1. 1 Economics & Evolution

2. 2 Cournot Game 2 players Each chooses quantity q i ≥ 0 Player i’s payoff is: q i (1- q i –q j ) Inverse demand (price) No cost Player 1’s best response (given q 2 ): BR 1 BR 2

3. 3 q1q1 q2q2 BR 1 BR 2 Nash (Cournot) Equilibrium ( ⅓, ⅓ )

4. 4 A Dynamic Process

5. 5 q1q1 q2q2

6. 6 A steady state: 4 - 1 = 3 2 - 2 = 0

7. 7 Define: The difference equation: becomes or

8. 8 Convergence to Nash Equilibrium

9. 9 q1q1 q2q2 A B C

10. 10 Change the model: Players are very rarely allowed to revise their strategy. At period t, player i is allowed to choose his best response with probability p, where p ~ 0. The players alternate:

11. 11 q1q1 q2q2

12. 12 Continuous Time Let the time interval between periods approach 0. A change of notation: In time Δt the individual advances only part of the path

13. 13 Continuous Time Let the time interval between periods approach 0. A change of notation:

14. 14 Continuous Time

15. 15

16. 16

17. 17 Cournot model, Three Firms discrete time +

18. 18 Cournot model, Three Firms discrete time does not converge !!!!! Continuous time Similarly for i = 2, 3. Add the three equations:

19. 19 Cournot model, Three Firms Continuous time

20. 20 Cournot model, Three Firms Continuous time

21. 21 Cournot model, Three Firms Continuous time

22. 22 Fictitious Play Two players repeatedly play a normal form game Each player observes the frequencies of the strategies played by the other player in the past ( (( (fictitious mixed strategy) Each player chooses a best response to the fictitious mixed strategy of his opponent.

23. 23 Updating the frequencies Updating the frequencies : or or :

24. 24 AB A 0, 23, 0 B 2, 11, 3 Let p(t),q(t) be the frequencies of the second strategy played by played 1,2 Analysis of the stage-game: q 0 1 (A) (B) (A) p 1 (B) An Example  BR 1  BR 2 Nash Equilibrium p = q = 1/2

25. 25 AB A 0, 23, 0 B 2, 11, 3 q 0 1 (A) (B) (A) p 1 (B) An Example As long as ( p(t), q(t) ) i s in the first quadrant, the best responses are: ( B, A ).

26. 26 AB A 0, 23, 0 B 2, 11, 3 q 0 1 (A) (B) (A) p 1 (B) An Example the best responses are: ( B, A ) p(t) i ncreases, q(t) decreases (with t )

27. 27 AB A 0, 23, 0 B 2, 11, 3 q 0 1 (A) (B) (A) p 1 (B) An Example

28. 28 AB A 0, 23, 0 B 2, 11, 3 q 0 1 (A) (B) (A) p 1 (B) An Example

29. 29 AB A 0, 23, 0 B 2, 11, 3 q 0 1 (A) (B) (A) p 1 (B) An Example

30. 30 AB A 0, 23, 0 B 2, 11, 3 q(t) 0 1 1 - p(t) 1 An Example

31. 31 AB A 0, 23, 0 B 2, 11, 3 q(t) 0 1 1 - p(t) 1 An Example Best responses in this quadrant are (B, B ) p(t), q(t) i ncrease (with t )

32. 32 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example Best responses in the quadrant are: (B, A ) (B, B ) (A, B ) (A, A )

33. 33 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example Best responses in the quadrant are: (B, A ) (B, B ) (A, B ) (A, A )

34. 34 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example Does it converge? (B, A ) (B, B ) (A, B ) (A, A ) atat a t+1  0 converges !!! Does convergence mean that they play the equilibrium?

35. 35 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) What does an outside observer see? (B, A ) (B, B ) (A, B ) (A, A ) How much time is spent in each quadrant ???

36. 36 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time is spent in each quadrant (to be used later)

37. 37 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time is spent in each quadrant

38. 38 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time is spent in each quadrant

39. 39 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time spent in the first quadrant analogously: