Trade Information, Not Spetrum: A Novel TV White Spae Information Market Model Yuan Luo, Lin Gao, and Jianwei Huang 1 Abstrat In this paper, we propose a novel information market for TV white spae networks, where the spetrum database operator sells the information regarding TV white spae to seondary users. Different from the traditional spetrum market, the information market proesses the unique property of positive externality, as more users purhasing the information servie will inrease the value of the servie to eah buyer. We systematially haraterize the market equilibrium and the database operator s optimal information priing strategy. Speifially, we first study how the market share dynamially evolves over time and eventually onverge to a market equilibrium. We show that the market equilibrium inreases with the initial market share, and there exist several tipping points of the initial market share, around whih a slight hange will lead to a signifiant hange on the emerging market equilibrium. Based on the market equilibrium analysis, we further study the impat of the database operator s information priing strategy on the market equilibrium, and derive the optimal information prie that maximizes the database operator s revenue. Theoretial analysis and numerial result indiate that this is a promising business model for reating inentives for the database operator in TV white spae networks. I. INTRODUCTION A. Bakground and Motivation TV white spae networks [1] [3] an effetively improve the TV spetrum effiieny and alleviate the spetrum sarity, and thus is a promising approah to solve the spetrum shortage problem. In a TV white spae network, unliensed wireless devies (alled white spae devies, WSDs opportunistially exploit the unused or under-utilized hannels (alled TV white spae, TVWS 1 in the broadast television spetrum band. The suessful deployment of a TV white spae network requires many tehnial innovations, among whih an important one is to reliably detet the available hannel and aurately estimate the hannel quality at different times and loations. Most early studies on the hannel detetion and quality estimation foused on the spetrum sensing tehnique [4]. However, reent studies [5] pointed out that spetrum sensing alone is often ineffiient, due to the high operational ost as well as the low detetion performane. As an alternative, spetrum regulatories (suh as FCC in the US and Ofom in the UK advoate the use of a geo-loation white spae database [3]. In this database approah, unliensed WSDs obtain the hannel information via querying a geo-loation database, rather than sensing the wireless environment. Aordingly, the This work is supported by the General Researh Funds (Projet Number CUHK 412713 and CUHK 412511 established under the University Grant Committee of the Hong Kong Speial Administrative Region, China. Authors are with the network ommuniations and eonomis lab (NCEL, Dept. of Information Engineering, The Chinese University of Hong Kong, E-mail: {ly011, lgao, jwhuang}@ie.uhk.edu.hk 1 For onveniene, we will also refer to TV white spae as TV hannel or just hannel in this paper. database is required to house an up-to-date repository of TV liensees, and periodially update the hannel oupation by TV liensees. Suh a database-assistant TV white spae network arhiteture has also been widely supported by standards bodies and industrial organizations [6] [12]. While most prior studies foused on the tehnial issues suh as the design and deployment of the white spae database, there laks a proper business model that offers neessary eonomi inentives to the database operators. 2 Prior studies related to the business modeling of TV white spae network mainly foused on the spetrum market [13] [17], where the database operators, ating as brokers or agents, purhase hannels from the TV liensees, and then sell the purhased hannels to unliensed WSDs at a relatively higher prie. However, the TV spetrum market model may not be suitable due to some regulatory onsiderations. For example, TV white spaes sometimes are treated as the publi spetrum resoure by regulators, whose goal is to make more spetrum available for publi and shared usage. Beause of this, the TV spetrum may not always be traded in the spetrum markets like other liensed spetrum bands. To this end, a new business model without involving the trading of spetrum is desired. Spetrum Bridge, the world s first white spae database ertified by the FCC, proposed an alternative business model alled White Spae Plus [12]. The basi idea is to sell ertain advaned information regarding TV hannel to WSDs, suh that they an hoose and operate on the most desirable hannel. An example of suh information is the degree of interferene on every TV hannel, whih may ome from either the nearby TV stations or the unliensed WSDs operating on that hannel. This essentially leads to an information market, where WSDs purhase the information regarding hannel quality, instead of purhasing the hannel itself. Clearly, the suessful deployment of suh an information market requires (i an aurate model to evaluate the value of information for WSDs (buyers, and (ii a arefully designed priing plan for the database operator. However, none of these two issues has been onsidered in the urrent White Spae Plus. 3 This motivates us to study the information market model for white spae databases in this paper. B. Contributions In this paper, we model and study an information market for TV white spae network, where the database operator (seller sells the following information to WSD users (buyers: the interferene levels on TV hannels. We fous on designing the optimal information priing plan that maximizes the database 2 Suh inentives are neessary, for example, for overing the database operators apital expenditure (CapEX and operational expenditure (OpEX. 3 Currently Spetrum Bridge just offers a one year free trial to use this White Spae Plus servie [12].
2 operator s revenue. To ahieve this, we need to aurately evaluate the value of information to users, and the users reations under any information prie. Information Value. We propose a general framework for evaluating the value of information to users. Notie that the interferene on a hannel may ome from the nearby TV stations operating on that hannel or the nearby WSDs using that hannel. The database an (relatively preisely predit the interferene from TV stations, as it maintains a repository of TV liensees. However, it may not be able to predit the preise interferene from WSDs, either beause some users may not want to inform the database their hoies of hannels, or simply beause some users may interat with another database in the same area. Therefore, the overall interferene information that the database provides may not be aurate. This will affet the value of information for WSD users, whih in turn will affet the database operator s optimal priing plan. Market Equilibrium. After haraterizing the value of information to users, we are able to derive the stable market share (i.e., the perentage of users who purhase information from the database operator, alled the market equilibrium. In ontrast to traditional spetrum markets whih are usually ongestion-oriented (i.e., the more users purhasing and using spetrum, the less value of spetrum for users due to the potential o-interferene among users, we show that the information market has the nie property of positive externality [18] [20]. That is, the more users purhasing the information from the database, the higher value of the information for eah buyer. This is beause when more users purhase the information and reveal their hannel seletions to the database impliitly, the database an predit the interferene information more aurately. Due to the positive network externality, the market equilibrium inreases with the initial market share. Interestingly, there exist several ritial points (alled tipping points of the initial market share, around whih a slight hange will result in a signifiant hange on the emerging market equilibrium. Suh a Small Changes, Big Impat [21] phenomenon implies that the database needs to initialize the market with a large enough market share, so that the market an eventually evolve to a desirable market equilibrium. We propose a refund mehanism to motivate users to purhase the information at the initial stage, so as to suessfully pass the largest tipping point. This has a similar spirit of the Spetrum Bridge s urrent marketing strategy, whih offers one year free trial to use the White Spae Plus servie. Finally, based on the market equilibrium analysis, we derive the optimal information priing plan that maximizes the database revenue. In summary, our main ontributions are as follows. To the best of our knowledge, this is the first paper proposing and studying an information market for TV white spae networks. Compared with the spetrum market model, this model better satisfies the requirements from the regulators and the pratie of the industry. We propose a general framework to evaluate the value of information to WSD users. This framework onsiders both the potential error of the information and the heterogeneity of users. We haraterize the positive network externality of the information market, and study the market equilibrium systematially. Based on this, we further derive the optimal information priing plan that maximizes the database operator s revenue. Theoretial analysis and numerial result indiate that the database operator an make a signifiant profit from suh an information market. Thus, this is a promising business model to give the ommerial entities neessary inentives to operate and maintain white spae databases. The rest of the paper is organized as follows. In Setion II we present the system model. In Setions III and IV, we analyze the users best behaviors and the database operator s optimal information priing plan, respetively. We present the simulations in Setion V, and finally onlude in Setion VI. II. SYSTEM MODEL We onsider a TV white spae network, where a set N = {1,..., N} unliensed white spae devie users (endusers operate on idle TV hannels requested from a white spae database. Let K {1,..., K} denote the set of idle TV hannels in the area of the network. Eah end-user an only transmit on one of these hannels at any given time. We onsider a time-slotted system (onsistent with many TV white spae trial systems [11], where end-users interat with the database periodially to obtain the available hannel information, usually with a period ranging from several minutes to several hours (e.g., 2 hours suggested by Ofom [3]. For eah end-user n N, eah hannel k is assoiated with an interferene level, denoted by Z n,k, whih reflets the aggregate interferene from all other nearby devies (inluding TV stations and other end-users operating on this hannel. Due to the fast varying nature of wireless hannels and the unertainty of end-users ativities, the interferene Z n,k is a random variable. We assume that Z n,k is temporalindependene and frequeny-independene. That is, (i the interferene Z n,k on hannel k is independent identially distributed (i.i.d. aross time periods, and (ii the interferenes on different hannels, Z n,k, k K, are also i.i.d. in the same time period. 4 As we are talking about a general WSD n, we will omit the WSD index n in the notations (e.g., write Z n,k as Z k, whenever there is no onfusion aused. Let F Z ( and f Z ( denote the umulative distribution funtion (CDF and probability distribution funtion (PDF of Z k, k K. White Spae Database. Aording to the regulator s ruling (e.g., FCC [1], a white spae database needs to provide the following information to end-users: (i the list of all available TV hannels, (ii the maximum transmission power on every hannel, and (iii some other optional requirements. This is the basi servie that every database is required to provide to any interested user free of harge. Beyond the basi servie, the database an also provide an advaned servie to make profit, under the onstraint that it does not onflit with the basi servie. Motivated by the pratie of Spetrum Bridge [12], we onsider suh a senario 4 Note that the i.i.d. assumption is a reasonable approximation of the pratial senario, where all hannel quality distributions are the same but the realized instant qualities of different hannels are different (e.g., [22].
3 where the database provides the interferene level Z k of every available hannel k to those end-users who subsribe to this advaned servie 5. With this advaned information, end-users are able to pik and operate on the best available hannel. Aordingly, the database will harge a subsription fee (denoted by for suh an advaned servie. This onstitutes an information market. White Spae Devie Users (End-Users. Basially, after obtaining the available hannel list through the free basi servie, eah end-user has 3 hoies (denoted by l in terms of hannel seletion: (i l = a: subsribing to the advaned servie and pik the hannel with the minimum interferene, (ii l = s: sensing the available hannels to figure out the best one, or (iii l = b: randomly hoosing a hannel from the list of available hannels. Different hoies may bring different benefits and inur different osts for/on end-users. We assume that eah end-user is rational, and will hoose the strategy that maximizes its payoff. The payoff of an enduser is defined as the differene between (i the benefit (utility ahieved from transmitting data on the seleted hannel, and (ii the subsription fee (if hoosing to subsribe to the advaned servie or the sensing ost (if hoosing to sense the hannels. We onsider heterogeneous end-users, where different end-users value the same data transmission rate or utility differently (due to different wireless appliations. Let θ denote an end-user s evaluation for its ahieved utility. For the analytial onveniene, we assume θ is uniformly distributed in [0, 1]. The payoff of a type-θ end-user is defined as: θ g ( R [b], if l = b, Π EU = θ g ( R [s], if l = s, θ g ( (1 R [a], if l = a, where R [l] denotes the expeted data rate when the end-user hooses a strategy l {b, s, a}, and g( is the utility funtion of end-user, whih is a onavely inreasing funtion of R [l]. Here we assume that all end-users have the same sensing ost when l = s, and are harged by the same prie when l = a. In other words, the database is not allowed to engage in either QoS disrimination or prie disrimination for the simpliity of pratial implementation. Let us further assume that there is no sensing error. 6 Then, the end-user s expeted data rates under the strategies l = b (random seletion and l = s (sensing an be omputed by: R [b] = E Z [r(z] = z r(zdf Z(z, [ ( ] R [s] = E Z(1 r Z(1 = z r(zdf (2 Z (1 (z, where Z (1 min{z 1,..., Z K } denotes the minimum interferene on all hannels, F Z(1 (z = [1 F Z (z] K is the CDF of Z (1, and r( is the transmission rate funtion (e.g., the Shannon apaity. It is important to note that the enduser s expeted data rate R [a] under the strategy l = a (subsribing to the advaned servie depends on the auray of the advaned information the database offers. Intuitively, we have: (i R [a] = R [s] in the extreme ase that the database s information is fully aurate, and (ii R [a] = R [b] 5 Subsribe to the advaned servie is used throughout this paper to mean an end-user s behavior of purhasing the advaned information. 6 Our analysis an be diretly applied to the ase with sensing error [23]. in another extreme ase that the database does not have any aurate information regarding the interferenes. However, in the general ase where the database s information is partially aurate, R [a] is generally different from R [s] and R [b]. We will provide the detailed haraterization of R [a] in (7 after we define the auray of the database s information. Interferene Level (Information. For a partiular enduser, its experiened interferene Z k on a hannel k is the aggregate interferene from all other (nearby devies operating on hannel k, and usually onsists of three omponents: 1 U k : the interferene from liensed TV stations; 2 W k,m : the interferene from another end-user m operating on the same hannel k; 3 V k : any other interferene from outside systems. The total interferene on hannel k is Z k = U k + W k + V k, where W k m N k W k,m is the total interferene from all other end-users operating on hannel k (denoted by N k. Similar to Z k, we assume that U k, W k, W k,m, and V k are random variables with temporal-independene (i.e., i.i.d. aross time periods and frequeny-independene (i.e., i.i.d. aross hannels. We further assume that W k,m is user-independene, i.e., W k,m, m N k, are i.i.d. Let F U (, F W (, and F V ( denote the CDFs of U k, W k,m, and V k, respetively. It is important to note that different end-users may experiene different interferenes U k (from TV stations, W k,m (from another end-user, and V k (from outside systems on a hannel k, as we have omitted the end-user index n for all these notations for larity. Next let us disuss the above interferene omponents more detailedly. First, based on the knowledge about the loation and hannel oupany of TV stations, the database is able to ompute the interferene U k from TV stations to a partiular end-user (on hannel k. Seond, due to the lak of outside interferene soure information, the database annot ompute the interferene V k from outside systems aurately. Thus, the information about V k will not be inluded in the database s advaned information sold to end-users, whih redues the auray of the database s advaned information. Third, the omputation of the interferene W k,m from another end-user m (operating on hannel k is more ompliated. Notie that the database knows preisely the loation information of every end-user who requests TV hannels (as end-users are mandatorily required to report their loation information [1], [2]. Thus, the database an preisely ompute the interferene of one end-user to another one, if it is able to get the operational hannels of end-users. However, the database may or may not know the exat hannel seletion of an end-user, depending on whether the end-user subsribes to the advaned servie or not. Speifially, if an end-user subsribes to the advaned servie, the database an predit the end-user s hannel seletion, sine the end-user is fully rational and will always hoose the hannel with the minimum interferene level indiated by the database (in the advaned servie. However, if an end-user does not subsribe, the database annot predit its hannel seletion, sine the end-user s sensing result may not be the same as that provided from the database s advaned information (due to the missing of V k in the database s
4 information as we explained above, or the end-user may even hoose a hannel randomly. Let N k[a] denotes the set of end-users (operating on hannel k subsribing to the advane servie, and N k[x] denotes those not subsribing to the advane servie. That is, N k[a] Nk[x] = and N k[a] Nk[x] = N k. Then, for a partiular hannel k, the interferene known by the database (and thus will be inluded in its advaned information is X k U k + m N k[a] W k,m. (3 The interferene not known by the database (and thus will not be inluded in its advaned information is Y k V k + m N k[x] W k,m. (4 Thus, the total interferene level on hannel k is Z k X k + Y k = U k + V k + m N k W k,m. (5 Obviously, Y k and X k are also random variables with temporal-independene and frequeny-independene. Sine the database knows only X k, it will provide this information as the advaned servie to end-users. It is easy to see that the more end-users subsribing to the advaned servie, the more information the database an provide (and thus the more aurate the database s information will be. Based on the above, we an haraterize the auray of the database s information under different number of endusers subsribing to the advaned servie. Let η denote the perentage of end-users subsribing to the advaned servie, alled the market share of the database. By the assumption of the frequeny independene of Z k (i.e., the total interferenes on different hannels Z k, k K are i.i.d. 7, eah end-user will be assigned to eah hannel with an equal probability. 8 N Therefore, there are, on average, K end-users operating on eah hannel k, with N K η end-users subsribing to the advaned servie and N K (1 η end-users not subsribing to the advaned servie. That is, N k = N K, N k[a] = N K η, and N k[x] = N K (1 η.9 Then, by (3 and (4, we an immediately obtain the distributions of X k and Y k under any given market share η. Information Value. Now we evaluate the value of database s information {X k } k K to end-users, whih is refleted by the end-user s benefit (i.e., the utility g( in Eq. (1 that an be ahieved from utilizing this information. We first onsider an end-user s utility without this information (i.e., when not subsribing to the advaned servie. In this ase, end-users an deide either to randomly selet a hannel (l = b, or to sense all hannels for the best one (l = s. The 7 This assumption is used for analysis onveniene. Note that the atual distribution depends on the users subsription behaviors. The more detailed analysis and simulation verifiations will be left for our future work. 8 Eah end-user with the strategy l = s (sensing will be assigned to eah hannel with an equal probability (as suh end-users will hoose the best hannel with the lowest Z k, k K. Similarly, by the assumption of the frequeny independene of X k, eah end-user with the strategy l = a (subsribing will be assigned to eah hannel with an equal probability (as suh end-users will hoose the best hannel with the lowest X k, k K. Furthermore, eah end-user with the strategy l = b (randomly seleting will straightforwardly be assigned to eah hannel with an equal probability. 9 Note that the above disussion is from the aspet of expetation, and in a partiular time period, the realized numbers of end-users in different hannels may be different. end-user s expeted utilities under strategies l = b and l = s are, respetively, B g ( R [b] and S g ( R [s], (6 where R [b] and R [s] are the respetive data rates defined in (2. Obviously, B and S depend only on the distribution of the total interferene Z k, while not on the speifi distributions of X k and Y k. This implies that the auray of the database s information does not affet the utilities of those end-users not subsribing to the advaned servie. Then we onsider an end-user s expeted utility with this information (i.e., when subsribing to the advane servie, l = a. In this advaned servie, the database returns the interferene {X k } k K to end-users, together with the basi information suh as the available hannel list. For a rational end-user, it will always hoose the hannel with the minimum X k (sine {Y k } k K are i.i.d.. Let X (1 = min{x 1,..., X k } denote the minimum interferene provided by the database. Then, the atual interferene experiened by an end-user an be formulated as a random variable Z [a] = X (1 + Y. Aordingly, the end-user s expeted data rate and utility are R [a] = E Z[a] [ r ( Z[a] ] = z r(zdf Z [a] (z, A g ( R [a], (7 where F Z[a] (z is the CDF of Z [a]. It is easy to see that both R [a] and A in (7 depend on the distributions of both X k and Y k, and thus depend on the market share η. Therefore, we will also write A as A(η. We an further hek that A(η inreases with η, whih shows that the information market has the property of positive network externality. This is beause the more end-users subsribing to the advaned servie, the more aurate the database s information is, and further the more benefit for the end-users subsribing to the advaned servie. We further assume that A(η is onave in η, whih is verified by simulations. By (6 (7, we an find the following useful properties. Proposition 1. B and S in (6 do not depend on η; A(η in (7 monotonously inreases with η; B A(η S, η [0, 1]; A(1 = S, if there is no outside interferene {V k } k K ; A(0 = B, if there is no liensee interferene {U k } k K. Problem Formulation. We will study the optimal information prie that maximizes the database revenue Π DB, where Π DB = η N. Note that the market share η of the database is a funtion of the information prie, and thus an be written as η(. Moreover, the hange of η will affet the interferene X k (known by the database and Y k (not known by the database, whih in turn will affet the end-user s utility A(η ahieved from the advaned servie, and the end-user s deision. III. END-USER SUBSCRIPTION DYNAMICS AND MARKET EQUILIBRIUM In this setion, we will study the end-user subsription dynamis and the market equilibrium. Speifially, we will first study the end-user s best hoie under a given information
5 prie and initial market share. Then we will study how the enduser subsription dynamially evolves over time, and what is the eventual stable market share (market equilibrium. A. End-user s Best Choie We first onsider an end-user s best hoie under a partiular information prie and initial market share η 0 [0, 1]. As mentioned earlier, eah end-user has three hoies: (i subsribing to the advaned servie, i.e., l = a; (ii sensing hannels to figure out the best one, i.e., l = s; and (iii randomly hoosing a hannel from the available hannel set, i.e., l = b. The respetive payoffs under different hoies are given in (1, where g ( R [s] = S, g ( R[b] = B, g ( R[a] = A(η 0 ; moreover, B < A(η 0 < S. For onveniene, we will write A(η 0 as A in this subsetion. First, let us ompare the end-user s hoies b and s (under the basi servie. By (1, we an find that a type-θ end-user prefers the hoie b to s, if and only if θ B θ S. Thus, we immediately have the following proposition. Proposition 2. There exists a user-type threshold θ BS = S B, (8 suh that (i the end-users with type θ < θ BS prefer the hoie l = b (randomly hoosing a hannel to l = s (sensing hannels, and (ii the end-users with type θ > θ BS prefer the hoie l = s to l = b. Figure 1 illustrates the threshold θ BS, whih is denoted by the intersetion of the blue urve (the end-user s payoff when l = b and the red urve (the end-user s payoff when l = b. S B Notie that if > S B, then θ BS = > 1, whih implies that none of end-users will hoose s (as the end-user type θ is defined in [0, 1] due to the high sensing ost. Therefore, in the following analysis we will fous on the senario of S B. Next, let us ompare all of the three hoies of end-users. By (1, we an easily find that a type-θ end-user will hoose to subsribe to the advaned servie (l = a, if and only if θ A θ B, and θ A θ S. Then, we further have the following proposition. Proposition 3. There exist two user-type thresholds θ BA = A B, and θ AS = S A, (9 suh that (i the best hoie of end-users with type θ [θ BA, θ AS ] is l = a, (ii the best hoie of end-users with type θ [0, min{θ BA, θ BS }] is l = b, and (ii the best of hoie of end-users with type θ [max{θ BA, θ BS }, 1] is l = s. Figure 1 illustrates the thresholds θ BA and θ AS, whih are denoted by the intersetion of the green urve (the end-user s payoff when l = a and the blue urve, and the intersetion of the green urve and the red urve, respetively. It is easy to see that the end-users with type θ < θ BA (Region I prefer the hoie b, the end-users with type θ > θ BS (Region III prefer the hoie s, and the perentage of end-users subsribing to the advaned servie is η = θ AS θ BA, alled the derived market share. Notie that if >, then θ AS < 0, whih implies that none of end-users will subsribe to the advaned servie due to the high subsription fee. 0 End-user Payoff θ BA θ BS θ AS Region I Region II Region III Sensing (s Subsription (a 1 Random (b End-user Type Fig. 1. End-user payoff vs End-user type θ. In Region I, the best hoie of end-user is l = b (random hoie, in Region II, the best hoie of end-user is l = a (subsribing to advaned servie, and in Region III, the best hoie of end-user is l = s (sensing. To failitate the haraterization of the derived market share η(, we introdue the following two ritial pries: 10 BS = (A B S B, AS = (S A. (10 Then, under a partiular information prie and an initial market share, the derived market share η( is given by If > BS, then η( = 0 (as θ AS < θ BA ; If AS BS, then η( = θ AS θ BA = S A A B ; If 0 AS, then η( = 1 θ BA = 1 A B. Formally, we have the following derived market share. Proposition 4. The derived market share under prie is η( = max { min{θ AS, 1} θ BA, 0 }. (11 B. End-user Subsription Dynamis Eq. (11 shows that the derived market share η depends not only on the information prie, but also on the initial market share η 0 (as A is a funtion of η 0. Notie that the hanging of market share will affet the end-users evaluation for the database s information (i.e., A, and thus affet the endusers future subsribing deisions. Thus, the market share will dynamially evolve, until it reahes a stable market share (alled market equilibrium. Now we study suh an end-user subsription dynamis, and haraterize the market equilibrium. To haraterize suh a dynamis of end-user s subsription, we onstrut a virtual time-disrete system with slots t = 1, 2,..., T (eah with a suffiently small time period, and allow end-users hange their deisions in every time slot based on the new addressed market share. 11 At eah time slot t, eah end-user will form a belief, or expetation, on the urrent market share η, and thereby on the A(η, before it makes a subsription deision. If the end-users belief is higher than the real market share η, some end-users subsribing to the advaned servie will anel their subsription in the next time slot. If the end-users belief is lower than the real market share η, some end-users not subsribing to the advaned servie will start their subsription in the next time slot. Let us denote η t as the market share at time slot t. Based on the market share η t 1 in the previous slot t 1, every 10 Intuitively, the ritial prie BS orresponds to the ase that θ BA = θ AS (whih must be same as θ BS, i.e., three urves in Figure 1 interset at the same point. The ritial prie AS orresponds to the ase that θ AS = 1, i.e., the green and red urves in Figure 1 interset at θ = 1. 11 Notie that the addressed market share at eah slot in the virtual system orresponds to the end-user s belief of market share in the real system.
6 Dynamis of Market Share - η 0.4 0.2 0 0.2 0.4 0.6 0.8 η0 ηc 1 prie = 0 prie = A prie = B prie = C prie = D ηb 1 1 0 0.2 0.4 0.6 0.8 1 Market Share - η Fig. 2. Dynamis of η under different pries (low sensing ost: α = 0.2 end-user makes the subsription deision in the urrent slot t in a myopi way, that is, aiming at maximizing its expeted payoff in the urrent slot. Setion III-A explains how an enduser makes suh a deision. Speifially, by introduing the result in (11, we have the following derived market share at the beginning of the time slot t = 1,..., T : η t = max { min { S A(η t 1, 1} A(η t 1 B, 0}, (12 where η 0 is the initial market share at the beginning. Let η denote the hange (dynamis of the market share η between two suessive time slots t and t 1, i.e., ηa1 η r η = η t η t 1, (13 where η t is a funtion of η t 1 given in (12. Thus, η is also a funtion of η t 1 (and hene is a funtion of t. Note that a positive (or negative η implies that the market share η will inrease (or derease along the dynamis. An equilibrium is defined as suh a market share where no end-user has an inentive to hange its ation. Formally, Definition 1. A market share η is a market equilibrium if and only if η(η = 0. In the following analysis, we will study the equilibrium market share systematially. Speifially, we will show that under a given prie, there may be one or multiple tipping points of the initial market share, around whih a slight hange will leads to a signifiant hange on the market equilibrium. We will also show that under a given prie, there may be multiple equilibria, and whih will eventually emerge depends on the end-user s initial belief on the market share. Besides, some equilibria are stable in the sense that a small flutuation around these equilibria will not drive the market share away from the equilibria, while some equilibria are un-stable in the sense that a tiny flutuation on these equilibria will drive the market share to a different equilibrium. A key system parameter that affets the haraterization of the market equilibria is the sensing ost. Next we will onsider both low and high sensing ost. For onveniene, we denote the the magnitude of the end-user s sensing ost as α S B, where α [0, 1]. 1 Low Sensing Cost: We first onsider the senario with a low sensing ost (i.e., a small α. We illustrate the dynamis of the market share η (i.e., η in Figure 2, where eah urve denotes the dynamis η under different pries 0 A B C D, whih will be disussed one by one. ηb 2 ηc 2 ηb 3 ηa2 η1 (A = 0. The orresponding dynamis η (or η =0 is denoted by the top blue urve. We notie that η =0 is always larger than zero exept the last point η =0 (1 = 0. Thus, there is a unique equilibrium η 1 = 1. That is, if the database offers the advaned servie for free, then all end-users will subsribe to the advaned servie eventually. Moreover, this equilibrium is stable. We further notie that this blue urve is indifferentiable at a point η = η r, whih is a ritial point satisfies S A(η = 1. Speifially, (i before the ritial point (i.e., η η r, we have < 1, and thus the blue urve is haraterized by η = S A(η S A(η A(η B η; (ii after the ritial point (i.e., η η r, we have S A(η 1, and thus the blue urve is haraterized by η = 1 A(η B η. Intuitively, if S A(η 1, all endusers an ahieve a higher payoff by subsribing to advaned servie than sensing (i.e., the green urve is always higher than the red urve in Figure 1, and thus no end-user will hoose S A(η sensing. If < 1, some end-users an ahieve a higher payoff by sensing than by subsribing to the advaned servie (i.e., there is an intersetion of the green and red urves in Figure 1. This leads to the different haraterizations of η. When η η r, we have the following first-order derivative: d η dη = ( [S A(η] + 2 [A(η B] da(η 2 dη 1, (14 whih is negative initially, and then beomes positive with the inrease of η. This explains the shape of the blue urve before η r. When η η r, we have d η dη 0, and thus the blue urve dereases with η in this range. Notie that for every prie desribed below, there exists a similar ritial point η r (but with a different value. (B = A. The orresponding dynamis η (or η =A is denoted by the seond (green urve. This urve is below the blue urve (when = 0, sine η dereases with. For better illustration, we intentially hoose a prie A suh that the smallest point before the ritial point meets zero. In this ase, there are two equilibria: η A1 and η A2. We further notie that the equilibrium η A2 (illustrated by the green dot is stable, sine any flutuation of market share around η A2 will ome bak to η A2 eventually, whereas η A1 (illustrated by the gray dot is not stable, as a tiny inrease on η A2 will drive the market share to the larger equilibrium η A2. In this sense, η A2 is the tipping point. (C = B. The orresponding dynamis η (or η =B is denoted by the third (red urve. As the prie inreases to B, there are three equilibria η B1, η B2, and η B3, where η B1 and η B3 are stable, and η B2 is not. Note that η B2 is the tipping point, sine a tiny inrease on η B2 will drive the market share to the larger equilibrium η B3, while a tiny derease on η B2 will drive the market share to the smaller equilibrium η B1. (D = C. The orresponding dynamis η (or η =C is denoted by the forth urve. For better illustration, we intensionally hoose a prie C suh that the ritial point meets zero. There are two equilibria η C1 and η C2, where η C1 is stable, and η C2 is not. η C2 is the tipping point. (E = D. The orresponding dynamis η (or η =D is denoted by the fifth urve. For better illustration, we intensionally hoose a prie D suh that the initial point meets zero η(0 = 0. In this ase, η is always smaller than zero
7 (exept the initial point. Thus, there is an unique equilibria η 0 = 0 whih is stable. Lemma 1 summarizes the above disussions regarding the stable equilibrium, where A, B, C, D are the pries illustrated in Figure 2. Lemma 1 (Stable Equilibrium. The stable market equilibrium under the low sensing ost senario is given by if D, there is a unique stable equilibrium: η EQ = 0; if C < D, there is a unique stable equilibrium: η EQ = η C1, where η C1 is given by S A(η A(η B η = 0; if A < < C, there exist two stable equilibria: η B1 and η B3, whih are respetively given by S A(η A(η B η = 0, 1 A(η B η = 0; if A, there exists a unique stable equilibrum η EQ = η A2, whih is given by 1 A(η B η = 0. Lemma 1 illustrates that there may be multiple stable equilibria under a partiular prie. Next, we show whih stable equilibrium will eventually emerge depends on the initial market share (or the initial belief of the market share. Let us take the ase = B in Figure 2 as an illustration, where there are two stable equilibria η B1 and η B3. If the initial market state η 0 < η B1, then the market share will gradually inrease to η B1 as η > 0. Similarly, if η B1 < η 0 < η B3, then the market share will gradually derease to η B1 as η < 0. Only if η 0 > η B2, the highest stable equilibrium η B2 will emerge. Notie that given the prie, the database always prefers the highest stable equilibrium if multiple equilibria exist. Thus, some inentive mehanism is neessary to motivate more endusers subsribing to the advaned servie earlier, so as to onstrut a higher initial market share and ahieve a higher stable equilibrium. We will study this in Setion IV-B. 2 High Sensing Cost: The analysis for the high sensing ost ase is similar to that for the low sensing ost ase, but the detailed results are different due to the differene between the shapes of the dynamis η. Due to spae limit, we will leave the detailed analysis in our tehnial report [23]. IV. DATABASE OPTIMAL INFORMATION PRICING In this setion, we will study the optimal information priing strategy for the database operator to maximize its revenue, based on the market equilibrium analysis in the previous setion. In the following analysis, we first suppose that there is an effetive mehanism suh that the highest stable equilibrium will emerge if multiple equilibria exist, and derive the optimal priing strategy aordingly. Then we propose a refund mehanism for the database to ahieve this goal. A. Best Priing Deision By Lemma 1, we an easily find that (i if D, then η EQ = 0, (ii if C < D, then η EQ = η C1 whih is given by S A(η A(η B η = 0; and (iii if < C, then the highest stable equilibrium η EQ = η B3 or η A2, both given by 1 A(η B η = 0. We illustrate this stable market share and the database s revenue under different pries in Figure 3, where η C ( and η D ( are respetively given by 1 A(η B η = 0, S A(η A(η B η = 0. N η C N η D η = 0 0 C D Fig. 3. Database revenue under different pries 1 Low prie region: C. The database s revenue is: Π DB ( = η C ( N, whih is onave in the database s prie. Thus, by the KKT analysis, we have: = (1 η [A(η B ] (15 where η is the solution of A(η B + (1 η η 1 2η da(η dη = 0. For more details, please refer to [23]. 2 High prie region: C < D. The database s revenue is Π DB ( = η D ( N. The optimal prie in this ase is = A(η B S B [ (S A(η η ] (16 ( where η is solved by d A(η B dη S B [ η (S A(η η2 ] = 0. By omparing the optimal priing and the orresponding maximum revenue in different prie regions, we an obtain the database s optimal priing deision. Lemma 2 (Optimal Information Priing. The database s optimal priing deision is given by (15 or (16, depending on whih of η C ( and η D ( is the larger one. B. Refund Poliy Now we propose a mehanism to ensure the highest stable equilibrium. As mentioned previously, the emerging equilibrium depends on the initial market share η 0. Moreover, the larger the initial market share, the higher possibility the emerging of the highest stable equilibrium. Therefore, the main purpose of the mehanism is to motivate more end-users subsribing to the servie in the early stage, so as to onstrut a high enough initial market share. First, the end-user subsription dynamis in Setion III shows that if the initial market share η 0 = 1, then the market will always onverge to the highest stable equilibrium. This implies that we only need to find a mehanism suh that the initial market share is η 0 = 1. A natural approah is to provide the advaned servie for free for a ertain time (as Spetrum Bridge did to ahieve a larger initial market share. Although this approah an inrease the probability of the highest equilibrium, it will inur onsiderable revenue loss on the database, and moreover, it still annot guarantee the highest equilibrium (see the example in [23]. We propose a refund poliy. The basi idea is as follows. The database first announes a high enough hypotheti market share (e.g., η 0 = 1 to end-users, and then end-users deide whether to subsribe to the servie. Notie that end-users may not believe the market share announed by the database, sine the database may announe an inflated market share (to enlarge its revenue potentially. To avoid this, the database adopts the following refund poliy: Refund the subsription fee to an end-user who is not satisfied with the information obtained. Meanwhile, to avoid end-users frequently ask for refund (even when it is satisfied with the information, the database will adopt the following stopping-servie poliy: stop to serve an end-user who ask for refund for a ertain long time. Obviously, by this refund poliy, end-users will subsribe to the advaned servie without hesitation, as they are freely to ask for refund.
8 Market Share η% 100 80 60 40 20 α = 0.1 α = 0.15 α = 0.2 α = 0.25 Tipping (prie point when α=0.25 Database Expeted Revenue 20 15 10 5 α = 0.1 α = 0.2 α = 0.3 α = 0.4 Fig. 4. 0 0.01 0.02 0.03 0.04 0.05 Prie Market equilibrium under different information prie. Moreover, by adopting the stopping-servie poliy for a long enough time, end-users who are satisfied with the servie will not ask for refund, sine this will disontinue in a long time and inur a large loss. 12 This implies that the database does not loss any revenue, in ontrast to the previous free-serving poliy. V. SIMULATION RESULT In this setion, we use numerial results to evaluate the performane of the proposed information priing sheme. The following settings are used in our simulations: N = 80, K = 20, and U, V, and W follow the trunated normal distributions. The transmission data rate is defined by the Shannon apaity: r(z = log(1 + P Z. The end-user utility is simply defined as the expeted data rate: g(r = R. Market Equilibrium. Figure 4 illustrate the market equilibrium (i.e., the stable perentage of end-users subsribing to the advaned servie vs the information prie. We an see that the market share dereases with the prie. This means that less end-users are willing to purhase information under a higher prie. We an also see that the market share inreases S B with the end-user s sensing ost (reall that α =. This means that more end-users are willing to purhase information if the sensing is expensive. More interestingly, we an see that there exists the tipping prie point, at whih a slight hange will lead to a dramati derease on the market equilibrium. Database Revenue. Figure 5 illustrates the database s revenue under different liensee interferenes U (with a mean hanging from 10mw to 100mw. The mean value of V and W are fixed at 40mw and 10mw, respetively. We an see that the database s revenue inreases with the mean of the liensee interferene. This is beause the liensee interferene is known by the database, and a larger liensee interferene makes the database s information more valuable for the end-users. We an further see from eah bar group that the database s revenue inreases with the end-user s sensing ost. VI. CONCLUSION In this paper, we study a novel information market for the spetrum database in TV white spae networks, whih allows the database to sell information to end-users for revenue. We show that the information market has the property of positive network externality, and study the subsription dynamis and market equilibrium systematially. Based on this equilibrium 12 This is beause the interferene is temporal-independene (i.e., randomly hanging over different subsription periods, and thus end-users need to subsribe to the advaned servie periodially in order to get the updated information. Note that the interferene keeps unhanged within eah subsription period, otherwise the interferene information is meaningless. 0 10 20 30 40 50 60 70 80 90 100 Mean of Liensee Interferene (mw Database s revenue under different liensee interferenes. Fig. 5. analysis, we propose a refund poliy to guarantee the desirable market equilibrium, and further derive the database s optimal priing strategy. Our theoretial analysis and numerial result show that the information market an bring signifiant revenue for the database. There are some possible diretions to extend the results in this paper. A natural extension is to onsider the oligopoly market with multiple database, where different databases sell their respetive information to end-users. 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