a fixed and finite number of periods, n, what would the Subgame Perfect Nash Equilibrium be? Briefly explain your answer (10 points).
c) Suppose these two firms play this game infinitely. Let each of them use a grim strategy in which they both set pH unless one of them defects, in which case they set pL for the rest of the game. Suppose the discount factor is (See attached) Will it be worthwhile to cooperate? (10 points).
d) Find the range of values of for which this strategy is able to sustain cooperation between the two firms (10 points).

4. An economy has two types of workers: Qualified and Unqualified. The population consists of π Qualified and (1-π). Unqualified (with 0<π<1). There are two types of jobs: Productive and Unproductive; in an Unproductive job either type of worker produces x units of output. In a Productive job, a qualified worker produces y (with y>x) units of output and an unqualified worker produces zero. There is enough demand for workers that companies must pay for each type of job what they expect in revenue from the workers.
Companies must hire each worker without observing his type and pay him before knowing his actual output. But qualified workers can signal their qualifications by getting educated. For a qualified worker, the cost of getting educated to level n is n^2/2. While for an unqualified worker it is n^2. These costs are measured in the same units as output, and n can be any positive number.
a) What is the minimum level of n that will achieve separation? And in such a case, what would the wages offered to both types of workers? Note that these answers will be in function of x and y (15 points)
b) Now suppose the signal is made unavailable. Which kind of jobs will be filled by which kind of workers and at what wages? Who will gain and who will lose from this change? Note that these answers will be in function of x, y and π (10 points).

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