Q1 Paul and John are working on a project for which they will receive the same grade. The utility to each of them would be their grade minus their own effort. If Paul puts in x units of effort and John puts in y units of effort their grade will be equal to the higher of the two effort levels, x and y. So for a given choice of x and y Paul's utility is given by 2 max{x, y} - x while John gets a utility of 2 max{x, y} - y.
The possible choices for effort level are 1 and 2.
Write down the payoff matrix for this game.
a) Which outcomes are Pareto efficient?
b) Find the best response functions.
c) State all Nash Equilibria of the game.
Now suppose the utility functions for Paul and John are given by 2 min{x, y} - x and 2 min{x, y} - y.
In other words the common grade is equal to the lower of the two effort levels, x and y.
Again, write down the payoff matrix for this game.
a) Which outcomes are Pareto efficient?
b) Find the best response functions.
c) State all Nash Equilibria of the game.
Q2. Cournot Duopoly Bell and HPhi are the leading producers of abaci. The market demand function for abaci is given by
p = 90 - 2x. The number of abaci produced by Bell and HPhi are denoted by xB and xH , respectively. Remember, x = xB + xH . The cost functions for Bell and HPhi are 2xB and 4xH , respectively. The two firms can choose any positive level of production.
a) Write down the profit equations for Bell and HPhi.
b) Write down the best response functions.
c) What are the Nash Equilibria choices of xB and xH .
(P.S. Assume, for now, that fractions of an abacus can in fact be produced and sold!)
Q3. Three Player Game
There are three players in this game. Players 1 and 2 must each choose whether to engage or drop. Player 3 must choose whether to favour player 1 or player 2. If Players 1 and 2 both drop, all three players get a payoff of 1. If either Player 1 or Player 2 engages then the player favoured by Player 3 gets 3 and the other player gets 0, while Player 3 himself gets 0. All players make their choices simultaneously. Find all Nash Equilibria of this game.
Q4. Stackelberg Outcome
Suppose that demand for quantity x in a market is given by the price p = 17 - x and there are two firms,
Savannah and Frontier. Savannah has a marginal cost of 3 while Frontier has a marginal cost of 1.
a) What is the Stackelberg outcome if Savannah is the Stackelberg leader?
b) What is the Stackelberg outcome if Frontier is the Stackelberg leader?