1
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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Existence of Equilibrium
• 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
Existence of Equilibrium
• 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
Solu.on:
No Nash equilibria
Existence of Equilibrium
• 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
Existence of Equilibrium
• Mixed strategies – Player 2 selects LeL with probability p (where 0 ≤ p ≤ 1) – Player 1 selects Up with probability q (where 0 ≤ q ≤ 1)
Existence of Equilibrium
• 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{ }
Left Right Up A, a C, c
Down B, b D, d
Existence of Equilibrium
• 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
Existence of Equilibrium
Existence of Equilibrium
• Q: How do we make predicTons? – Find Nash equilibria in the new game
• Q: What procedure to we use? – Derive best-‐reply funcTons
Existence of Equilibrium
• 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
Existence of Equilibrium
• A useful “trick” – It turns out to be convenient to start out studying when the “pure strategies” are beber than one another
Existence of Equilibrium
• 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{ }
Left Right Up A, a C, c
Down B, b D, d
Existence of Equilibrium
• 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( )
Existence of Equilibrium
• Player 1 prefers Up (ie q=1) if
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( )
Existence of Equilibrium
• Player 1 prefers Up (ie q=1) if
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( )
If Up is beber than Down,
Then, Player 1 selects Up with probability one
Existence of Equilibrium
• Player 1 prefers Up (ie q=1) if
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( )
If Up is beber than Down,
Then, Player 1 selects Up with probability one
Player 1’s Best Reply (Optimal q for every p)
Existence of Equilibrium
• 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)
Player 1's Best Reply
Existence of Equilibrium
!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
Existence of Equilibrium
• 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)
Player 1's Best Reply
Existence of Equilibrium
!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)
Player 2's Best Reply
Existence of Equilibrium
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q
p
1
1
(d-b)(a-c)+(d-b)
Player 2's Best Reply
(C-D)(B-A)+(C-D)
Player 1's Best Reply
Existence of Equilibrium
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q
p
1
1
(d-b)(a-c)+(d-b)
Nash Equilibrium
(C-D)(B-A)+(C-D)
Existence of Equilibrium
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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
Existence of Equilibrium
• 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
Existence of Equilibrium
• 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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Cournot Duopoly Residual Demand
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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Cournot Duopoly Residual Demand
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 = q’1, then p = P(q’1+q2)
q’1 q2+q’1
P(q’1+q2) *
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Cournot Duopoly Residual Demand
Market clearing price Assume firm 2 will produce q2.
How will market price vary with q1?
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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Cournot Duopoly Residual Demand
Market clearing price Assume firm 2 will produce q2.
How will market price vary with q1?
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
Market clearing price
Quantity
Assume firm 2 will produce q2. How much will firm 1 produce?
D1 D
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Cournot Duopoly Best Reply
Market clearing price
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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Cournot Duopoly Best Reply
Assume firm 2 will increases production. How will firm 1 react?
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Cournot Duopoly Best Reply
Market clearing price
Quantity D +Δq2 -Δq1
If Firm 2 produces more, Firm 1 produces less
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Cournot Duopoly Best Reply
Market clearing price
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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Cournot Duopoly Best Reply
Market clearing price
Quantity
If q2 = 0 Then q1 = qm
D qm
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Cournot Duopoly Best Reply
Market clearing price
Quantity
If q2 = qc Then q1 = 0
D qc D1
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Cournot Duopoly Best Reply
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$%&
'()
−1mq1 = 0
Solve for q1 (best-reply function)
q1 =a − c( )m
2−
12q2
Similarly
q2 =a − c( )m
2−
12q1
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Derive best-‐reply funcTons q1
q2
a − c( )m2
Firm 1's best-reply function
q1 =a − c( )m
2−
12q2
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Derive best-‐reply funcTons q1
q2
a − c( )m2
Firm 1's best-reply function
q1 =a − c( )m
2−
12q2
Firm 2's best-reply function
q2 =a − c( )m
2−
12q1
a − c( )m2
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Compute equilibrium quanTTes q1
q2
q1*
q2*
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Compute equilibrium quanTTes Equilibrium
q1 =a − c( )m
2−
12q2
q2 =a − c( )m
2−
12q1
Find q1*
q1* =
a − c( )m2
−12
a − c( )m2
−12q1
*"#$
%&'
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
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