Game Theory Lecture Jan 18

1. Game Theory Lecture Jan 18

2. In Bertrand’s model of oligopoly • Each firm chooses its quantity as the best response to the quantity chosen by the other(s). • Each firm chooses its price as the best response to the price chosen by the other(s). • The firms set quantities sequentially. The second firm’s quantity is the best response to the first firm’s quantity. • The first firm sets a quantity. The second firm follows by setting a price. • The firms jointly set the price that maximizes industry profits.

3. Ex 42.2 (a joint project) Two players, choose effort levels x1 and x2 between 0 and 1. Cost of effort to player i is xi2 Part a) Total output is 3x1x2. They divide output equally. Payoff to Player 1 is:

4. Find Player 1’s best response • Maximize

5. Player 1’s reaction function

6. The two reaction functions

7. Reaction Function Graph

8. Part b • Total output is 4x1x2 and is split between two players. • Cost of effort xiis c(xi)=xi • Payoff to player 1 is:

9. Find Player 1’s best response function • Take derivative: What does it tell us?

11. Ex 58.2 (Cournot) • Two firms, linear inverse demand function P=a-Q=a-q1-q2. Firms have constant marginal costs, c1and c2 • Profit function for firm 1 is:

12. Reaction Functions

13. Graph reaction functions

14. Algebra

15. Ex 59.1 (Cournot 2) • Linear inverse demand P=a-Q • Quadratic Cost function: C(qi)=qi2 • What is profit function for firm 1?

16. Bertrand model • Each player does best response to other’s price. • Constant marginal cost • Buyers will purchase only from seller with lowest price. If prices are equal, demands are split. • What can be an equilibrium?

17. Ex 69.1 Bertrand with fixed costs

18. Mixed Strategies Matching pennies

19. Player 2’s best response • If Player 1 plays heads with probability p>1/2, what is Player 2’s best response? • What if p<1/2? • What if p=1/2

20. Both players’ best responses

21. Hide and seek game