Instructor: Martin J. Osborne

Time: 1 hour 50 minutes. Answer all questions. The numbers in brackets at the start of each question are the numbers of points the questions are worth. **To obtain credit, you must give arguments to support your answers.**

- Consider the following strategic game.
*X**Y**Z**X*2,2 0,2 3,1 3,0 0,0 4,0 4,2 0,1 5,1 *Y**Z*- (5) Is any action of either player strictly dominated?
- (5) Is any action of either player weakly dominated?
- (5) Find the Nash equilibria of the game. Is any equilibrium strict?

- (5) Is any action of either player strictly dominated?
- Consider a situation in which two players receive the amounts of
*money*in the following table.*X**Y**X*$5,$0 $50,$40 $10,$9 $80,$100 *Y*Each player cares not about the amount of money she gets, but about the

*difference*between the amount of money she gets and the amount of money the other player gets. (That is, she cares about the amount of money she gets minus the amount of money the other player gets.)- (5) Model this situation as a strategic game.
- (5) Find the Nash equilibria.

- (5) Model this situation as a strategic game.
- (20) Find the Nash equilibria of the following strategic game.
- Players
- Two people.
- Actions
- Each person's set of actions is the set of nonnegative numbers.
- Preferences
- Person 1's preferences are represented by the payoff function

and person 2's preferences are represented by the payoff function*u*_{1}(*a*_{1},*a*_{2}) =*a*_{1}(2*a*_{2}*a*_{1})*u*_{2}(*a*_{1},*a*_{2}) =*a*_{2}(2 2*a*_{1}*a*_{2})

- (20) In Cournot's model of oligopoly with two firms, the payoff function of each firm
*i*is

where_{i}(*q*_{1},*q*_{2}) =*q*_{i}*P*(*q*_{1}+*q*_{2})*C*_{i}(*q*_{i}),*P*is the inverse demand function and*C*_{i}is firm*i*'s cost function.Find the Nash equilibrium (equilibria?) of the strategic game that models this situation when the inverse demand function is linear, given by

the cost function of firm 1 is*P*(*Q*) =*Q*if *Q*0 if *Q*> ,*C*_{1}(*q*_{1}) =*c**q*_{1}, the cost function of firm 2 is*C*_{2}(*q*_{2}) =*q*_{2}^{2}, and*c*< (3/4).(Carefully present the steps in your argument. Throughout your analysis, ignore the case in which the total output is so high that the price is zero--just assume that the price is always given by

*P*(*Q*) =*Q*.) - Consider a variant of Bertrand's model of duopoly in which some consumers always shop at the same firm. These "regular" customers do not compare firms' prices, but they do change the amount they buy according to the price "their" firm charges.
Specifically, the total demand facing firm

*i*is the sum of*R*_{i}(*p*_{i}) (the demand from*i*'s regular customers) and the demand from customers who compare prices, which is the same as it is in Bertrand's model (*D*(*p*_{i}) if*p*_{i}<*p*_{j}, (1/2)*D*(*p*_{i}) if*p*_{i}=*p*_{j}, and 0 if*p*_{i}>*p*_{j}).The functions

*R*_{1},*R*_{2}, and*D*are continuous and decreasing (a higher price reduces demand in each case), and*R*_{1}(*c*) > 0,*R*_{2}(*c*) > 0, and*D*(*c*) > 0. The cost function of firm*i*is defined by*C*_{i}(*q*_{i}) =*c**q*_{i}for*i*= 1, 2.- (5) Write down firm
*i*'s payoff function in the strategic game that models this situation. - (5) Is the pair (
*c*,*c*) of prices a Nash equilibrium? - (10) Does the game have any (other?) Nash equilibrium (
*p*,*p*) in which both firms charge the same price?

- (5) Write down firm
- (15) Consider the variant of Hotelling's model of electoral competition in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Either find all the Nash equilibria of the game or show that it has no Nash equilibria.