Competition with Shared Spectrum

Transcription

1 ometition with Shared Sectrum hang Liu S eartment Northwestern University, vanston, IL 628 mail: changliu22@u.northwestern.edu Randall. erry S eartment Northwestern University, vanston, IL 628 mail: rberry@eecs.northwestern.edu bstract Sharing sectrum has emerged as a romising way to meet the exloding demand for wireless data services. In this aer, we consider a scenario in which sectrum is shared between a rimary and multile secondary service roviders, all of which are cometing for a common ool of customers. We adot a similar model to one used in earlier work to study cometition with unlicensed sectrum, in which users select service roviders based on the sum of a congestion cost and the rice announce by the service rovider. ometition with unlicensed sectrum was shown to otentially decrease social welfare. In contrast, with shared sectrum, we show here that social welfare is always non-decreasing, although the welfare of the rimary rovider can decrease. Various models of user demand and congestion costs are considered. I. INTROUTION One of the biggest current challenges in wireless networks is to meet the skyrocketing demand for wireless data services. ue in art to the high cost of clearing sectrum bands, sharing has been ut forth as a romising aroach for making more sectrum available to meet this demand. Indeed, in the U.S., romoting greater sharing is a central recommendation of the 22 reort from the President s ouncil of dvisors on Science and Technology (PST) [] and has been the subject of several residential memoranda [2], [3]. Sharing has already been adoted in the TV white saces [4] and is current being considered for federal sectrum such as the 3.5 GHz band [5]. Sharing has also been suggested for other commercial bands including those used to offer cellular services, e.g. [6], [7]. Sharing of such commercial bands is the focus of this aer. There has much research on sectrum sharing in recent years, in articular on the rimary-secondary aroach to sharing in which a rimary licensed holder has riority access to the sectrum while other secondary devices are allowed to share the sectrum rovided that they do not degrade the rimary s service. This includes work on using sectrum sensing (e.g. [8]) or market-based aroaches (e.g. [9]) to ensure the rimary has accetable erformance. s in [7], we consider such rimary-secondary sharing for a model where the rimary is a commercial cellular rovider. Other service roviders may offer service in the same band as secondary users, where all roviders (rimary and secondary) comete for a common ool of customers. s in the TV white-saces, we assume that secondary access is oen, meaning that any firm can offer This research is suorted in art by NSF grants NS-47786, SS and SS such secondary service. Such a olicy lowers barriers to enter the market, since secondary roviders do not need to acquire any sectrum license, and thus has the otential of increasing cometition and imroving overall welfare. However, since secondary access is not limited, there is also a risk that the sectrum becomes overly congested, leading to a tragedy of the commons. We resent an analytical model to study such trade-offs. Our aroach is based on the framework in [], [2], which considered similar questions for a model of cometition among service roviders with both licensed and unlicensed sectrum. This was in turn used models for rice cometition with congestible resources develoed in the oerations and economics literature (e.g., [3], [4]). In this framework, service roviders comete for customers by announcing service rices. ustomers in turn select roviders based on a delivered rice, which consists of the announced service rice lus a congestion cost, where the congestion cost reflects the quality of service obtained from a rovider. In [], licensed sectrum was not shared while unlicensed sectrum was shared equally among all roviders, i.e., no rovider was a rimary user. main result in [] was to show that adding unlicensed sectrum to an existing allocation of licensed sectrum among incumbent service roviders may lead to a decrease in overall social welfare. The reason for this was that when faced with cometition from the unlicensed band, licensed roviders have an incentive to increase their rices to drive traffic to the unlicensed band and increase its congestion, thus decreasing overall welfare. Here, instead of a searate unlicensed band, we consider secondary roviders that can oerate in the same band as the rimary using some form of sharing technology. We abstract this by modeling the congestion cost costs in the band differently for rimary and secondary firms. Namely, a rimary firm s congestion only deends on the number of customers it serves, while a secondary firm s congestion deends on the total customer mass served in the band by all firms. For such a model we characterize the equilibria of the resulting rice setting game among the service roviders and study the imact of sharing on the overall social welfare, the consumer welfare, and the service rovider s rofits. Interestingly, unlike the model in [], here, sharing never decreases overall welfare, This can be contrasted with licensed secondary access aroaches, in which only holders of secondary licenses can access the sectrum, e.g., [].

2 but it can decrease the rofits of the rimary service rovider. In addition to rimary-secondary sharing, this tye of model is also alicable in other settings. For examle, it may aly to sharing done as in the lower two tiers of the three-tier hierarchy roosed in the PST reort [], where at the highest tier riority is given to federal systems; the second tier corresonds to licensed secondary use, and the third tier is oen access. lso the form of sharing between rimary and secondary users could involve sharing raw sectrum or could also involve some form of infrastructure sharing as in [5], though in such cases the rimary could likely exert more control over secondary users than we consider here. The rest of the aer is organized as follows. Our models for rice cometition are described in Section II. We then resent an analysis of the resulting welfare with and without sharing for various models of demand and congestion costs in Sections III - V. Finally, we conclude in Section VI. II. OMPTITION MOL s in [], we consider a wireless service market in which a set of service roviders (SPs) comete for a common ool of customers by setting rices for their services. We focus on a model in which there is one incumbent SP. Without sharing, this firm acts as a monoolist. With sharing, the SP will be the rimary sectrum user and there will also be a set of N > secondary SPs, denoted by F, F 2,..., F N, who share the rimary s sectrum subject to not degrading the erformance seen by its customers. ach SP announces a rice for service, which we denote by and S i, for the rimary SP and the ith secondary SP, resectively. The SPs then serve all customers that accet their rice. Thus if x customers accet rice, the SP s rofit is simly x. The service that SP s offer is also characterized by a congestion cost, g(x), which models various congestion effects such as increased interference or queueing delays that arise as a SP serves more customers in a given area. In the case of the rimary SP, the congestion cost seen by its customers is given by g(x ), where x denotes the mass of customers it is serving. ustomers of each secondary SP F, F 2..., F N will encounter a congestion cost of g S (X), where X = x + N k= is the overall mass of customers being served in the band, with x S k being the number of customers served by secondary SP F k. This models the fact that the rimary SP does not see any degradation from the secondaries, while the secondary firms do exerience degradation due to the rimary as well as each other. In general, we assume that g(x) is an increasing, convex function as shown in Figure, though for much of our analysis, for simlicity, we focus on the case where this is a linear function, i.e., g(x) = x, where reresents the bandwidth or caacity of the sectrum band. For the secondary users we consider two different tyes of congestion costs: a model of erfect sharing, in which g S (x) = g(x) and a x S k model of degraded sharing, where g S (x) = g(x/α) where < α < is the degradation factor. Note as α aroaches, the degraded model aroaches the model with erfect sharing. For α <, secondary firms see greater congestion for the same customer mass. This can model a case where the secondary users exerience additional congestion due for examle to time sent sensing the medium to avoid interfering with the rimary or being required to use a lower ower level than the rimary. We assume a single mass of infinitesimal customers, and we normalize the total customer mass to be. ustomers select SPs based on the overall delivered rice for each SP s service, which consists of the sum of the SP s announced rice and the congestion cost of their service. Secifically, the delivered rice of the rimary SP is + g(x ), while the delivered rice of F i is S i + g S (X). ach customer in the market selects the service rovider who has the lowest delivered rice. If more than one service rovider has the same delivered rice, a customer chooses one of the low rice SPs uniformly at random. ustomer demand is modeled by an inverse demand function P (q), which will be a non-increasing function that indicates the delivered rice at which q customers are willing to ay for service. s shown in Fig., this will in general be a concave function. However, we again focus on two secial cases. First, in Section III, we assume that customers are homogeneous in their demand, i.e., all customers are willing to ay u to the same same delivered rice, in which case P (q) has a box shae. Then in Section IV, we consider a simle case of heterogeneous demand, where P (q) = q for q between and. The case of a general concave decreasing demand is briefly considered in V. We view the cometition among the rimary firm, the secondary firms, as well as the customers as a ricing game where firms set rices, (, S ), simultaneously. ustomers then choose one of the firms offering the lowest delivered rice. Thus, given a set of rices, (, S ), the customers receiving service must satisfy the following Wardro equilibrium conditions [6]: + g(x ) P (X), (with equality if x > ) S i + g S (X) = P (X), for i N with x i >, S i + g S (X) P (X), i N. These conditions secify that the delivered rice of all roviders serving customers are equal and no greater than P (X). We define a (ure strategy) Nash equilibrium of the overall ricing game to be a set of rices (, S ) and demands (x, x S ), which satisfy these Wardro equilibrium conditions and also have the roerty that no SP can increase its rofit by unilaterally changing its rice. Given such an equilibrium, the firm rofit, f c is defined by the sum of the rofits made by all SPs. The welfare of the xth consumer served is the difference between that consumers value for the service given by P (x) and the delivered rice it ays for service; consumers that are not served receive zero

3 P(x*) g(x) rice g(x) charging a rice higher than some other SP would serve no customers and thus make no rofit. Further, if this common rice is greater than zero, then each secondary SP has an incentive to decrease its rice as this will enable it to serve the total mass of customers being served by all secondary SPs at the common non-zero rice. Hence, the only ossible equilibrium is for all secondary SPs to set a rice of S i =. It follows from this lemma and the Wardro equilibrium conditions that with secondary firms in the market, the number of customers served in the entire market must be given by the value x such that g(x) g S (x ) = P (x ). () x X x* Fig.. Illustration of ricing game with rimary and secondary firms with shared sectrum welfare. The total consumer welfare, S c, is the integral of this over all consumers. The social welfare, S, of the entire economy is the sum of the firm rofit and the consumer welfare, i.e., S = S c + f c. Next we give some reliminary discussions about such a game both without and with sectrum sharing.. Without Sectrum Sharing Without sectrum sharing, the rimary firm is a monoolist in the market. s there is no cometition with other firms, it sets the rice to maximize its own rofit, i.e. it solves: max subject to x + g(x) = P (x), x. Under our assumtions, this will be a convex roblem with a unique solution.. With Sectrum Sharing With sectrum sharing, the rimary firm now faces cometition from the secondary firms, these secondary firms all see greater congestion and so must offer lower rices. Indeed, rovided there are at least 2 secondary firms, the following lemma shows that this rice must be zero. 2 Lemma 2.: In cometition with shared sectrum and at least 2 secondary SPs, all secondary SPs will charge zero rice to customers, i.e., S i = for all i. similar results was derived in [] for SPs offering service using unlicensed sectrum and this result can be derived similarly. First note that all secondary SPs must charge the same rice in any equilibrium since all of their customers exerience the same congestion, g S (X). If not, then any SP 2 Here we are not modeling any fixed costs for offering service. If such costs were included, then the rice would be equal to the fixed cost of service. This is the intersection of g S (x) and P (x) as shown in Figure. The resulting delivered rice in the market is then fixed at P (x ). III. WLFR NLYSIS WITH LINR ONGSTION N HOMOGNOUS MN In this section we analyze models with linear congestion costs, g(x) = x/ and homogeneous customer demands given by {, x P (x) =, otherwise. Here, we have set the rice that customers are willing to ay to be ; similar results hold for other values. For this setting, we analyze the social welfare, S, consumer welfare, S c, and firm rofit, f c both with and without sharing. The main result from our analysis is summarized in the following theorem: Theorem 3.: When >, with erfect sectrum sharing, social welfare and consumer welfare are always greater than without sharing while firm rofit shrinks. When <, social welfare, consumer welfare and firm rofit are the same with and without sharing. The roof of this theorem follows from the analysis in the following two sections. Subsequently we will show an analogous result for the case of degraded sharing.. Without Sectrum Sharing Without sharing sectrum, the rimary SP s rofit maximization roblem in this case is: max x subject to + x/ =, x. To solve this otimization, we first obtain from the equality constraint that = P (x) x/. Using this, the objective function can be re-written as P (x)x x 2 /, which we can then otimize subject to x >, the results of this are summarized in the following lemma. Lemma 3.2: For a model with with linear congestion, homogeneous demand and no-sharing, the equilibrium outcome is as follows:

4 rice rice g(x)=x/ g(x)=x/ g(x) g(x) x X x X= Fig. 2. Sharing and non-sharing case when with linear congestion and homogeneous inverse-demand Fig. 3. Non-sharing case when > with linear congestion and homogeneous inverse-demand i) When 2, the rimary serves a mass of x = /2 customers at a rice of = /2, resulting in S = f c = /4 and S c =. i) When > 2, the rimary serves the entire customer mass at a rice of = /, resulting in S = f c = / and S c =. s stated in this lemma, the outcome in this scenario can be naturally divided into two cases deending on the sectrum bandwidth. When is small ( < 2), the monooly firm will not serve all of the customers because the congestion cost is too high. For <, as shown in Figure 2, it could not serve these customers even if it announced a rice of zero; for 2, as in Figure 3, it could serve all the customers with a small enough rice, but this would not maximize its rofit. The otimal rice in this regime is /2 and as increases, the SP gains more rofit by the increase in number of customers. However, when > 2 is large enough as in Figure 3, the monooly SP will serve the entire market and so will increase its rofits as increases by increasing its rice. In either case, the delivered rice will be equal to, the maximum the customers are willing to ay and so consumer welfare will be zero and social welfare will be equal to the rofit of the SP. n examle of how these quantities vary with is shown by the dashed curves in Figure 5.. Perfect Sectrum Sharing Next, we turn to the case of erfect sharing, i.e., g S (x) = g(x). In this case, as noted after Lemma 2., the delivered rice is uniquely determined by the congestion and inverse demand functions. This gives the rimary firm less freedom to choose its rice. For examle in Figure 4, these curves intersect at oint, which means the rimary firm can only charge a rice u to the value. To otimize its rofit, the rimary tries to maximize the area of the rectangle in this figure, where the height of this rectangle is the rice and the width is the number of customers it serves. rice x X= g(x)=x/ Fig. 4. Perfect sharing case when > with linear congestion and homogeneous inverse-demand When, the rimary s otimization roblem becomes max x subject to + x/ =, x. In this case, shairng does not further constrain the delivered rice and so the otimization is the same as in the monooly case discussed above. When >, the delivered rice is constrained and so the rimary s roblem becomes max subject to x + x/ = /, x. The following result summarizes the solution to these otimization roblems:

5 Social Welfare Firm Profit onsumer Welfare.5 With Sharing W/O Sharing Fig. 5. omarison of erfect sharing and non-sharing cases on welfare with linear congestion and homogeneous inverse-demand Lemma 3.3: With erfect sharing, linear congestion costs and homogeneous demand, the equilibrium outcomes are characterized as follows: i) When, the rimary serves a mass of x = /2 customers at a rice of = /2, resulting in S = f c = /4 and S c = ; ii) When >, the rimary serves a mass of x = /2 at a rice of = /(2), resulting in f c = /(4), S c = and S = 3 4. s seen from Figure 5, social welfare with erfect sharing always increases when increases. For <, social welfare increases linearly with and is the same as in the non-sharing case. When >, social welfare with sharing is always greater than in the monooly case. Thus when the rimary firm s bandwidth is large enough relative to the demand (bigger than in this case), sharing sectrum will always benefit in terms of overall social welfare. lso, when >, consumer welfare with sharing becomes ositive. This is a dramatic change from the non-sharing case where consumer welfare stays zero no matter how large is. With sharing, cometition limits the rice a rimary can charge thus reventing the SPs from extracting all of the welfare. With sharing, as increases, not only does congestion become less, but the ca on the delivered rice also becomes smaller, both leading to increases in consumer welfare. s we can see, the cost of this increased social welfare and consumer welfare comes from the rofit of rimary firm. With the growth of, the rofit of firm first increases but once exceeds, it decreases due to the ceiling on the delivered rice in the market.. egraded Sectrum Sharing Now we turn to the case where sharing is degraded, i.e., g S (x) = x α, where the degradation factor α satisfies < α <. Under this assumtion, for a given α, with sharing, the delivered rice is now fixed by the intersection of g S (x) and P (x), unless α is too small so that the congestion cost seen by any secondary user exceeds the user s willingness to ay, in which case the secondary SPs will have no customers. The next lemma summarizes the deendence of the equilibrium outcome on both α and. Lemma 3.4: With degraded sectrum sharing, linear congestion and homogeneous demand, the equilibrium outcome s deendence on α is as follows: i) When < α min{/2, /}, no sharing occurs; ii) When min{/2, /} < α min{/, }, S = f c = /4 and S c =. iii) When / < α <, S = + 4α 2 α, f c = 4α 2, S c = α. Proof: We will rove this by considering each of the three cases given in the theorem seerately. ase i): < α min{/2, /}. Let x M be the otimal customer mass served by the rimary without sharing. From Lemma 3.2 we have that x M = min{/2, }. For α in this range it follows that g S (x M ), which means that if the rimary continues serving the same number of customers as it did without sharing, then the congestion of any secondary SP will exceed any user s willingness to ay. Thus, with degraded sharing, the rimary can continue serving the same mass of customers at the same rice, and no sharing will occur. ase ii): min{/2, /} < α min{/, }. Note that this range is non-emty only when < 2, in which case it becomes /2 α < /. For < 2, from Lemma 3.2, the rimary will not serve the entire market without sharing, but only serves a mass of x M = /2 customers. For α in the given range, it also follows that g S (x M ) < and g S () = α >, which means that if the rimary continues serving x M = /2 customers, the secondary bands will attract new customers, but will become so congested that it will not constrain the rimaries delivered rice and all new secondary users will receive zero welfare (see Figures 6 and 7). So although more users aear in the network, the rimary firm can still act like a monoolist and the welfare and firm rofit are the same as in the no sharing case. Note that in this case even if > as in Figure 7, the entire market is not served; this differs from the model with erfect sharing in which when > all users are served. ase iii): / < α <. In this case, g S () < and so sharing constrains the total delivered rice to be no more than

6 rice rice gs(x)=x/α g(x)=x/ gs(x)=x/α g(x)=x/ x X=x* x x X=x*= Fig. 6. n examle of case (ii) for degraded sharing with linear congestion and homogeneous inverse-demand with <. The rimary SP serves x customers and the total mass of customers served in the whole market is X = x which is less than. rice gs(x)=x/α x X=x* g(x)=x/ Fig. 7. n examle of case (ii for degraded sharing the linear congestion and homogeneous inverse demand where >. ven in this case the total customer mass served is less than. g S () (see Figure 8). This in turn constrains the rice the rimary can charge and so the rimary is now faced with the following otimization roblem: max subject to x + x/ = /α, x. Referring to Figure 9, this corresonds to maximizing the area of the square that lies below the line G and above g(x). The solution to this is for the rimary SP to charge a rice of = /2α as to maximize its rofit and serve x = /2α of the overall users. The entire market is served and all consumers receive a welfare of α. The rimary Fig. 8. n examle of case (iii) for degraded sharing with linear congestion and homogeneous inverse-demand. firms rofit is given by x = 4α 2. dding these gives the indicated total welfare. omaring Lemma 3.4 to Lemma 3.2, we have the following analog to Theorem 3.. Theorem 3.5: When > /α, with degraded sectrum sharing, social welfare and consumer welfare are always greater than without sharing while firm rofit shrinks. When < < /α, social welfare, consumer welfare, and firm rofit are the same with and without sharing. This shows that the benefits of sharing deend on the roduct of the degradation factor α and the available bandwidth, the large is the smaller the degradation factor that can be allowed and still see an increase in welfare. lot of how social welfare, firm rofit, and consumer welfare deend on α for various fixed values of is shown in Figure. Note change in the degradation factor or has a relative small effect on firm rofit, while changing has a larger effect on consumer welfare. When is small, α has no effect on consumer welfare which is zero. For in (, 2], α needs to be large enough to increase consumer welfare. When is large enough, it dominates α as well as the delivered rice, in which case, social welfare and consumer welfare will always grow with α while firm rofits shrinks. IV. WLFR NLYSIS WITH LINR ONGSTION N HTROGNOUS MN Next, we consider a model in which there is heterogeneous demand, i.e., different customers are willing to ay different amounts for service. We model this by making using a linear inverse demand function given by P (x) = x as shown in Figure. 3 Thus the only way the entire market can be served 3 The analysis can be easily generalize to any linear demand of the form P (x) = a bx, where a and b are ositive constants.

7 rice rice H gs(x)=x/α F G F g(x)=x/ g(x)=x/ J G I x X=x*= Fig. 9. egraded sharing for > 2, /2 < α with linear congestion and homogeneous inverse-demand. Fig.. x X Non-sharing with linear congestion and linear inverse-demand Social Welfare Firm Profit onsumer Welfare α.5 =.5, <<= =.25, <<=2 =5, > α α Fig.. The effect of α on social welfare(to), firm rofit(middle) and consumer welfare(bottom). is if the delivered rice goes was zero, and as the delivered rice increases to one, there are fewer and fewer customers willing to ay for service. For simlicity, we still assume linear congestion g(x) = x/. s in the revious section, we again analyze the social welfare, consumer welfare and firm rofit both with and without sharing. The main result of this analysis is summarized in the following theorem: Theorem 4.: With erfect sharing, linear congestion and linear inverse demand, social welfare and consumer welfare are always greater than without sharing while firm rofits shrink. omared to the case of homogeneous demand, now both social and consumer welfare always increase (regardless of ). s in the case of homogeneous demand, we will rove this in the following two sections by characterizing the equilibrium both with and without sharing. We will then generalize this for imerfect sharing.. Without Sectrum Sharing gain, without sharing the rimary firm is a monoolist and simly sets the rice to maximize its rofit. With linear inverse demand, the corresonding otimization is given by: max x subject to + x/ = x, x. Grahically, as shown in Figure, this corresonds to maximizing the area off the rectangle which is contained within the triangle F G formed by the inverse demand and the congestion cost. y again, solving for in terms of x, this can be written as an otimization over the single variable x, whose solution is given by: x =, and = /2. (2) 2( + ) The firms rofit f c is then given by the roduct x and referring to Figure, the consumer welfare,s c is given by the area of the triangle F. Summing these gives the total welfare. The result of these calculations is summarized in the following lemma. Lemma 4.2: For a model with linear congestion, linear heterogeneous demand, and no sectrum sharing, the equilibrium outcome is for the rimary to set the rice and number of

8 customers served as in (2) resulting in: S = + 3/22 4( + ) 2, f c = 4( + ), 2 S c = 8( + ) 2. Note that the number of customers served in (2) is a strictly increasing function of, and as becomes arbitrarily large it converges to /2. In other words, for any finite, the rimary will always serve less than half of the market. lso note here that the rice charged is always /2, regardless of the value of, while with homogeneous demand, the rice increases with for large enough.. Perfect Sectrum Sharing With erfect sectrum sharing, as discussed earlier, the number of customers served in any equilibrium is given by the value x satisfying (), which in this case corresonds to x = +. omaring to (2), it can be seen that x is exactly double the number of customers served by the rimary without sharing. This corresonds to an uer bound on the delivered rice of P (x ) = +. Given this bound the otimization faced by the rimary firm is now: max x subject to + x/ = /( + ), x. The following lemma summarizes the results of this otimization. Lemma 4.3: With erfect sharing, linear congestion costs, and linear inverse demand, the equilibrium outcome is for the rimary to set the rice and number of customers served as resulting in: x = 2( + ), and = 2( + ) 3 2 S = 4( + ) 2, 2 f c = 4( + ) 2, 2 S c = 2( + ) 2. omaring (3) with (2) it can seen that the rimary serves the same number of customers both with and without sharing, but at a lower rice with sharing. This leads to a decrease in firm rofit and an increase in customer welfare. Namely, referring to Figure 2, firm rofit will decrease by the area (3) Fig. 2. H rice F I G g(x)=x/ x X Perfect sharing with linear congestion and linear inverse-demand I H and the welfare of the customers it serves will increase by the same amount. ustomer welfare will also increase due to the new customers being served by the secondary SPs, which is given by the the area of the triangle I G in Figure 2. omaring the areas of these regions with the customer welfare without sharing (given by the triangle F ), we have the following lemma: Lemma 4.4: With erfect sectrum sharing, linear congestion and linear demand, consumer welfare is four times that without sharing. It follows from the above discussion that overall welfare must increase with sharing as stated in Theorem 4... egraded Sectrum Sharing We next consider degraded sharing with linear inverse demands. gain, this means that g S (x) = x/α, for < α <. s in the case of homogeneous demand, this changes the limit on the delivered rice introduced by sharing (see Figure 3). The resulting equilibrium is summarized in the following lemma. Lemma 4.5: With degraded sectrum sharing, linear congestion, and linear inverse demand, the equilibrium outcome s deendence on α is as follows: i) When < α 2+, no sharing occurs and the rimary serves the same customers at the same rice as in the non-sharing case. ii) When 2+ α 2, again no customers are served by secondary roviders, but the rimary serves x customers, resulting in S = α( α) + 2 α2 2 (α + ) 2, α( α) f c = (α + ) 2, α 2 2 S c = 2(α + ) 2.

9 iii) When 2 < α, the rimary serves more customers than without sharing and the secondary SPs also serve some customers, resulting in S = + 2α2 2 4(α + ) 2, f c = 4(α + ) 2, S c = α 2 2 2(α + ) 2. Proof: The first case, where no sharing occurs, again corresonds to the case where g S (x ) > P (x ), where x is the number of customers served by the rimary without sharing as in (2). In the second and third cases, the rimary SP must also account for the new constraint on the delivered rice given by + x = α +. For α > /2, this constraint will be tight and the constraint given by the inverse demand (+ x = x) will not be tight, resulting in the rimary serving a customer mass of x = 2(α + ). (4) This corresonds to case (iii) in the lemma and using this the indicated quantities can then be calculated. In case (ii) both the new constraint on the delivered rice and the constraint due to the inverse demand are tight, in which case the rimary serves x customers at a rice of = α α +, from which again the indicated quantities can be calculated. Note that in case (iii) from (4), it can be seen that the rimary serves more customers with degraded sharing than without sharing and the number of customers served increases as sharing becomes more degraded. However once α becomes smaller than 2, the constraint given by the inverse demand becomes tight and the number of customers served then decreases with α. In both cases, since the delivered rice is smaller than without sharing, customer welfare must increase and firm rofit must decrease. If the rimary continued serving the same amount of customers as without sharing, x M, then in either case (ii) or (iii), as in the case of erfect sharing, the total welfare must increase. Since in these cases, the rimary always serves more than x M customers, it follows that the customer welfare will be the same as if it continued serving x M customers (i.e., it will always be the area of the triangle F H in Figure 3). Further, since the rimary is maximizing its rofit, it must be that its rofit in the equilibrium is greater than if it continued to serving x M customers and so again total welfare must increase. n examle of the social welfare, consumer welfare and firm rofit as a function of under different scenerios is shown in Figure 4. Fig. 3. Social Welfare Firm Profit onsumer Welfare.4.2 rice F H G gs(x)=x/α I g(x)=x/ x X x egraded sharing with linear congestion and linear inverse-demand With Sharing (α=) With Sharing (α=.6) With Sharing (α=.2) W/O Sharing Fig. 4. Imact of and α on welfares with linear congestion and linear inverse-demand V. GNRL ONGSTION N GNRL MN In this section we make a few comments about the more general scenario, where the congestion is any increasing, convex, differentiable function g(x) and the inverse demand function P (x) is decreasing, concave and differentiable. dditionally, we assume that g(x) decreases oint-wise as bandwidth increases. Under these assumtions, the ricing roblem faced by the rimary will always be a convex roblem with a unique solution. The following theorem shows that in such a general setting again the corresonding equilibrium consumer welfare and social welfare are non-decreasing as the bandwidth

10 is increases. 4 Theorem 5.: With erfect sharing, social welfare and consumer welfare are non-decreasing functions of bandwidth. Proof: onsider two choices of bandwidth <, with corresonding congestion functions g(x) and g (x) as shown in Figure 5. Let x and x be the overall customer mass served in the market with erfect sharing, where clearly x < x, i.e. greater bandwidth means more customers are served. The deliver rice in these two cases will be P (x ) and P (x ), with P (x ) > P (x ) since P (x) is decreasing (these corresond the oints and H in Figure 5). From this it is clear that the consumer welfare increases with the bandwidth, since more customers are being served at a lower delivered rice. To see the result for social welfare. Referring to Figure 2, note that welfare for and is given by the area of the regions K N and K H F G M, resectively. To comare these two areas, we searate each of them into two ieces using the same segment HM. It is easy to see that the area above segment HM is exactly consumer welfare for which is larger than the corresonding area with. Next we turn to the ortion of the regions under the segment HM. For this corresonds to the rectangle area H I, and for, it corresonds to the rectangle area H F G. s rectangle H F G is the otimal rofit for the rimary firm, it is the largest rectangle within the region H O M and so it must have a larger area than that of rectangle H I. Thus the welfare for must be greater than that with, comleting the roof. Note in this theorem we did not say anything about the rofit of the rimary firm. s we have seen in the revious sections it can decrease with increasing, but whether this occurs for a secific family of congestion costs aears to deend on the details of how the costs scale with. The next result, shows that in general, consumer welfare and social welfare increase with sharing while firm rofits decrease. Lemma 5.2: With shared sectrum, social welfare and consumer welfare are greater than that without sharing while firm rofit is less. Proof: For the change in social welfare and consumer welfare with bandwidth, the roof is similar to Thereom 5. and is illustrated in Figure 6, while the roof that the rimary firm rofit decreases follows from the same argument as given to show this for linear inverse demands at the end of the revious section. Sharing can results in either the rimary serving more customers or fewer customers, the next theorem characterizes when this occurs deending of the inverse demand P (x). Theorem 5.3: When the firm shares sectrum with secondary firms, the customer mass x comared with nonsharing case will increase if P (x ) + P (x )x < P (x ) decrease if P (x ) + P (x )x > P (x ) 4 We state for erfect sharing, a similar result holds for degraded sharing K H O rice x I J G F x x* N x* M g(x) g (x) Fig. 5. hange in congestion function with resect to bandwidth with shared sectrum in a general setting. The red lines indicate the case with less bandwidth, while the green lines show the case with more bandwidth. K L H O rice N G F x x x* M g(x) Fig. 6. omarison between with and without sharing in a general setting. Red lines indicate the monooly case without sharing, and green lines show the case with erfect sharing. Proof: The difference between with and without erfect sharing is the constraint on the delivered rice. That is, Without sharing + g(x ) = P (x ) With sharing + g(x ) = P (x + x i ) To maximize firm rofit, i. e., x, we can again solve for the rice in terms of the customers served in each case, and substitute this into the objective, giving a function of only x. ifferentiating this with resect to x, and setting it equal to zero we obtain P (x ) + P (x )x = g(x ) + g (x )x

11 without sharing and P (x ) = g(x ) + g (x )x. with sharing. We know that once congestion function and demand functions are fixed, P (x ) is a constant. The righthand side of two revious equations is identical are strictly increasing in x as g(x) is increasing and convex. Hence, comaring the left-hand side of the equations will secify which case has the larger value of x, comleting the roof. VI. ONLUSIONS We have studied a stylized model for sharing a licensed band of sectrum among a rimary and multile secondary service roviders, all of who seek to serve a common ool of customers. We have shown that for this model, consumer welfare and overall social welfare never decreases with sharing comared to without sharing, while the rofit of the rimary firm never increases with sharing and may decrease. Further we have shown that as the bandwidth of the shared band increases, overall welfare and consumer welfare increases, while the rimaries rofits may either increase of decrease. We note that these results are quite different from that observed in [], where adding a searate band of shared unlicensed sectrum to a market consisting of a single rimary license holder was shown to otentially decrease social welfare. Further, in [] the social welfare could decrease as the bandwidth of the unlicensed band was increased. In [] as in the model considered here, the addition of sharing to the market otentially laces a limit on the delivered rice that the rimary can charge. However, when this sharing is in a searate unlicensed band, the rimary can increase this limit by shifting more traffic to the unlicensed band and causing it to be more congested. Such an action is exactly the cause of the decrease in social welfare. Under the rimary-secondary sharing model considered in this aer, the rimary can not effect this limit on the delivered rice, since it just deends on the total traffic served by it and the secondary. This gives the rimary less flexibility and revents social welfare from decreasing, but also leads to a greater decrease in the rimary s rofits due to sharing. This suggests that if limited sectrum is available for sharing, there might be advantages to allocating the sectrum to an incumbent and sharing it under a rimarysecondary model as oosed to simly making it unlicensed, though we leave a detailed comarison of these regimes to future work. We also considered a model with degraded sharing in which secondary users incurred an additional overhead, leading them to exerience greater congestion. We showed that if this degradation was too high, then even if sharing was allowed, no secondary service roviders would enter the market. This suggests that incumbents who do not want to share may have an incentive to lace overly burdensome requirements on secondary users, and a good olicy should should seek to minimize these overheads. Here, we only modeled degradation due to sharing on the art of the secondary roviders, another otentially interesting future direction would be to consider sharing which also degrades the rimaries erformance, due for examle to imerfect sensing on the the art of secondary users. Here we simly comared a scenario with and without sharing but did not address the rimary rovider s incentives to share. Given that the rimary s rofits decrease, it would clearly not have an incentive to share unless it was required by olicy or received some comensation for doing this. Further, since the secondary users rofits are cometed away, the rimary could not hoe to receive any comensation from them. However, since overall welfare does increase with sharing, this could suggest a olicy in which the government collects a ortion of the added consumer welfare (e.g. via taxes) and comensates the rimary for allowing sharing. In the model considered here, secondary sharing was oen, which resulted in the secondary SPs not receiving any rofits. n alternative would be for secondary access to also be licensed. If a single secondary SP was licensed, then they would be able to sustain a ositive rice and rofit. In this case, the ricing game between the rimary and secondary SP becomes more comlicated, another toic we leave for future work. lso, here we did not consider the investment decisions made by the rimary or secondary firms. Such issues could be introduced here using a similar model as in [2], where they were considered for the case of unlicensed sectrum. Finally, secondary SPs could serve different market segments than a rimary firm and thus not be direct cometitors leading to different models of cometition that could be studied. RFRNS [] PST, Realizing the Full Potential of Government-Held Sectum to Sur conomic Growth. President s ouncil of dvisors on Science and Technology, July 22. 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