Duopoly Competition in Dynamic Spectrum Leasing and ... - IEEE Xplore

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Duopoly Competition in Dynamic Spectrum Leasing and Pricing Lingjie Duan, Member, IEEE, Jianwei Huang, Senior Member, IEEE, and Biying Shou Abstract—This paper presents a comprehensive analytical study of two competitive secondary operators’ investment (i.e., spectrum leasing) and pricing strategies, taking into account operators’ heterogeneity in leasing costs and users’ heterogeneity in transmission power and channel conditions. We model the interactions between operators and users as a three-stage dynamic game, where operators simultaneously make spectrum leasing decisions in Stage I, and pricing decisions in Stage II, and then users make purchase decisions in Stage III. Using backward induction, we are able to completely characterize the dynamic game’s equilibria. We show that both operators’ investment and pricing equilibrium decisions process interesting threshold properties. For example, when the two operators’ leasing costs are close, both operators will lease positive spectrum. Otherwise, one operator will choose not to lease and the other operator becomes the monopolist. For pricing, a positive pure strategy equilibrium exists only when the total spectrum investment of both operators is less than a threshold. Moreover, two operators always choose the same equilibrium price despite their heterogeneity in leasing costs. Each user fairly achieves the same service quality in terms of signal-to-noise ratio (SNR) at the equilibrium, and the obtained predictable payoff is linear in its transmission power and channel gain. We also compare the duopoly equilibrium with the coordinated case where two operators cooperate to maximize their total profit. We show that the maximum loss of total profit due to operators’ competition is no larger than 25 percent. The users, however, always benefit from operators’ competition in terms of their payoffs. We show that most of these insights are robust in the general SNR regime. Index Terms—Cognitive radio, spectrum trading, dynamic spectrum leasing, spectrum pricing, multistage dynamic game, subgame perfect equilibrium

Ç 1

INTRODUCTION

W

IRELESS spectrum is often considered as a scarce resource, and thus has been tightly controlled by the governments through static license-based allocations. However, several recent field measurements show that many spectrum bands are often underutilized even in densely populated urban areas [2]. To achieve more efficient spectrum utilization, secondary users may be allowed to share the spectrum with the licensed primary users. Various dynamic spectrum access mechanisms have been proposed along this direction. One of the proposed mechanisms is dynamic spectrum leasing,1 where a spectrum owner dynamically transfers and trades the usage right of temporarily unused part of its licensed spectrum to secondary network operators or users in exchange for monetary compensation [3], [4], [5], [6], [7]. In this paper, we

1. “Dynamic” here means that spectrum leasing can be flexibly done at a short time scale between any two involved parties.

. L. Duan is with the Engineering Systems and Design Pillar, Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682. E-mail: [email protected]. . J. Huang is with the Network Communications and Economics Laboratory, Department of Information Engineering, The Chinese University of Hong Kong, Ho-Sin Hang Engineering Building, Shatin, N.T., Hong Kong 999077. E-mail: [email protected]. . B. Shou is with the Department of Management Sciences, City University of Hong Kong, Room 7519, Academia Building Tat Chee Avenue, Kowloon, Hong Kong 99907. E-mail: [email protected]. Manuscript received 2 Nov. 2010; revised 29 Aug. 2011; accepted 16 Sept. 2011; published online 10 Oct. 2011. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TMC-2010-11-0500. Digital Object Identifier no. 10.1109/TMC.2011.213. 1536-1233/12/$31.00 ß 2012 IEEE

study the competition of two secondary operators under the dynamic spectrum leasing mechanism. Our study is motivated by the successful operations of mobile virtual network operators (MVNOs) in many countries today.2 An MVNO does not own wireless spectrum or even the physical infrastructure. It provides services to end-users by long-term spectrum leasing agreements with a spectrum owner. MVNOs are similar to the “switchless resellers” of the traditional landline telephone market. Switchless resellers buy minutes wholesale from the large long distance companies and resell them to their customers. It is shown by Lev-Ram [9] and Dewenter and Haucap [10] that it can be more efficient for the spectrum owner to hire an MVNO as intermediary to retail its spectrum resource, as MVNO has a better understanding of local user population and local users’ demand. However, an MVNO is often stuck in a long-term leasing contract with a spectrum owner and cannot make flexible spectrum leasing and pricing decisions to match the dynamic demands of the users. The secondary operators considered in this paper do not own wireless spectrum either. Compared with a traditional MVNO, the secondary operators can dynamically adjust their spectrum leasing and pricing decisions to match the users’ demands that change with users’ channel conditions. This paper studies the competition between secondary operators in spectrum acquisition and pricing to serve a common pool of secondary users. To abstract the interactions among operators, we focus on two operator case 2. Since the late 1990s, there have been over 400 mobile virtual network operators owned by over 360 companies worldwide as of February 2009 [8]. Published by the IEEE CS, CASS, ComSoc, IES, & SPS

DUAN ET AL.: DUOPOLY COMPETITION IN DYNAMIC SPECTRUM LEASING AND PRICING

(i.e., duopoly) and will study multiple operator case (i.e., oligopoly) in our future work. The secondary operators will dynamically lease spectrum from spectrum owners, and then compete to sell the resource to the secondary users to maximize their individual profits. We would like to understand how the operators make the equilibrium investment (leasing) and pricing (selling) decisions, considering operators’ heterogeneity in leasing costs and wireless users’ heterogeneity in transmission power and channel conditions. We adopt a three-stage dynamic game model to study the (secondary) operators’ investment and pricing decisions as well as the interactions between the operators and the (secondary) users.3 From here on, we will simply use “operator” to denote “secondary operator” and “users” to denote “secondary users.” In Stage I, the two operators simultaneously lease spectrum (bandwidth) from the spectrum owners with different leasing costs. In Stage II, the two operators simultaneously announce their spectrum retail prices to the users. In Stage III, each user determines how much resource to purchase from which operator. Each operator wants to maximize its profit, which is the difference between the revenue collected from its users and the cost paid to the spectrum owner. Key results and contributions of this paper include .

.

.

An appropriate wireless spectrum sharing model. We assume that heterogeneous users share the spectrum using orthogonal frequency division multiplexing (OFDM) technology. Then, a user’s achievable rate and thus its spectrum demand depend on its allocated bandwidth, maximum transmission power, and channel condition. This model is more suitable to our problem than the generic economic models used in related literature [6], [11], [12], [13]. It can also provide more engineering insights on how different wireless network parameters in the spectrum sharing model (e.g., users’ various wireless characteristics) impact the operators’ leasing and pricing decisions. Symmetric pricing structure. We show the two operators always choose the same equilibrium price, even when they have different leasing costs and make different investment decisions. Moreover, this price is independent of users’ transmission powers and channel conditions.4 Threshold structures of investment and pricing equilibrium. We show that both operators’ investment and pricing equilibrium decisions process interesting threshold properties. For example, when the two operators’ leasing costs are close, both operators will lease positive spectrum. Otherwise, one operator will choose not to lease and the other operator becomes the monopolist. For pricing, a positive pure strategy equilibrium exists only when the total spectrum investment of both operators is less than a threshold.

3. Here, the word “dynamic” in “dynamic game” is different from that in footnote 1. It refers to the interactions between operators and users over time. 4. Such independency is good for the development of spectrum market, since a user does not need to worry about how variations of user population and wireless characteristics change its performance in spectrum trading.

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Fair service quality achieved by users. We show that each user achieves the same signal-to-noise ratio (SNR) that is independent of the users’ population and wireless characteristics. . Impact of competition. We show that the operators’ competition leads to a maximum 25 percent loss of their total profit compared with a coordinated case. The users, however, always benefit from the operators’ competition by achieving better payoffs. Next, we briefly discuss the related literature. In Section 2, we describe the network model and game formulation. In Section 3, we analyze the dynamic game through backward induction and calculate the duopoly leasing/pricing equilibrium. We discuss various insights obtained from the equilibrium analysis in Section 4. In Section 6, we show the impact of duopoly competition on the total operators’ profit and the users’ payoffs. We conclude in Section 7 together with some future research directions. .

1.1 Related Work Recently, researchers have started to study the economic aspect of dynamic spectrum access, such as the secondary operators’ strategies of spectrum acquisition from spectrum owners and service provision to the users. For example, several auction mechanisms have been proposed for the spectrum owner to allocate spectrum [14], [15], [16], [17]. Auction is a good choice when a spectrum owner does not have a good estimation of how much the spectrum is worth to the users, and thus relies on the bids of the users to determine the pricing and resource allocation. When a spectrum owner has the complete network information, however, he can simply price the spectrum accordingly and lease to the users. For operators’ service provision, most related results looked at the pricing interactions between network operators and the secondary users [6], [11], [12], [13], [17], [18], [19]. There are few papers that jointly studied resource investment and service pricing decisions for intermediary secondary operator(s) [6], [20], [21] as this paper. Such joint optimization is very important since different investment amounts at the beginning weigh heavily on later pricing and service accommodation capabilities of operators. Jia and Zhang [6] only used a generic economic model for total users’ spectrum demand without much wireless details. Niyato et al. [20] did not explicitly analyze the joint optimization over leasing and pricing resource and extensive simulations are mainly used instead. Also, Duan et al. [21] only considered a single operator case without discussing operator competition. In multiple competitive operator case, the operators’ strategy spaces are often coupled and is hard to analyze over multiple stages. The key difference of this work here is that we present a comprehensive analytical study that characterizes both the competitive operators’ equilibrium investment and pricing decisions, with heterogeneous leasing costs for the operators and an appropriate wireless spectrum sharing model for the users.

2

NETWORK AND GAME MODEL

We consider two operators (i; j 2 f1; 2g and i 6¼ j) and a set K ¼ f1; . . . ; Kg of users in an ad hoc network as shown in

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TABLE 1 Key Notations

Fig. 1. Network model for the secondary network operators.

Fig. 2. Three-stage dynamic game: the duopoly’s leasing and pricing, and the users’ resource allocation.

Fig. 1. The operators obtain wireless spectrum from different spectrum owners with different leasing costs, and compete to serve the same set K of users. Each user has a transmitter-receiver pair. We assume that users are equipped with software-defined radios and can transmit in a wide range of frequencies as instructed by the operators, but do not have the capability of spectrum sensing in cognitive radios.5 Such a network structure puts most of the implementation complexity for dynamic spectrum leasing and allocation on the operators, and thus is easier to implement than a “full” cognitive radio network especially for a large number of users. A user may switch among different operators’ services (e.g., WiMAX, 3G) depending on operators’ prices. It is important to study the competition among multiple operators as operators are normally not cooperative. The interactions between the two operators and the users can be modeled as a three-stage dynamic game, as shown in Fig. 2. Operators i and j first simultaneously determine their leasing bandwidths in Stage I, and then simultaneously announce the prices to the users in Stage II. Finally, each user chooses to purchase bandwidth from only one operator to maximize its payoff in Stage III. The key notations of the paper are listed in Table 1. Some are explained as follows: . .

Leasing decisions Bi and Bj : leasing bandwidths of operators i and j in Stage I, respectively. Costs Ci and Cj : the fixed positive leasing costs per unit bandwidth for operators i and j, respectively. These costs are determined by the negotiation between the operators and their own spectrum suppliers.

5. Spectrum sensing is the most important functionality of cognitive radios, which enables users to actively monitor the external radio environments to communicate efficiently without interfering primary users. The capability of spectrum sensing includes comprehensive monitoring of frequency spectrum, user behavior, and network state over time.

.

.

Pricing decisions pi and pj : prices per unit bandwidth charged by operators i and j to the users in Stage II, respectively. The User k’s demand wki or wkj : the bandwidth demand of a user k 2 K from operator i or j. A user can only purchase bandwidth from one operator.

2.1 Users’ and Operators’ Models OFDM has been proposed as a promising physical layer choice for dynamic spectrum sharing [22], [23]. We assume that the users share the spectrum using OFDM to avoid mutual interferences. The main analysis in this paper assumes that users are located close-by, and thus no two users will transmit over the same channel (also called subcarriers in the OFDM literatures [24], [25]). We will relax this assumption later on (Appendix E, which can be found on the Computer Society Digital Library at http:// doi.ieeecomputersociety.org/10.1109/TMC.2011.213) and show that our results can be extended to the case with spectrum spatial reuse. If a user k 2 K obtains bandwidth wki from operator i, then it achieves a data rate (in nats) of [26] P max hk rk ðwki Þ ¼ wki ln 1 þ k ; ð1Þ n0 wki where Pkmax is user k’s maximum transmission power, n0 is the noise power density, hk is the channel gain between user k’s transmitter and receiver. The channel gain hk is independent of the operator, as the operator only sells bandwidth and does not provide a physical infrastructure.6 Here, we assume that user k spreads its power Pkmax across the entire allocated bandwidth wki . To simplify later discussions, we let gk ¼ Pkmax hk =n0 ; 6. We also assume that the channel condition is independent of transmission frequencies, such as in the current 802.11d/e standard [27] where the channels are formed by interleaving over the tones. In other words, each user experiences a flat fading over the entire spectrum.


thus gk =wki is the user k’s SNR. The rate in (1) is calculated based on the Shannon capacity. To obtain closed-form solutions, we first focus on the high SNR regime where SNR 1. This will be the case where a user has limited choices of modulation and coding schemes, and thus cannot decode a transmission if the SNR is below some threshold. In the high SNR regime, the rate in (1) can be approximated as gk rk ðwki Þ ¼ wki ln : ð2Þ wki Although the analytical solutions in Section 3 are derived based on (2), we will show later in Section 5 that all major engineering insights remain unchanged in the general SNR regime. If a user k purchases bandwidth wki from operator i, it receives a payoff of gk pi wki ; uk ðpi ; wki Þ ¼ wki ln ð3Þ wki which is the difference between the data rate and the payment. The payment is proportional to price pi announced by operator i. This linear pricing scheme has been widely used in the literature [28], [29]. For an operator i, its profit is the difference between the revenue and the total cost, i.e., i ðBi ; Bj ; pi ; pj Þ ¼ pi Qi ðBi ; Bj ; pi ; pj Þ Bi Ci ;

ð4Þ

where Qi ðBi ; Bj ; pi ; pj Þ and Qj ðBi ; Bj ; pi ; pj Þ are realized demands of operators i and j. The concept of realized demand will be defined later in Definition 4.

3

BACKWARD INDUCTION OF THE THREE-STAGE GAME

A common approach of analyzing dynamic game is backward induction to find the subgame perfect equilibrium (SPE) [30]. Subgame perfect equilibrium (or simply, equilibrium) represents a Nash equilibrium of every subgame of the original game. In this paper, we start with Stage III and analyze the users’ behaviors given the operators’ investment and pricing decisions. Then, we look at Stage II and analyze how operators make the pricing decisions taking the users’ demands in Stage III into consideration. Finally, we look at the operators’ leasing decisions in Stage I knowing the results in Stages II and III. Throughout the paper, we will use “bandwidth,” “spectrum,” and “resource” interchangeably. In the following analysis, we only focus on pure strategy SPE and rule out mixed SPE in the multistage game.7 Such a methodology has been widely used in the literature [31], [32]. Following the definition in [31], we use conditionally SPE to denote an SPE with pure strategies only, where the network’s pure strategies constitute a Nash equilibrium in every subgame. The concept of conditionally SPE is motivated by the concept of SPE but rules out mixed strategies. In Section 3.2, we will show that a conditionally 7. For interested readers, we have provided some preliminary analysis of mixed strategy SPE in Appendix F, available in the online supplemental material.

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SPE will not include any investment decisions ðBi ; Bj Þ in the medium investment regime in Stage I. Otherwise, there is no pure strategy Nash equilibrium for pricing in Stage II, and it will not be a conditionally SPE.8 Following very similar statements in [31], we list several reasons to focus on conditionally SPE in this paper without considering mixed strategies First, we want to emphasize the result that a pure strategy pricing equilibrium may not exist in Stage II, as this result highlights the very important Edgeworth paradox for the medium investment regime (which will be introduced in Section 3.2). Such result reveals the special structure of our problem and leads to important engineering insights for practical network design. . Second, a standard criticism of mixed strategy equilibrium is that they impose very large informational burdens on users [30]. If operators choose prices according to mixed strategies, users need to consider price distributions (from which the final prices will be drawn by operators) when they choose which operator to purchase from. When the operators’ leasing costs change over time, the leasing amounts and the corresponding mixed pricing strategies can also be time varying. Given all these complexities, it is unlikely that end users will have the computational capacities and willingness to calculate the “equilibrium choices” in real spectrum market. In other words, the analysis results when allowing mixed strategies may not be very relevant for engineering practice. . Third, two operators need to run the randomization procedure in the pricing stage of each time slot if they adopt mixed pricing strategies. However, such randomization over time may be too complicated to implement in practice in a short time scale [34]. In the following analysis, we derive the conditionally SPE, which is also referred to as equilibrium for simplicity. .

3.1 Spectrum Allocation in Stage III In Stage III, each user needs to make the following two decisions based on the prices pi and pj announced by the operators in Stage II: 1. Which operator to choose? 2. How much to purchase? If a user k 2 K obtains bandwidth wki from operator i, then its payoff uk ðpi ; wki Þ is given in (3). Since this payoff is concave in wki , the unique demand that maximizes the payoff is wki ðpi Þ ¼ arg max uk ðpi ; wki Þ ¼ gk expðð1 þ pi ÞÞ: wki 0

ð5Þ

Demand wki ðpi Þ is always positive, linear in gk , and decreasing in price pi . Since gk is linear in channel gain hk and transmission power Pkmax , then a user with a better 8. If we do not focus on the concept of conditionally SPE, there may be an SPE with mixed strategies. For example, in the pricing subgame in Stage II, mixed pricing strategy Nash equilibrium can exist in the medium investment regime, which is supported by our analysis in Appendix F, available in the online supplemental material, and [33].

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channel condition or a larger transmission power has a larger demand. It is clear that wki ðpi Þ is upper bounded by gk expð1Þ for any choice of price pi 0. In other words, even if operator i announces a zero price, user k will not purchase infinite amount of resource since it cannot decode the transmission if SNRk ¼ gk =wki is low. Equation (5) shows that every user purchasing bandwidth from operator i obtains the same SNR

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Notice that both KR i and Qi depend on prices ðpi ; pj Þ in Stage II and leasing decisions ðBi ; Bj Þ in Stage I. Calculating the Realized Demands also requires considering two different pricing cases: 1.

Different prices (pi < pj ). The Preferred Demands are Di ðpi ; pj Þ ¼ G expðð1 þ pi ÞÞ and Dj ðpi ; pj Þ ¼ 0. a.

gk ¼ expð1 þ pi Þ; SNRk ¼ wki ðpi Þ and obtains a payoff linear in gk uk ðpi ; wki ðpi ÞÞ

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If Operator i has enough resource ði:e:; Bi Di ðpi ; pj ÞÞ. All Preferred Demand will be satisfied by operator i. The Realized Demands are Qi ¼ minðBi ; Di ðpi ; pj ÞÞ ¼ G expðð1 þ pi ÞÞ; Qj ¼ 0:

¼ gk expðð1 þ pi ÞÞ: b.

3.1.1 Which Operator to Choose? Next, we explain how each user decides which operator to purchase from. The following definitions help the discussions. Definition 1. The Preferred User Set KPi includes the users who prefer to purchase from operator i. Definition 2. The Preferred Demand Di is the total demand from users in the preferred user set KPi , i.e., X gk expðð1 þ pi ÞÞ: ð6Þ Di ðpi ; pj Þ ¼

If Operator i has limited resource ði:e:; Bi < Di ðpi ; pj ÞÞ. Since operator i cannot satisfy the Preferred Demand, some demand will be satisfied by operator j if it has enough resource. Since the realized demand Qi ðBi ; Bj ; pi ; pj Þ ¼ P P Bi ¼ k2KRi gk expðð1 þ pi ÞÞ, then gk ¼ k2KR i Bi expð1 þ pi Þ.9 The remaining users want to purchase bandwidth from operator j with a total demand of ðG Bi expð1 þ pi ÞÞ expðð1 þ pj ÞÞ. Thus, the Realized Demands are

k2KPi ðpi ;pj Þ

Qi ¼ minðBi ; Di ðpi ; pj ÞÞ ¼ Bi ;

The notations in (6) imply that both set KPi and demand Di only depend on prices ðpi ; pj Þ in Stage II and are independent of operators’ leasing decisions ðBi ; Bj Þ in Stage I. Such dependance can be discussed in two cases: 1.

Different prices (pi < pj ). Every user k 2 K prefers to purchase from operator i since uk ðpi ; wki ðpi ÞÞ > uk ðpj ; wkj ðpj ÞÞ: We have KPi ¼ K and KPj ¼ ;, and

2.

Di ðpi ; pj Þ ¼ G expðð1 þ pi ÞÞ and Dj ðpi ; pj Þ ¼ 0; P where G ¼ k2K gk represents the aggregate wireless characteristics of the users. This notation will be used heavily later in the paper. Same prices (pi ¼ pj ¼ p). Every user k 2 K is indifferent between the operators and randomly chooses one with equal probability. In this case, Di ðp; pÞ ¼ Dj ðp; pÞ ¼ G expðð1 þ pÞÞ=2:

Now, let us discuss how much demand an operator can actually satisfy, which depends on the bandwidth investment decisions ðBi ; Bj Þ in Stage I. It is useful to define the following terms: Definition 3. The Realized User Set KR i includes the users whose demands are satisfied by operator i. Definition 4. The Realized Demand Qi is the total demand of users in the Realized User Set KR i , i.e., X Qi Bi ; Bj ; pi ; pj ¼ gk expðð1 þ pi ÞÞ: k2KR i ðBi ;Bj ;pi ;pj Þ

! G Bi expð1 þ pi Þ : Qj ¼ min Bj ; exp 1 þ pj

2.

Same prices (pi ¼ pj ¼ p). Both operators will attract the same Preferred Demand G expðð1 þ pÞÞ=2. The Realized Demands are Qi ¼ min Bi ; Di ðp; pÞ þ max Dj ðp; pÞ Bj ; 0 G ¼ min Bi ; 2 expð1 þ pÞ G þ max Bj ; 0 ; 2 expð1 þ pÞ Qj ¼ min Bj ; Dj ðp; pÞ þ maxðDi ðp; pÞ Bi ; 0Þ G ¼ min Bj ; 2 expð1 þ pÞ G Bi ; 0 : þ max 2 expð1 þ pÞ

3.2 Operators’ Pricing Competition in Stage II In Stage II, the two operators simultaneously determine their prices ðpi ; pj Þ considering the users’ preferred demands in Stage III, given the investment decisions ðBi ; Bj Þ in Stage I. An operator i’s profit is defined earlier in (4). Since the payment Bi Ci is fixed at this stage, operator i’s profit maximization problem is equivalent of maximization of its 9. In this paper, we consider a large number of users and each user is nonatomic (infinitesimal). Thus, an individual user’s demand is infinitesimal to an operator’s supply and we can claim equality holds for Qi ¼ Bi .


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The operators’ profits in Stage II are G II;i ðBi ; Bj Þ ¼ Bi ln 1 Ci ; Bi þ Bj II;j ðBi ; Bj Þ ¼ Bj ln

. Fig. 3. Pricing equilibrium types in different (Bi ; Bj ).

revenue pi Qi . Note that users’ total demand Qi to operator i depends on the received power of each user (product of its transmission power and channel gain). We assume that an operator i knows users’ transmission powers and channel conditions. This can be achieved similarly as in today’s cellular networks, where users need to register with the operator when they enter the network and frequently feedback the channel conditions. Thus, we assume that an operator knows the user population and user demand. Game 1 (Pricing game). The competition between the two operators in Stage II can be modeled as the following game: . . .

Players: two operators i and j. Strategy space: operator i can choose price pi from the feasible set P i ¼ ½0; 1Þ. Similarly for operator j. Payoff function: operator i wants to maximize the revenue pi Qi ðBi ; Bj ; pi ; pj Þ. Similarly for operator j.

At an equilibrium of the pricing game, ðpi ; pj Þ, each operator maximizes its payoff assuming that the other operator chooses the equilibrium price, i.e., pi ¼ arg max pi Qi ðBi ; Bj ; pi ; pj Þ; i ¼ 1; 2; i 6¼ j: pi 2P i

In other words, no operator wants to unilaterally change its pricing decision at an equilibrium. Next, we will investigate the existence and uniqueness of the pricing equilibrium. First, we show that it is sufficient to only consider symmetric pricing equilibrium for Game 1. Proposition 1. Assume both operators lease positive bandwidth in Stage I, i.e., minðBi ; Bj Þ > 0. If pricing equilibrium exists, it must be symmetric pi ¼ pj . The proof of Proposition 1 is given in our online technical report [35]. The intuition is that no operator will announce a price higher than its competitor to avoid losing its Preferred Demand. This property significantly simplifies the search for all possible equilibria. Next, we show that the symmetric pricing equilibrium is a function of (Bi ; Bj ) as shown in Fig. 3. Theorem 1. The equilibria of the pricing game are as follows: .

Low investment regime (Bi þ Bj G expð2Þ as in region (L) of Fig. 3). There exists a unique nonzero pricing equilibrium G 1: ð7Þ pi ðBi ; Bj Þ ¼ pj ðBi ; Bj Þ ¼ ln Bi þ Bj

.

G Bi þ Bj

ð8Þ

1 Cj :

ð9Þ

Medium investment regime (Bi þ Bj > G expð2Þ and minðBi ; Bj Þ < G expð1Þ as in regions (M1)(M3) of Fig. 3). There is no pricing equilibrium. High investment regime (minðBi ; Bj Þ G expð1Þ as in region (H) of Fig. 3). There exists a unique zero pricing equilibrium pi ðBi ; Bj Þ ¼ pj ðBi ; Bj Þ ¼ 0; and the operators’ profits are negative for any positive values of Bi and Bj .

Proof of Theorem 1 is given in Appendix A, available in the online supplemental material. Intuitively, higher investments in Stage I will lead to lower equilibrium prices in Stage II. Theorem 1 shows that the only interesting case is the low investment regime where both operators’ total investment is no larger than G expð2Þ, in which case there exists a unique positive symmetric pricing equilibrium. Notice that the same prices at equilibrium do not imply the same profits, as the operators can have different costs (Ci and Cj ) and thus can make different investment decisions (Bi and Bj ) as shown next. Note that our equilibrium results in medium investment regime are consistent with the well-known Edgeworth paradox [36] in economics. Edgeworth paradox describes a situation where two players cannot reach a state of equilibrium with pure strategies. Each operator faces capacity constraints when determining pricing decisions in Stage II. The choice of both operators charging zero prices is not an equilibrium in the medium investment regime, since at least one operator can raise its price and obtain nonzero revenue. Nor is the case where one operator charges less the other an equilibrium, since the lower price operator can profitably raise its price toward the other. Nor is the case where both operators charge the same positive price, since at least one operator can lower its price slightly and increase its profit. The above nonequilibrium cases will not happen in the low investment regime where operators have very limited resources. This is because that in the low investment regime no operator can satisfy the whole demand alone, and thus it is possible for the two operators to share the market at the equilibrium. Also, these nonequilibrium cases will not happen in the high investment regime, where both two operators have more resources than users’ total demand even the price is zero. In this regime, we can ignore the resource constraints (similar to the Bertrand competition) and the zero price equilibrium is the same as the Bertrand paradox [37]. In the Bertrand paradox, either operator deviating from zero price cannot attract any demand from its competitor who can already serve all users.

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Theorem 2. The duopoly investment (leasing) equilibria in Stage I are summarized as follows: .

Low costs regime (0 < Ci þ Cj 1, as region (L) in Fig. 4). There exists infinitely many investment equilibria characterized by Bi ¼ G expð2Þ; Bj ¼ ð1 ÞG expð2Þ;

ð10Þ

where can be any value that satisfies Cj 1 Ci :

Fig. 4. Leasing equilibrium types in different (Ci ; Cj ).

3.3 Operators’ Leasing Strategies in Stage I In Stage I, the operators need to decide the leasing amounts ðBi ; Bj Þ to maximize their profits. Based on Theorem 1, we only need to consider the case where the total bandwidth of both the operators is no larger than G expð2Þ. We emphasize that the analysis of Stage I is not limited to the case of low investment regime; we actually also consider the medium investment regime and the high investment regime. The key observation is that an SPE will not include any investment decisions (Bi ; Bj ) in the medium investment regime, as it will not lead to a pricing equilibrium in Stage II. Moreover, any investment decisions in the high investment regime lead to zero operator revenues and are strictly dominated by any decisions in low investment regime. After the above analysis, the operators only need to consider possible equlibria in the low investment regime in Stage I.

The operators’ profits are LI;i ¼ Bi ð1 Ci Þ; LI;j ¼ Bj ð1 Cj Þ; .

.

Players. Two operators i and j. Strategy space. The operators will choose ðBi ; Bj Þ from the set B ¼ fðBi ; Bj Þ : Bi þ Bj G expð2Þg. Notice that the strategy space is coupled across the operators, but the operators do not cooperate with each other. Payoff function. The operators want to maximize their profits in (8) and (9), respectively.

At an equilibrium of the investment game, ðBi ; Bj Þ, each operator has maximized its payoff assuming that the other operator chooses the equilibrium investment, i.e., Bi ¼ arg

max

G 0Bi expð2Þ Bj

II;i ðBi ; Bj Þ; i ¼ 1; 2; i 6¼ j:

To calculate the investment equilibria of Game 2, we can first calculate operator i’s best response given operator j’s (not necessarily equilibrium) investment decision, i.e., Bi ðBj Þ ¼ arg

max

G 0Bi expð2Þ Bj

II;i ðBi ; Bj Þ; i ¼ 1; 2; i 6¼ j:

By looking at operator i’s profit in (8), we can see that a larger investment decision Bi will lead to a smaller price. The best choice of Bi will achieve the best tradeoff between a large bandwidth and a small price. After obtaining best investment responses of duopoly, we can then calculate the investment equilibria, given different costs Ci and Cj .

where “L” denotes the low costs regime. High comparable costs regime (Ci þ Cj > 1 and jCj Ci j 1, as region (HC) in Fig. 4). There exists a unique investment equilibrium ð1 þ Cj Ci ÞG Ci þ Cj þ 3 exp ; ð12Þ Bi ¼ 2 2 Bj ¼

ð1 þ Ci Cj ÞG Ci þ Cj þ 3 exp : 2 2

ð13Þ

The operators’ profits are 1 þ Cj Ci 2 Ci þ Cj þ 3 ; ¼ G exp HC I;i 2 2

Game 2 (Investment Game). The competition between the two operators in Stage I can be modeled as the following game: . .

ð11Þ

HC I;j ¼

.

1 þ Ci Cj 2

2

Ci þ Cj þ 3 ; G exp 2

where “HC” denotes the high comparable costs regime. High incomparable costs regime (Cj > 1 þ Ci or Ci > 1 þ Cj , as regions (HI) and (HI 0 ) in Fig. 4). For the case of Cj > 1 þ Ci , there exists a unique investment equilibrium with Bi ¼ G expðð2 þ Ci ÞÞ;

Bj ¼ 0;

i.e., operator i acts as the monopolist and operator j is out of the market. The operators’ profits are HI I;i ¼ G expðð2 þ Ci ÞÞ;

HI I;j ¼ 0;

where “HI” denotes the high incomparable costs. The case of Ci > 1 þ Cj can be analyzed similarly. The proof of Theorem 2 is given in Appendix B, which is available in the online supplemental material. Let us further discuss the properties of the investment equilibrium in three different costs regimes.

3.3.1 Low Costs Regime (0 < Ci þ Cj 1) In this case, both the operators have very low costs. It is the best response for each operator to lease as much as possible. However, since the strategy set in the Investment Game is


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TABLE 2 Operators’ and Users’ Behaviors at the Equilibria (Assuming Ci Cj )

coupled across the operators (i.e., B ¼ fðBi ; Bj Þ : Bi þ Bj G expð2Þg), there exist infinitely many ways for the operators to achieve the maximum total leasing amount G expð2Þ. We can further identify the focal point, i.e., the equilibrium that the operators will agree on without prior communications [30]. The details can be found in our online technical report [35].

3.3.2 High Comparable Costs Regime (Ci þ Cj > 1 and jCj Ci j 1) First, the high costs discourage the operators from leasing aggressively; thus, the total investment is less than G expð2Þ. Second, the operators’ costs are comparable, and thus the operator with the slightly lower cost does not have sufficient power to drive the other operator out of the market. 3.3.3 High Incomparable Costs Regime (Cj > 1 þ Ci or Ci > 1 þ Cj ) First, the costs are high and thus the total investment of two operators is less than G expð2Þ. Second, the costs of the two operators are so different that the operator with the much higher cost is driven out of the market. As a result, the remaining operator thus acts as a monopolist.

4

EQUILIBRIUM SUMMARY

Based on the discussions in Section 3, we summarize the equilibria of the three-stage game in Table 2, which includes the operators’ investment decisions, pricing decisions, and the resource allocation to the users. Without loss of generality, we assume Ci Cj in Table 2. The equilibrium for Ci > Cj can be described similarly. Several interesting observations are as follows: Observation 1. The operators’ equilibrium investment decisions Bi and Bj are linear in the users’ aggregate wireless characteristics ! X X max G ¼ gk ¼ Pk hk n0 : k2K

k2K

This shows that the operators’ total investment increases with the user population, users’ channel gains, and users’ transmission powers. Observation 2. The symmetric equilibrium price pi ¼ pj does not depend on users’ wireless characteristics. Observations 1 and 2 are closely related. Since the total investment is linearly proportional to the users’ aggregate characteristics, the “average” equilibrium resource allocation per user is “constant” and does not depend on the user population. Since resource allocation is determined by the price, this means that the price is also independent of the user population and wireless characteristics. Observation 3. The operators can determine different equilibrium leasing and pricing decisions by observing some linear thresholds in Figs. 3 and 4. For equilibrium investment decisions in Stage I, the feasible set of investment costs can be divided into three regions by simple linear thresholds as in Fig. 4. As leasing costs increase, operators invest less aggressively; as the leasing cost difference increases, the operator with a lower cost gradually dominates the spectrum market. For the equilibrium pricing decisions, the feasible set of leasing bandwidths is also divided into three regions by simple linear thresholds as well. A meaningful pricing equilibrium exists only when the total available bandwidth from the two operators is no larger than a threshold (see Fig. 3). Observation 4. Each user k’s equilibrium demand is positive, linear in its wireless characteristic gk , and decreasing in the price. Each user k achieves the same SNR independent of gk , and obtains a payoff linear in gk . Observation 4 shows that the users receive fair resource allocation and service quality. Such allocation does not depend on the wireless characteristics of the other users. Observation 5. In the High Incomparable Costs Regime, users’ equilibrium SNR increases with the costs Ci and Cj , and the equilibrium payoffs decrease with the costs.

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Fig. 5. Transition matrix of Ci ðtÞ and Cj ðtÞ over time slots.

As the costs Ci and Cj increase, the pricing equilibrium (pi ¼ pj ) increases to compensate the loss of the operators’ profits due to increased costs. As a result, each user will purchase less bandwidth from the operators. Since a user spreads its total power across the entire allocated bandwidth, a smaller bandwidth means a higher SNR but a smaller payoff. Finally, we observe that the users achieve a high SNR at the equilibrium. The minimum equilibrium SNR that users achieve among the three costs regime is expð2Þ. In this case, the ratio between the high SNR approximation and Shannon capacity, lnðSNRÞ= lnð1 þ SNRÞ, is larger than 94 percent. This validates our assumption on the high SNR regime. The next section, on the other hand, shows that most of the insights remain valid in the general SNR regime.

4.1

How Network Dynamics Affect Equilibrium Decisions Our analysis so far has not considered network dynamics, as we have focused on a single time slot where an operator knows users’ channel conditions through proper feedback mechanisms. In this section, we will look at how the equilibrium results in Table 2 change over multiple time slots with the network dynamics. Note that operators still have the complete network information in each time slot. Users are myopic in the sense that they do not take into account the effect of time-varying network parameters on future prices when they determine bandwidth demand in the current time slot. First, we consider the case where the spectrum available for leasing changes over time. Intuitively, when a spectrum owner faces a strong demand from its own primary users, it will have less spectrum resource for the virtual operator and will set a higher leasing cost. Here, we look at the case where operators’ leasing costs Ci and Cj change over time according to some Markov decision processes. We write two costs as Ci ðtÞ and Cj ðtÞ to emphasize their dependancies in time. For the illustration purpose, we consider three possible values for both Ci ðtÞ and Cj ðtÞ: 0.4, 0.8, and 2, and the transition probabilities (same for two operators) are shown in Fig. 5. Fig. 6 shows how costs Ci ðtÞ and Cj ðtÞ, equilibrium leasing decisions Bi and Bj , and pricing decisions pi and pj change over time. Here, we represent a price N=A in Table 2 as a zero price. This means that whenever we see a zero price in the figure, the corresponding operator does not participate in the game and the other operator becomes the monopoly in the market. We observe that as an operator’s leasing cost increases, its leasing amount

Fig. 6. Costs, equilibrium bandwidth and pricing decisions as functions of time slots. Here, we fix to be 0.5.

decreases. The operator with a lower cost will lease more and will become the monopolist if its cost is much lower than its competitor (i.e., with jCj ðtÞ Ci ðtÞj > 1 in the high incomparable costs regime). In this case, its competitor decides not to lease. As costs increase, operators’ symmetric prices tend to increase to compensate the costs. When two costs are low (with Ci ðtÞ þ Cj ðtÞ 1), both operators announce the same high price. Second, we can consider the dynamics of users’ channel gains and their population over time. Users’ channel gains may follow, for example, different Rayleigh distributions. Also, there can be users departing or entering the network over different time slots. As a result, users’ aggregate wireless characteristics G will change over time. Table 2 has clearly shown that operators’ leasing amounts and profits will change proportionally to G. But the equilibrium prices will not be affected, since operators will balance their leasing amounts with users’ demands. For the sake of space, we will not show additional plots for this case.

5

EQUILIBRIUM ANALYSIS UNDER THE GENERAL SNR REGIME

In Sections 3 and 4, we computed the equilibria of the three-stage game based on the high SNR assumption in (2), and obtained five important observations (Observations 1-5). The high SNR assumption enables us to obtain closed-form solutions of the equilibria analysis and clear engineering insights. In this section, we further consider the more general SNR regime where a user’s rate is computed according to (1) instead of (2). We will follow a similar backward induction analysis, and extend Observations 1, 2, 4, 5, and pricing threshold structure of Observation 3 to the general SNR regime. We first examine the pricing equilibrium in Stage II. Theorem 3. Define Bth :¼ 0:462G. The pricing equilibria in the general SNR regime are as follows: .

Low investment regime (Bi þ Bj Bth as in region (L) of Fig. 7). There exists a unique pricing equilibrium


Fig. 7. Pricing equilibrium types in different (Bi ; Bj ) regions for general SNR regime.

pi ðBi ; Bj Þ ¼ pj ðBi ; Bj Þ G G : ¼ ln 1 þ Bi þ Bj Bi þ Bj þ G The operators’ profits in Stage II are

.

i ðBi ; Bj Þ ¼ G G Ci ; Bi ln 1 þ Bi þ Bj Bi þ Bj þ G

ð15Þ

j ðBi ; Bj Þ ¼ G G Cj : Bj ln 1 þ Bi þ Bj Bi þ Bj þ G

ð16Þ

High investment regime (Bi þ Bj > Bth as in region (H) of Fig. 7). There is no pricing equilibrium.

Proof of Theorem 3 is given in Appendix C, available in the online supplemental material. This result is similar to Theorem 1 in the high SNR regime, and shows that the pricing equilibrium in the general SNR regime still has a threshold structure in Observation 3. Unlike Theorem 1, here we only have two investment regimes. The “high investment regime” in Theorem 1 is gone, and the “medium investment regime” in Theorem 1 corresponds to the high investment regime here. Intuitively, the high SNR assumption in Section 3 requires each user to demand relatively small amount of bandwidth to spread its transmission power efficiently; thus, the users’ total demand is not elastic to prices and is always upper bounded by G expð1Þ in Fig. 3. But in the general SNR case, users’ demands are elastic to prices and is no longer upper bounded. Hence, we only have two regimes here. For more details, please refer to Appendix C, which is available in the online supplemental material. Based on Theorem 3, we are ready to prove Observations 1, 2, 4, and 5 in the general SNR regime. Theorem 4. Observations 1, 2, 4, and 5 in Section 4 still hold for the general SNR regime. Proof of Theorem 4 is given in Appendix D, available in the online supplemental material.

6

IMPACT OF OPERATOR COMPETITION

We are interested in understanding the impact of operator competition on the operators’ profits and the users’ payoffs. As a benchmark, we will consider the coordinated case where both operators jointly make the investment and

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Fig. 8. The three-stage Stackelberg game for the coordinated operators.

pricing decisions to maximize their total profit. In this case, there does not exist competition between the two operators. However, it is still a Stackelberg game between a single decision maker (representing both operators) and the users. Then, we will compare the equilibrium of this Stackelberg game with that of the duopoly game as in Section 4.

6.1 Maximum Profit in the Coordinated Case We analyze the coordinated case following a three-stage model as shown in Fig. 8. Compared with Fig. 2, the key difference here is that a single decision maker makes the decisions for two operators in both Stages I and II. In other words, the two operators coordinate with each other. Again, we use backward induction to analyze the threestage game. The analysis of Stage III in terms of the spectrum allocation among the users is the same as in Section 3.1 (still assuming the high SNR regime), and we focus on Stages II and I. Without loss of generality, we assume that Ci Cj . In Stage II, the decision maker maximizes the following total profit T by determining prices pi and pj : T ðBi ; Bj ; pi ; pj Þ ¼ i ðBi ; Bj ; pi ; pj Þ þ j ðBi ; Bj ; pi ; pj Þ; where i ðBi ; Bj ; pi ; pj Þ is given in (4) and j ðBi ; Bj ; pi ; pj Þ can be obtained similarly. Theorem 5. In Stage II, the optimal pricing decisions for the coordinated operators are as follows: .

.

If Bi > 0 and Bj ¼ 0, then operator i is the monopolist and announces a price G co 1: ð17Þ pi ðBi ; 0Þ ¼ ln Bi Similar result can be obtained if Bi ¼ 0 and Bj > 0. If minðBi ; Bj Þ > 0, then both operators i and j announce the same price G co 1: ðB ; B Þ ¼ p ðB ; B Þ ¼ ln pco i j i j i j Bi þ Bj

Proof of Theorem 5 can be found in our online technical report [35]. Theorem 5 shows that both operators will act together as a monopolist in the pricing stage. Now, let us consider Stage I, where the decision maker determines the leasing amounts Bi and Bj to maximize the total profit max T ðBi ; Bj Þ ¼

Bi ;Bj 0

co max Bi ðpco i ðBi ; Bj Þ Ci Þ þ Bj ðpj ðBi ; Bj Þ Cj Þ;

Bi ;Bj 0

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co where pco i ðBi ; Bj Þ and pj ðBi ; Bj Þ are given in Theorem 5. In this case, operator j will not lease (i.e., Bco j ¼ 0) as operator i can lease with a lower cost. Thus, the optimization problem in (18) degenerates to

max T ðBi Þ ¼ max Bi ðpco i ðBi ; 0Þ Ci Þ: Bi 0

This leads to the following result:

co Bco i ðCi ; Cj Þ ¼ G expðð2 þ Ci ÞÞ; Bj ðCi ; Cj Þ ¼ 0;

THC ðCi ; Cj Þ

4

Tco ðCi ; Cj Þ ¼ G expðð2 þ Ci ÞÞ:

¼

TL ðCi ; Cj ; Þ ¼ ½ð1 Ci Þ þ ð1 Þð1 Cj ÞG expð2Þ; where can be any real value in the set of ½Cj ; 1 Ci . Each choice of corresponds to an investment equilibrium and there are infinitely many equilibria in this case as shown in Theorem 2. The minimum profit ratio between the duopoly case and the coordinated case optimized over is 4

T RL ðCi ; Cj Þ ¼

TL ðCi ; Cj ; Þ : 2½Cj ;1Ci Tco ðCi ; Cj Þ min

Since TL ðCi ; Cj ; Þ is increasing in , the minimum profit ratio is achieved at

This means ð20Þ

Although (20) is a nonconvex function of Ci and Cj , we can numerically compute the minimum value over all possible values of costs in this regime min

ðCi ;Cj Þ:01;0Cj Ci 1

¼ T RHC ð2

T RHC ðCj Ci Þ

pffiffiffi 3Þ ¼ 0:773:

6.2.3 High Incomparable Costs Regime (Cj Ci > 1) In this case, only one operator leases a positive amount at the duopoly equilibrium and achieves the same profit as in the coordinated case. The profit ratio is 1. We summarize the above results as follows: Theorem 7 (Operators’ Profit Loss). Comparing with the coordinated case, the operator competition leads to a maximum total profit loss of 25 percent in the low costs regime. Since low leasing costs lead to aggressive leasing decisions and thus intensive competitions among operators, it is not surprising to see that the maximum profit loss happens in the low cost regime. For detailed discussions on the relationship between the profit loss and the costs, see our online technical report [35].

6.3 Impact of Competition on the Users’ Payoffs Theorem 8. Comparing with the coordinated case, users obtain the same or higher payoffs under the operators’ competition.

¼ Cj :

T RL ðCi ; Cj Þ ¼ ½Cj ð1 Ci Þ þ ð1 Cj Þ2 expðCi Þ:

1 þ ðCj Ci Þ2 Ci þ Cj þ 3 : G exp ¼ 2 2

T RHC ðCi ; Cj Þ ¼

and the total profit is

6.2.1 Low Costs Regime (0 < Ci þ Cj 1) First, the total equilibrium leasing amount in the duopoly case is Bi þ Bj ¼ G expð2Þ, which is larger than the total leasing amount G expðð2 þ Ci ÞÞ in the coordinated case. In other words, operator competition leads to a more aggressive overall investment. Second, the total profit at the duopoly equilibria is

NOVEMBER 2012

And the profit ratio is

ð19Þ

6.2 Impact of Competition on the Operators’ Profits Let us compare the total profit obtained in the competitive duopoly case (Theorem 2) and the coordinated case (Theorem 6).

NO. 11,

which is greater than G expðð2 þ Ci ÞÞ of the coordinated case. Again, competition leads to a more aggressive overall investment. Second, the total profit of duopoly is

Bi 0

Theorem 6. In Stage I, the optimal investment decisions for the coordinated operators are

VOL. 11,

Ci þ Cj þ 3 Bi þ Bj ¼ G exp 2

T RL ðCi ; Cj Þ ¼ lim T RL ð; 0:5 þ Þ ¼ 0:75: !0

This means that the total profit achieved at the duopoly equilibrium is at least 75 percent of the total profit achieved in the coordinated case under any choice of cost parameters in the Low Costs Regime.

6.2.2 High Comparable Costs Regime (Ci þ Cj > 1 and Cj Ci 1) First, the total duopoly equilibrium leasing amount is

By substituting (19) into (17), we obtain the optimal price in the coordinated case as 1 þ Ci . This means that user k’s payoff equals to gk expðð2 þ Ci ÞÞ in all three costs regimes. According to Table 2, users in the duopoly competition case have the same payoffs as in coordinated case in the high incomparable costs regime. The payoffs are larger in the other two costs regimes with the competitor competition. The intuition is that operator competition in those two regimes leads to aggressive investments, which results in lower prices and higher user payoffs.

7

CONCLUSION AND FUTURE WORK

Dynamic spectrum leasing enables the secondary network operators to quickly obtain the unused resources from the


primary spectrum owner and provide flexible services to secondary end-users. This paper studies the competition between two secondary operators and examines the operators’ equilibrium investment and pricing decisions as well as the users’ corresponding achieved service quality and payoffs. We model the economic interactions between the operators and the users as a three-stage dynamic game. Our appropriate OFDM-based spectrum sharing model captures the wireless heterogeneity of the users in terms of maximum transmission power levels and channel gains. The two operators engage in investment and pricing competitions with asymmetric costs. We have discovered several interesting features of the game’s equilibria. For example, the operators can determine different equilibrium leasing and pricing decisions by observing some linear thresholds. We also study the impact of operator competition on operators’ total profit loss and the users’ payoff increases. Compared with the coordinated case where the two operators cooperate to maximize their total profit, we show that at the maximum profit loss due to competition is no larger than 25 percent. We also show that the users always benefit from competition by achieving the same or better payoffs. Although we have focused on the high SNR regime when obtaining closed-form solutions, we show that most engineering insights summarized in Section 4 still hold in the general SNR regime. Due to the page limit, more detailed discussions and all proofs can be found in the online technical report [35]. There are several ways to extend the results here .

.

.

.

First, we can consider the case where the operators can also obtain resource through spectrum sensing [21], [38]. Compared with leasing, sensing is cheaper but the amount of useful spectrum is less predictable due to the primary users’ stochastic traffic. With the possibility of sensing, we need to consider a fourstage dynamic game model. Second, we can consider the case where users might experience different channel conditions when they choose different providers, e.g., when they need to communicate with the base stations of the operators. Competition under such channel heterogeneity has been partially considered by Gajic et al. [39] without considering the cost of spectrum acquisition. Third, we can consider a more general and realistic model for user’s transmission. For example, a user’s payoff and demand are affected by the received signal strength indicator (RSSI) to its receiver and whether the transmitter is able to transmit the data to the receiver with the allocated bandwidth and time given. Also, we can consider detailed signal fading and its effect on users’ changing demand, and different allocated frequencies may have different power requirements. Fourth, we can study the multiple-operator (oligopoly) case, where the analysis becomes much more complicated without closed-form solutions. For example, when we do backward induction analysis at Stage II, all possible combinations of multiple

.

8

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operators’ leasing decisions in Stage I need to be considered. Nevertheless, we can still infer some intuitions about the oligopoly case based on our duopoly analysis. For example, operators’ competition will be more severe and their equilibrium symmetric prices will be closer to leasing costs with oligopoly. Also, operators will be more conservative in leasing decisions in Stage I, since each operator is expected to serve fewer users in Stage III. Finally, each operator’s profit will decrease in the number of operators. It will be useful to verify these intuitions and discover additional new insights in the oligopoly case. Finally, we can consider the case where the operator improves its profit by price differentiation, i.e., charging different users different prices based on their channel conditions and transmission power. The key issue is to achieve the best tradeoff between pricing complexity and profit improvement. A similar tradeoff has been studied for a monopoly network service provider in Li et al. [40].

NOTICE FOR APPENDICES OF PROOFS

In accordance with TMC’s publication guidelines, the appendices for this paper are provided in a separate supplemental material file that can be found on the Computer Society Digital Library at http://doi.ieeecomputersociety. org/10.1109/TMC.2011.213. Interested readers can also find the proofs in our online technical report [35].

ACKNOWLEDGMENTS This work was supported by the General Research Funds (Project Numbers CUHK 412710, CUHK 412511, and CityU 144209) established under the University Grant Committee of the Hong Kong Special Administrative Region, China. This work was also partially supported by grants from the City University of Hong Kong (Project Numbers 7002517 and 7008116). Part of the results appeared in IEEE DySPAN, Singapore, April 2010 [1]. Jianwei Huang was the corresponding author for this paper. Lingjie Duan was previously with the Network Communications and Economics Laboratory, Department of Information Engineering, The Chinese University of Hong Kong.

REFERENCES [1] [2] [3]

[4] [5]

L. Duan, J. Huang, and B. Shou, “Competition with Dynamic Spectrum Leasing,” Proc. IEEE Symp. New Frontiers in Dynamic Spectrum Access Networks (DySPAN), Apr. 2010. M.A. McHenry and D. McCloskey, “Spectrum Occupancy Measurements: Chicago, Illinois, Nov. 16-18, 2005,” technical report, Shared Spectrum Company, 2005. S.K. Jayaweera and T. Li, “Dynamic Spectrum Leasing in Cognitive Radio Networks via Primary-Secondary User Power Control Games,” IEEE Trans. Wireless Comm., vol. 8, no. 6, pp. 3300-3310, June 2009. Q. Zhao and B. Sadler, “A Survey of Dynamic Spectrum Access: Signal Processing, Networking, and Regulatory Policy,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 78-89, May 2007. O. Simeone, I. Stanojev, S. Savazzi, Y. Bar-Ness, U. Spagnolini, and R. Pickholtz, “Spectrum Leasing to Cooperating Secondary Ad Hoc Networks,” IEEE J. Selected Areas in Comm., vol. 26, no. 1, pp. 203-213, Jan. 2008.

1718

[6] [7] [8] [9] [10] [11]

[12] [13]

[14] [15]

[16]

[17] [18] [19] [20]

[21] [22] [23] [24] [25]

[26]

[27] [28] [29] [30]


J. Jia and Q. Zhang, “Competitions and Dynamics of Duopoly Wireless Service Providers in Dynamic Spectrum Market,” Proc. ACM MobiHoc, pp. 313-322, 2008. J.M. Chapin and W.H. Lehr, “Time-Limited Leases in Radio Systems,” IEEE Comm. Magazine, vol. 45, no. 6, pp. 76-82, June 2007. “The MVNO Directory 2009,” http://www.mvnodirectory.com, 2009. M. Lev-Ram, “As Mobile ESPN Falls, a Helio Rises,” CNN Business 2.0 Magazine, http://money.cnn.com/2006/10/17/ magazines/business2/espnmobile_mvno.biz2/index.htm, 2006. R. Dewenter and J. Haucap, “Incentives to Lease Mobile Virtual Network Operators (MVNOs),” Proc. 34th Research Conf. Comm. Information Internet Policy, Oct. 2006. D. Niyato and E. Hossain, “Competitive Pricing for Spectrum Sharing in Cognitive Radio Networks: Dynamic Game, Inefficiency of Nash Equilibrium, and Collusion,” IEEE J. Selected Areas in Comm., vol. 26, no. 1, pp. 192-202, Jan. 2008. Y. Xing, R. Chandramouli, and C. Cordeiro, “Price Dynamics in Competitive Agile Spectrum Access Markets,” IEEE J. Selected Areas in Comm., vol. 25, no. 3, pp. 613-621, Apr. 2007. O. Ileri, D. Samardzija, T. Sizer, and N.B. Mandayam, “Demand Responsive Pricing and Competitive Spectrum Allocation via a Spectrum Server,” Proc. IEEE Int’l Symp. New Frontiers in Dynamic Spectrum Access Networks (DySPAN), pp. 194-202, 2005. J. Huang, R.A. Berry, and M.L. Honig, “Auction-Based Spectrum Sharing,” ACM/Springer Mobile Networks and Applications J., vol. 24, no. 5, pp. 1074-1084, 2006. S. Wang, P. Xu, X. Xu, S.-J. Tang, X.-Y. Li, and X. Liu, “ODA: Truthful Online Double Auction for Spectrum Allocation in Wireless Networks,” Proc. IEEE Int’l Symp. New Frontiers in Dynamic Spectrum Access Networks (DySPAN), 2010. X. Wang, Z. Li, P. Xu, Y. Xu, X. Gao, and H. Chen, “Spectrum Sharing Cognitive Radio Networks - An Auction-Based Approach,” IEEE Trans. System, Man and Cybernetics Part B: Cybernetics, vol. 40, no. 3, pp. 587-596, June 2010. S. Sengupta and M. Chatterjee, “An Economic Framework for Dynamic Spectrum Access and Service Pricing,” IEEE/ACM Trans. Networking, vol. 17, no. 4, pp. 1200-1213, Aug. 2009. P. Maille and B. Tuffin, “Analysis of Price Competition in a Slotted Resource Allocation Game,” Proc. IEEE INFOCOM, 2008. G.S. Kasbekar and S. Sarkar, “Spectrum Pricing Games with Bandwidth Uncertainty and Spatial Reuse in Cognitive Radio Networks,” Proc. ACM MobiHoc, Sept. 2010. D. Niyato, E. Hossain, and Z. Han, “Dynamic Spectrum Access in IEEE 802.22-Based Cognitive Wireless Networks: A Game Theoretic Model for Competitive Spectrum Bidding and Pricing,” IEEE Wireless Comm., vol. 16, no. 2, pp. 16-23, Apr. 2009. L. Duan, J. Huang, and B. Shou, “Cognitive Mobile Virtual Network Operator: Investment and Pricing with Supply Uncertainty,” Proc. IEEE INFOCOM, 2010. T.A. Weiss and F.K. Jondral, “Spectrum Pooling: An Innovative Strategy for the Enhancement of Spectrum Efficiency,” IEEE Comm. Magazine, vol. 42, no. 3, pp. S8-S14, Mar. 2004. H.A. Mahmoud and H. Arslan, “Sidelobe Suppression in OFDMBased Spectrum Sharing Systems Using Adaptive Symbol Transition,” IEEE Comm. Letters, vol. 12, no. 2, pp. 133-135, Feb. 2008. J. Huang, V. Subramanian, R. Agrawal, and R. Berry, “Downlink Scheduling and Resource Allocation for OFDM Systems,” IEEE Trans. Wireless Comm., vol. 8, no. 1, pp. 288-296, Jan. 2009. J. Huang, V. Subramanian, R. Agrawal, and R. Berry, “Joint Scheduling and Resource Allocation in Uplink OFDM Systems for Broadband Wireless Access Networks,” IEEE J. Selected Areas in Comm., vol. 27, no. 2, pp. 226-234, Feb. 2009. J. Bae, E. Beigman, R.A. Berry, M.L. Honig, and R. Vohra, “Sequential Bandwidth and Power Auctions for Distributed Spectrum Sharing,” IEEE J. Selected Areas in Comm., vol. 26, no. 7, pp. 1193-1203, Sept. 2008. IEEE 802.16 Working Group on Broadband Wireless Access Standards, IEEE 802.16e-2005 and IEEE Std 802.16-2004/Cor1-2005, http:// www.ieee802.org/16, 2005. F. Kelly, “Charging and Rate Control for Elastic Traffic,” European Trans. Telecomm., vol. 8, no. 1, pp. 33-37, 1997. S. Shakkottai and R. Srikant, “Economics of Network Pricing with Multiple ISPs,” IEEE/ACM Trans. Networking, vol. 14, no. 6, pp. 1233-1245, Dec. 2006. D. Fudenberg and J. Tirole, Game Theory. MIT Press, 1991.

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[31] R. Gibbens, R. Mason, and R. Steinberg, “Internet Service Classes under Competition,” IEEE J. Selected Areas in Comm., vol. 18, no. 12, pp. 2490-2498, Dec. 2000. [32] D. Abreu, “On the Theory of Infinitely Repeated Games with Discounting,” Econometrica: J. Econometric Soc., vol. 56, pp. 383-396, 1988. [33] P. Dasgupta and E. Maskin, “The Existence of Equilibrium in Discontinuous Economic Games, I: Theory,” The Rev. of Economic Studies, vol. 53, no. 1, pp. 1-26, 1986. [34] C. Courcoubetis and R. Weber, Pricing Communication Networks, vol. 2. Wiley, 2003. [35] L. Duan, J. Huang, and B. Shou, “Competition with Dynamic Spectrum Leasing,” technical report, http://arxiv.org/abs/ 1003.5517, 2010. [36] E. Rasmusen, Games and Information: An Introduction to Game Theory. Wiley-Blackwell, 2007. [37] A. Mas-Colell, M.D. Whinston, and J.R. Green, Microeconomic Theory. Oxford Univ. Press, 1995. [38] L. Duan, J. Huang, and B. Shou, “Investment and Pricing with Spectrum Uncertainty: A Cognitive Operators Perspective,” IEEE Trans. Mobile Computing, vol. 10, no. 11, pp. 1590-1604, Nov. 2011. [39] V. Gajic, J. Huang, and B. Rimoldi, “Competition of Wireless Providers for Atomic Users: Equilibrium and Social Optimality,” Proc. Allerton Conf., Sept. 2009. [40] S. Li, J. Huang, and S.-Y.R. Li, “Revenue Maximization for Communication Networks with Usage-Based Pricing,” Proc. IEEE Globecom, Dec. 2009. [41] P. Gupta and P. R.Kumar, “The Capacity of Wireless Networks,” IEEE Trans. Information Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000. [42] M. Luby, “A Simple Parallel Algorithm for the Maximal Independent Set Problem,” Proc. 17th Ann. ACM Symp. Theory of Computing, pp. 1-10, 1985. [43] N. Alon, L. Babai, and A. Itai, “A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem,” J. Algorithms, vol. 7, no. 4, pp. 567-583, 1986. [44] A. Schrijver, Theory of Linear and Integer Programming, vol. 11. Wiley, 1986. [45] O. Giel and I. Wegener, Evolutionary Algorithms and the Maximum Matching Problem. Springer, 2003. [46] D. Koller and N. Megiddo, “Finding Mixed Strategies with Small Supports in Extensive Form Games,” Int’l J. Game Theory, vol. 25, no. 1, pp. 73-92, 1996. Lingjie Duan received the BE degree in electrical engineering from the Harbin Institute of Technology, China, in 2008, and the PhD degree in information engineering from The Chinese University of Hong Kong in 2012. He is an assistant professor in the Engineering Systems and Design Pillar at Singapore University of Technology and Design. During 2011, he was a visiting scholar in the Department of Electrical Engineering and Computer Sciences at the University of California at Berkeley. His research interests are in the areas of resource allocation and game theoretical analysis of communication networks, with current emphasis on cognitive radio networks and small cell networks. He has served on the technical program committees (TPC) for multiple top conferences (e.g., IEEE VTC, IEEE PIMRC, IEEE WCNC, and ACM MobiArch). He is a member of the IEEE.


Jianwei Huang received the BS degree in electrical engineering from the Southeast University, Nanjing, Jiangsu, China, in 2000, and the MS and PhD degrees in electrical and computer engineering from the Northwestern University, Evanston, Illinois, in 2003 and 2005, respectively. He is an assistant professor in the Department of Information Engineering at the Chinese University of Hong Kong. He worked as a postdoctoral research associate in the Department of Electrical Engineering at Princeton University from 2005-2007 and as a summer intern at Motorola, Arlington Heights, Illinois, in 2004 and 2005. He currently leads the Network Communications and Economics Laboratory (ncel.ie.cuhk.edu.hk), with a main research focus on nonlinear optimization and game theoretical analysis of communication networks, especially on network economics, cognitive radio networks, and smart grid. He is the recipient of the IEEE Marconi Prize Paper Award in Wireless Communications in 2011, the IEEE GlobeCom Best Paper Award in 2010, the IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award in 2009, the Asia-Pacific Conference on Communications Best Paper Award in 2009, and the Walter P. Murphy Fellowship at Northwestern University in 2001. He has served as an editor of the IEEE Transactions on Wireless Communications, the IEEE Journal on Selected Areas in Communication - Cognitive Radio Series, a guest editor of the IEEE Journal on Selected Areas in Communications special issue on economics of networks and wireless systems, the lead guest editor of the IEEE Journal of Selected Areas in Communications special issue on game theory in communication systems, and guest editor of several other journals including Wireless Communications and Mobile Computing (Wiley), the Journal of Advances in Multimedia, and the Journal of Communications. Dr. Huang served as the chair of the IEEE ComSoc Multimedia Communications Technical Committee (MMTC) from 2012-2014, steering committee member of the IEEE Transactions on Multimedia and IEEE ICME from 2012-2014, vice-chair of the IEEE MMTC from 2010-2012, director of the IEEE MMTC e-letter in 2010, and chair of meetings and conference committees of the IEEE ComSoc Asia Pacific Board from 2012-2013. He also served as a TPC cochair of the IEEE GlobeCom Selected Areas of Communications Symposium (Game Theory for Communications Track) 2013, a publicity cochair of the IEEE Communications Theory Workshop 2012, a TPC cochair of the International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (IEEE WiOpt) 2012, a TPC cochair of the IEEE ICCC Communication Theory and Security Symposium 2012, a student activities cochair of IEEE WiOpt 2011, a TPC cochair of the IEEE GlobeCom Wireless Communications Symposium 2010, a TPC cochair of the International Wireless Communications and Mobile Computing (IWCMC) Mobile Computing Symposium 2010, and a TPC cochair of the International Conference on Game Theory for Networks (GameNets) 2009. He is a TPC member of leading conferences such as INFOCOM (2009-2013), MobiHoc (2009, 2012), ICC, GlobeCom, DySPAN, WiOpt, NetEcon, and WCNC. He is a senior member of the IEEE.

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Biying Shou received the BE degree from Tsinghua University and the MS and PhD degrees from the Northwestern University. She is an assistant professor of management sciences at the City University of Hong Kong. Her main research interests include operations and supply chain management, game theory, and economics of wireless networks. She has published in leading journals including Operations Research, Production and Operations Management, Naval Research Logistics, and the IEEE Transactions on Mobile Computing.

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