Nonlinear Pricing of Information Goods

1. Nonlinear Pricing of Information Goods ArunSundararajan New York University, Leonard N. Stern School of Business Associate Professor of Information, Operations and Management Sciences Management Science, Vol. 50, No. 12 (Dec., 2004), pp. 1660-1673

2. 描述故事，從provider角度給出合理的切入點說明provider的strategy應為何(formulation, figure) • 有了transaction fee之後故事要重跑

3. Ground Truth Customer type: θ Customer distribution: F(θ) Provider Pricing Scheme: - Usage-based menu: (q(θ), τ(θ)) - Fixed-fee: T Customer Utility function: U(q, θ) Question: How does the provider set his parameters to maximize his revenue (or profit)

4. Assume that U(q, θ) and F(θ) are known to the provider • Assume that the customers are rational and would choose the best selection • Provider use some techniques to set the menu (q(θ), τ(θ)) and the fixed-fee T to maximize his profit

5. Monopoly firm • Selling information goods • Either a homogeneous good (ex. bandwidth) or a bundle of related heterogeneous quantity units (ex. a library of MP3 songs) • Variable costs of production are zero • That is, the provider could provide unlimited goods to customer (fixed-fee pricing scheme) • While the spectrum is limited, the fixed-fee pricing scheme is inappropriate, and the usage-based pricing scheme should be under some constraints • Better suits in the cloud service case • Discuss more later

6. Customers are heterogeneous • Indexed by θ in this paper • Indexed by two parameters μ & λ in previous work • We can use this feature to simplify our work • Note that we should examine the properties of the utility functions though

7. Customer type distribution F(θ) and their utility function are known • Standard assumption in many works • In our previous work, we didn’t mention the distribution F(θ) • But by revelation principle, we can assume that each MVNO would truly report his type (want to adjust μ & λ to θ)

8. Also, we use acceptance function A(b, p) and W2P and tW2P from the demand functionrather than using the utility function • 從[*]懷疑 U(q, θ) = ∫p(q, θ), where p(q, θ) is the demand function • 所以W2P, tW2P, utility function之間應該是可以彼此取代的 • A(b, p)的值介於[0, 1]之間，如何與utility function做轉換會是一個問題 • Lim U(q, θ) = v(θ) 可能可以派上用場 • 而且U只與q有關而A與b, p有關 [*] Eric Maskin and John Riley, “Monopoly with Incomplete Information,” The RAND Journal of Economics, Vol. 15, No. 2 (Summer, 1984), pp. 171-196

9. Pricing schemes are usage-based and fixed-fee in this paper • Fixed-fee is not appropriate in our work due to we can’t provide unlimited spectrum • Usage-based scheme should under some constraint • Σbi* ≦ B, bi* means the optimal amount that MVNOi would choose in the optimal menu (bi*, pi*) • 但是因為有bi, min & bi, max，在沒有分析之前不能確定bi*會落在區間內

10. 如果將utility function改寫成有jump的形式，雖然能表達在[bi, min, bi, max]之外MVNOi絕對不會選擇的特性，但不連續卻可能會破壞原有optimal menu的性質 • 或許bi, max是不必要的，因為如果我們採用這篇paper的假設，U會抵達一個定值，則應該就是在bi, max達到定值，而bi*不會＞bi, max，可以讓效用函數在bi, max沒有jump而為連續 bi, min bi, max

11. Which type of the customer is not known to the provider in this paper • Only know the distribution F(θ) • In our work, each MVNO’s type is known to the MNO • The bandwidth and price is set specifically to the MVNO (?) • Or the (bi*, pi*) MVNOi accept is just the same as the one MVNOi would choose if the menu is provided • 後者比較不會引起customer反感

12. Fairness didn’t be mentioned in this paper • 冀望由於utility function為concave所以high-value customer不會很快拿到大部分的資源(從resource allocation來看) • 但是當願意付的錢差距很大時資源還是會集中到high-value customer手中 • 防止starvation倒是可以做到 • 但economic fairness如何以數學式表達出來

13. 如果economic fairness可以表示出來則可以這樣做 • Define a fairness indicator • 多目標決策(first max the indicator, then max the profit) • 這樣可以將fairness考慮進去

14. 2010/4/27 • 這篇paper我們可以利用的東西： • Theta function(theta underline corresponds to discount MVNO, theta bar corresponds to business MVNO) • Transaction cost(提出一段說明transaction cost在我們的論文中為什麼需要考慮) • U(q(theta), theta)(想一想告訴老師跟demand curve的關係)(這篇論文user utility只與quantity有關)(如果考慮demand curve裡的price的話是否我們必須思考U需不需要多加一個price的argumentp(q, theta)或是不需要，因為q與p是根據demand curve，所以是否會增加一個constraint: user‘s willingness topay) • v(theta)(跟原本的bmin, bmax有何關係) • tau(theta) (MNO’smenu) • MNO • Assumptions • Finite capacity • MVNO要提出bi, min, bi, max • Copy (3), q要constaint在bi, min, bi, max之間 • i直接對應到 theta • MNO藉由市場調查得知utility • 先給usage-basedmenu (q(theta), tau(theta))，再找出T*使profit會最大 (next page)

15. 2010/4/27 • How to construct the menu • 得知市場上的utility function U(q, theta), eg: • 得知transaction cost function, eg: C(q) = K + cq • 得知theta的distribution, eg: F(theta) = theta

16. Find q0(theta) • Find thetak

17. Final menu

18. 剛剛的menu沒有考慮quantity的constraint • 但我們的有，所以當MVNO request時，MNO可能沒辦法滿足 • 掛牌=>有不滿足(超過B)=>修正重新掛牌

19. U(q(theta), theta)(想一想告訴老師跟demand curve的關係)(這篇論文user utility只與quantity有關)(如果考慮demand curve裡的price的話是否我們必須思考U需不需要多加一個price的argumentp(q, theta)或是不需要，因為q與p是根據demand curve，所以是否會增加一個constraint: user‘s willingness topay) • 參考不同的preference的寫法 • in [*] • in this paper [*] Eric Maskin and John Riley, “Monopoly with Incomplete Information,” The RAND Journal of Economics, Vol. 15, No. 2 (Summer, 1984), pp. 171-196

20. 所以U(q(theta), theta)與demand function:p(q, theta)的關係應為 • Demand function代表customer心中的價值，積分後恰好能代表”滿意程度”，或說是我們故意如此定義utility function • U需不需要多加一個argument p(q, theta)? • 確實U是由p定義的，可是theta應該就能充分表達了不同theta會有不同的demand function，在此前提下只用q & theta不無不可。當然用U(q, p(q’, theta))可能會比較精確

21. W2Pi(b) ≡ p(q, theta) • tW2Pi(b) ≡ U(q, theta) • 只是名稱的不同而已，實際上還是指相同的意義，應該談不上是constraint

22. v(theta)(跟原本的bmin, bmax有何關係) • U(q, theta) = v(theta), for q ≧ bmax • MVNO不願意為超過bmax的頻寬付錢 =>p(b, theta) = 0 for b≧bmax • 在bmax之後utility function不會有任何增加 =>reach a limit, v(theta)<= 用式子寫出來 • v(theta) 與 bmin 沒關係，bmin與U的起點比較有關係 bi, min bi, max

23. Demand curve有最大最小，map到price要大於c。 • 不知道buyer的demand，所以用bmin, bmaxbound起來。

24. 2010/5/3 • 把所有的參數整理、統一 • 做實驗時把F(theta)轉換成discrete(一個type 有兩個MVNO)(uniform => extreme) 最後會不會converge？(先follow前面的model，再看看要不要將quantity constraint加入修正) • 證明可以用continuous，real world(實驗)都是discrete

25. Notations • We use a parameter θ to describe the MVNO type • θ is bounded by [θL, θU] • We use θ to present the relation between the discount MVNO and the business MVNO • The concept between discount and business MVNO is relative • For all θa, θb belongs to [θL, θU], if θa < θb, relatively, θa can be viewed as a discount MVNO and θb can be viewed as a business MVNO

26. The MVNOs follow a continuous distribution • MNO wouldn’t know what the exact type an MVNO is • But MNO can know the distribution by the historical information • The distribution is presented bya cdf F(θ) and a pdf f(θ) • There might be some assumptions on the distribution

27. For each MVNO • MVNO has a demand function p(b’, θ) where b’ represents the bandwidth MVNO holds so far • The utility function U(b, θ) can be defined by • Note: 這邊我將p(b’, θ)與U(b, θ)的第一個參數設為不同的符號，雖然都是代表bandwidth但似乎應用情境不同，我不確定這樣子是否合理

28. Assumption • p(b’, θ) ≧ 0, that is, U(b, θ) is nondecreasing in b • p(b’, θ) is nonincreasing in b’ (decreasing for b’ ≦ bmax, the highest amount MVNO demands, and p(b’, θ) = 0 for b’ ＞ bmax), and U(b, θ) = v(θ) = U(bmax, θ) for b ＞bmax (在bmax之後U不再成長) • p(b’, θ) is strictly increasing in θ • U(b, θ) is strictly increasing in θ • MVNO has no idea about the marginal cost c and has nothing to do with c (如果限制p(b’, θ) ≧ c 會使U的形狀不同) • Note: 待補：一次、二次連續可微(可能要到後面有需要時再回來補充)，bmin的補充(初步先不考慮)

29. For MNO • The good MNO provides is the bandwidth and the bandwidth capacity is denoted as B • MNO should provide a menu of bandwidth-price pair (b(θ), τ(θ)) to MVNOs • (0, 0) should be included • The menu is continuous in θ (這句的用法不確定)(But in practice and experiment, we can just assign some specific θ value to generate a finite set of bandwidth-price pair)

30. Assumption • MVNO會從MNO提供的menu中選擇最適合的bandwidth-price pair • That is, [Incentive Compatibility (IC)]: for each θ, • And [Individually Rational (IR)] : for each θ, • Note: 所以menu才會是depends on θ，因為我們假設type θ的MVNO一定會選擇(b(θ), τ(θ))這組pair。當然menu必須經過精心設計，滿足[IC], [IR]。 • Also, The menu should be defined properly such that

31. The total cost of MNO is denoted as C(b) = K + cb • K is the fixed cost • c is the variable cost • Assumption • Constant variable cost c • If not constant, c(b) should be used • Note: 雖然[Nonlinear Pricing of Information Goods]假設usage-based有transaction cost但沒有production cost，但在[Monopoly with Incomplete Information]只有用到marginal cost，所以我在這邊就打算直接考慮total cost

32. Example • MVNOs are uniform distributed, that is, F(θ) = θ, f(θ) = 1 • Constant variable cost c = 10 • Utility function

33. And using Proposition 3, we get • b(θ) = 40θ – 20 • 40θ – 20 ≦ 20θ + 10 => θ≦ 1.5 • θK = (K + 200) / 400, θK = 0.5 if K = 0 • τ(θ) = -400θ2+ 1200θ – 500 • The menu • (0, 0) for θ < 0.5 • (40θ – 20, -400θ2+ 1200θ – 500) for θ≧ 0.5

34. If there are 10 MVNOs, indexed by 0.1, 0.2, …, 1.0 • And MNO only provides {(b(θ), τ(θ)): θ = 0.2, 0.4, 0.6, 0.8, 1.0}, that is {(0, 0), (4, 76), (12, 204), (20, 300)} • The MVNOs would choose

35. The max total bandwidth would be 4 + 12 + 12 + 20 + 20 = 68 • Is 68 ≦ B? • Can’t verify since the capacity constraint didn’t mention • And if there are more MVNOs, such as indexed by 0.05, 0.1, …, 0.95, 1.0, the required bandwidth definitely greater than 68 • The capacity constraint seems useless

36. Consider the capacity constraint • Using the Lagrangian form and the first-order necessary condition, we get (λ: L. Multiplier)

37. It seems that we can derive b(θ) which contains λ and then substitute into the capacity constraint to get λ, and thus we get b(θ) without other variables (I’m not sure whether it is sound) • But again, the required bandwidth would increase if the number of the MVNOs increases, under the same distribution but different size • I guess that the capacity constraint shouldn’t be that simple, but should consider N, the number of the MVNOs, that is

38. Problems • However, since the difference between the reality and the expectation might exist, how can we provide a proper menu satisfying the capacity constraint?