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Game Theory Assignment 2 (final)

1. There are two effort level for the agent and they are eH,eL. For the agent the cost for high effort is 500 and the cost for low effort is 0. The reservation utility for the agent is 500. When the agent is working for the principal, there can be three levels of profit and they are (x1x2x3) = ($1,000,000,$4,000,000,$9,000,000). The principal is risk neutral and the agent is risk averse. The agent’s utility from the wage, w, would be /W. The agent’s total utility would be u = /W - c(eK) where k = H,L. When the effort level is high the probabilities that each result would happen are . When the effort level is low they are .

(1) In the ideal case where the principal can observe the effort level of the agent, what will be the wage system like? In other words, what will be (w1w2w3)?

(2) If the principal cannot observe the effort level of the agent, what will be (w1w2w3)? And what will be the effort level of the agent in that case? What will be the profit of the principal?

2. Two players play the following game for infinite times.

For the player to continue to cooperate what would be the ranges of their discount factor, , respectively?

3. (professor – TA game) Professor Hahn can give a TA scholarship to Gong Yi for maximum 2 years. At the beginning of each year professor Hahn decides whether he will give a scholarship to Gong Yi or not. Gong Yi can get a scholarship in t=2, only if he gets it in t=1. Basically, professor Hahn and Gong Yi will play the following game twice.

Gong Yi can be a Hard working type with probably 0.3 and can be a Lazy type with probability 0.7 Professor Hahn does not know Gong Yi’s type. If Gong Yi is hard working, it will be X=6 and Gong Yi will always work if he gets a scholarship. If Gong Yi is lazy, it will be X= 2. There is no time discount for t=2.

Find out a Perfect Bayesian Equilibrium of the game.

4. Consider a duopoly with a Cournot competition. The demand of the market is Q=3-p. Both firm 1 and firm 2’s marginal costs can take two values. For firm 1 it can be MC=5/4 with probability 1/3 and MC=3/4 with probability 2/3. For firm 2 it can be MC=5/4 with probability 2/3 and MC=3/4 with probability 1/3. Each firm knows its own MC but does not know the MC of the other firm. (But the probabilities are known to everyone.) What will be he Bayesian equilibrium of the game?

l 15 points for each questions