Economics & Evolution

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

Nash (Cournot) Equilibrium ( ⅓ , ⅓ )

A Dynamic Process

q1 q2

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

Define: The difference equation: becomes or

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

q1 q2 C B A

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:

q1 q2

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

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

Continuous Time

Cournot model, Three Firms discrete time +

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

Cournot model, Three Firms Continuous time

Cournot model, Three Firms Continuous time

Cournot model, Three Firms Continuous time

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.

Updating the frequencies: or:

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)

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 ).

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)

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

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

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

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

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 )

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:

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:

 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 !!!

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 ???

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)

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

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

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: