1. Economics & Evolution

2. Cournot Game 2 players Each chooses quantity qi ≥ 0
Player i’s payoff is: qi(1- qi –qj )
Player 1’s best response (given q2 ):
Inverse demand
(price)
No cost
BR1
BR2

3. Nash (Cournot) Equilibrium ( ⅓ , ⅓ )

4. A Dynamic Process

5. q1 q2

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

7. Define:
The difference equation:
becomes or

8. Do the players' beliefs make sense ?
Convergence to
Nash Equilibrium
Do the players' beliefs make sense ?

9. q1 q2 C B A

10. Do the players' beliefs make sense ?
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. q1 q2

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

14. Continuous Time

17. Cournot model, Three Firms
discrete time +

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

19. Cournot model, Three Firms
Continuous time

20. Cournot model, Three Firms
Continuous time

21. Cournot model, Three Firms
Continuous time

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. Updating the frequencies:
or:

24. An Example A B
0 , 2
3 , 0
2 , 1
1 , 3
Let p(t),q(t) be the frequencies of the second strategy played by played 1,2
Analysis of the stage-game: q 1
(A)
(B)
BR1
 BR2
Nash Equilibrium p = q = 1/2 p 1
(B)

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

26. p(t) increases, q(t) decreases (with t)
An Example q 1
(A)
(B) A B
0 , 2
3 , 0
2 , 1
1 , 3 p 1
(B)
the best responses are: ( B , A )
p(t) increases, q(t) decreases (with t)

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

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

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

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

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

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

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

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

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

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

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

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

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