Answer all the questions
Consider the game of Chicken that we discussed in class: Two players, James and Dean, each has two strategies: Swerve and Straight. If they both choose Swerve, both get 0 as payoff. If they both choose Straight, both get -2 as payoff. If one chooses Swerve, the other chooses Straight, then the player chooses Straight gets 1 and the player chooses Swerve gets -1.a. Assume the game of Chicken is played sequentially with James moves first, and Dean moves second. Draw the game tree for this sequential game. Finding the rollback equilibrium of this sequential game. (10 pt)b. Again, assume the game of Chicken is played sequentially with James moves first, and Dean moves second. Draw the payoff table for this sequential game. Finding the Nash equilibria of this sequential game. Are rollback equilibrium and Nash equilibria the same? If not, why? (15 pt)2. Consider the following gamesa. Two players, each with two strategies:LEFT ROWENA TOP 80,20 BOTTOM 10,90COLINRIGHT 5,95 85,15Does this game have pure strategy Nash equilibrium? Does this game have mixed strategy Nashequilibrium? (15 pt)b. Two players, one with two strategies, one with three strategies:LEFT ROWENA TOP 80,20 BOTTOM 10,90COLIN CENTER 50,50 25,75RIGHT 5,95 85,15Find the mixed strategy Nash equilibrium of the above game. (15 pt) c. Two players, each with three strategies:COLIN CENTER3. Consider a used car market. There are two types of used cars, orange and lemon. Orange cars owner value their own cars at $10,000. Lemon cars owner value their own cars at $2,000. Buyers are willing pay $13,000 for orange cars and $3,000 for lemon cars. There are limited stocks of both types of cars. Thus, buyers have to compete with each other and pay their full willingness to pay. Owners of used car know the quality of their car, but potential buyers don’t.a. Suppose signaling is not available, what is the minimum value of the fraction of oranges in the population, f, such that the market for oranges does not collapse? (15 pt)b. Suppose now signaling and screening is available. Owners of used car can get certificate to show that their car is orange. For owners of orange cars, the cost of getting the certificate is $𝑥. For owners of lemon cars, the cost of getting the certificate is $𝑥2 (they have to bribe the person). Finding the range of 𝑥 to achieve separation. (Hint: You need to check both Incentive Compatibility condition and Participation condition in this question. Car Owners’ alternative choice is not selling the car and keeping it.) (15 pt)LEFT 80,20 55,45 10,90RIGHT 5,95 25,75 85,15ROWENA TOP MIDDLEBOTTOMFind the mixed strategy Nash equilibrium of the above game. (15 pt)50,50 65,35 25,75
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