Wireless Network Pricing Chapter 5: Monopoly and Price Discriminations


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1 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 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
2 The Book EBook freely downloadable from NCEL website: http: //ncel.ie.cuhk.edu.hk/content/wirelessnetworkpricing Physical book available for purchase from Morgan & Claypool ( and Amazon ( Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
3 Chapter 5: Monopoly and Price Discriminations Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
4 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 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) Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
5 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 The service provider charges a single optimized price to all the consumers. Part II: Price Discrimination The service provider charges different prices for different units of products or to different consumers. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
6 Section 5.1 Theory: Monopoly Pricing Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
7 What is Monopoly? Etymology suggests that a monopoly is a single seller, i.e., the only firm in its industry. Question: Is Apple a monopoly? 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. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
8 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 downwardsloping 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. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
9 Profit Maximization Problem P: the market price that a monopolist chooses; Q D(P): the downwardsloping 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. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
10 Profit Maximization Problem The firstorder condition: dπ(p) dp The optimality condition: = Q + P dq dp = 0 P Q Q P + 1 = 0 P is a very small change in price, and Q is the corresponding change in demand quantity. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
11 Demand Elasticity Price Elasticity of Demand (defined in Section 3.2.5) η Q/Q P/P = P Q Q P 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 Q units of product? Answer: P = P Q η Q Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
12 Demand Elasticity The monopolist s total profit change by selling additional Q units of product: π P Q P Q = P Q P Q η Q Q ( = P Q 1 1 ) η P Q is the profit gain that the monopolist achieves, by selling additional Q units of product at price P; P Q is the profit loss that the monopolist suffers, due to the decrease of price (by P ) for the previous Q units of product. We ignore the higher order term of P Q. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
13 Demand Elasticity Monopolist s Total Profit Change: π = P Q ( 1 1 ) η If η > 1, then π > 0. This implies that the monopolist has incentive to decrease the price when η > 1. If η < 1, then π < 0. This implies that the monopolist has incentive to increase the price when η < 1. If η = 1, then π = 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) Equivalent to the previous firstorder condition. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
14 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 π = C Q, or equivalently, ( P Q 1 1 ) = C Q. η Hence at the optimal price, we have η = 1 1 C/P > 1 Recall that 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. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
15 Demand Elasticity When Q = 1, then ) π = P (1 1 η is called the marginal revenue (MR); C is the marginal cost (MC). In general, MC may not be a constant. Hence the optimal production quality equalizes MR and MC. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
16 Section 5.2 Theory: Price Discriminations Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
17 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: 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 Firstdegree price discrimination Seconddegree price discrimination Thirddegree price discrimination Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
18 An Illustrative Example How would the monopolist increase his profit via price discrimination? MR: the marginal revenue curve; MC: the marginal cost curve; Demand: the downwardsloping demand curve; Price MC P π + π π Demand C 0 0 Q Q MR Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
19 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 +. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
20 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; Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
21 First Degree Price Discrimination With the firstdegree 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. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
22 Illustration of First Degree Price Discrimination The consumer is willing to pay a maximum price P 1 for the first product, P 2 for the second product, and so on. Under the firstdegree price discrimination, the consumer is charged by P 1 for the first product, P 2 for the second product, and so on. The monopolist captures all the market surplus (shadow area). Price 1 2 P 3 4 P 5. Demand C MC Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
23 Second Degree Price Discrimination With the seconddegree 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 seconddegree price discrimination can be viewed as a limited version of the firstdegree price discrimination (where a different price is set for every different unit). The seconddegree 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). Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
24 Illustration of Second Degree Price Discrimination Under this seconddegree price discrimination, the monopolist offers a bundle of prices {P 1, P, P 2 } with P 1 > P > P 2. P1 is the unit price for the first block (the first Q 1 units) of products; P is the unit price for the second block (from Q 1 to Q ) of products; P 2 is the unit price for the third block (from Q to Q 2 ). The monopolist s profit is illustrated by the shadow area, and the consumer surplus is δ 1 + δ + δ 2. Price MC δ P 1 1 P δ 2 δ 2 Demand 0 Q 1 Q Q 2 MR Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
25 First vs. Second Degree Price Discrimination Under the seconddegree price discrimination {P 1, P, P 2 }: The monopolist s profit is illustrated by the shadow area, and the consumer surplus is δ 1 + δ + δ 2. Under the firstdegree price discrimination: The monopolist charges a different price D(Q) for each unit of product; The monopolist captures all the market surplus (the shadow area + δ 1 + δ + δ 2, and the consumer achieves zero surplus. Price MC δ P 1 1 P δ 2 δ 2 Demand 0 Q 1 Q Q 2 MR Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
26 Single Monopoly Pricing vs. Second Degree Price Discriminations Under the seconddegree price discrimination {P 1, P, P 2 }: The monopolist s profit is illustrated by the shadow area, and the consumer surplus is δ 1 + δ + δ 2. Under the monopoly pricing without price discrimination: The optimal monopoly price is P and the demand is Q ; The monopolist s profit is P Q Q MC(Q)dQ, and the consumer 0 surplus is δ 1 + δ + (P 1 P ) Q 1. Price MC δ P 1 1 P δ 2 δ 2 Demand MR 0 Q 1 Q Q 2 Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
27 Second Degree Price Discrimination Comparison of different pricing strategies When there is a single price, the seconddegree price discrimination degrades to the monopoly pricing; When the price bundle curve approximates to the inverse demand curve P(Q), the seconddegree price discrimination converges to the firstdegree price discrimination. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
28 Third Degree Price Discrimination Limitation of First and SecondDegree Price Discriminations 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? ThirdDegree Price Discrimination Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
29 Third Degree Price Discrimination With the thirddegree price discrimination (or multimarket price discrimination), the monopolist specifies different prices for different consumer groups (with different total demand curves). Example: The Disney Park offers different ticket prices to three player groups: children, adults, and elders. Thirddegree price discrimination usually occurs when 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? Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
30 By Age Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
31 By Time Kindle 2 02/2009: $399 07/2009: $299 10/2009: $259 06/2010: $189 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
32 Even More Dynamic Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
33 More Innovative Ways Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
34 Third Degree Price Discrimination Consider a simple scenario: 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 {P 1, P 2 } to maximize his profit? 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? Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
35 Third Degree Price Discrimination The monopolist s profit π(p 1, P 2 ) under prices {P 1, P 2 } is π(p 1, P 2 ) P 1 Q 1 + P 2 Q 2 C(Q 1 + Q 2 ) The firstorder condition: π(p 1, P 2 ) P i = Q i + P i dq i dp i C (Q 1 + Q 2 ) dq i dp i = 0 Q i D i (P i ) is the demand curve in market i; η i P i dq i Q i dp i is the price elasticity of demand in market i; C (Q 1 + Q 2 ) is the marginal cost (MC) of the monopolist; Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
36 Third Degree Price Discrimination The optimality condition: C (Q 1 + Q 2 ) = P i + Q i dp i dq i = P i ( 1 1 ) η i Under the optimal prices (P1, P 2 ), the marginal revenues (MR) in all markets are identical, and are equal to the marginal cost (MC): P 1 ( 1 1 ) η 1 = P 2 ( 1 1 ) η 2 P i (1 1 η i ) is the marginal revenue (MR) of the monopolist in market i; Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
37 Third Degree Price Discrimination The optimal prices (P1, P 2 ) satisfy ( P1 1 1 ) ( = P2 1 1 ) η 1 η 2 If η1 η 2, then P1 P 2. That is, the monopolist will charge different prices when two markets have different price elasticities. If η1 > η 2, then P1 < P 2. That is, the market with the higher price elasticity will get a lower optimal price. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
38 Third Degree Price Discrimination Graphic Interpretation of Optimal Prices (P1, P 2 ) 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 C 0 MR D 1 D 2 MR 1 MR 2 Q 1 Q 2 Q 1 + Q 2 Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
39 Third Degree Price Discrimination Graphic Interpretation of Optimal Prices (P1, P 2 ) Market 1: the demand is Q1, the marginal revenue equals C 0 ; Market 2: the demand is Q2, the marginal revenue equals C 0 ; Total market demand is Q 1 + Q 2, and the marginal cost is C 0 ; C 0 is at the intersection of MC and MR curves. Price MC P2 P1 C 0 MR D 1 D 2 MR 1 MR 2 Q 1 Q 2 Q 1 + Q 2 Quantity Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
40 Third Degree Price Discrimination Necessary conditions to make the thirddegree price discrimination applicable and profitable: 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. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
41 Section 5.3: Cellular Network Pricing Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
42 Network Model A cellular operator with B Hz of bandwidth Sell bandwidth to multiple users Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
43 TwoStage 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 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
44 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) ( θ = log 2 (1 + SNR) = log Ph ) n 0 When allocated bandwidth b, the user achieves a data rate of θb Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
45 User s Spectrum Efficiency Different users have different spectrum efficiencies 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] Divided by the maximum value of θ among all users 0 1 θ Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
46 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 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
47 User s Demand in Stage II Payoff maximization problem max b 0 π(θ, b, p) = max (ln(1 + θb) pb) b 0 Concave maximization problem user s optimal demand { 1 b (θ, p) = p 1 θ, if p θ, 0, otherwise. User s maximum payoff π(θ, b (θ, p), p) = { ( ) ln θ p 1 + p θ, if p θ, 0, otherwise. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
48 User Separation Based on Spectrum Efficiency No service Cellular service 0 p 1 θ Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
49 Users Total Demand Price p max θ [0,1] θ = 1 If p > 1, the total user demand will be 0 Total user demand Q(p) = Decreasing in p. 1 p ( 1 p 1 ) dθ = 1 θ p 1 + ln p Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
50 Operator s Optimal Pricing Operator s revenue maximization problem max min (pb, pq(p)) 0<p 1 pb is increasing in p pq(p) is decreasing in p: dpq(p) dp = ln p < 0 We can show that at the optimal price p, p B = p Q(p ). Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
51 Operator s Optimal Pricing Operator s revenue maximization problem max min (pb, pq(p)) 0<p 1 pb is increasing in p pq(p) is decreasing in p: dpq(p) dp = ln p < 0 We can show that at the optimal price p, p B = p Q(p ). The optimal price p is the unique solution of B = ln p p B 0 p 1 B p 0 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
52 Section 5.4: Partial Price Differentiation Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
53 Network Model One wireless service provider (SP) A set of I groups of users, where each group i I has N i homogenous users Same utility function u i (s i ) = θ i ln(1 + s i ) Groups have decreasing preference coefficients: θ1 > θ 2 > > θ I The SP s decision for each group i Admit n i N i users Charge a unit price p i (per unit of resource) Subject to total resource limit: i n is i S Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
54 TwoStage Decision Process Stage I: (Service provider s pricing and admission control) The SP decides price p i and n i for each group i Stage II: (Users demands) Each user in group i decides the demand s i 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 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
55 Complete Price Differentiation: Stage II Each (admitted) group i user chooses s i to maximize payoff maximize s i 0 (θ i ln(1 + s i ) p i s i ) The unique optimal demand is ( ) ( ) + si θi θi (p i ) = max 1, 0 = 1 p i p i Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
56 Complete Price Differentiation: Stage I SP performs admission control n and determines prices p: n i p i s i maximize n,p 0,s 0 i I ( ) + θi subject to s i = 1, i I, p i n i {0,..., N i }, i I, n i s i S. i I The Stage II s user responses are incorporated Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
57 Complete Price Differentiation: Stage I SP performs admission control n and determines prices p: n i p i s i maximize n,p 0,s 0 i I ( ) + θi subject to s i = 1, i I, p i n i {0,..., N i }, i I, n i s i S. i I The Stage II s user responses are incorporated This problem is challenging to solve due to nonconvex objectives, integer variables, and coupled constraint. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
58 Complete Price Differentiation: Stage I The admission control and pricing can be decoupled At the unique optimal solution Do not reject any user Charge prices such that users perform voluntary admission control: there exists a group threshold K cp and λ cp with { pi θi λ =, i K cp ; θ i, i > K cp. and The choice of λ satisfies { si θ i = λ 1, i K cp ; 0, i > K cp. K cp i=1 ( ) θi n i λ 1 = S Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
59 Complete Price Differentiation: Optimal Solution IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 22, NO. 3, JUNE 2014 ing to the must hold g function be viewed lution for al solution lem. Note following Fig. 2. Sixgroup example for the effective market: The willingness to pay decreases from group 1 to group 6. The effective market threshold can be obtained (12) Effective by Algorithm market: 1 andincludes is 4 thisgroups example. receiving positive resources and users in these groups as effective users. Anexampleofthe effective market is illustrated in Fig. 2. Now let us solve the admission control subproblem. Denote the objective (10) as,by(15),then.wefirst relax the integer do Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
60 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 Less users will be served Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
61 Effectiveness of Complete Price Differentiation G Θ, N, S Case Case N 1 N 2 N Case 3 0 N 1 N 2 N N 1 N 2 N 3 S Relative Revenue Gain = R Complete Price Differentiation R Single Price R Single Price Relative revenue gain of price differentiation reaches the maximum if The high willingnesstopay users are minority, and Total resource S is limited Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
62 Partial Price Differentiation The most general case SP can charge J prices to I groups, where J I Complete price differentiation: J = I Single pricing: J = 1 How to divide I groups into J clusters, and optimize the J prices? Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
63 Partial Price Differentiation a={a j i, j J, i I}: binary variables defining the partition a j i = 1 group i is in cluster j Revenue optimization problem: maximize {n i,p i,s i,p j,a j i } i,j n i p i s i i I ( ) + θi subject to s i = 1, i I, p i n i {0,..., N i }, i I, n i s i S, i I p i = a j i pj, j J a j i = 1, a j i {0, 1}, i I. j J Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
64 ThreeLevel Decomposition Level1 (Cluster Partition): partition I groups into J clusters Level2 (InterCluster Resource Allocation): allocate resources among clusters (subject to the total resource constraint) Level3 (IntraCluster Pricing and Resource Allocation): optimize pricing and resource allocations within each cluster Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
65 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 : Level3: maximize n i p j s i n i,s i,p j i C j ( ) + θi subject to s i = p j 1, i C j, n i N i, i C j, i C j n i s i s j. Equivalent to a single pricing problem Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
66 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 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73 fi a
67 Level 2: Resource Allocation Among Clusters For a fixed partition a Consider the resource allocation among clusters: Level2: maximize s j 0 subject to R j (s j, a) j J s j S. j J Solving Level 2 and Level 3 together is equivalent of solving a complete price differentiation problem Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
68 Level1: Cluster Partition Level1: maximize a j i {0,1}, i,j subject to R pp (a) a j i = 1, i I. j J Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
69 How to Perform Cluster Partition in Level 1 Naive exhaustive search leads to formidable complexity for Level 1 Groups I = 10 I = 100 I = 1000 Clusters J = 2 J = 3 J = 2 J = 2 Combinations Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
70 How to Perform Cluster Partition in Level 1 Naive exhaustive search leads to formidable complexity for Level 1 Groups I = 10 I = 100 I = 1000 Clusters J = 2 J = 3 J = 2 J = 2 Combinations Do we need to check all partitions? Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
71 Fig. 6. Decomposition and simplification of the general problem: The problem is shown in the lefthand side. After simplifications in Sections VB and VC, the problem will be reduced to structure in righthand side. Property threelevel ofdecomposition An Optimal structure Partition of the Will the following partition ever be optimal? th th L Fig. 7. Example of coupling thresholds. C The equality (a) in (26) means that each cluster can be equivalently treated as a group with homogeneous users with the same willingness to pay. We name this equivalent group as Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73 o th a
72 Fig. 6. Decomposition and simplification of the general problem: The problem is shown in the lefthand side. After simplifications in Sections VB and VC, the problem will be reduced to structure in righthand side. Property threelevel ofdecomposition An Optimal structure Partition of the Will the following partition ever be optimal? th th L Fig. 7. No. Example of coupling thresholds. C The equality (a) in (26) means that each cluster can be equivalently treated as a group with homogeneous users with the same willingness to pay. We name this equivalent group as Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73 th a o
73 Fig. 6. Decomposition and simplification of the general problem: The problem is shown in the lefthand side. After simplifications in Sections VB and VC, the problem will be reduced to structure in righthand side. Property threelevel ofdecomposition An Optimal structure Partition of the Will the following partition ever be optimal? th th L Fig. 7. No. Example of coupling thresholds. C We prove that group indices in the effective market are consecutive. The equality (a) in (26) means that each cluster can be equivalently treated as a group with homogeneous users with the same willingness to pay. We name this equivalent group as Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73 th a o
74 Reduced Complexity of Cluster Partition in Level I The search complexity reduces to polynomial in I. Groups I = 10 I = 100 I = 1000 Clusters J = 2 J = 3 J = 2 J = 2 Combinations Reduced Combos Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
75 Relative Revenue Gain G G Complete price differentiation Five Prices Four Prices Three Prices Two Prices S S A total of I = 5 groups Plot the relative revenue gain of price differentiation vs. total resource Maximum gains in the small plot J = 3 is the sweet spot Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
76 Section 5.5: Chapter Summary Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
77 Key Concepts Theory Monopoly pricing and the demand elasticity Firstdegree price discrimination Seconddegree price discrimination Thirddegree price discrimination Application Cellular Network Pricing Partial Price Discrimiation Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
78 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 , November 2013 S. Li and J. Huang, Price Differentiation for Communication Networks, IEEE Transactions on Networking, vol. 22, no. 2, pp , June Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 28, / 73
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