Cournot Models of Duopoly and Oligopoly

Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Outline�

• Experiment #3
• Review
• Cournot models of duopoly and oligopoly

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Experiment #3: Contest game

• Veconlab login:
• http://veconlab.econ.virginia.edu/login.htm

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• Class experiment:
• Two players each choose an effort level between 0 and 100
• The cost of effort is 1
• The prize value is 100
• The probability of winning the prize is proportional to the relative efforts by both players
• The payoff is the prize of 100 if you win (0 otherwise) minus cost of effort

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Review�

• DEFINITION: Nash equilibrium
• In the normal-form game {S₁, S₂,..., S_n, u₁, u₂,..., u_n}, a combination of strategies (s₁*,…, s_n*) is a Nash equilibrium if for every player i

u_i(s₁*,...,s_i-1*,s_i*,s_i+1*,...,s_n*) ≥ u_i(s₁*,...,s_i-1*,s_i,s_i+1*,...,s_n*) for all s_i∈ S_i

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• That is s_i* solves the following problem: Maximize u_i(s₁*,...,s_i-1*,s_i,s_i+1*,...,s_n*) subject to s_i∈ S_i

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Review�

• DEFINITION: Best response
• In the normal-form game {S₁, S₂,..., S_n, u₁, u₂,..., u_n}, if players 1, 2, ..., i-1, i+1, ..., n choose strategies, s₁,s₂,...,s_i-1,s_i+1,...,s_n, respectively, then player i‘s best response function is defined by:

B_i(s₁,...,s_i-1,s_i+1,...,s_n) = {s_i∈ S_i: u_i(s₁,...,s_i-1,s_i,s_i+1,...,s_n) ≥ u_i(s₁,...,s_i-1,s_i', s_i+1,...,s_n), for all s_i'∈ S_i }

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Review�

• DEFINITION: Best response (alternative)
• In the normal-form game {S₁, S₂,..., S_n, u₁, u₂,..., u_n}, if players 1, 2, ..., i-1, i+1, ..., n choose strategies, s₁,s₂,...,s_i-1,s_i+1,...,s_n, respectively, then player i‘s strategy s_i ∈ B_i(s₁,...,s_i-1,s_i+1,...,s_n) if and only if it solves (or it is an optimal solution to)

Maximize u_i(s₁,...,s_i-1,s_i',s_i+1,...,s_n) subject to s_i'∈ S_i

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Review�

• DEFINITION: Nash Equilibrium
• In the normal-form game {S₁, S₂,..., S_n, u₁, u₂,..., u_n}, a combination of strategies (s₁*,…, s_n*) is a Nash equilibrium if for every player i

s_i* ∈ B_i(s₁*,...,s_i-1*,s_i+1*,...,s_n*)

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• Nash equilibrium is a set of strategies, one for each player, such that each player’s strategy is best for her, given that others are playing their best strategies

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Review�

• Best response:
• What is Prisoner 1’s best response if Prisoner 2 chooses NC, C?
• What is Prisoner 2’s best response if Prisoner 1 chooses NC, C?

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• Therefore, (C, C) is a Nash equilibrium in Prisoners’ Dilemma

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Cournot model of duopoly�

• Augustin Cournot introduced first formal model of oligopoly in 1838
• In his model oligopoly firms choose how much to produce at same time
• As in prisoners' dilemma game, firms are playing non-cooperative game of imperfect information: each firm chooses its output level before knowing what other firm will choose

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Cournot model of duopoly�

• A product is produced by two firms: firm 1 and firm 2

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• The quantities are denoted by q₁ and q₂, respectively. Each firm chooses the quantity (any positive number) without knowing the other firm’s choice

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• The payoff of each firm depends on the market price and the cost of production

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• The market price is P(Q) = a - Q, where a is a constant number and Q=q₁+q₂

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• The cost to firm i of producing quantity q_i is C_i(q_i) = cq_i

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Cournot model of duopoly�

• Set of players: {Firm 1, Firm 2}
• Sets of strategies: S₁=[0, +∞), S₂=[0, +∞)
• Payoff functions: u₁(q₁, q₂)=q₁{a-(q₁+q₂)}-q₁c,

u₂(q₁, q₂)=q₂{a-(q₁+q₂)}-q₂c

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Cournot model of duopoly�

• How to find a Nash equilibrium?
• Find the quantity pair (q₁*, q₂*) such that q₁* is firm 1’s best response to firm 2’s quantity q₂* and q₂* is firm 2’s best response to firm 1’s quantity q₁*

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• q₁* solves � Maximize u₁(q₁, q₂) = q₁{a-(q₁+q₂)}-q₁c Subject to 0 ≤ q₁ ≤ +∞

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• q₂* solves� Maximize u₂(q₁, q₂) = q₂{a-(q₁+q₂)}-q₂c Subject to 0 ≤ q₂ ≤ +∞

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Review�

• Find a maximum of a function f(x):
• 1) Compute f'(x)
• 2) Compute f''(x) and check whether f''(x) ≤ 0
• 3) If f''(x) ≤ 0 then solve f'(x) = 0
• 4) The solution is a maximum

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Cournot model of duopoly: Nash equilibrium�

• Finding the best response of firm 1:

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• Maximize u₁(q₁, q₂) = q₁{a-(q₁+q₂)}-q₁c �Subject to 0 ≤ q₁ ≤ +∞

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• FOC: u₁'(q₁, q₂) = a - 2q₁ - q₂ - c = 0�Solution: q₁ = (a - q₂ - c)/2 (best response of firm 1)

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• SOC: u₁''(q₁, q₂) = - 2 < 0

Solution: concave, therefore maximum

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Cournot model of duopoly: Nash equilibrium�

• Finding the best response of firm 2:

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• Maximize u₂(q₁, q₂) = q₂{a-(q₁+q₂)}-q₂c �Subject to 0 ≤ q₂ ≤ +∞

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• FOC: u₂'(q₁, q₂) = a - q₁ - 2q₂ - c = 0�Solution: q₂ = (a - q₁ - c)/2 (best response of firm 2)

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• SOC: u₂''(q₁, q₂) = - 2 < 0

Solution: concave, therefore maximum

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Cournot model of duopoly: Nash equilibrium�

• Finding the Nash equilibrium:
• Find the quantity pair (q₁*, q₂*) such that q₁* is firm 1’s best response to firm 2’s quantity q₂* and q₂* is firm 2’s best response to firm 1’s quantity q₁*

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• The quantity pair (q₁*, q₂*) is a Nash equilibrium if� q₁* = (a - c - q₂*)/2 (best response of firm 1)� q₂* = (a - c - q₁*)/2 (best response of firm 2)

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• Solving these two equations gives us� q₁* = q₂* = (a-c)/3

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Cournot model of duopoly: Nash equilibrium�

• Best response functions:
• Firm 1’s best function to firm 2’s quantity q₂*: q₁ = (a - c - q₂)/2
• Firm 2’s best function to firm 1’s quantity q₁*: q₂ = (a - c - q₁)/2

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Nash Equilibrium

Cournot model of duopoly: Nash equilibrium�

• What is the total industry output?

Q* = q₁* + q₂* = 2(a-c)/3

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• What is the market price?

P(Q*) = a - Q* = a - 2(a-c)/3 = c + (a-c)/3

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• What is individual firm’s profit?

u₁(q₁*, q₂*) = q₁*{a-(q₁*+q₂*)}-q₁*c = (a-c)²/9

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• What happens to Q* and P(Q*) when a and c changes?

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Experiment #2: Cournot duopoly�

• Class experiment:
• The market price is P(Q) = 100 - Q, where Q = q₁ + q₂
• The cost to firm i of producing quantity q_i is C_i(q_i) = q_i

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• Given these parameters, at Nash equilibrium:
• Individual output: q₁* = q₂* = (100 - 1)/3 = 33
• The market price: P(Q*) = (100 + 2)/3 = 34
• Individual firm’s profit: u₁(q₁*, q₂*) = (100 - 1)²/9 = 1089

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Experiment #2: Results (2020 CWRU)�

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Cournot model of oligopoly: N firm case�

• A product is produced by n firms: firm 1 to firm n

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• Firm i’s quantity is denoted by q_i. Each firm chooses the quantity without knowing the other firms’ choices

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• The payoff of each firm depends on the market price and the cost of production

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• The market price is P(Q) = a - Q, where a is a constant number and Q = q₁+q₂+...+q_n

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• The cost of producing quantity q_i is C_i(q_i) = cq_i

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Cournot model of oligopoly: N firm case�

• The normal-form representation:

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• Set of players: {Firm 1,…, Firm n}
• Sets of strategies: S_i=[0, +∞) for all i=1, 2, ..., n
• Payoff functions: u_i(q₁ ,...,q_n)=q_i{a-(q₁+...+q_n)}-q_ic for i=1, 2, ..., n

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Cournot model of oligopoly: N firm case�

• Finding a Nash equilibrium (q₁*, ..., q_n*):
• q₁* solves max u₁(q₁,q₂, ...,q_n)=q₁{a-(q₁+q₂+...+q_n)}-q₁c
• q₂* solves max u₂(q₁,q₂, ...,q_n)=q₂{a-(q₁+q₂+...+q_n)}-q₂c
• etc ….

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• The Nash equilibrium of this game is:

q₁* = q₂* = … = q_n* = (a-c)/(n+1)

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• The total industry output is:

Q* = q₁* + q₂* + … + q_n* = n(a-c)/(n+1)

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• The market price is:

P(Q*) = a - Q* = c + (a-c)/(n+1)

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Comparison�

• The aggregate quantities are

Monopoly (n = 1): Q_M* = 1(a - c)/2

Cournot duopoly (n = 2): Q_CD* = 2(a - c)/3

Perfect competition (n → ∞): Q_PC* = (a - c)

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• The market prices are

Monopoly (n = 1): P_M* = c + (a - c)/2

Cournot duopoly (n = 2): P_CD* = c + (a - c)/3

Perfect competition (n → ∞): P_PC* = c

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Contests and tournaments?�

• Next Time!

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Thank you!

Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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References�

• Watson, J. (2013). Strategy: An Introduction to Game Theory (3rd Edition). Publisher: W. W. Norton & Company. (Chapters 10)

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