Wireless Network Pricing
Chapter 6: Oligopoly Pricing
Jianwei Huang & Lin Gao
Network Communications and Economics Lab (NCEL)
Information Engineering Department
The Chinese University of Hong Kong
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
1 / 69
The Book
E-Book freely downloadable from NCEL website: http:
//ncel.ie.cuhk.edu.hk/content/wireless-network-pricing
Physical book available for purchase from Morgan & Claypool
(http://goo.gl/JFGlai) and Amazon (http://goo.gl/JQKaEq)
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
2 / 69
Section 6.2
Theory: Oligopoly
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
30 / 69
Oligopoly
In this part, we consider three classical strategic form games to
formulate the interactions among multiple competitive entities
(Oligopoly):
I
I
I
The Cournot Model
The Bertrand Model
The Hotelling Model
Our purpose in this part is to illustrate
I
I
(a) Game Formulation: the translation of an informal problem
statement into a strategic form representation of a game;
(b) Equilibrium Analysis: the analysis of Nash equilibrium when a
player can choose his strategy from a continuous set.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
31 / 69
The Cournot Model
The Cournot model describes interactions among firms that compete
on the amount of output they will produce, which they decide
independently of each other simultaneously.
Key features
I
I
I
I
I
At least two firms producing homogeneous products;
Firms do not cooperate, i.e., there is no collusion;
Firms compete by setting production quantities simultaneously;
The total output quantity a↵ects the market price;
The firms are economically rational and act strategically, seeking to
maximize profits given their competitors’ decisions.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
32 / 69
The Cournot Model
Example: The Cournot Game
I
I
Two firms decide their respective output quantities simultaneously;
The market price is a decreasing function of the total quantity.
Game Formulation
I
I
I
The set of players is I = {1, 2},
The strategy set available to each player i 2 I is the set of all
nonnegative real numbers, i.e., qi 2 [0, 1),
The payo↵ received by each player i is a function of both players’
strategies, defined by
⇧i (qi , q i ) = qi · P(qi + q i )
F
F
ci · q i
The first term denotes the player i’s revenue from selling qi units of
products at a market-clearing price P(qi + q i );
The second term denotes the player i’s production cost.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
33 / 69
The Cournot Model
Consider a linear cost: P(qi + q i ) = a
(qi + q i )
Equilibrium Analysis
I
Given player 2’s strategy q2 , the best response of player 1 is:
q1⇤ = B1 (q2 ) =
I
q2
2
c1
,
Given player 1’s strategy q1 , the best response of player 2 is:
q2⇤ = B2 (q1 ) =
I
a
a
q1
2
c2
,
A strategy profile (q1⇤ , q2⇤ ) is an Nash equilibrium if every player’s
strategy is the best response to others’ strategies:
q1⇤ = B1 (q2⇤ ), and q2⇤ = B2 (q1⇤ )
I
Nash Equilibrium:
q1⇤ =
Huang & Gao ( c NCEL)
a + c1 + c2
3
c1 , q2⇤ =
Wireless Network Pricing: Chapter 6
a + c1 + c2
3
c2
November 15, 2016
34 / 69
The Cournot Model
Illustration of Equilibrium
I
Geometrically, the Nash equilibrium is the intersection of both players’
best response curves.
q2
a
1
2 (a
c1
c2 )
B1 (q2 )
Nash Equilibrium
B2 (q1 )
0
Huang & Gao ( c NCEL)
1
2 (a
c1 )
Wireless Network Pricing: Chapter 6
a
c2
q1
November 15, 2016
35 / 69
The Bertrand Model
The Bertrand model describes interactions among firms (sellers) who
set prices independently and simultaneously, under which the
customers (buyers) choose quantities accordingly.
Key features
I
I
I
I
At least two firms producing homogeneous products;
Firms do not cooperate, i.e., there is no collusion;
Firms compete by setting prices simultaneously;
Consumers buy products from a firm with a lower cost (price).
F
I
If firms charge the same price, consumers randomly select among them.
The firms are economically rational and act strategically, seeking to
maximize profits given their competitors’ decisions.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
36 / 69
The Bertrand Model
Example: The Bertrand Game
I
I
Two firms decide their respective prices simultaneously;
The consumers buy products from a firm with a lower price.
Game Formulation
I
I
I
The set of players is I = {1, 2},
The strategy set available to each player i 2 I is the set of all
nonnegative real numbers, i.e., pi 2 [0, 1),
The payo↵ received by each player i is a function of both players’
strategies, defined by
⇧i (pi , p i ) = (pi
F
F
ci ) · Di (p1 , p2 )
ci is the unit producing cost;
Di (p1 , p2 ) is the consumers’ demand to player i:
(i) Di (p1 , p2 ) = 0 if pi > p i ; (ii) Di (p1 , p2 ) = D(pi ) if pi < p i ;
(iii) Di (p1 , p2 ) = D(pi )/2 if pi = p i .
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
37 / 69
The Bertrand Model
Equilibrium Analysis
I
Given player 2’s strategy p2 , the best response of player 1 is to select a
price p1 slightly lower than p2 under the constraint that p1 c1 :
p1⇤ = B1 (p2 ) = max{c1 , p2
I
Given player 1’s strategy p1 , the best response of player 2 is to select a
price p2 slightly lower than p1 under the constraint that p2 c2 :
p2⇤ = B2 (p1 ) = max{c2 , p1
I
✏}
✏}
Both players will gradually decrease their prices, until one player gets to
his producing cost. Therefore, the Nash equilibrium is
8 ⇤
⇤
>
< p1 = c2 ✏, p2 = c2 , if c1 < c2
p1⇤ = c1 , p2⇤ = c1 ✏, if c1 > c2
>
: ⇤
p1 = p2⇤ = c,
if c1 = c2 = c
F
Here we assume that the demand is always inelastic, hence each seller
wants to set the price as high as possible.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
38 / 69
The Bertrand Model
Illustration of Equilibrium
I
Geometrically, the Nash equilibrium is the intersection of both players’
best response curves.
p2
c2
B2 (p1 )
Nash Equilibrium
B1 (p2 )
0
Huang & Gao ( c NCEL)
c1
Wireless Network Pricing: Chapter 6
p1
November 15, 2016
39 / 69
The Hotelling Model
The Hotelling model studies the e↵ect of locations on the price
competition among two or more firms.
Key features
I
I
I
I
Two firms at di↵erent locations sell the homogeneous good;
The customers are uniformly distributed between two firms.
Customers incur a transportation cost as well as a purchasing cost.
The firms are economically rational and act strategically, seeking to
maximize profits given their competitors’ decisions.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
40 / 69
The Hotelling Model
Example: The Hotelling Game
I
I
Two firms at di↵erent locations decide their respective prices
simultaneously;
The consumers buy products from a firm with a lower total cost,
including both the transportation cost and the purchasing cost.
Game Formulation
I
I
I
The set of players is I = {1, 2}, each locating at one end of the
interval [0, 1];
The strategy set available to each player i 2 I is the set of all
nonnegative real numbers, i.e., pi 2 [0, 1);
The payo↵ received by each player i is a function of both players’
strategies, defined by
⇧i (pi , p i ) = (pi
F
F
ci ) · Di (p1 , p2 )
ci is the unit producing cost;
Di (p1 , p2 ) is the ratio of consumers coming to player i.
Huang & Gao ( c NCEL)
Wireless Network Pricing: Chapter 6
November 15, 2016
41 / 69
The Hotelling Model
Consumer Demand: Di (p1 , p2 )
I
Under price profile (p1 , p2 ), the total cost of a consumer at location
x 2 [0, 1] buying products from player 1 or 2 is
C1 (x) = p1 + w · x, and C2 (x) = p1 + w · (1
I
x)
Under (p1 , p2 ), two players receive the following consumer demand:
D1 (p1 , p2 ) =
Huang & Gao ( c NCEL)
p2
p1 + w
p1
, D2 (p1 , p2 ) =
2w
Wireless Network Pricing: Chapter 6
p2 + w
2w
November 15, 2016
42 / 69
The Hotelling Model
Equilibrium Analysis
I
Given player 2’s strategy p2 , the best response of player 1 is
p1⇤ = B1 (p2 ) =
I
Given player 1’s strategy p1 , the best response of player 2 is
p2⇤ = B2 (p1 ) =
I
p 2 + w + c1
2
p 1 + w + c2
2
Nash Equilibrium:
p1⇤ =
Huang & Gao ( c NCEL)
3w + c1 + c2
c1
3w + c1 + c2
c2
+ , p2⇤ =
+ .
3
3
3
3
Wireless Network Pricing: Chapter 6
November 15, 2016
43 / 69
The Hotelling Model
Illustration of Equilibrium
I
Geometrically, the Nash equilibrium is the intersection of both players’
best response curves.
p2
Nash Equilibrium
B1 (p2 )
w +c1
2
B2 (p1 )
0
Huang & Gao ( c NCEL)
w +c2
2
Wireless Network Pricing: Chapter 6
p1
November 15, 2016
44 / 69