An Integrated Spectrum and Information Market for Green Cognitive Communications
A database-assisted TV white space network can achieve the goal of green cognitive communication by effectively reducing the energy consumption in cognitive communications. The success of such a novel network relies on a proper business model that provides substantial incentives for all parties involved. In this paper, we propose an integrated spectrum and information market for a database-assisted TV white space network, where the geo-location database acts as an online platform providing services to both spectrum market and information market. We model the interactions among the database operator, the spectrum licensee, and unlicensed users as a three-stage sequential decision process. Specifically, Stage I characterizes the negotiation between the database and the licensee, in terms of the commission for the licensee to use the spectrum market platform, Stage II models the pricing decisions of the database and the licensee, and Stage III characterizes the subscription behaviors of unlicensed users. Analyzing such a three-stage model is challenging due to the co-existence of both positive and negative network externalities in the information market. Nevertheless of this, we are able to explicitly characterize the impact of network externalities on the equilibrium behaviors of all parties involved. We analytically show that the licensee can never get a market share larger than half in the integrated market. Our numerical results further show that the proposed integrated market can outperform a pure information market in terms of network profit up to .
With the explosive growth of mobile smartphones and bandwidth-hungry wireless applications, the corresponding energy consumption due to telecommunication industry is increasing at a unprecedented speed of per annum . Moreover, according to the Climate Group SMART 2020 Report , the information and communication technology (ICT) infrastructures account for of global energy consumption and of global CO emissions. Hence, energy optimization of wireless communications, ranging from equipment manufacturing to core functionalities, becomes increasingly important for protecting our environment, coping with global warming, and facilitating sustainable development.
Cognitive communication has been viewed as a promising paradigm for achieving energy-efficient communications. The key idea is to allow the cognitive radio device to adapt its configuration and transmission decision to the real-time radio environment. Hence, a cognitive radio device can select the best reconfiguration operation that balances the energy consumption and communication quality. Obviously, the success of cognitive communication system greatly relies on the accurate detection of radio environment (e.g., locating the idle channels and figuring out the allowable transmission power to minimize interference to existing users). If a mobile device is fully responsible for the continuous and accurate detection of radio environment, it would consume a significant amount of energy. The higher accuracy, the higher computational burden on a mobile device, and thus the higher energy consumption.
In order to reduce energy consumption and guarantee the performance of cognitive communication, some spectrum regulators (such as FCC in the USA and Ofcom in the UK), together with standards bodies and industrial organizations,111 Example include IEEE 802.22 WRAN standard (http://www.ieee802.org/22/) and the industrial companies such as Google (http://www.google.org/spectrum/whitespace/), Microsoft (http://whitespaces.msresearch.us/), and Spectrum Bridge (http://www.spectrumbridge.com/). have advocated a database-assisted TV white space network architecture. In such a network, a white space database (called geo-location database) assists unlicensed wireless devices (called white space devices, WSDs) opportunistically exploit the under-utilized UHF/VHF frequency band, which is originally assigned for broadcast television services (hereafter called TV channels) [4, 5]. The main reason for choosing the UHF/VHF frequency band to support cognitive communications is two-fold. First, this band is largely under-utilized by the TV broadcast services. Second, this low-frequency band can support long-distance wireless communications with low transmission power (hence low energy consumption), comparing with the current spectrum band used by cellular and WiFi networks.
In such a database-assisted network, the geo-location database houses a global repository of TV licensees, and updates the licensees’ channel occupations periodically. Each WSD obtains the available TV channel information via querying a geo-location database (via some existing communication networks such as cellular or Wi-Fi networks), rather than having to directly sense the TV channels which lead to significant energy consumption. In other words, WSDs are mainly responsible for performing the necessary local computations (e.g., identifying their current locations), and databases are responsible for performing intensive data processing (e.g., computing the available TV channels for each WSD based on the channel availabilities and WSD location information). Such a network architecture can effectively reduce the overall energy consumptions and lead to a green communication ecosystem.
According to existing related regulations, the geo-location databases are operated by third-party companies (instead of directly by the regulators or TV license owners). These database operators such as SpectrumBridge, Microsoft, and Google need to cover its capital expense (CapEx) and operating expense (OpEx) through a properly designed business model. Existing models related to the database-assisted network can be categorized into two classes: Spectrum Market and Information Market.
The spectrum market (e.g., [6, 7, 8]) focuses on the trading of licensed TV channels, which are registered to some TV licensees but are under-utilized by the licensees. Hence, the licensees can temporarily lease the under-utilized (licensed) TV channels to WSDs which are able to enjoy an exclusive usage right during a short time period. This will generate some additional revenue for the licensees. The database serves as a market platform facilitating such a spectrum market.222For example, it can act as a spectrum broker or agent, purchase spectrum from licensees and then resell the purchased spectrum to unlicensed users. Spectrum Bridge, the world first certified geo-location database, provides such a database-provided spectrum market platform called SpecEx .
The information market has been modeled and analyzed for the unlicensed TV channels (i.e., TV white spaces) in our early work [10, 11]. The unlicensed TV channels are those not registered to any TV licensee at a particular location (for example, outside the official coverage range of the TV towers), hence are the public resources at that location. The spectrum regulators can assign the unlicensed TV channels for the public and shared usage among unlicensed WSDs, and usually do not allow direct trading of such channels in a spectrum market. As these channels will be used by WSDs in a shared manner (in contrast to the exclusive usage in the spectrum market), the communication quality in these unlicensed TV channels is usually not guaranteed. Notice that the database knows more advanced information regarding the quality of unlicensed TV channels than unlicensed users.333For example, based on the knowledge about the network infrastructures of TV licensees and their licensed channels, the database can predict the average interference (from licensed devices) on each TV channel at each location. Hence, it can sell this information to the unlicensed users through an information market, which not only improves the unlicensed users’ expected communication quality, but also provides additional profit to the database. A commercial example of information market is White Space Plus , again operated by Spectrum Bridge.
All of above listed works considered the spectrum market and information market separately and independently. In practice, however, the licensed TV channels and unlicensed TV channels often co-exist at a particular location. Some users may prefer to lease the licensed TV channels for the exclusive usage, while other users may prefer to share the free unlicensed TV channels with others (and purchase advanced information if needed). Hence, a joint formulation and optimization of both spectrum market and information market is important for the practical large scale deployment of the database-based TV white space network. However, none of the existing work on economics of TV white space networks [6, 7, 8] looked at the interaction between spectrum market and information market. This motivates our study of an integrated spectrum and information market for such a database-assisted TV white space network.
In this paper, we model and study an integrated spectrum and information market for a database-assisted TV white space network, in which the geo-location database serves as (i) a spectrum market platform for the trading of (under-utilized) licensed TV channels between spectrum licensees and unlicensed users444For convenience, we will call “white space devices” as “unlicensed users” in this paper., and (ii) an information market platform for the trading of information (regarding the unlicensed TV channels) between the database itself and unlicensed users. Unlicensed users can choose to lease the licensed TV channels from licensees (via the database) for the exclusive usage, or to share the free unlicensed TV channels with others and purchase the advanced information from the information market if needed. Figure 1 illustrates such an integrated market.
To understand the market dynamics and equilibrium behaviors in such an integrated market, we formulate the interactions among the geo-location database (operator), the spectrum licensee, and unlicensed users as a three-stage hierarchical model:
I-B1 Stage I: Commission Negotiation (Section Vi)
In Stage I, the database and the spectrum licensee negotiate regarding the commission for the licensee to use the spectrum market platform. Specifically, the database provides a platform facilitating the under-utilized licensed TV channels trading between the licensee and unlicensed users. Such a database-assisted licensed spectrum leasing has been widely adopted in the current Licensed Shared Access (LSA) and Authorized Shared Access (ASA) system. In return, the database takes some commission charge from each successful transaction between the spectrum licensee and unlicensed users. We consider two different commission charging schemes: (i) revenue sharing scheme (RSS), where the licensee shares a fixed percentage of revenue with the database, and (ii) wholesale pricing scheme (WPS), where the database charges a fixed wholesale price from each successful transaction of the licensee. We assume that both the database and spectrum licensee have market powers, and study the equilibrium commission charge decisions under both schemes using the Nash bargaining theory .
I-B2 Stage II: Price Competition Game (Section V)
In Stage II, the database and the spectrum licensee compete with each other for selling information or channels to unlicensed users. The spectrum licensee decides the price of the licensed TV channels, and the database decides the price of the advanced information (regarding the unlicensed TV channels). We analyze such a price competition game, and show that it is a supermodular game  with nice properties.
I-B3 Stage III: User Behavior and Market Dynamics (Section Iv)
In Stage III, unlicensed users decide the best subscription decisions, given the database’s information price and the licensee’s spectrum price. Note that the users need to consider both negative network externality (due to congestion and interference) and positive network externality (due to the quality of information provided by the database) of the information market. We will show how the market dynamically evolves based on users’ choices, and what the market equilibrium point is.
In our model, Stage I and Stage II focus on cooperation and competition, respectively. In Stage I, we study the bargaining process between the database and the licensee, and analyze how they reach an collaborative agreement on the leasing service. In Stage II, we study the pricing strategies of the database and the licensee who target at different user groups. The licensee targets at those users who choose the licensed channels for the exclusive usage, while the database targets at those users who choose the unlicensed channels. The goal of the cooperation is to make the pie larger, and the goal of competition is to decide the way to divide the pie .
The timescales of three stages are as follows.
The bargaining in Stage I is performed at a large time scale, e.g., once every year;
The price competition in Stage II occurs at a medium time scale, e.g., once every month;
The user subscription in Stage III changes at a small time scale, e.g., once every day.
This three-stage hierarchical order of decision making can be explained as follows. In Stage I, the licensee and the database would not frequently re-negotiate the new terms, as it would consume time and resources to reach an new agreement and change the resource based on the new agreement. They will only negotiate again when the network resources or network infrastructure change. In Stage II, the achieved price equilibrium depends on the database’s and the licensee’s costs for providing the advanced service or leasing service to users. In practice, these costs will not change frequently (e.g., change per one month), and neither will the price equilibrium. In Stage III, a user’s subscription decision depends on his instantaneous preference for data rate (i.e., the user’s type in our model). We divide the whole time period into multiple frames, each lasting for certain time (e.g., one day), and assume that the distribution of users’ preference remains the same across different time frames.
We summarize the main contributions as follows.
Novelty and Practical Significance: We propose and study the first integrated spectrum and information market in the literature, for promoting the unlicensed spectrum access to both licensed and unlicensed TV channels. The proposed model captures both the positive and negative network externaltiy of the TV white space network.
Market Equilibrium Analysis: We characterize the sufficient condition under which the proposed integrated market has a unique (user subscription) market equilibrium. We prove that the unique equilibrium is stable, in the sense that a small fluctuation on the equilibrium will drive the market back to the equilibrium.
Competition Between the Database and the Licensee: We study the price competition between the database and the licensee and prove the existence and uniqueness of the price equilibrium. Key contributions of our analysis involves the transformation of the price competition game into an equivalent market share competition game, and the demonstration of the existence and uniqueness of the equilibrium of the transformed game using supermodular game theory.
Nash Bargaining on the Commission Charge: We adopt the Nash bargaining framework to achieve a fair and Pareto-efficient commission charge between the database and the licensee, under both revenue sharing scheme (RSS) and wholesale pricing scheme (WPS).
Observations and Insights: We show that in this integrated spectrum and information market, the market share equilibrium of the licensee is always no more than half. In terms of the network profit, our proposed integrated market scheme can improve up to comparing with a pure information market, and the gap with the coordinated benchmark is less than . When further comparing our two proposed commission charging schemes, we show that the revenue sharing scheme (RSS) always outperforms wholesale pricing scheme (WPS) in terms of social welfare maximization. In terms of maximizing database’s and licensee’s own profit, the result will depend on the level of network externality. When the negative network externality is dominant, RSS is a better choice for the database, while WPS is a better choice fort the licensee. When the positive network externality is dominant, WPS is a better choice for both the database and the licensee.
The rest of the paper is organized as follows. In Section II, we review the related literature. In Section III, we present the system model. In Sections IV-VI, we analyze the market equilibrium in Stage III, the price competition game game in Stage II, and the commission bargaining solution in Stage I, respectively. In Section VII, we provide the simulation results. Finally, we conclude in Section VIII.
Ii Related Work
Most of the existing studies on green cognitive communications aimed at addressing the technical issues. For example, Hafeez and Elmirghani in  presented a new licensed shared access spectrum sharing scheme to increase the energy efficiency in a network. Palicot in  demonstrated how to achieve green radio communications by employing cognitive radio technology. Ji et al. in  proposed a platform to explore TV white space in order to achieve green communications in cognitive radio network. Successful commercialization of new green cognitive technology, however, not only relies on sound engineering, but also depends on the proper design of a business model that provides sufficient incentives to the involved parties such as spectrum licensees and the network operators. The joint study of technology and business issues is relatively under explored in the current green cognitive radio literature.
A common approach for studying market price competition is to model and analyze it as a non-cooperative game. For example, Niyato et al. in  proposed an iterative algorithm to achieve the Nash equilibrium in the competitive spectrum trading market. Min et al. in  studied two wireless service providers’ pricing competition by considering spectrum heterogeneity. Zhu et al. in  studied pricing competition among macrocell service providers via a two-stage multi-leader-follow game. In the above literature, the market is assumed to be associated with the negative network externality or non-externality. Luo et al. in  studied the price competition in the information market of TV white space, where the information market is only associated with the positive network externality. In our work, the integrated market is associated with both the positive and negative network externality. Our numerical results show that the database benefits from the positive network externality, while the licensee benefits from the negative network externality. Furthermore, which commission charging scheme is better for the database or the licensee depends on what kind of network externality is dominant in the network. This makes our market analysis quite different with the above works.
Iii System Model
We consider a database-assisted TV white space network, with a geo-location database (or database for short) and a set of unlicensed users (or users for short). There exist some unlicensed TV channels, which can be used by unlicensed users freely in a shared manner (e.g., using CDMA). Meanwhile, there is a spectrum licensee, who owns some licensed channels and wants to lease the under-utilized channels to users for additional revenue.555In case there are multiple spectrum licensees, we assume that they are coordinated by the single representative. We will study the more general issue of licensee competition in the future work. Different from the unlicensed TV channels, the licensed TV channels can be used by users in an exclusive manner (with the permission of the licensee). Therefore, users can enjoy a better performance (e.g., a higher data rate or a lower interference) on the licensed TV channels.
Iii-a Services Offered by the Database
Iii-A1 Basic Service
Regulators such as Ofcom in UK and FCC in US require a database to provide an unlicensed user with the following information [4, 5]: (i) the list of unlicensed TV channels, and (ii) each channel’s transmission constraints such as a user’s maximum allowable transmission power. Any user can have this basic (information) service for free.
Iii-A2 Advanced Service
Beyond the basic information, the database can also provide certain advanced information regarding the quality of TV channels (as SpectrumBridge did in ), as long as it does not conflict with the free basic service. pWe refer to such additional service as the advanced (information) service. Appendix A illustrates an example of the advanced information in terms of the interference level on each channel. With the advanced information, the user is able to choose a channel with the highest quality (e.g., with the lowest interference level). Hence, the database can sell this advanced information to users for profit. This leads to an information market.
Iii-A3 Leasing Service
As mentioned previously, the database can also serve as a spectrum market platform for the trading of licensed channels between the spectrum licensee and users, which we call the leasing service (as SpectrumBridge did in ). In return, the database will charge commission to the spectrum licensee when a trading happens. Through using the database as the trading platform, the licensees received the aggregation benefit , comparing with the case that they try to directly reach leasing agreement with users. Specifically, due to the database’s proximity to both spectrum licensees and the users, the database’s spectrum market platform can aggregate users demand and licensees spectrum, provide trust between participants, and match users and licensees. Hence, the licensees can save time and market efforts in identifying the potential buyers. We consider two different commission charging schemes: (i) revenue sharing scheme (RSS), where the licensee shares a fixed percentage of revenue with the database, and (ii) wholesale pricing scheme (WPS), where the database charges a fixed wholesale price from each successful transaction, regardless of the licensee’s revenue from that transaction.666Both commission charging schemes are widely used in practice. The revenue sharing scheme is widely used in retail markets such as [24, 25, 26]. The wholesale pricing scheme is widely used in many newsvendor models such as [20, 27, 28].
Iii-B A User’s Choices
Users can choose either to purchase the licensed channel from the licensee for the exclusive usage, or to share the unlicensed TV channels with others (with and without advanced information). We assume that all licensed and unlicensed TV channels have the same bandwidth (e.g., 8MHz in UK), and each user only needs one channel (either licensed or unlicensed) at a particular time.
Formally, we denote as the strategy of a user, where
: Choose the basic service (i.e., share unlicensed channels with others, without the advanced information);
: Choose the advanced service (i.e., share unlicensed channels with others, with the advanced information).
: Choose the leasing service (i.e., lease the licensed channel for the exclusive usage).
We further denote , , and as the expected utilities that a user can achieve from choosing the basic service (), the advanced service (), and the leasing service (), respectively. Here, , , and denote the fractions of users choosing the basic service, the advanced service, and the leasing service, respectively. For convenience, we refer to , , and as the market shares of the basic service, the advanced service, and the leasing service, respectively. Obviously, and . As explained in Section III-C, the values of and depend on all users’ choices, while the value of is independent of market share. The payoff of a user is defined as the difference between the achieved utility and the cost (i.e., the information price when choosing the advanced service, or the leasing price if choosing the leasing service). Let denote the user’s evaluation for the achieved utility777We consider a simplified but rather general model, where the QoS of a user is a linear function of the expected data rate., denote the leasing price of the licensee, and denote the (advanced) information price of the database. Then, the payoff of a user with an evaluation factor is
A rational user will choose a strategy to maximize its payoff. Note that users are heterogeneous in terms of , which characterizes how different users evaluate the same data rate. For example, for a user who wants to send a plain text email, his evaluation for a small data rate may be similar to that for a high data rate, as a small data rate is enough for finish sending the email. This can be captured by a small . On the other hand, if the email contains a large attachment and needs to be sent within a short amount of time (such as tens of seconds), then the user has a much higher valuation for a high data rate and hence a large . Similarly, for a user who is watching a high-quality online video, his evaluation for a small data rate may be much lower than that for a high data rate. This can also be captured by a large .
Let denote the energy consumption cost of the database for providing the advance service, and let denote the energy consumption cost of the licensee for providing the leasing service. For the rest of the paper, we will also use ”operational cost” to refer to these costs.
We further denote as the revenue sharing percentage under RSS, and as the wholesale price under WPS. We assume a unit population of agents., (i.e., the total number of users is equal to ). Then, the payoffs (profits) of the spectrum licensee and the database , which are defined as the difference between the revenue obtained by providing the services and the cost, are given as follow. The payoffs under the RSS scheme (Scheme I) are
and under the WPS scheme (Scheme II) are
Iii-C Positive and Negative Network Externalities
Network externalities arise when a user’s experiencing of consuming a service/product depends on the behavior of other users in the same network . In the integrated market that we study, there coexist two types of network externalities. The negative network externality characterizes the user performance degradation due to an increased level of congestion. The positive network externality corresponds to the increasing quality of the (advanced) information as more users purchase the information. Next we analytically quantify these two network externalities. As , sometimes we also denote the total fraction of users using unlicensed TV channels as in the rest of the paper.
We first have the following observations for a user’s expected utility of three strategy choices:
is a constant independent of , , and . This is because a user can access to the licensed channel exclusively, hence the communication performance on such a channel does not depend on the choices of others.
is non-increasing in (the total fraction of users using unlicensed TV channels) due to the congestion effect. This is because the unlicensed TV channels must be used in a shared manner, hence more users using these channels increases the level of congestion (interference) and reduces the performance of each user. We denote the congestion effect caused by users using the same unlicensed TV channels as negative network externality.
is non-increasing in , due to the negative network externality. This is because the unlicensed TV channels are shared by users, independent of their choices of subscribing to the database’s advanced service or not.
is non-decreasing in . As more users subscribe to the database’s advanced service, the more information (e.g., users channel choices and transmission power levels) the database knows. In such case, the database can estimate more accurate channel information, which benefits the users who subscribe to the advanced service. More detailed explanation is provided in Appendix A. Such benefit that increases with the users choosing the advanced service is called the positive network externality.
We write as a non-increasing function of , i.e.,
We write as the combination of a non-increasing function of and a non-decreasing function of , i.e.,
Function reflects the congestion effect, and is identical for and (as users experience the same congestion effect in both basic and advanced services). Function reflects the performance gain induced by the advanced information, i.e., the (advanced) information value.
We have the following natural assumption:
The expected utilities achieved by choosing different services satisfy
Comparing with unlicensed TV channels, there is no congestion on the licensed TV channels. Hence, the expected utility provided by the leasing service is larger than that provided by the advanced service (i.e., and ). As advanced information provides benefit to the users, we have . 888If we set , no one will choose the leasing service even the leasing service is free of charge. In this case, our integrated model degenerates to the pure information market that is analyzed in . If we set , then no user will choose the advanced service even the database provides the advanced service for free. In this case, our integrated model degenerates to a simpler market, where the licensee provides the leasing service and the database provides the basic service only. The analysis of such a model is simpler than the most general case that we consider here.
We further introduce the following assumptions on functions and .
Function is non-negative, non-increasing, convex, and continuously differentiable.
Function is non-negative, non-decreasing, concave, and continuously differentiable.
Due to the increasing marginal performance degradation under congestion, we assume that function is non-increasing and convex. Such assumption is widely used to model the network congestion effect in wireless networks (see, e.g., [29, 30] and references therein). Because of the diminishing marginal performance improvement induced by the advanced information, we assume that function is non-decreasing and concave. Note that the above generic functions and can potentially model a wide range of scenarios where advanced information has different meanings. We will provide more detailed discussions in the Appendix A.
Iii-D Three-Stage Interaction Model
Based on the above discussion, an integrated spectrum and information market involves the interactions among the database, the spectrum licensee, and the users. Hence, we formulate the interactions as a three-stage hierarchical model as shown in Figure 2.
|Stage I: Commission Negotiation|
|The database and the spectrum licensee negotiate the commission charge details (i.e., under RSS or under WPS).|
|Stage II: Price Competition Game|
|The database determines the information price ;|
|The spectrum licensee determines the channel price .|
|Stage III: User Behavior and Market Dynamics|
|The users determine and update their choices according to best responses;|
|The market evolves to an equilibrium point.|
Specifically, Stage I captures the negotiation process between the database and the spectrum licensee, who negotiate the commission charge details of the spectrum market platform, i.e., the revenue sharing factor under RSS, or the wholesale price under WPS. Stage II studies the price competition between the database and the spectrum licensee, where the database determines the advanced information price , and the spectrum licensee determines the leasing licensed channel price . Stage III focuses on the subscription behaviors of users, where each user makes his best choice, and dynamically updates the subscription choice based on the current market shares.
In the following sections, we will use backward induction to analyze this three-stage interaction model.
Iv Stage III – User Behavior and Market Equilibrium
In this section, we study the user behavior and market dynamics in Stage III, given the database’s information price and the licensee’s channel price (in Stage II). In the following, we first discuss the user’s best response choice, then show how the user behavior dynamically evolves, finally discuss how the market converges to an equilibrium point.
Iv-a User’s Best Response
Equations (1), (4) and (5) show that users’ choices of services depend on the current market shares of different services. Hence given the market shares, users can compute their best response choices, which in turn will affect the market shares. Next we will characterize such a process in details.
For convenience, we introduce a virtual time-discrete system with slots , where users change their decisions at the beginning of every slot, based on the market shares at the end of the previous time slot. 999 The main purpose of introducing the virtual time-discrete system, similar as the best response iterative algorithm in classic game theoretic analysis, is to characterize the relation between the price and the market equilibrium, and to facilitate the calculation of database’s optimal price strategy later. Such an analysis technique (i.e., using a dynamic system to understand the outcome of a one-shot system) has been extensively adopted in the existing literature, e.g., [31, 32]. Let denote the market shares at the end of slot satisfying , where is the market share feasible set defined as . For convenience, we assume that is uniformly distributed in for all users.101010Uniform assumption has been frequently used in the past networking literature (e.g., [31, 29, 33]), and the relaxation to more general distribution often does not change the main insights. As each user will choose a strategy that maximizes its payoff defined in (1), a type- user’s best response is
where and . 111111Here we have written as . For convenience, we will use (the leasing service’s market share) and (the advanced service’s market share) to represent the market state, since the basic service’s market share can be directly computed with and .
To better illustrate the above best response, we introduce the following notations:
The above three notations denote three user type thresholds. For example, is defined as the type of user who is indifferent between choosing the leasing service and the basic service (i.e., both services provide the user with the same expected payoff). Hence, a user with a type would achieve a higher expected payoff when choosing the leasing service than choosing the basic service. Similarly, a user with the type threshold is indifferent between the basic service and advanced service, and a user with the type threshold is indifferent between the leasing service and advanced service. Combining these three user types thresholds together, we can compute the market share of each service.
Figure 3 illustrates several possible relationships among , , and . Intuitively, the users with low values of are more willing to choose the free basic service. The users with medium values of are willing to choose the advanced service, in order to achieve a relatively large utility with a relatively low service cost. The users with high values of are more willing to choose the leasing service so that they can obtain a large utility. Notice that we have if the information price is high or the information value is low. In this case, users will not choose the advanced service, as shown in the bottom subfugure in Figure 3.
Next we characterize the market shares in slot due to users’ best responses. This can help us understand the user behaviour dynamics and market evolutions in the next subsection. As we assume that is uniformly distributed in , the newly derived market shares in slot , given any market shares at the end of , are
If , then and ;
If , then and .
We summarize this in Lemma 1.
Given any pair of market shares at the end of slot , the derived pair of market shares in slot are given by
where , , and are given in (7).
Under the assumption that all users update the best responses once synchronously, we can get the results in Lemma 1. The detailed proof can be found in Appendix A-B. Since , , and are functions of market shares , the derived market shares in slot are also functions of , and hence can be written as and .
Iv-B Market Dynamics and Equilibrium
When the market shares change, the users’ payoffs (on the advanced service and basic service) change accordingly, as both and change. As a result, users will update their best responses repeatedly, hence the market shares will evolve dynamically until reaching a stable point (called market equilibrium). In this subsection, we will study such a market dynamics and equilibrium, given fixed prices (which are decided in Stage II).
Base on the analysis in Section IV-A, let denote the initial market shares in slot and denote the market shares derived at the end of slot (which serve as the initial market shares for the next slot ). We further denote and as the changes (dynamics) of market shares between two successive time slots, e.g., and , that is,
where are the derived market share in slot , and can be computed by Lemma 1.
Obviously, if both and are zero in a slot , i.e., and , then users will no longer change their strategies in the future. This implies that the market achieves the market equilibrium. Formally,
Definition 1 (Market Equilibrium).
A pair of market shares is a market equilibrium, if and only if
Next, we study the existence and uniqueness of the market equilibrium, and further characterize the market equilibrium analytically. These results will help us analyze the price competition game in Stage II (Section V).
Proposition 1 (Existence and Uniqueness).
Given any feasible price pair , there exists at least one market equilibrium . Furthermore, the market equilibrium is unique and the market dynamics globally converges to it starting from any initial point if
where , and is the first-order derivative of with respect to .
We prove the convergence of this market dynamics based on the contraction mapping theorem , with the detailed proof in Appendix A-C. A practical implication of condition (11) is that the existence of a unique equilibrium requires the information value (which corresponds to positive network externality) increases slowly with . Note that the condition (11) is sufficient but not necessary for the uniqueness. Our numerical simulations show that the market converges to a unique equilibrium for a wide range of prices, even when the condition (11) is violated. Nevertheless, the sufficient condition in (11) suggests that there could be multiple equilibrium points if the impact of positive network externality is significant.
Suppose the uniqueness condition (11) is satisfied. We characterize the unique equilibrium by the following theorem.
Theorem 1 (Market Equilibrium).
Suppose the uniqueness condition (11) holds. Then, for any feasible price pair ,
If , there is a unique market equilibrium satisfies
If , there is a unique market equilibrium satisfies
A practical implication of Theorem 1 is that the information price should not be too high or the information value (i.e., ) should be large enough; otherwise, no users will choose the advance service at the equilibrium.
V Stage II – Price Competition Game Equilibrium
In this section, we study the price competition between the database and the spectrum licensee in Stage II, given the commission negotiation solution in Stage I. Based on the analysis of Stage III in Section IV, the database and spectrum licensee are able to predict the user behavior and market equilibrium in Stage III when making their pricing decisions. We will analyze the pricing equilibrium under both the revenue sharing scheme (RSS) and the wholesale price scheme (WPS).
Definition 2 (Price Competition Game).
For the rest of this section, we assume that condition (8) holds. Then, we write the unique market equilibrium in Stage III as functions of prices , i.e., and . Intuitively, we can interpret and as the demand functions of the licensee and the database, respectively.
V-a Revenue Sharing Scheme — RSS
We fist study the game equilibrium under RSS, where the licensee shares a fixed percentage of revenue with the database. By (2), the payoffs of the licensee and the database can be written as:
Definition 3 (Price Equilibrium).
A price pair is a Nash equilibrium, if
It is difficult to analytically characterize the market equilibrium under a particular price pair . We tackle this challenge by transforming the original price competition game (PCG) into an equivalent market share competition game (MSCG). In such a case, the market shares are the strategies of the database and the licensee, while the prices are the functions of the market shares.
A key observation of such a transformation is that, under the uniqueness condition (11), there is a one-to-one correspondence between the market equilibrium and the prices . Because of this, once the licensee and the database choose the prices , they have equivalently chosen the market shares . Substitute and into (13), we can derive the inverse function of (13), where prices are functions of market shares defined on , i.e.,121212Note that we omit the trivial case in (12), where the database has a zero market share in terms of the advanced service. In this case, the licensee can determine the market share splitting (between leasing service and basic service from the database) by optimizing his leasing price .
The payoffs of the database and the licensee can be rewritten as:
Definition 4 (Market Share Competition Game).
The equivalent Market Share Competition Game (MSCG) is defined as follows.
Players: The database and the spectrum licensee;
Strategies: Market share for the database, and for the licensee, where ;
Payoffs: Payoffs are defined in (17).
Definition 5 (Market Share Equilibrium).
Market shares is a Market Share Equilibrium if
We first show that the equivalence between the original PCG and the above MSCG.
Proposition 2 (Equivalent Games).
If is a Market Share Equilibrium of MSCG, then given by (16) is a Price Equilibrium of PCG.
We next characterzie the Market Share Equilibrium of the MSCG. We first give the following proposition which bounds the market shares maximizing the database’s and the licensee’s expected payoffs in (18).
Proposition 3 (Boundary of Market Share Equilibrium).
For any that is a solution of (18), we have and .
Proposition 3 shows that the licensee will achieve an equilibrium market share that is smaller than half. Intuitively, the objective of the licensee is to maximize its own profit, and a larger market share of licensee corresponds to a smaller market price, which does not necessarily increase the licensee’s profit. Another key insight of Proposition 3 is that considering (18) alone is enough to guarantee the feasibility constraint of in Definition 5. Hence, we can study the existence and uniqueness of the MSCG market share equilibrium by analyzing (18) only.
Theorem 2 (Existence and Uniqueness of Market Share Equilibrium).
Given the commission charge , there exists at least one Market Share Equilibrium of MSCG . Furthermore, a sufficient condition for the uniqueness of the market share equilibrium is
For a supermodular game with a unique Nash equilibrium, several commonly used updating rules are guaranteed to converged to the NE . In this paper, we use the best response algorithm as in . Due to the space limit, we provide the detailed algorithm in Appendix A-I.
Once we obtain the Market Share Equilibrium of MSCG, we can compute the Price Equilibrium of the original PCG by (16)
V-B Wholesale Pricing Scheme — WPS
We now study the game equilibrium under WPS, where the database charges a fixed wholesale price from each successful transaction of the licensee. By (61), the payoffs of the licensee and the database can be written as:
With the similar analysis used in Section V-A, we can transform the Price Competiton Game (PCG) into an equivalent Market Share Competition Game (MSCG), and show that the MSCG is a supermodular game. Our key results about the existence and uniqueness of game equilibrium are as follows.
Given the wholesale price , there exists a unique Market Share Equilibrium and for the MSCG, and thus a unique Price Equilibrium, if the following conditions are satisfied
The detailed discussions in Appendix H.
Although we use the similar method to derive the Nash equilibrium of the PCG game under both RSS and WPS, these two equilibria are quite different. Intuitively, under RSS, the objective of the database is consistent with that of the licensee. This can be shown by the common term in both players’ payoffs given in (14). Hence, the database becomes less aggressive in competing with spectrum licensee under RSS that under WPS.
To emphasize the fact that the equilibrium payoffs in Stage II depend on (under RSS) or (under WPS), we will write the equilibrium payoff of the database as under RSS and under WPS. Similarly, we will write the equilibrium payoff of the licensee as under RSS and under WPS.
Vi Stage I – Commission Bargaining Solution
In this section, we study the commission negotiation among the database and the spectrum licensee in Stage I, based on their predictions of the price equilibrium in Stage II and the market equilibrium in Stage III131313In our proposed bargaining model, the database and the licensee only conduct the bargaining if this leads to a payoff increase for both sides. Otherwise they can choose not to bargain and do not cooperate in Stage I. Furthermore, based on the discussions in the previous paragraph, we can see that the licensee does not have the full market power. Hence a bargaining model is suitable for such a market. .
Specifically, we want to find a feasible revenue sharing percentage under RSS, or a feasible wholesale price under WPS, that is satisfactory for both the database and the spectrum licensee. This is motivated by the fact that both the database operator (e.g., Google, Microsoft, and SpectrumBridge) and spectrum licensees (e.g., ATT, Verizon, and China Mobile) have considerable market power, and one side can not determine or alone. We formulate the commission negotiation problem as a bargaining problem, and solve it using the Nash bargaining theory .
Vi-a Revenue Sharing Percentage -Bargaining under RSS
We first study RSS, where we want to determine the revenue sharing percentage .
We first derive the database’s and the licensee’s payoffs when reaching an agreement and when not reaching any agreement. This allows us to characterize the payoff improvements due to successful bargaining.
Formally, when reaching an agreement , the database’s and the licensee’s payoffs are and derived in Section V-A, respectively. When not reaching any agreement (reaching the disagreement), the licensee’s profit is , and the database’s profit is , where and are the database’s optimal price and the corresponding market share in the pure information market.141414Such an optimal price and the corresponding market share can be derived in the same way as in Section V-A, by simply setting . Then, the Nash bargaining solution is the solution of the following optimziation problem,
Analytically solving (22) may be difficult, if we are not able to analytically characterize and . Nevertheless, we notice that the bargaining variable lies in a closed and bounded range of , and the objective function of (22) is bounded. Hence, there exists at least one optimal solution for (22). As our numerical results show that the obejctive function of (22) is approximately quadratic in , the optimal solution is unique and can be found by using one-dimensional search methods (e.g., ).
Vi-B Wholesale Price -Bargaining under WPS
We now study WPS, where the database charges the spectrum licensee a fixed wholesale price for each successful transaction of the latter.
When reaching an agreement , the database’s and the licensee’s payoffs are and derived in Section V-B, respectively. When not reaching any agreement (reaching the disagreement), the licensee’s profit is , and the database’s profit is , which is same as that under RSS (Section VI-A). The Nash bargaining problem is
We further notice that that we can restrict the bargaining variable within a closed and bounded set, say , while not affecting the optimality of the solution. This is no user will choose the leasing service when , in which case the spectrum licensee will get a zero payoff. This is certainly not an optimal solution of (23). Similar as Section VI-A, there would exist a unique solution for (23) as shown in our simulations, and which can be found effectively through one-dimensional numerical search.
Vii Simulation Result
We evaluate the system performance (e.g., the database’s profit, the network profit, and the social welfare) achieved under both revenue sharing scheme (RSS) and wholesale price scheme (WPS) through extensive numerical studies. We will focus on the impact of system parameters (i.e., the degree of network externality and the licensee’s operating cost) on system performance.
Vii-a Simulation Setting
As a concrete example, we choose