1 Economics & Evolution. 2 Cournot Game 2 players Each chooses quantity q i ≥ 0 Player i’s...

1

Economics & Evolution

2

Cournot Game

• 2 players

• Each chooses quantity qi ≥ 0

• Player i’s payoff is: qi(1- qi –qj )

Inverse demand(price)

No cost

• Player 1’s best response (given q2 ):

11 1 2 q 1 2q 1 - q - q ' = 1 - 2q - q = 0

21

1 - qq =

2similarly, 1

2

1 - qq =

2BR1 BR2

3

q1

q2

21

1 - qq =

2BR1

12

1 - qq =

2BR2

Nash (Cournot) Equilibrium ( ⅓ , ⅓ )

4

tt+1 21

1 - qq =

2

tt+1 12

1 - qq =

2

A Dynamic ProcessA Dynamic Process

t = 1, 2, 3, ..... di.. sc.. rete

t+1 t1 0 11 1= -

1 1 02 2q q

t+1 t1 1t+1 t2 2

1 0 1q q1 1= -

1 1 02 2q q

5

q1

q2

t+11q

tq

t+12q

t+1q

6

t+1 t1 0 11 1= -

1 1 02 2q q

A steady state:

1 0 11 1= -

1 1 02 2q q

0 1 11 1=

1 0 12 2I + q

2 1 1=

1 2 1q

-12 1 2 -11

=1 2 -1 23

2 1 2 -1

1 2 -1 2

4 - 1 = 3

2 - 2 = 0

7

2 1 1=

1 2 1q

-12 1 2 -11

=1 2 -1 23

2 -1 1 11 1 1= =

-1 2 1 13 3 3q e

Define:

t tt t 1

3

1= -

3d q e q d e

t+1 t0 11 1 1 1

= -1 03 2 2 3

d e e d e

The difference equation:

becomes

t+1 t0 11 1= -

1 02 2q e q

t+1 t0 11= -

1 02d dor

8

t+1 t0 11= -

1 02d d

t+1 t+11 2t+1 t+12 1

d d1= -

2d d

,t+1 t t+1 t1 2 2 1

1 1d = d d = d

2 2

tid 0

t t 1 1

3 3q d e e Convergence to

Nash Equilibrium

9

q1

q2

A

Show that if then for all 0 t:q A t q A

B

C

Show that if then t t+1q B q C

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:

For odd 's:

For even 's:

t t -1 t t -11 1 2 2 2

t t -1 t t -11 1 2 2 1

q = BR q , q = q

q = q , q = BR q

t

t

11

q1

q2

0q1q

2q4q

For odd 's:

For even 's:

t t -1 t t -11 1 2 2 2

t t -1 t t -11 1 2 2 1

q = BR q , q = q

q = q , q = BR q

t

t

.

:For even odd

t -21 2

t t -1 t -21 1 1 2

t t -12 2 1 2

t - 1

=

q

q = q BR q ,

q = BR q BR BR

t

.

t -22

t -22t

2

1 - q1 - 1+ q2 =

2 4q =1

3

12

1 1 1 2 1t + Δt t + Δt × t tq = q BR q q

Continuous Time

Let the time interval between periods approach 0.tq tqA change of notation:

1q 1 2BR q

In time Δt the individual advances only part of the path

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1 1 1 2 1t + Δt t + Δt × t tq = q BR q q

Continuous Time

Let the time interval between periods approach 0.

1 11 2 1

t + Δt - t= t t

Δt

q qBR q q t 0

•

1 1 2 1q = BR q t q t

tq tq

similarly • 1

2 2

1-q tq = q t

2

A change of notation:

21

1-q tq t

2

21

1-q tq t

2

14

• 2

1 1 t1-q t

q = q2

• 1

2 2

1-q tq = q t

2

Continuous Time

• •

1 2 1 21

- -2

t tq q = q q

denote: 1 2t t d t q q

1

2t d(t)= - d

1

2t

d(t) = -d

1- t2td = Ce

•

1

21 1

1 1 t2

t1+ -q t

qq2t

qq

=

1- t2•

11

3 t2

1+q = q2Ce

1- t2•

11

+ t32

1+q q = Ce2

ln

1= -

2d t

15

1- t2•

11

+ t32

1+q q = Ce2

16

1- t2•

11

+ t32

1+q q = Ce2

1- t2• 3 3t t

2 21

1+

3 t2

1+ Ceq q e = e2

1- t2•

11

+3 t2

1+ Ceq q =2 X

1- t2e

2

3 3t t t2 21 t 1 Cq e = e e

2

2

3 3t t t2 21 t 1 Cq e = e e + K

3

2

1 3- t t2 2

1 t 1 Cq = e + Ke3

13

17

t tt+1 2 31

1 - q - qq =

2

1 1 1 2 3π q 1 - q - q - q=

Cournot model, Three Firms

discrete time

11 2 3

1

π1 - 2q - q - q

q=

t tt+1 1 32

1 - q - qq =

2

t tt+1 1 23

1 - q - qq =

2

t+1 t+1 t+1 t t t1 2 3 1 2 3

3q + q + q = q + q + q

2+

t+1 t3Q = Q

2

3Q = Q ,

2

3Q =

4

t t t+1 t t+1 t3 3 3 3d = Q - d + = - - d d = -d

4 4 2 4

18


discrete time

t+1 td = -d 3

4 ttttd = -1 C, Q -1 C

does not converge !!!!!

Continuous time

t

2 31 1 1

1 - q t - qq t + Δt = q t + Δt - q t

2

2 31 1=

1-q t -q tq -q t

2

Similarly for i = 2, 3. Add the three equations:

=3Q - 2Q t2

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Continuous time

=3Q - 2Q t2

+ =3Q 2Q2 X 2te

•

2t 2t=3Qe e2

2t 2t= +3Qe e K4

-2t= +3Q Ke4

2 31 1=

1-q t -q tq -q t

2

1 31

21 1-=

1- -q t -q tq

q t 1 q t+ q2 2

t

1 1= -1-Q t 1q q t

2 2

-2t

1 1

-= -

31- Ke 14q q t2 2

20


Continuous time

-2t

1 1

-= -

31- Ke 14q q t2 2

21


Continuous time

-2t

1 1

-= -

31- Ke 14q q t2 2

-2t

1 1

-= -

1 Ke 14q q t2 2

-2t1 1+ =

1 1q q t -Ce2 8 X

1t2e

1t

21

1 32 2

- tq e =

t1 e -Ce8

1 32 2

1t - t21 = +

t1q e e Ce4

-2t1 = +

1q Ce4

14

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Fictitious PlayFictitious 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)(fictitious mixed strategy)

• Each player chooses a best response to the fictitious mixed strategy of his opponent.

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Updating the frequenciesUpdating the frequencies:

t t t1 2 n(f , f , ........, f )+ (0,0, ......,0,1,0, ....,0)

t + 1t

oror:

t t t1 2 n(f , f , ........, f )+ (0,0, ......,0,1,0, ....,0)Δ

t + Δt

24

A B

A 0 , 2 3 , 0

B 2 , 1 1 , 3

Let p(t),q(t) be the frequencies of the

second second strategy played by played 1,2

Analysis of the stage-game:

q

0

1

(A)

(B)

(A) p 1(B)

An ExampleAn Example

BRBR11

BRBR22

Nash Equilibrium p = q = 1/2

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3 q

0

1

(A)

(B)

(A) p 1(B)


As long as ( p(t) , q(t) ) is in the first quadrant,

the best responses are: ( B , A ).

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3 q

0

1

(A)

(B)

(A) p 1(B)


the best responses are: ( B , A )

p(t) increases, q(t) decreases (with t)

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3 q

0

1

(A)

(B)

(A) p 1(B)


λtp t + λτp t + τ =

λ t + τ

τ 1 - p ttp t + τ

p t + τ - = - =t +

p tτ

p tt + τ

=t p 1 - p t

tp t +λ λτp t + τ =

λ t + τ

=ap t 1 -t

•

=tp 1

tp t + τp t + τ =

t + τ

tp = t - a

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3 q

0

1

(A)

(B)

(A) p 1(B)


=ap t 1 -t

tq t + 0 τq tq t + τ - = - = -

tq t

+ τq t

t + τ

=t q -q t

λtq t + 0q t + τ =

λ t + τ

tq = 0

=bq tt

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3 q

0

1

(A)

(B)

(A) p 1(B)


=ap t 1 -t

=bq tt

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

q(t)

0

1

1 - p(t) 1


=ap t 1 -t

=bq tt

=1- p t a/t = const.

b/tq t

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

q(t)

0

1

1 - p(t) 1


=ap t 1 -t

=bq tt

=1- p t a/t = const.

b/tq t

Best responses in this quadrant are (B , B )

p(t) , q(t) increase (with t )

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


Best responses in the quadrant are:

(B , A )(B , B )

(A , B )

(A , A )

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


Best responses in the quadrant are:

(B , A )(B , B )

(A , B )

(A , A )

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


Does it converge? (B , A )(B , B )

(A , B )

(A , A )

at

at+1

t t

t+1

a a + 1/2=

a 1/2

t+1 t

1 1= 1+

2a 2adefine: t

t

1 s =

2a

t+1 ts = 1+ st 0s = t + s

t0

1a =

2 t + s0

converges !!!

Does convergence mean that Does convergence mean that they play the equilibrium?they play the equilibrium?

35

A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


(B , A )(B , B )

(A , B )

(A , A )

What does an outside observer see?What does an outside observer see?

(B , A )

(B , B )

(A , B )

(A , A )

How much time is spent in each quadrant ???How much time is spent in each quadrant ???

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


(B , A )(B , B )

(A , B )

(A , A )

time is spent in each quadranttime is spent in each quadrant

0 0

1t , p ,

2

1 1

1t , ,q

2,2 2

1t , p

2

t+1 t

1 1= 1+

2a 2a 1 0

1 1= 1+

2 1/2 - q 2 1/2 - p

2 1

1 1= 1+

2 p - 1/2 2 1/2 - q

(to be used later)

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


(B , A )(B , B )

(A , B )

(A , A )


0 0

1t , p ,

2

1 1

1t , ,q

2,2 2

1t , p

2 =

ap t 1 -t =

bq tt

00

=ap t 1 -t

00a = t 1 - p

0

0 =bq tt

0tb =2

0 0=

t 1 - pp t 1 -

t 0=

tq t

2t

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


(B , A )(B , B )

(A , B )

(A , A )


0 0

1t , p ,

2

1 1

1t , ,q

2,2 2

1t , p

2 0 0=

t 1 - pp t 1 -

t 0=

tq t

2t

0

11

0= =

t 1 - p1p t 1 -2 t

1 0 0t = 2t 1 - p

1 0 0 0-t t = 2 1 - p - 1 t

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A B

A 0 , 2 3 , 0B 2 , 1 1 , 3

0

1

1


(B , A )(B , B )

(A , B )

(A , A )

0 0

1t , p ,

2

1 1

1t , ,q

2,2 2

1t , p

2

1 0 0 0-t t = 2 1 - p - 1 t

time spent in the time spent in the firstfirst quadrantquadrant

0 0=

t 1 - pp t 1 -

t 0=

tq t

2t

1 1=

t 1 - pp t 1 -

t 1 1t 1 - q

q t = 1 -t

analogously:

1 Economics & Evolution. 2 Cournot Game 2 players Each chooses quantity q i ≥ 0 Player i’s...

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Transcript of 1 Economics & Evolution. 2 Cournot Game 2 players Each chooses quantity q i ≥ 0 Player i’s...