1. Consider country A. In this country, 20% of the people are from ethnicity α, 40% from ethnicity β, and the other 40% are from ethnicity γ. In the same country, 1/3 of the people from ethnicity α speak language BLUE, 2/3 of the people from ethnicity β speak language BLUE, and 1/5 of the people from ethnicity γ speak language BLUE.

a) Suppose a person is speaking language BLUE. What is the probability that this person is from ethnicity γ? Provide your answer in fractions and not in decimal places. Failure to do so will result in the loss of points.

2. Consider the model by Hollyer and Rosendorff (2011). Write down the autocrat's expected utility function for e=0; this is the level of effort put into deposing the government. Interpret this function in 30 words or less.

3. You are an advisor to the ruler of a big country that is surrounded by smaller nations. Your boss is considering whether to invade one of the smaller neighboring nations because its citizens have demonstrated capacity for independent thought. You explain to your boss that he can do that and obtain a single payoff of $100. You also explain to your boss that he can also encourage the citizens of the small nation to continue developing independent thought and with it a dynamic population that might buy your country's products. If this is the case, he will obtain a payoff of $10 in this current year (i.e. period 0), $10 next year (i.e. period 1), $10 in two year time (i.e. period 2), $10 in three years time (i.e. period 3), and so on for an infinite number of years (i.e. infinite periods). The discount factor of your boss is δ.

a) Under what conditions do you recommend the promotion of independent thought in the small nation as opposed to an invasion?

4. Following Downs et al., the utilities for pairs of countries engaged in trade depend on levels of protection. Consider the following utility function for country A, where PA is level of protection in country A and PB is level of protection in country B:

U_A(P_A, P_B )= -(P^B - P₀^B ) - (P^A - P₀^A)² + (1/2)(P^B - P₀^B )(P^A - P₀^A)+ (P^B - P₀^B )².

Assume that the utility function for country B is:

U_B(P_A, P_B )= -(P^A - P₀^A ) - (P^B - P₀^B)² + (1/2)(P^A - P₀^A )(P^B - P₀^B)+ (P^A - P₀^A )².

Assume that the non-cooperative tariffs (i.e. the status quo) are P₀^A = P₀^B = 1 and cooperative tariffs are P_A = P_B = 1/2. Countries can either choose to implement non-cooperative tariffs or cooperative tariffs.

a) Write down the matrix with payoffs for both players.

b) What is the Nash equilibrium in pure strategies?