Games & Oligopoly, Continuedstennek.se/onewebmedia/4 - Games & Oligopoly, Continued (slides).ppt.pdf · Existence’of’Equilibrium’ • If’game’has’ – Finitely’many’players’

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Games & Oligopoly, Continued

Johan Stennek

Agenda

•  Games: Mixed strategy equilibrium

•  Oligopoly: Cournot model

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Mixed Strategies and Existence of Equilibrium

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Existence of Equilibrium

•  If game has –  Finitely many players

–  Each player has finitely many strategies

•  Then, game has at least one Nash equilibrium –  Possibly in mixed strategies

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•  Example –  2 players –  Player 1 has two pure strategies: Up and Down –  Player 2 has two pure strategies: LeL and Right –  Player 1’s Payoffs: B > A, C > D, –  Player 2’s Payoffs: a > c, d > b

Left Right Up A, a C, c

Down B, b D, d

Exercise:

Find the Nash equilibria


•  Example –  2 players –  Player 1 has two pure strategies: Up and Down –  Player 2 has two pure strategies: LeL and Right –  Player 1’s Payoffs: B > A, C > D, –  Player 2’s Payoffs: a > c, d > b


Down B, b D, d

Solu.on:

No Nash equilibria


•  Game in mixed strategies –  Let us now define a new game, which acknowledges that people may randomize their choices if they want to.

•  Q: New game –  Players: Same as before

–  Strategies: All possible probability distribuTons over “pure strategies”

–  Payoffs: Expected payoff


•  Mixed strategies –  Player 2 selects LeL with probability p (where 0 ≤ p ≤ 1) –  Player 1 selects Up with probability q (where 0 ≤ q ≤ 1)


•  Expected uTlity p*q = Prob (Up & Left)

U1 q, p( ) = A ⋅ p ⋅q + B ⋅ p ⋅ 1− q( ) + C ⋅ 1− p( ) ⋅q + D ⋅ 1− p( ) ⋅ 1− q( )

Wherep = Prob Left{ }q = Prob Up{ }


Down B, b D, d


•  Game in mixed strategies – Players: 1 and 2

– Strategies: p in [0, 1] and q in [0, 1]

– Payoffs: U1(p,q); U2(p,q)

q

p

1

1

Mixed strategies



•  Q: How do we make predicTons? – Find Nash equilibria in the new game

•  Q: What procedure to we use? – Derive best-‐reply funcTons


•  NoTce: “the pure strategies are sTll there” – Player 2 going Right corresponds to p = 0 – Player 2 going LeL corresponds to p = 1 – Player 1 going Down corresponds to q = 0 – Player 1 going Up corresponds to q = 1


•  A useful “trick” –  It turns out to be convenient to start out studying when the “pure strategies” are beber than one another


•  Expected uTlity of pure strategies

U1 p,1( ) = A ⋅ p + C ⋅ 1− p( ) q = 1⇔ "Up"

U1 p,0( ) = B ⋅ p + D ⋅ 1− p( ) q = 0⇔ "Down"

p = Prob Left{ }


Down B, b D, d


•  Player 1 prefers Up (ie q=1) if

!U1 Up( ) > !U1 Down( )

⇔ A ⋅ p + C ⋅ 1− p( ) > B ⋅ p + D ⋅ 1− p( )

⇔ p <C − D( )

B − A( ) + C − D( )



q

p

1

1(C-D)(B-A)+(C-D)

Player 1's Best Reply

!U1 Up( ) > !U1 Down( )

⇔ p <C − D( )

B − A( ) + C − D( )



q

p

1

1(C-D)(B-A)+(C-D)


!U1 Up( ) > !U1 Down( )

⇔ p <C − D( )

B − A( ) + C − D( )

If Up is beber than Down,

Then, Player 1 selects Up with probability one



q

p

1

1(C-D)(B-A)+(C-D)


!U1 Up( ) > !U1 Down( )

⇔ p <C − D( )

B − A( ) + C − D( )

If Up is beber than Down,

Then, Player 1 selects Up with probability one

Player 1’s Best Reply (Optimal q for every p)


•  Player 1 prefers Down (ie q=0) if

!U1 Up( ) < !U1 Down( )

⇔ A ⋅ p + C ⋅ 1− p( ) < B ⋅ p + D ⋅ 1− p( )

⇔ p >C − D( )

B − A( ) + C − D( )

q

p

1

1(C-D)(B-A)+(C-D)



!U1 Up( ) < !U1 Down( )

⇔ p >C − D( )

B − A( ) + C − D( )If Up is worse than Down,

Then, Player 1 selects Up with probability zero


•  Player 1 indifferent if

!U1 Up( ) = !U1 Down( )

⇔ A ⋅ p + C ⋅ 1− p( ) = B ⋅ p + D ⋅ 1− p( )

⇔ p =C − D( )

B − A( ) + C − D( )

q

p

1

1(C-D)(B-A)+(C-D)



!U1 Up( ) = !U1 Down( )

⇔ p =C − D( )

B − A( ) + C − D( )If Up and Down equally good,

Then, Player 1 selects Up with any probability

q

p

1

1

(d-b)(a-c)+(d-b)



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q

p

1

1

(d-b)(a-c)+(d-b)


(C-D)(B-A)+(C-D)



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q

p

1

1

(d-b)(a-c)+(d-b)

Nash Equilibrium

(C-D)(B-A)+(C-D)


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Exercise

Exercise

•  Bable of the sexes –  Two spouses want to go out, either to see a football game or a theater play

–  The man enjoys football (but not theater)

–  The woman enjoys theater (but not football)

–  They both enjoy each other’s company


•  Payoff matrix – Man is player one –  v = value of preferred alternaTve (0 is value of other) –  t = value of being together –  Assume t > v.

Football Theater Football v+t, t v, v Theater 0, 0 t, v+t


•  To do –  Define the game in mixed strategies –  Find the man’s best-‐reply funcTon. Display in diagram –  Same for woman –  Find equilibria – Which is more plausible?

Football Theater Football v+t, t v, v Theater 0, 0 t, v+t

Cournot Model (AlternaTve to Bertrand)

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QuanTty CompeTTon

•  Bertrand model – Firms set prices

•  Cournot model – Firms chose quanTTes – Then price is set to clear the market

•  Note 1: Difference mabers (contrast to monopoly)

•  Note 2: Two different interpretaTons

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QuanTty CompeTTon

•  First interpretaTon: Two-‐stage game –  Stage 1: Firms chose capaciTes: k1, k2

–  Stage 2: Firms set prices: p1, p2

•  Note: – Under some condiTons p1 = p2 = P(k1 +k2)

– Then study choice of capacity (= quanTty)

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QuanTty CompeTTon •  Second interpretaTon

–  Stage 1: Firms produce: q1, q2 –  Stage 2: Firms bring produce to aucTon: p = P(q1+q2)

•  Example –  Fishing village

•  Note –  Pricing decision is delegated –  But equilibrium price affected by amount produced –  No Tme to react –  We omit the issue why p = P(q1+q2)

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Cournot Duopoly Game

•  Players – Firms: 1 and 2

•  Strategies – ProducTon: q1 and q2

•  Payoffs – Profits: πi(q1, q2) = P(q1 + q2)qi – C(qi)

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Exogenous condiTons

•  Simplify 1: Technology – Constant marginal cost – Firms have same marginal cost

•  Simplify 2: Demand – Firms’ goods homogenous – Market demand: Linear

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Simplified payoff funcTon

•  Profits

π i q1,q2( ) = α − β ⋅ q1 + q2( )⎡⎣ ⎤⎦ ⋅qi – c ⋅qi

Cournot Duopoly: Graphical SoluTon

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Cournot Duopoly Residual Demand

Market clearing price

q1

Assume firm 2 will produce q2. How will market price vary with q1?

q2 D

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Market clearing price Assume firm 2 will produce q2.

How will market price vary with q1?

q2

P(0+q2) *

D q1 0

If q1 = 0, then p = P(0+q2)

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q2

P(0+q2) *

D q1 0

If q1 = q’1, then p = P(q’1+q2)

q’1 q2+q’1

P(q’1+q2) *

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q2

P(0+q2) *

D q1 0

Two point on residual demand

q’1 q2+q’1

P(q’1+q2) *

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q2

P(0+q2)

D q1 0 q’1 q2+q’1

P(q’1+q2)

D1

D1 is a parallel shift of D by q2 units

*

*

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Cournot Duopoly Best Reply


Quantity

Assume firm 2 will produce q2. How much will firm 1 produce?

D1 D

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Quantity

Assume firm 2 will produce q2. How much will firm 1 produce?

q*1

P(q2+ q*1)

D1 D

c

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Assume firm 2 will increases production. How will firm 1 react?

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Quantity D +Δq2 -Δq1

If Firm 2 produces more, Firm 1 produces less

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Quantity D +Δq2 -Δq1

Note: P(q1 + q2) is reduced Hence: q1 + q2 is increased Hence: q1 reduced by less than q2 increased

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Quantity

If q2 = 0 Then q1 = qm

D qm

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Quantity

If q2 = qc Then q1 = 0

D qc D1

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q1

q2

qm

qc

q*1(0) = qm

q*1(qc) = 0

Negative slope

Less than -1

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Cournot Duopoly Equilibrium

q1

q2

qm

qc qm

qc

* qn

qn

Firm 2’s best reply Equilibrium: Both a doing their best, given what the other does 2qn

qm < 2qn < qc

q1+q2=qc

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Cournot Duopoly

•  Conclusions – Effect of concentraTon

•  pm = p(qm) > p(qn + qn) = pd

– QuanTty compeTTon vs price compeTTon •  Cournot price higher than Bertrand price •  Details maber

Cournot Duopoly: AnalyTcal SoluTon

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Cournot Duopoly

•  Technology –  Constant marginal costs, c

•  Demand (linear) –  Individual demand: q = a – p –  Number of consumers: m – Market demand: Q = m*(a – p)

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Cournot Duopoly

•  Exercise: –  Solve the model

•  Steps: 1. Define the game 2. Compute best-‐reply funcTons 3. Find equilibrium quanTTes 4. Find equilibrium price 5. Check if Cournot price is lower than monopoly price

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Define the game

•  Players –  Firms 1 and 2

•  Strategies –  Firms choose quanTTes q1 and q2 (any posiTve real number)

•  Payoffs –  Profits –  Need to specify how the firms’ profits depend on the two quanTTes (strategy profile)

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Define the game

•  CriTcal assumpTon in Cournot –  First, firms choose quanTTes q1 and q2 –  Then, price is set to clear the market

•  To find market clearing price, use indirect market demand funcTon – Market demand: Q = m*(a – p) –  Indirect market demand: p = a – Q/m = a – (q1+q2)/m

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Define the game

Profit

Π1 = p − c( ) q1

Market-clearing price given by inverse demand

Π1 q1,q2( ) = a − 1m

q1 + q2( ) − c⎛⎝⎜

⎞⎠⎟q1

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Derive best-‐reply funcTons

Profit

Π1 = p − c( ) q1

Market-clearing price given by inverse demand

Π1 = a − 1m

q1 + q2( ) − c#$%

&'(q1

First-order condition

∂Π1

∂q1

= a − 1m

q1 + q2( ) − c#$%

&'(

−1mq1 = 0

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Derive best-‐reply funcTons

First-order condition

∂Π1

∂q1

= a − 1m

q1 + q2( ) − c$%&

'()

−1mq1 = 0

Solve for q1 (best-reply function)

q1 =a − c( )m

2−

12q2

Similarly

q2 =a − c( )m

2−

12q1

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Derive best-‐reply funcTons q1

q2

a − c( )m2

Firm 1's best-reply function

q1 =a − c( )m

2−

12q2

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Derive best-‐reply funcTons q1

q2

a − c( )m2


q1 =a − c( )m

2−

12q2


q2 =a − c( )m

2−

12q1

a − c( )m2

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Compute equilibrium quanTTes q1

q2

q1*

q2*

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Compute equilibrium quanTTes Equilibrium

q1 =a − c( )m

2−

12q2

q2 =a − c( )m

2−

12q1

Find q1*

q1* =

a − c( )m2

−12

a − c( )m2

−12q1

*"#$

%&'

Solve for q1*

q1* =

a − c( )m3

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Compute equilibrium quanTTes q1

q2

q1*

q2*

Equilibrium

q1* =

a − c( )m3

q2* =

a − c( )m3

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Compute equilibrium price

Equilibrium price

p* = a − 1m

q1* + q2

*( )

p* = a − 1m

a − c( )m3

+a − c( )m

3"#$

%&'

p* =a + 2c

3

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Compare with monopoly

Question: Effect of competition on price?

p* =a + 2c

3

pm = a + c2

Answer: Duopoly price lower

p* < pm

a + 2c3

<a + c

2

c < a

•  Bertrand duopoly – With different costs

•  Cournot oligopoly – With n firms – Duopoly with differenTated goods

Problem Set

Documents

Games & Oligopoly, Continuedstennek.se/onewebmedia/4 - Games & Oligopoly, Continued (slides).ppt.pdf · Existence’of’Equilibrium’ • If’game’has’ – Finitely’many’players’