Wireless Network Pricing
Chapter 5: Monopoly and Price Discriminations
Jianwei Huang & Lin Gao
Network Communications and Economics Lab (NCEL)
Information Engineering Department
The Chinese University of Hong Kong
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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)
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Chapter 5: Monopoly and Price Discriminations
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Focus of This Chapter
Key Focus: This chapter focuses on the problem of profit
maximization in a monopoly market, where one service provider
(monopolist) dominates the market and seeks to maximize its profit.
Theoretic Approach: Price Theory
I
Price theory mainly refers to the study of how prices are decided and
how they go up and down because of economic forces such as changes
in supply and demand (from Cambridge Business English Dictionary)
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Price Theory
Mainly follow the discussions in “Price Theory and Applications” by
B. Peter Pashigian (1995) and Steven E. Landsburg (2010)
Part I: Monopoly Pricing
I
The service provider charges a single optimized price to all the
consumers.
Part II: Price Discrimination
I
The service provider charges different prices for different units of
products or to different consumers.
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Section 5.1
Theory: Monopoly Pricing
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What is Monopoly?
Etymology suggests that a “monopoly” is a single seller, i.e., the only
firm in its industry.
I
Question: Is Apple a monopoly?
F
F
It is the only firm that sells iPhone;
It is not the only firm that sells smartphones.
The formal definition of monopoly is based on the monopoly power.
Definition (Monopoly)
A firm with monopoly power is referred to as a monopoly or monopolist.
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What is Monopoly Power?
Monopoly power (or market power) is the ability of a firm to affect
market prices through its actions.
Definition (Monopoly Power)
A firm has monopoly power, if and only if
(i) it faces a downward-sloping demand curve for its product, and
(ii) it has no supply curve.
(i) implies that a monopolist is not perfectly competitive. That is, he is
able to set the market price so as to shape the demand.
(ii) implies that the market price is a consequence of the monopolist’s
actions, rather than a condition that he must react to.
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Profit Maximization Problem
P: the market price that a monopolist chooses;
Q , D(P): the downward-sloping demand curve that the monopolist
faces;
Definition (Monopolist’s Profit Maximization Problem)
The monopolist’s choice of market price P to maximize his profit (revenue)
π(P) , P · Q = P · D(P).
Here we tentatively assume that there is no production cost, hence
profit = revenue.
In general, profit = revenue - cost.
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Profit Maximization Problem
The first-order condition:
dQ
dπ(P)
=Q +P ·
=0
dP
dP
The optimality condition:
P · 4Q
+1=0
Q · 4P
I
4P is a very small change in price, and 4Q is the corresponding
change in demand quantity.
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Demand Elasticity
Price Elasticity of Demand (defined in Section 3.2.5)
η,
I
4Q/Q
P · 4Q
=
4P/P
Q · 4P
The ratio between the percentage change of demand and the
percentage change of price.
A Closely Related Question: Under a particular price P and demand
Q = D(P), how much should the monopolist lower his price to sell
additional 4Q units of product?
⇒ Answer:
4P =
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P
· 4Q
Q ·η
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Demand Elasticity
The monopolist’s total profit change by selling additional 4Q units of
product:
4π , P · 4Q − |4P| · Q
P
= P · 4Q − · 4Q · Q
Q ·η
1
= P · 4Q · 1 −
|η|
I
I
I
P · 4Q is the profit gain that the monopolist achieves, by selling
additional 4Q units of product at price P;
|4P| · Q is the profit loss that the monopolist suffers, due to the
decrease of price (by |4P|) for the previous Q units of product.
We ignore the higher order term of 4P · 4Q.
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Demand Elasticity
Monopolist’s Total Profit Change:
1
4π = P · 4Q · 1 −
|η|
I
I
I
If |η| > 1, then 4π > 0. This implies that the monopolist has incentive
to decrease the price when |η| > 1.
If |η| < 1, then 4π < 0. This implies that the monopolist has incentive
to increase the price when |η| < 1.
If |η| = 1, then 4π = 0. This implies that the monopolist has no
incentive to increase or decrease the price when |η| = 1 (assuming no
producing cost).
The price under |η| = 1 is the optimal price (if no producing cost)
I
Equivalent to the previous first-order condition.
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Demand Elasticity
Now consider the production cost, where profit = revenue - cost.
Suppose the unit producing cost is C . Then, the optimal price is
given by 4π = C · 4Q, or equivalently,
1
= C · 4Q.
P · 4Q · 1 −
|η|
Hence at the optimal price, we have
|η| =
1
>1
1 − C /P
Recall that
I
I
When |η| > 1, we say that the demand curve is elastic.
When |η| < 1, we say that the demand curve is inelastic.
Theorem
A monopolist always operates on the elastic portion of the demand curve.
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Demand Elasticity
When 4Q = 1, then
I
4π = P · 1 −
I
C is the marginal cost (MC). In general, MC may not be a constant.
1
|η|
is called the marginal revenue (MR);
Hence the optimal production quality equalizes MR and MC.
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Section 5.2
Theory: Price Discriminations
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What is Price Discrimination?
Price discrimination (or price differentiation) is a pricing strategy
where products are transacted at different prices in different markets
or territories.
Examples of Price Discriminations:
I
I
Charge different prices to the same consumer, e.g., for different units of
products;
Charge uniform but different prices to different groups of consumers for
the same product.
Types of Price Discrimination
I
I
I
First-degree price discrimination
Second-degree price discrimination
Third-degree price discrimination
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An Illustrative Example
How would the monopolist increase his profit via price discrimination?
I
I
I
MR: the marginal revenue curve;
MC: the marginal cost curve;
Demand: the downward-sloping demand curve;
Price
MC
P∗
π+
π?
π∗
Demand
C0
MR
0
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Q∗
Q?
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Without Price Discrimination
Without price discrimination, the monopolist charges a single
monopoly price to all consumers:
The optimal production quality (and demand) is Q ∗ , which equalizes
MC and MR;
The optimal monopoly price is P ∗ , which is determined by the Q ∗
and the demand curve.
The monopolist’s profit (=revenue - cost) is π ∗ , and the consumer
surplus is π + .
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With Price Discrimination
With price discrimination, the monopolist can charge different prices
to different consumers:
For example, the monopolist can charge each consumer the most that
he would be willing to pay for each product that he buys;
With the same demand Q ∗ , the monopolist’s profit is π ∗ + π + , and
the consumer surplus is 0;
When the demand increases to Q ? , the monopolist’s profit is
π ∗ + π + + π ? , and the consumer surplus is 0;
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First Degree Price Discrimination
With the first-degree price discrimination (or perfect price
discrimination), the monopolist charges each consumer the most that
he would be willing to pay for each product that he buys.
The monopolist captures all the market surplus, and the consumer
gets zero surplus.
It requires that the monopolist knows exactly the maximum price that
every consumer is willing to pay for each product, i.e., the full
knowledge about every consumer demand curve.
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Illustration of First Degree Price Discrimination
The consumer is willing to pay a maximum price P1 for the first
product, P2 for the second product, and so on.
Under the first-degree price discrimination, the consumer is charged
by P1 for the first product, P2 for the second product, and so on.
The monopolist captures all the market surplus (shadow area).
Price
P1
P2
P3
P4
P5
...
Demand
C
0
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MC
1
2
3
4
5
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Second Degree Price Discrimination
With the second-degree price discrimination (or declining block
pricing), the monopolist offers a bundle of prices to each consumer,
with different prices for different blocks of units.
The second-degree price discrimination can be viewed as a limited
version of the first-degree price discrimination (where a different price
is set for every different unit).
The second-degree price discrimination can be viewed as a generalized
version of the monopoly pricing (as it degrades to the monopoly
pricing when the number of prices is one).
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Illustration of Second Degree Price Discrimination
Under this second-degree price discrimination, the monopolist offers a
bundle of prices {P1 , P ∗ , P2 } with P1 > P ∗ > P2 .
I
I
I
I
P1 is the unit price for the first block (the first Q1 units) of products;
P ∗ is the unit price for the second block (from Q1 to Q ∗ ) of products;
P2 is the unit price for the third block (from Q ∗ to Q2 ).
The monopolist’s profit is illustrated by the shadow area, and the
consumer surplus is δ1 + δ ∗ + δ2 .
Price
MC
P1
P∗
P2
δ1
δ∗
δ2
Demand
MR
0
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Q1
Q∗
Q2
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First vs. Second Degree Price Discrimination
Under the second-degree price discrimination {P1 , P ∗ , P2 }:
I
The monopolist’s profit is illustrated by the shadow area, and the
consumer surplus is δ1 + δ ∗ + δ2 .
Under the first-degree price discrimination:
I
I
The monopolist charges a different price D(Q) for each unit of product;
The monopolist captures all the market surplus (the shadow area +
δ1 + δ ∗ + δ2 , and the consumer achieves zero surplus.
Price
MC
P1
P∗
P2
0
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δ1
δ∗
Q1
δ2
Q∗
Demand
Q2
MR
Quantity
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Single Monopoly Pricing vs. Second Degree Price
Discriminations
Under the second-degree price discrimination {P1 , P ∗ , P2 }:
I
The monopolist’s profit is illustrated by the shadow area, and the
consumer surplus is δ1 + δ ∗ + δ2 .
Under the monopoly pricing without price discrimination:
I
I
The optimal monopoly price is P ∗ and the demand is Q ∗ ;
R Q∗
The monopolist’s profit is P ∗ · Q ∗ − 0 MC (Q)dQ, and the consumer
surplus is δ1 + δ ∗ + (P1 − P ∗ ) · Q1 .
Price
MC
P1
P∗
P2
0
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δ1
δ∗
Q1
δ2
Q∗
Demand
Q2
MR
Quantity
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Second Degree Price Discrimination
Comparison of different pricing strategies
I
When there is a single price, the second-degree price discrimination
degrades to the monopoly pricing;
I
When the price bundle curve approximates to the inverse demand curve
P(Q), the second-degree price discrimination converges to the
first-degree price discrimination.
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Third Degree Price Discrimination
Limitation of First- and Second-Degree Price Discriminations
I
Needs the full or partial demand curve information of every individual
consumer, and benefits from this information by charging the consumer
different prices for different units of products.
How should the monopolist discriminates the prices, if he does not
know the detailed demand curve information of each individual
consumer, but knows from experience that different groups of
consumers have different total demand curves?
→ Third-Degree Price Discrimination
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Third Degree Price Discrimination
With the third-degree price discrimination (or multi-market price
discrimination), the monopolist specifies different prices for different
consumer groups (with different total demand curves).
I
Example: The Disney Park offers different ticket prices to three player
groups: children, adults, and elders.
Third-degree price discrimination usually occurs when
I
I
the monopolist faces multiple identifiably groups of consumers with
different total demand curves;
the monopolist knows the total demand curve of every consumer group
(but not the individual demand curve of each consumer.
How to Identify Customers?
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By Age
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By Time
Kindle 2
I
I
I
I
02/2009:
07/2009:
10/2009:
06/2010:
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$399
$299
$259
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Even More Dynamic
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More Innovative Ways
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Third Degree Price Discrimination
Consider a simple scenario:
I
I
I
Two groups (markets) of consumers:
The total demand curve in each market i ∈ {1, 2} is Di (P);
The monopolist decides the price Pi for each market i.
Key problem: How should the monopolist set the prices {P1 , P2 } to
maximize his profit?
I
I
I
Whether to charge the same price or different prices in different
markets (groups)?
Which market should get the lower price if the monopolist charges
different prices?
What is the relationship between the prices of two markets?
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Third Degree Price Discrimination
The monopolist’s profit π(P1 , P2 ) under prices {P1 , P2 } is
π(P1 , P2 ) , P1 · Q1 + P2 · Q2 − C (Q1 + Q2 )
The first-order condition:
dQi
∂π(P1 , P2 )
dQi
= Q i + Pi ·
− C 0 (Q1 + Q2 ) ·
=0
∂Pi
dPi
dPi
I
I
I
Qi , Di (Pi ) is the demand curve in market i;
Pi dQi
is the price elasticity of demand in market i;
ηi , Q
i dPi
0
C (Q1 + Q2 ) is the marginal cost (MC) of the monopolist;
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Third Degree Price Discrimination
The optimality condition:
1
dPi
= Pi · 1 −
C (Q1 + Q2 ) = Pi + Qi ·
dQi
|ηi |
0
⇒ Under the optimal prices (P1∗ , P2∗ ), the marginal revenues (MR) in
all markets are identical, and are equal to the marginal cost (MC):
1
1
∗
∗
P1 · 1 −
= P2 · 1 −
|η1 |
|η2 |
I
Pi · 1 − |η1i | is the marginal revenue (MR) of the monopolist in
market i;
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Third Degree Price Discrimination
The optimal prices (P1∗ , P2∗ ) satisfy
1
1
P1∗ · 1 −
= P2∗ · 1 −
|η1 |
|η2 |
I
I
If |η1 | =
6 |η2 |, then P1∗ 6= P2∗ . That is, the monopolist will charge
different prices when two markets have different price elasticities.
If |η1 | > |η2 |, then P1∗ < P2∗ . That is, the market with the higher price
elasticity will get a lower optimal price.
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Third Degree Price Discrimination
Graphic Interpretation of Optimal Prices (P1∗ , P2∗ )
I
I
I
I
Di: the demand curve in market i;
MRi: the marginal revenue curve in market i;
MR (the blue curve): the overall marginal revenue curve (summing
MR1 and MR2 horizontally);
MC (the red curve): the marginal cost curve;
Price
MC
P2∗
P1∗
C0
MR
MR1
Q1
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D1
MR2
Q2 Q1 + Q2
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Third Degree Price Discrimination
Graphic Interpretation of Optimal Prices (P1∗ , P2∗ )
I
I
I
I
Market 1: the demand is Q1 , the marginal revenue equals C0 ;
Market 2: the demand is Q2 , the marginal revenue equals C0 ;
Total market demand is Q1 + Q2 , and the marginal cost is C0 ;
C0 is at the intersection of MC and MR curves.
Price
MC
P2∗
P1∗
C0
MR
MR1
Q1
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D1
MR2
Q2 Q1 + Q2
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Third Degree Price Discrimination
Necessary conditions to make the third-degree price discrimination
applicable and profitable:
I
I
I
Monopoly power: The firm must have the monopoly power to affect
market price (there is no price discrimination in perfectly competitive
markets).
Market segmentation: The firm must be able to split the market into
different groups of consumers, and also be able to identify the type of
each consumer.
Elasticity of demand: The price elasticities of demand in different
markets are different.
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Section 5.3: Cellular Network Pricing
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Network Model
A cellular operator with B Hz of bandwidth
Sell bandwidth to multiple users
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Two-Stage Decision Process
Stage I: (Operator pricing)
The operator decides price p and announces to users
Stage II: (Users’ demands)
Each user decides how much bandwidth b to request
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User’s Spectrum Efficiency
h: a user’s (average) channel gain between him and the base station
P: a user’s transmission power density (per unit bandwidth)
θ: a user’s spectrum efficiency (data rate per unit bandwidth)
Ph
θ = log2 (1 + SNR) = log2 1 +
n0
When allocated bandwidth b, the user achieves a data rate of θb
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User’s Spectrum Efficiency
Different users have different spectrum efficiencies
I
I
Due to different values of P and h
Indoor users often have a smaller h than outdoor users
Normalize the range of θ to be [0,1]
I
Divided by the maximum value of θ among all users
0
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User’s Utility and Payoff
A user’s utility when allocated bandwidth b
u(θ, b) = ln(1 + θb)
A user’s payoff under linear pricing p:
π(θ, b, p) = ln(1 + θb) − pb
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User’s Demand in Stage II
Payoff maximization problem
max π(θ, b, p) = max (ln(1 + θb) − pb)
b≥0
b≥0
Concave maximization problem ⇒ user’s optimal demand
1 1
if p ≤ θ,
∗
p − θ,
b (θ, p) =
0,
otherwise.
User’s maximum payoff
(
∗
π(θ, b (θ, p), p) =
ln
θ
p
− 1 + pθ , if p ≤ θ,
0,
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User Separation Based on Spectrum Efficiency
No service
0
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Cellular service
p
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θ
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Users’ Total Demand
Price p ≤ maxθ∈[0,1] θ = 1
I
If p > 1, the total user demand will be 0
Total user demand
Z
Q(p) =
p
I
1
1 1
−
p θ
dθ =
1
− 1 + ln p
p
Decreasing in p.
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Operator’s Optimal Pricing
Operator’s revenue maximization problem
max min (pB, pQ(p))
0<p≤1
I
I
pB is increasing in p
pQ(p) is decreasing in p:
dpQ(p)
= ln p < 0
dp
I
We can show that at the optimal price p ∗ , p ∗ B = p ∗ Q(p ∗ ).
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Operator’s Optimal Pricing
Operator’s revenue maximization problem
max min (pB, pQ(p))
0<p≤1
I
I
pB is increasing in p
pQ(p) is decreasing in p:
dpQ(p)
= ln p < 0
dp
I
We can show that at the optimal price p ∗ , p ∗ B = p ∗ Q(p ∗ ).
The optimal price p ∗ is the unique solution of
B=
I
I
1
− 1 + ln p ∗
p∗
B→0⇒p→1
B →∞⇒p→0
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Section 5.4: Partial Price Differentiation
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Network Model
One wireless service provider (SP)
A set of I groups of users, where each group i ∈ I has
I
I
I
Ni homogenous users
Same utility function ui (si ) = θi ln(1 + si )
Groups have decreasing preference coefficients: θ1 > θ2 > · · · > θI
The SP’s decision for each group i
I
I
I
Admit ni ≤ Ni users
Charge a unit price pi (per unit P
of resource)
Subject to total resource limit:
i ni si ≤ S
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Two-Stage Decision Process
Stage I: (Service provider’s pricing and admission control)
The SP decides price pi and ni for each group i
Stage II: (Users’ demands)
Each user in group i decides the demand si
Complete price differentiation: charge up to I different prices
Single pricing (no price differentiation): charge one price
Partial price differentiation: charge J prices with 1 ≤ J ≤ I
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Complete Price Differentiation: Stage II
Each (admitted) group i user chooses si to maximize payoff
maximize (θi ln(1 + si ) − pi si )
si ≥0
The unique optimal demand is
+
θi
θi
− 1, 0 =
−1
si∗ (pi ) = max
pi
pi
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Complete Price Differentiation: Stage I
SP performs admission control n and determines prices p:
X
maximize
ni pi si
n,p≥0,s≥0
i∈I
+
θi
−1
, i ∈ I,
subject to si =
pi
ni ∈ {0, . . . , Ni } , i ∈ I,
X
ni si ≤ S.
i∈I
I
The Stage II’s user responses are incorporated
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Complete Price Differentiation: Stage I
SP performs admission control n and determines prices p:
X
maximize
ni pi si
n,p≥0,s≥0
i∈I
+
θi
−1
, i ∈ I,
subject to si =
pi
ni ∈ {0, . . . , Ni } , i ∈ I,
X
ni si ≤ S.
i∈I
I
The Stage II’s user responses are incorporated
This problem is challenging to solve due to non-convex objectives,
integer variables, and coupled constraint.
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Complete Price Differentiation: Stage I
The admission control and pricing can be decoupled
At the unique optimal solution
I
I
Do not reject any user
Charge prices such that users perform voluntary admission control:
there exists a group threshold K cp and λcp with
√
θi λ∗ , i ≤ K cp ;
∗
pi =
θi ,
i > K cp .
and
( q
si∗
I
=
θi
λ∗
− 1, i ≤ K cp ;
i > K cp .
r
!
θi
−1 =S
λ∗
0,
∗
The choice of λ satisfies
cp
K
X
i=1
c
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Complete Price Differentiation: Optimal Solution
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 22, NO. 3, JUNE 2014
ding to the
must hold
ng function
Fig. 2. Six-group example for the effective market: The willingness to pay decreases from group 1 to group 6. The effective market threshold can be obtained
by Algorithm
1 andincludes
is 4 in thisgroups
example. receiving positive resources
Effective
market:
(12)
be viewed
olution for
mal solution
blem. Note
e following
and users in these groups as effective users. An example of the
effective market is illustrated in Fig. 2.
Now let us solve the admission control subproblem
.
Denote the objective (10) as
, by (15), then
c
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5
Septemberdo28, 2016
. WeChapter
first relax
the integer
57 / 73
Single Pricing (No Price Differentiation)
Problem formulation similar as the complete price differentiation case
Key difference: change the same price p to all groups
Similar optimal solution structure
Effective market is no larger than the one under complete price
differentiation
I
Less users will be served
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Effectiveness of Complete Price Differentiation
GΘ, N, S
80
Case 1
60
0.15
40
Case 2
20
0
80
60
N1 N2 N3
Case 3
40
80
20
60
0.10
0
N1 N2 N3
40
20
0
0.05
N1 N2 N3
S
200
Relative Revenue Gain =
400
600
800
1000
RComplete Price Differentiation − RSingle Price
RSingle Price
Relative revenue gain of price differentiation reaches the maximum if
I
I
The high willingness-to-pay users are minority, and
Total resource S is limited
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Partial Price Differentiation
The most general case
SP can charge J prices to I groups, where J ≤ I
I
Complete price differentiation: J = I
I
Single pricing: J = 1
How to divide I groups into J clusters, and optimize the J prices?
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Partial Price Differentiation
a={aij , j ∈ J , i ∈ I}: binary variables defining the partition
I
aij = 1 ⇒ group i is in cluster j
Revenue optimization problem:
X
maximize
ni pi si
{ni ,pi ,si ,p j ,aij }∀i,j
i∈I
+
θi
, ∀ i ∈ I,
−1
si =
pi
ni ∈ {0, . . . , Ni }, ∀ i ∈ I,
X
ni si ≤ S,
subject to
i∈I
pi =
X
aij p j ,
j∈J
X
aij = 1, aij ∈ {0, 1}, ∀ i ∈ I.
j∈J
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Three-Level Decomposition
Level-1 (Cluster Partition): partition I groups into J clusters
Level-2 (Inter-Cluster Resource Allocation): allocate resources among
clusters (subject to the total resource constraint)
Level-3 (Intra-Cluster Pricing and Resource Allocation): optimize
pricing and resource allocations within each cluster
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Level 3: Pricing and Resource Allocation in Single
Cluster
∆
Given a fixed partition a and a cluster resource allocation s = {s j }j∈J
Solve the pricing and resource allocation problems in cluster C j :
Level-3:
maximize
ni ,si ,p j
X
ni p j si
i∈C j
subject to
si =
θi
−1
pj
+
∀ i ∈ Cj ,
,
ni ≤ Ni , ∀ i ∈ C j ,
X
ni si ≤ s j .
i∈C j
Equivalent to a single pricing problem
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NUMERICAL EXAMPLES FOR FEASIBLE SET SIZE
Level 3: Effective Market in a Single Cluster
re
Fig. 5. Illustrative example of Level 3: The cluster contains four
groups—groups 4, 5, 6 and 7—and the effective market contains groups
.
4 and 5, thus
The
problem is a combinatorial optimization problem
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fi
a
Level 2: Resource Allocation Among Clusters
For a fixed partition a
Consider the resource allocation among clusters:
Level-2:
maximize
s j ≥0
subject to
X
R j (s j , a)
j∈J
X
s j ≤ S.
j∈J
Solving Level 2 and Level 3 together is equivalent of solving a
complete price differentiation problem
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Level-1: Cluster Partition
Level-1:
maximize
Rpp (a)
subject to
X
aij ∈{0,1},∀i,j
aij = 1, i ∈ I.
j∈J
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How to Perform Cluster Partition in Level 1
Naive exhaustive search leads to formidable complexity for Level 1
Groups
Clusters
Combinations
c
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I = 10
J=2 J=3
511
9330
I = 100
J=2
6.33825 × 1029
Wireless Network Pricing: Chapter 5
I = 1000
J=2
5.35754 × 10300
September 28, 2016
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How to Perform Cluster Partition in Level 1
Naive exhaustive search leads to formidable complexity for Level 1
Groups
Clusters
Combinations
I = 10
J=2 J=3
511
9330
I = 100
J=2
6.33825 × 1029
I = 1000
J=2
5.35754 × 10300
Do we need to check all partitions?
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Fig. 6. Decomposition and simplification of the general
problem: The
problem is shown in the left-hand
three-level
structure
of the
Property
ofdecomposition
An Optimal
Partition
side. After simplifications in Sections V-B and V-C, the problem will be reduced
to structure in right-hand side.
th
th
L
Will the following partition ever be optimal?
Fig. 7. Example of coupling thresholds.
C
th
The equality (a) in (26) means that each cluster can be equiv- a
alently treated as a group with
homogeneous users with the
c
Huangsame
& Gao (willingness
NCEL)
Pricing:
Chapter this
5
September 28,
2016
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toWireless
payNetwork
. We
name
equivalent
group
as
Fig. 6. Decomposition and simplification of the general
problem: The
problem is shown in the left-hand
three-level
structure
of the
Property
ofdecomposition
An Optimal
Partition
side. After simplifications in Sections V-B and V-C, the problem will be reduced
to structure in right-hand side.
th
th
L
Will the following partition ever be optimal?
Fig. 7. Example of coupling thresholds.
No.
C
th
The equality (a) in (26) means that each cluster can be equiv- a
alently treated as a group with
homogeneous users with the
c
Huangsame
& Gao (willingness
NCEL)
Pricing:
Chapter this
5
September 28,
2016
68 / 73 o
toWireless
payNetwork
. We
name
equivalent
group
as
Fig. 6. Decomposition and simplification of the general
problem: The
problem is shown in the left-hand
three-level
structure
of the
Property
ofdecomposition
An Optimal
Partition
side. After simplifications in Sections V-B and V-C, the problem will be reduced
to structure in right-hand side.
th
th
L
Will the following partition ever be optimal?
Fig. 7. Example of coupling thresholds.
No.
C
th
The equality (a) in (26) means that each cluster can be equiv- a
alently treated as a group with
homogeneous users with the
c
Huangsame
& Gao (willingness
NCEL)
Pricing:
Chapter this
5
September 28,
2016
68 / 73 o
toWireless
payNetwork
. We
name
equivalent
group
as
We prove that group indices in the effective market are consecutive.
Reduced Complexity of Cluster Partition in Level I
The search complexity reduces to polynomial in I .
Groups
Clusters
Combinations
Reduced Combos
c
Huang & Gao (NCEL)
I = 10
J=2 J=3
511
9330
9
36
I = 100
J=2
6.33825 × 1029
99
Wireless Network Pricing: Chapter 5
I = 1000
J=2
5.35754 × 10300
999
September 28, 2016
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Relative Revenue Gain
G
G
0.14
0.20
Complete price differentiation
Five Prices
0.12
0.10
Four Prices
0.08
0.15
Three Prices
0.06
0.04
Two Prices
0.02
S
0.10
10
20
30
40
50
0.05
S
100
200
300
400
500
A total of I = 5 groups
Plot the relative revenue gain of price differentiation vs. total resource
Maximum gains in the small plot
I
J = 3 is the sweet spot
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Section 5.5: Chapter Summary
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Key Concepts
Theory
I
I
I
I
Monopoly pricing and the demand elasticity
First-degree price discrimination
Second-degree price discrimination
Third-degree price discrimination
Application
I
I
Cellular Network Pricing
Partial Price Discrimiation
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References and Extended Reading
L. Duan, J. Huang, and B. Shou, “Economics of Femtocell Service
Provisions,” IEEE Transactions on Mobile Computing, vol. 12, no. 11,
pp. 2261 - 2273, November 2013
S. Li and J. Huang, “Price Differentiation for Communication Networks,”
IEEE Transactions on Networking, vol. 22, no. 2, pp. 703 - 716, June 2014
http://ncel.ie.cuhk.edu.hk/content/wireless-network-pricing
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