Game Theory
Part B
(20 pts total): Consider the following two-player game. Answer the following two questions. Each are worth 10 pts.
Player 2
𝒍 𝒄 𝒓
Player 1 T 1, 1 0, 1 3, 1
M 1, 0 2, 2 1, 3
D 1, 3 3, 1 2, 2
B.1. Are there any strictly dominated strategies? Are there any weakly dominated strategies? If so, explain what dominates what and how.
B.2. After deleting any strictly or weakly dominated strategies, are there any strictly or weakly dominated strategies in the `reduced’ game? If so, explain what dominates what and how. What is left?
Part C
35 pts total: Li has invited Bart to her party. Li must choose whether or not to hire a DJ. Simultaneously, Bart must decide whether or not to go the party. Bart likes Li but he hates DJs – he even hates other people dancing!
Bart’s payoff from going to the party is 4 if there is no DJ, but 0 if there is a DJ there. Bart’s payoff from not going to the party is 3 if there is no DJ at the party, but 1 if there is a DJ at the party.
Li likes DJs – she especially likes Bart’s reaction to them – but does not like paying for a DJ. Li’s payoff if
Bart comes to the party is 4 if there is no DJ, but 8 – 𝑥 if there is a DJ (𝑥 is the cost of a DJ). Li’s payoff if
Bart does not come to the party is 2 if there is no DJ, but 3 – 𝑥 if there is a DJ there.
Answer the following four questions based on the description above.
C.1. Write down the payoff matrix of this game. (5 PTS)
C.2. Suppose 𝑥 = 0. Identify any dominated strategies. Categorize them and explain why these strategies are dominated. Then find the Nash Equilibrium. What are the Nash Equilibrium Payoffs? (10 PTS)
C.3. Suppose 𝑥 = 2. Identify any dominated strategies. Categorize them and explain why these strategies are dominated. Then find the Nash Equilibrium. What are the Nash Equilibrium Payoffs? (10 PTS)
C.4. Suppose 𝑥 = 5. Identify any dominated strategies. Categorize them and explain why these strategies are dominated. Then find the Nash Equilibrium. What are the Nash Equilibrium Payoffs? (10 PTS)
Part D – 30 pts total:
Address the following questions using the decision tree provided below. Show your calculations clearly, including the equations you used.
D.1 Assuming that you have an exponential utility function of the form 𝑢(𝑤) = 1 − 𝑒−𝑤/𝜌 and a risk tolerance 𝜌 of $5m, calculate the certainty equivalent of this decision. Note: Recall that for the given utility function, 𝑤(𝑢) = −𝜌 ∗ ln (1 − 𝑢) . (10pts)
D.2 Assuming that you have an exponential utility function of the form 𝑢(𝑤) = 1 − 𝑒−𝑤/𝜌 and a risk tolerance 𝜌, plot the certainty equivalents of Glare and Titanium for varying values of 𝜌. Then, identify the critical value 𝜌 that renders the certainty equivalent of Glare and Titanium indifferent.
Note: Recall that for the given utility function, 𝑤(𝑢) = −𝜌 ∗ ln (1 − 𝑢) . (10pts)
D.3 If you were risk neutral and there was a clairvoyant, what would be the value of perfect information for chance node A? (Note: you need to recalculate the certainty equivalent of the decision assuming risk neutrality) (10pts)
