ECON20120
Homework 1
Answer all questions.
Question 1 Consider the two-player, normal form. game displayed below, with a > b > c.
Find the range of values of x so that (U, L) is a Nash equilibrium for this game.
Question 2 Consider the following two player game:
Let p be the probability attached to strategy C by player 1 and q the probability attached to strategy A by player 2. Find all mixed strategy Nash equilibria of this game.
Question 3 Consider the two-player, normal form. game displayed below. Find the probability that player 1 will play U in a mixed strategy Nash equilibrium.
Question 4 Two duopolists face an inverse demand curve p = 10−Q, where Q is the aggregate output of both firms. Both firms have constant marginal cost c1 = 2, c2 = 3. Find the equilibrium price, total quantity and profits in this market.
Question 5 There are two firms operating in the market for designer cars. They compete in quantities, choosing their output independently from each other. The market demand is given by P = 6 − Q, where P is the market price and Q = q1 + q2 is total output, with qi measuring the number of cars produced by firm i = 1, 2. Both firms operate with constant marginal cost c = 2, so for each firm, profits are given by πi(q1, q2) = P qi − 2qi
, i = 1, 2.
(i) Find all the Nash equilibria in the game where the two firms choose simultaneously their outputs.
(ii) Suppose now that the two firms only produce entire units and due to deadlines for timely delivery, firms can produce at most 2 cars each; that is, qi ∈ {0, 1, 2} for i = 1, 2. Write down the matrix form. that represents this situation. Find all the Nash equilibria in the game where the two firms choose simultaneously their outputs.