Introduction Mathematical Economics
Max Batstra
March 29, 2026
1
,Week 1,2: Bertrand & Cournot Model
We have two general market structures:
• Oligopoly:
Several decision makers in 1 model
• Monopoloy:
One firm in the market
In perfect competition competitive firms are to small to affect the market price. Firms can compete on
both the longrun and the shortrun and use different instruments to compete in both these time spans:
• Short run:
– Static (choices are made once):
∗ Price competition =⇒ Bertrand
∗ Quantity competition =⇒ Cournot
– Dynamic (choices are remade every period):
∗ Sequential move games
• Longer run ( = Spatial competition):
– Product characteristics: most of the time firms will compete using location
• Long run (= Innovation)
– Process innovation (Improve production process)
– Product innovation (Invent/ design new products)
The Bertrand Model
In the Bertrand model, firms compete in price. In general we use the following info:
• n competing firms, each firm denoted by i = 1, . . . , n
• Firm i charges price pi and has unit cost ci
D(pi )
if pi < p−i
• Demand firm i: Di (pi , p−i ) = k1 D(pi ) if pi = pj for k ≤ n firms
0 if pi > pj
Y (pi − ci )D(pi )
if pi < p−i
• Profit function firm i: 1
(pi , p−i ) = (pi −ci )Di (pi , p−i ) = (pi − ci ) · k D(pi ) if pi = pj for k ≤ n firms
i
0 if pi > pj
2
,As a result competing firms will set pi = min{c1 , . . . , cn } to avoid undercutting. We consider two cases:
• Symmetric situation: Cost per unit of production is ci = c for all i = 1, . . . , n =⇒ All firms
i = 1, . . . , n set pi = c
Q
– We obtain equilibrium pi = c and i (pi , p−i ) = 0 for all i = 1, . . . , n
• Asymmetric situation: Cost per unit of procedure is not equal for all firms. This allows firms to
compete in price as they’re able to undercut firms with higher unit costs, therefore allowing them to
grab a bigger share of the market demand. Suppose firm i has the lower unit cost, i.e. ci < c−i < pm ,
then
m
– Suppose firm i has the lower unit cost, i.e. ci < cQ
−i < p , then we have equilibrium:
∗
pi = min{c−i } − ε > ci for small ε > 0 and i (pi , p−i ) ≈ (min{c−i } − ci )D(min{c−i }) and
Q
−i (pi , p−i ) = 0
Special cases:
- Capacity constraint: This will lead to positive profits
- Dynamic market: Firms are able to set prices more than once =⇒ Week 9
- Product differentiation: Identical goods sold at different locations =⇒ Week 7,8
Cournot Model
In the Cournot model, firms compete in quantities. The Cournot outcome can be obtained in a two-stage
game:
1. Firms choose qi simultaneously
• Compute FOC for firm i and j =⇒ Ri (qj ) and Rj (qi ) result
• Substitute reaction curves in FOCs and Q solve:
δ ∗ δ ∗
Q
δqi i (qi , R j (qi )) = 0 =⇒ q i and δqj j (Ri (qj ), qj ) = 0 =⇒ qj result.
Q∗ Q∗
2. Optimal price and profit are determined by optimal quantities qi∗ , qj∗ : P (qi∗ , qj∗ ), and ∗ ∗
i (qi , qj ),
∗ ∗
j (qi , qj )
In general it holds that:
• Cournot output > Monopoly output
• Cournot price < Monopoly price
• Cournot profit < Monopoly profit
3
,Week 3,4 Non-Cooperative Game Theory
Definition 1. [Normal-form game]
Each game can be written in it’s normal-form, which is given by:
(N, (Xi )i∈N , (πi )i∈N )
Here:
- N = {1, . . . , n} denotes the set of players
- Xi denotes the Strategy profile of player i, which is a set containing all possible actions he can play.
Qn
- X = i=1 Xi denotes the set of strategy profiles
- πi : X → R denotes the set of payoffs for player i (Note X as we have to consider the actions of other
players to find the utility of player i)
- πi (x) denotes the realization of the utility of player i if strategy profile x = (x1 , . . . , xn ) is played
Q
- X−i = j̸=i Xj denotes the set of strategy profiles of other players than player i
Definition 2. [Dominant Strategy]
For player i, the strategy x∗i ∈ Xi is called a dominant strategy if:
∀xi ∈ Xi and ∀x−i ∈ X−i : πi (x∗i , x−i ) ≥ πi (xi , x−i )
Interpretation: Playing x∗i gives player i the highest payoff, regardless of what other players play.
Definition 3. [Best Reply Strategy]
For player i, the strategy x∗i ∈ Xi is called a best reply strategy if:
given x−i ∈ X−i , ∀xi ∈ Xi : πi (x∗i , x−i ) ≥ πi (xi , x−i )
Interpretation: Given the strategies x−i ∈ X−i chosen by other players −i, the strategy x∗i gives player i
the highest possible payoff.
Definition 4. [Nash Equilibrium]
A strategy profile x∗ ∈ X is called a Nash Equilibrium if:
∀i ∈ N, ∀xi ∈ Xi : πi (x∗ ) ≥ π(xi , x∗−i )
Lemma 1. Let every player i ∈ N have dominant strategy x∗i ∈ Xi . Then the profile x∗1 , . . . , x∗n is a Nash
Equilibrium
Proof
Since for every i it holds that if he plays dominant strategy x∗i , that:
∀xi ∈ Xi and ∀x−i ∈ X−i : πi (x∗i , x−i ) ≥ πi (xi , x−i )
it thus must also hold that if players play x∗ = (x∗1 , . . . , x∗n ) that:
∀i ∈ N, ∀xi ∈ Xi : πi (x∗ ) ≥ π(xi , x∗−i )
Thus the strategy profile x∗ = (x∗1 , . . . , x∗n ) is a Nash equilibrium
4
, Definition 5. [Bimatrix Game]
A bimatrix game is defined as follows:
- N = {1, 2}, thus there are two players, hence the name bimatrix as we’ll have two pay-off matrices.
- X1 is the strategy profile of player 1, containing actions M1 = {1, . . . , m1 }
- X2 is the strategy profile of player 2, containing actions M2 = {1, . . . , m2 }
n o
- ∆M1 = p ∈ RM
P
+ | i∈M1 pi = 1 denotes the mixed strategy of player 1, where for i = 1, . . . , m1 , pi
1
is the probability of player 1 playing action i. Note that p is a vector.
n o
- ∆M2 = q ∈ RM
P
+ | i∈M2 qi = 1 denotes the mixed strategy of player 2, where for i = 1, . . . , m2 , qi
2
is the probability of player 2 playing action i. Note that q is a vector.
- A is the payoff matrix of player 1
- B is the payoff matrix of player 2
- ai,j = (A)i,j is the payoff to player 1 when playing action i and player 2 playing action j
- bi,j = (B)i,j is the payoff to player 2 when playing action j and player 1 playing action i
In bimatrix games, we can find two types of Nash equilibria:
• Pure Strategy Nash equilibrium
– These are strategies of the form {C, C} or {AB, CD}. Each player in the bimatrix game chooses a
specific strategy, and both players do not have an incentive to deviate from their chosen strategy
(i.e. the probability of playing that strategy is 1), given the strategies of the other player.
• Mixed Strategy Nash equilibrium
nP o
m1 Pm2
– These strategies are of the form i=1 pi Ai , j=1 qj Bj , where Ai is action type i of player 1
and Bj is action type j of player 2
Algorithm finding Pure Strategy Nash equilibrium
1. In the pay-off matrix (A, B), for each action Bi for i ∈ M2 , underscore the payoff that corresponds to
the best response of player 1 against action Bi
2. In the pay-off matrix (A, B), for each action Ai for i ∈ M1 , draw a line | next to the payoff that
corresponds to the best response of player 2 against action Ai
3. The strategy pairs where both an underscore and line | are drawn are pure strategy Nash equilibria
D C
Consider bimatrix game with payoff matrix: D (−3, −3) (−10, −1)
C (−1, −10) (−6, −6)
5
Max Batstra
March 29, 2026
1
,Week 1,2: Bertrand & Cournot Model
We have two general market structures:
• Oligopoly:
Several decision makers in 1 model
• Monopoloy:
One firm in the market
In perfect competition competitive firms are to small to affect the market price. Firms can compete on
both the longrun and the shortrun and use different instruments to compete in both these time spans:
• Short run:
– Static (choices are made once):
∗ Price competition =⇒ Bertrand
∗ Quantity competition =⇒ Cournot
– Dynamic (choices are remade every period):
∗ Sequential move games
• Longer run ( = Spatial competition):
– Product characteristics: most of the time firms will compete using location
• Long run (= Innovation)
– Process innovation (Improve production process)
– Product innovation (Invent/ design new products)
The Bertrand Model
In the Bertrand model, firms compete in price. In general we use the following info:
• n competing firms, each firm denoted by i = 1, . . . , n
• Firm i charges price pi and has unit cost ci
D(pi )
if pi < p−i
• Demand firm i: Di (pi , p−i ) = k1 D(pi ) if pi = pj for k ≤ n firms
0 if pi > pj
Y (pi − ci )D(pi )
if pi < p−i
• Profit function firm i: 1
(pi , p−i ) = (pi −ci )Di (pi , p−i ) = (pi − ci ) · k D(pi ) if pi = pj for k ≤ n firms
i
0 if pi > pj
2
,As a result competing firms will set pi = min{c1 , . . . , cn } to avoid undercutting. We consider two cases:
• Symmetric situation: Cost per unit of production is ci = c for all i = 1, . . . , n =⇒ All firms
i = 1, . . . , n set pi = c
Q
– We obtain equilibrium pi = c and i (pi , p−i ) = 0 for all i = 1, . . . , n
• Asymmetric situation: Cost per unit of procedure is not equal for all firms. This allows firms to
compete in price as they’re able to undercut firms with higher unit costs, therefore allowing them to
grab a bigger share of the market demand. Suppose firm i has the lower unit cost, i.e. ci < c−i < pm ,
then
m
– Suppose firm i has the lower unit cost, i.e. ci < cQ
−i < p , then we have equilibrium:
∗
pi = min{c−i } − ε > ci for small ε > 0 and i (pi , p−i ) ≈ (min{c−i } − ci )D(min{c−i }) and
Q
−i (pi , p−i ) = 0
Special cases:
- Capacity constraint: This will lead to positive profits
- Dynamic market: Firms are able to set prices more than once =⇒ Week 9
- Product differentiation: Identical goods sold at different locations =⇒ Week 7,8
Cournot Model
In the Cournot model, firms compete in quantities. The Cournot outcome can be obtained in a two-stage
game:
1. Firms choose qi simultaneously
• Compute FOC for firm i and j =⇒ Ri (qj ) and Rj (qi ) result
• Substitute reaction curves in FOCs and Q solve:
δ ∗ δ ∗
Q
δqi i (qi , R j (qi )) = 0 =⇒ q i and δqj j (Ri (qj ), qj ) = 0 =⇒ qj result.
Q∗ Q∗
2. Optimal price and profit are determined by optimal quantities qi∗ , qj∗ : P (qi∗ , qj∗ ), and ∗ ∗
i (qi , qj ),
∗ ∗
j (qi , qj )
In general it holds that:
• Cournot output > Monopoly output
• Cournot price < Monopoly price
• Cournot profit < Monopoly profit
3
,Week 3,4 Non-Cooperative Game Theory
Definition 1. [Normal-form game]
Each game can be written in it’s normal-form, which is given by:
(N, (Xi )i∈N , (πi )i∈N )
Here:
- N = {1, . . . , n} denotes the set of players
- Xi denotes the Strategy profile of player i, which is a set containing all possible actions he can play.
Qn
- X = i=1 Xi denotes the set of strategy profiles
- πi : X → R denotes the set of payoffs for player i (Note X as we have to consider the actions of other
players to find the utility of player i)
- πi (x) denotes the realization of the utility of player i if strategy profile x = (x1 , . . . , xn ) is played
Q
- X−i = j̸=i Xj denotes the set of strategy profiles of other players than player i
Definition 2. [Dominant Strategy]
For player i, the strategy x∗i ∈ Xi is called a dominant strategy if:
∀xi ∈ Xi and ∀x−i ∈ X−i : πi (x∗i , x−i ) ≥ πi (xi , x−i )
Interpretation: Playing x∗i gives player i the highest payoff, regardless of what other players play.
Definition 3. [Best Reply Strategy]
For player i, the strategy x∗i ∈ Xi is called a best reply strategy if:
given x−i ∈ X−i , ∀xi ∈ Xi : πi (x∗i , x−i ) ≥ πi (xi , x−i )
Interpretation: Given the strategies x−i ∈ X−i chosen by other players −i, the strategy x∗i gives player i
the highest possible payoff.
Definition 4. [Nash Equilibrium]
A strategy profile x∗ ∈ X is called a Nash Equilibrium if:
∀i ∈ N, ∀xi ∈ Xi : πi (x∗ ) ≥ π(xi , x∗−i )
Lemma 1. Let every player i ∈ N have dominant strategy x∗i ∈ Xi . Then the profile x∗1 , . . . , x∗n is a Nash
Equilibrium
Proof
Since for every i it holds that if he plays dominant strategy x∗i , that:
∀xi ∈ Xi and ∀x−i ∈ X−i : πi (x∗i , x−i ) ≥ πi (xi , x−i )
it thus must also hold that if players play x∗ = (x∗1 , . . . , x∗n ) that:
∀i ∈ N, ∀xi ∈ Xi : πi (x∗ ) ≥ π(xi , x∗−i )
Thus the strategy profile x∗ = (x∗1 , . . . , x∗n ) is a Nash equilibrium
4
, Definition 5. [Bimatrix Game]
A bimatrix game is defined as follows:
- N = {1, 2}, thus there are two players, hence the name bimatrix as we’ll have two pay-off matrices.
- X1 is the strategy profile of player 1, containing actions M1 = {1, . . . , m1 }
- X2 is the strategy profile of player 2, containing actions M2 = {1, . . . , m2 }
n o
- ∆M1 = p ∈ RM
P
+ | i∈M1 pi = 1 denotes the mixed strategy of player 1, where for i = 1, . . . , m1 , pi
1
is the probability of player 1 playing action i. Note that p is a vector.
n o
- ∆M2 = q ∈ RM
P
+ | i∈M2 qi = 1 denotes the mixed strategy of player 2, where for i = 1, . . . , m2 , qi
2
is the probability of player 2 playing action i. Note that q is a vector.
- A is the payoff matrix of player 1
- B is the payoff matrix of player 2
- ai,j = (A)i,j is the payoff to player 1 when playing action i and player 2 playing action j
- bi,j = (B)i,j is the payoff to player 2 when playing action j and player 1 playing action i
In bimatrix games, we can find two types of Nash equilibria:
• Pure Strategy Nash equilibrium
– These are strategies of the form {C, C} or {AB, CD}. Each player in the bimatrix game chooses a
specific strategy, and both players do not have an incentive to deviate from their chosen strategy
(i.e. the probability of playing that strategy is 1), given the strategies of the other player.
• Mixed Strategy Nash equilibrium
nP o
m1 Pm2
– These strategies are of the form i=1 pi Ai , j=1 qj Bj , where Ai is action type i of player 1
and Bj is action type j of player 2
Algorithm finding Pure Strategy Nash equilibrium
1. In the pay-off matrix (A, B), for each action Bi for i ∈ M2 , underscore the payoff that corresponds to
the best response of player 1 against action Bi
2. In the pay-off matrix (A, B), for each action Ai for i ∈ M1 , draw a line | next to the payoff that
corresponds to the best response of player 2 against action Ai
3. The strategy pairs where both an underscore and line | are drawn are pure strategy Nash equilibria
D C
Consider bimatrix game with payoff matrix: D (−3, −3) (−10, −1)
C (−1, −10) (−6, −6)
5
