two-sided matching based cooperative spectrum...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMC.2016.2557794, IEEETransactions on Mobile Computing

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Two-sided Matching Based CooperativeSpectrum Sharing

Lin Gao, Member, IEEE, Lingjie Duan, Member, IEEE, and Jianwei Huang, Fellow, IEEE

Abstract—Dynamic spectrum access (DSA) can effectively improve the spectrum efficiency and alleviate the spectrum scarcity, byallowing unlicensed secondary users (SUs) to access the licensed spectrum of primary users (PUs) opportunistically. Cooperativespectrum sharing is a new promising paradigm to provide necessary incentives for both PUs and SUs in dynamic spectrum access.The key idea is that SUs relay the traffic of PUs in exchange for the access time on the PUs’ licensed spectrum. In this paper, weformulate the cooperative spectrum sharing between multiple PUs and multiple SUs as a two-sided market, and study the marketequilibrium under both complete and incomplete information. First, we characterize the sufficient and necessary conditions for themarket equilibrium. We analytically show that there may exist multiple market equilibria, among which there is always a unique Pareto-optimal equilibrium for PUs (called PU-Optimal-EQ), in which every PU achieves a utility no worse than in any other equilibrium.Then, we show that under complete information, the unique Pareto-optimal equilibrium PU-Optimal-EQ can always be achieveddespite the competition among PUs; whereas, under incomplete information, the PU-Optimal-EQ may not be achieved due to themis-representations of SUs (in reporting their private information). Regarding this, we further study the worse-case equilibrium for PUs,and characterize a Robust equilibrium for PUs (called PU-Robust-EQ), which provides every PU a guaranteed utility under all possiblemis-representation behaviors of SUs. Numerical results show that in a typical network where the number of PUs and SUs are different,the performance gap between PU-Optimal-EQ and PU-Robust-EQ is quite small (e.g., less than 10% in the simulations).

Index Terms—Cooperative Spectrum Sharing, Game Theory, Two-Sided Matching, Market Equilibrium

F

1 INTRODUCTION

1.1 Background and Motivation

Wireless spectrum is becoming increasingly congestedand scarce with the explosive development of wirelessdevices and services. Dynamic spectrum access (DSA) isa promising approach to increase the spectrum efficiencyand alleviate the spectrum scarcity, by allowing unli-censed secondary users (SUs) to opportunistically accessto the spectrum licensed to primary users (PUs) [1]-[3].To successfully implement DSA, it is important to offernecessary incentives for PUs to open their spectrum forSUs’ utilizations, and for SUs to access the PUs’ spectrumdespite of the potential costs [4]-[10].

Cooperative spectrum sharing (also called cooperativespectrum leasing [11]) has been proposed as an effectiveapproach to offer crucial incentives for both PUs andSUs in DSA [11]-[17]. The basic idea is that SUs relaythe traffic of PUs in exchange for the opportunities toaccess the PUs’ licensed spectrum. With the cooperativespectrum sharing, PUs can increase their transmissionrates through the cooperative relay of SUs, and SUs can

• Lin Gao is with the School of Electronic and Information Engineering,Harbin Institute of Technology Shenzhen Graduate School, China, Email:[email protected]; Lingjie Duan is with Engineering Systems andDesign Pillar, Singapore University of Technology and Design (SUTD),Singapore, Email: lingjie [email protected]; Jianwei Huang (correspond-ing author) is with the Network Communications and Economics Lab,Department of Information Engineering, The Chinese University of HongKong, Email: [email protected].

• This work is supported by the General Research Funds (Project NumberCUHK 412713) established under the University Grant Committee of theHong Kong Special Administrative Region, China.

PU2

SU2SR2

ST2

SR1 ST1

PU1

SU1

PU3PT1

PT2

PT3

PR1

PR2

PR3

SU3

ST3 SR3

PU BroadcastingPhase I

SU RelayingPhase II

SU TransmittingPhase III

Phase I

Phase IPhase

II

Phase III

Phase IPhase I Phase II

Phase III

Phase I

Phase I

Phase II

Phase III

SR4

ST4

SU4

Fig. 1. Illustration of cooperative spectrum sharing.

obtain transmission opportunities on the PUs’ licensedspectrum. Thus, it will lead to a win-win situationfor PUs and SUs.1 Figure 1 illustrates an example ofcooperative spectrum sharing between 3 PUs and 4 SUs,2

where each SU i cooperates with PU i for i ∈ {1, 2, 3},and SU 4 does not cooperate with any PU. SUs relaytraffic for intended PUs according to a pre-defined co-operative protocol, such as amplified-and forward (AF)and decode-and-forward (DF). Hence, each cooperationframe is divided into 3 phases: in Phase I, each PU i’stransmitter (PTi) broadcasts its messages to its receiver(PRi) as well as to the cooperating SU i; in Phase II,each SU i’s transmitter (STi) forwards the received PUi’s signal to the PU i’s receiver PRi; and in Phase III,each SU i transmits its own messages (from STi to SRi)

1. Cooperative spectrum sharing can also be viewed as an enhancedcooperative relay scheme. From the perspective of cooperative relays(e.g., D2D relays in 5G system [28]), it can provide the necessaryincentives for SUs to relay the traffic of PUs.

2. Here each PU i is a dedicated transceiver pair {PTi, PRi}, andeach SU i is also a dedicated transceiver pair {STi, SRi}.



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on the PU i’s spectrum.While there are some prior works considering the co-

operative spectrum sharing problem [11]-[17], all of theseworks focused on the interactions between one PU andone or multiple SUs. Our work advances the research inthis area by analyzing the interaction between multiplePUs and multiple SUs as shown in Figure 1. This morepractical scenario is significantly more challenging toanalyze, as not only SUs compete with each other for thePUs’ licensed spectrums, but PUs also compete with eachother for the SUs’ collaborations. An important questionarising in such a multi-PU multi-SU scenario is:• Which PU cooperates with which SU, and what is the

resource exchange (i.e., the PU’s spectrum access timereward and the SU’s relay effort) between them?

As the traditional matching problem usually studies thebinary pairing of users without considering the resourceexchange details, our problem is actually an extendedmatching problem.

1.2 Solution and Contribution

In this work, we formulate the cooperative spectrumsharing problem between multiple PUs and multiple SUsas a two-sided matching market, with PUs as one sideand SUs as the other side. A PU is matched to an SUmeans that the PU cooperates with the SU under certainresource exchange agreement. Accordingly, a matchingbetween PUs and SUs defines not only their collabo-rative relationships, but also the resource exchange be-tween each pair of matched PU and SU. An equilibrium isdefined as a stable matching, from which neither PUs norSUs have the incentive to deviate. For such a two-sidedmarket, we want to answer the following questions:• what is the market equilibrium, and which equilibrium

will emerge in different information scenarios?Note that both problems are challenging due to thefollowing reasons. First, finding the stable matching of atwo-sided matching market is well-known an NP-hardproblem, and thus mathematically intractable. Second,in some two-sided matching markets including the onein this paper, there may be an infinite number of marketequilibria, and analytically characterizing these marketequilibria is challenging. Third, different informationscenarios (depending on how much information thatPUs know) may lead to different market equilibria.Characterizing the equilibrium in different informationscenarios is also a challenging problem.

We will study the market equilibrium in two differentinformation scenarios systematically: complete and incom-plete information. In the former case, each PU knowsthe whole network information, while in the latter case,each PU knows only its local network information (seeSection 3.F for details). To the best of our knowledge, thisis the first paper that systematically studies cooperativespectrum sharing between multiple PUs and multipleSUs by using the matching theory. The main results andcontributions of this paper are summarized as follows.

• Two-sided Market Model and Solution Technique:We formulate the cooperative spectrum sharing be-tween multiple PUs and multiple SUs as a two-sidedmatching market, and comprehensively study themarket equilibrium (stable matching) in both com-plete and incomplete information scenarios.

• Equilibrium Analysis: We characterize the sufficientand necessary conditions for market equilibrium,and show that there may be an infinite numberof market equilibria. We further show that theseis always a unique Pareto-optimal equilibrium forPUs (PU-Optimal-EQ) , in which every PU achieves autility no worse than that in any other equilibrium.

• Complete and Incomplete Information: We showthat under complete information, the unique Pareto-optimal equilibrium for PUs (PU-Optimal-EQ) canalways be achieved despite of the competition a-mong PUs. Under incomplete information, however,the PU-Optimal-EQ may not be achieved due to themis-representations of SUs. To this end, we charac-terize a Robust equilibrium for PUs (PU-Robust-EQ),which provides every PU a guaranteed utility underall possible mis-representation behaviors of SUs.

• Performance Evaluation: Numerical studies showthat each PU’s utility increases (or decreases) withthe number of SUs (or PUs) in both PU-Optimal-EQ and PU-Robust-EQ. In many practical networkscenarios where the numbers of PUs and SUs areoften different, the performance gap between PU-Optimal-EQ and PU-Robust-EQ is quite small (e.g.,less than 10% in the simulations). This implies thatthe equilibrium performance depends more on themarket structure than on the information scenario.

The rest of this paper is organized as follows. InSection 2, we review the related literature. In Section 3,we present the system model. In Section 4, we providethe two-sided market formulation. In Section 5, we usea simplified example to illustrate the equilibrium. InSection 6, we study the equilibria of a general model.In Section 7, we show which equilibria will emerge indifferent information scenarios. We provide numericalresults in Section 8, and finally conclude in Section 9.

2 LITERATURE REVIEW

A. Cooperative Spectrum SharingMost prior work on cooperative spectrum sharing fo-cused on the interactions between one PU and one ormultiple SUs [11]-[17]. Some assumed that the PU hascomplete information of SUs (e.g., SUs’ relay channelgains and sensitivities to power consumption) [11]-[13],while others considered that the PU has limited infor-mation about SUs [14]-[16]. These works used eitherStackelberg game or contract theory to model and ana-lyze the problem. However, both approaches are difficultto be extended to the scenario of multiple PUs andmultiple SUs. A closely related paper that considers theinteractions between multiple PUs and multiple SUs is



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[17], where authors proposed a distributed algorithm toreach a stable matching that is weak Pareto optimal. Ourwork generalizes the result of [17] in the following way:(i) we analytically show that there are multiple (possiblyinfinite) stable matchings including the Pareto optimalone studied in [17], (ii) we characterize the necessary andsufficient conditions for all possible stable matchings,and (iii) we propose distributed algorithms convergingto different stable matchings, given different informationavailable to PUs.

B. Two-sided Matching MarketIn economics, two-sided matching market is an effectiveframework and widely-used for studying the interac-tions between two disjoint player sets in a two-sidedmarket setting. In such markets, the matching theo-ry can systematically capture not only the cooperativeinteractions between users in different sides, but alsothe competitive interactions between users on the sameside. Most early results in this area focused on thematching under complete information, without consid-ering the incentive issues under incomplete information.The first basic two-sided market models were proposedby Gale and Shapley [18]. Shapley and Shubik [19]and Thompson [20] studied the more general modelswith additive and transferable utilities. Crawford andKnoer [21] studied the two-sided labor models withnontransferable utilities. Some later results [22] studiedthe incentive issue under incomplete information. Ourwork differs from the above works: we consider notonly the binary matching decision, but also the detailedresource exchange between each pair of matched users.

It is notable that our analysis techniques and engi-neering insights are applicable to other wireless networkproblems that can be modeled as a two-sided market.The equilibrium obtained in this paper provides a quitegeneral characterization of the network stable outcomein a wide range of large complicated networks.

3 SYSTEM MODEL

We consider a DSA network with a setM , {1, ...,M} ofPUs and a set N , {1, ..., N} of SUs. Each PU m ∈M (orSU n ∈ N ) is a dedicated transceiver pair {PTm,PRm}(or {STn,SRn}) as illustrated in Figure 1. A PU has theexclusive usage right of a licensed frequency band (witha normalized bandwidth). We assume that the frequencybands of different PUs are non-overlapping, and thus thereis no interference among PUs’ transmissions. As in manyexisting literature [11]-[17], we further assume that thereexists a common control channel for the communicationsand interactions between PUs and SUs.

On one hand, PUs may suffer from low transmissionrates or high outage probabilities due to the poor chan-nel conditions between their transmitters and receiverscaused by, for example, the long-distance attenuation,shadowing, and fading. Thus, PUs want to employSUs as relays to improve their transmission rates. On

Phase I Phase IIIPhase II

0.5*Tm 0.5*Tm

Time#T-1 #T+1Frame #T

mnt

Fig. 2. Frame structure of cooperative spectrum sharing.

the other hand, SUs need spectrum resources for theirown transmissions, but cannot access the PUs’ licensedspectrum without PUs’ permissions. Thus, we expect awin-win situation where SUs relay PUs’ traffic in orderto get free access time on the PUs’ spectrums.

To facilitate the practical implementation, we assumethat one PU can choose at most one SU to relay its trafficat a particular time (but can change the relaying SU atdifferent time). This is motivated by existing results thatchoosing the most appropriate relay is usually sufficientto achieve the optimal (or close-to-optimal) performance[23]. Moreover, this can be implemented more easily inpractice, as there is no need to consider the coordinationamong multiple relays. We further assume that each SUcan serve at most one PU at a particular time (but canchange the serving PU at different time). Therefore, akey question arising in such a scenario is: who will relaywhose traffic, and how?

3.1 Cooperative Spectrum Sharing Protocol

We first consider the cooperative spectrum sharing pro-tocol between a particular PU m and SU n (assuming SUn cooperates with PU m). As shown in Figure 1, eachcooperation period (frame) includes 3 phases: Phases Iand II for the cooperative communication between PUm and SU n (each with a fixed period of Tm/2),3 andPhase III for SU n’s own transmission (with a period oftmn ). Thus, the total length of one cooperation frame isTm + tmn . For clarity, we illustrate the structure of such a3-phase transmission frame in Figure 2.

Obviously, the cooperative communication protocolused in Phases I and II plays an important role in thecooperative spectrum sharing. In this paper, we adoptthe widely-used Amplified-and-Forward (AF) protocol [24]in Phases I and II.4 For the analytical convenience, weassume the AWGN channel in the physical links. Notethat the channel model can be easily extended to fast fad-ing channels (e.g., Rayleigh channel), by considering theaverage transmission rate or average outage probability.We further assume that each pair of transmitter andreceiver know the channel gain between them through,for example, measuring the received signal strength.

For convenience, we list the key notations in Table 1,where m is the PU index and n is the SU index. Themeaning of each notation will be explained later.

3. The total cooperative transmission period Tm is determined bythe physical layer specifications. Different PUs may have different Tm.

4. Note that our analysis can be easily applied to other coopera-tive protocols (e.g., Decode-and-Forward and Compress-and-Forward)with minor modifications on the utility functions of PUs and SUs.



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TABLE 1Key Notations

Tm PU m’s cooperative communication time (Phases I and II);Cn SU n’s sensitivity of unit power consumption;Gm PU m’s direct channel (PTm to PRm) gain;Gn(m) SU n’s direct channel (STn to SRn) gain on PU m’s band;Gm,n 1st relay channel (PTm to STn) gain of PU m and SU n;Gn,m 2nd relay channel (STn to PRm) gain of PU m and SU n;pmn The relay power of SU n for PU m’s traffic;tmn The dedicated access time on PU m’s band for SU n;Πmn The utility of PU m when cooperating with SU n;

∆mn The utility of SU n when cooperating with PU m;

3.2 PU and SU Modeling

We now define the PU’s and SU’s utilities achieved in thecooperative spectrum sharing. Suppose SU n cooperateswith PU m in the following way: SU n relays the PU m’ssignal with a power pmn , and PU m rewards a dedicatedaccess time tmn on its spectrum to SU n.

With the AF protocol, the PU m’s transmitter PTmbroadcasts the data (with unit power) in Phase I, andSU n normalizes the received signal rn,m by a factor√pmn /|rn,m| and forwards it to PRm in Phase II. The

receiver PRm combines the received signals in PhasesI and II with a maximal ratio combining. Essentially, wecan view the transmission during Phases I and II as asingle-input, two-output complex Gaussian noise chan-nel. Accordingly, the transmission rate during Phases Iand II achieved from i.i.d. complex Gaussian inputs isgiven by the Shannon-Hartley theorem [24]:

Rmn = 12 · log(1 + κmd + κmn ), (1)

where κmd is the signal-to-noise ratio (SNR) on the PUm’s direct channel (PTm to PRm), and κmn is the SNR onthe relay channel of SU n. For convenience, we normalizethe PU’s direct transmission power into 1. Then, we haveκmd =

G2m

σ2 , where Gm is the channel gain of the PU m’sdirect channel and σ2 is the noise power, and

κmn =pmn G

2m,nG

2n,m

pmn G2n,m+G2

m,n+σ2 · 1

σ2 , (2)

where Gm,n is the channel gain of the channel betweenPTm and STn (called the 1st relay channel), and Gn,m isthe channel gain of the channel between STn and PRm

(called the 2nd relay channel). Obviously, the SNR κmnincreases with the SU’s relay power pmn .

Since the cooperative period occupies Tm

Tm+tmnfraction

of the total frame, PU m’s effective transmission rate dur-ing the entire frame (when cooperating with SU n) is

R̃mn = Tm

Tm+tmn· 12 · log(1 + κmd + κmn ). (3)

We define PU m’s utility as the increase of its transmis-sion rate when cooperating with an SU n, i.e.,

Πmn , R̃

mn −Rmd =

Tm·log(1+κmd +κm

n )2·(Tm+tmn ) −Rmd , (4)

where Rmd = log(1 + κmd ) = log(1 + G2m/σ

2) is PU m’sdirect transmission rate without any relay.

Similarly, the SU n’s transmission rate in Phase III onthe PU m’s frequency band is given by [24]:

Rn(m) , log(1 +

G2n(m)

σ2

), (5)

where Gn(m) is the SU n’s direct channel (STn to SRn)gain on the PU m’s frequency band. Note that each SU nmay have different direct channel gains Gn(m) on differ-ent PU m’s band, due to the frequency selective fading.Moreover, there is no interference between SUs’ trans-missions as they operate on non-overlapping bands.

Since the SU n’s transmission occupies tmnTm+tmn

fractionof each frame, its equivalent transmission rate during theentire frame (when cooperating with PU m) is

R̃n(m) =tmn

Tm+tmn· log

(1 +

G2n(m)

σ2

). (6)

The SU n’s total energy consumption during the entireframe (when cooperating with PU m) is

Sn(m) ,Tm

2 · pmn + tmn · 1, (7)

where the first term is due to relaying PU m’s traffic,and the second term is due to SU n’s own traffic (witha normalized unit transmission power).

We define the SU n’s utility (when cooperating withPU m) as the difference between the achieved transmis-sion rate and the incurred power cost, denoted by

∆mn , R̃n(m) − CnSn(m)

Tm+tmn=

tmn (Rn(m)−Cn)−pmn CnTm2

Tm+tmn, (8)

where Sn(m)

Tm+tmnis the average transmission power, and Cn

is its sensitivity for unit power consumption.

4 TWO-SIDED MARKET FORMULATION

Before the detailed modeling of the spectrum sharingmarket, we first provide the definition of a basic two-sided matching market (also called two-sided market).

Definition 1 (Two-sided Matching Market): A two-sidedmatching market is a market consisting of two disjointsets of users, where an user on one side can be matchedwith only one user on the other side.

A one-to-one (binary) matching (or just matching) in atwo-sided market is defined as follows.

Definition 2 (Matching): A one-to-one matching be-tween two disjoint sets M and N can be representedby a one-to-one correspondence µ(·), where m ∈ M ismapped to n ∈ N (i.e., µ(m) = n) if and only if n is alsomapped to m (i.e., µ(n) = m).

For notational convenience, we will also write µ(m) asµm and µ(n) as µn when there is no confusion caused.

4.1 Two-sided Market ModelingIn our model, each PU has different preferences over SUsdepending on the locations of SUs, and wants to selectthe most efficient SU as relay; each SU also has differentpreferences over PUs depending on its channel gains onPUs’ bands, and want to obtain the dedicated spectrumaccess time from the most “generous” PU. Thus, we can



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PU m

SU nSRn

STn Gn,m

Gn(m)

GmGm,n

PTm PRm

Fig. 3. Illustration of the key network information.

formulate this model as a two-sided market, with PUs onone side and SUs on the other side. A PU is matchedto an SU (or equivalently, an SU is matched to a PU)means that the PU cooperates with the SU, that is, the SUrelays traffic for the PU, and the PU rewards dedicatedspectrum access time to the SU. Clearly, in this two-sided market, we need to consider not only the binarymatching between PUs and SUs (given in Definition 2),but also the detailed resource exchange between eachpair of matched PU and SU.5

More specifically, for each pair of matched PU and SU,the PU’s and SU’s utilities depend not only on the SU’srelay power (in Phase II), but also on the PU’s spectrumaccess time reward (in Phase III). In addition, an increasein one user’s utility will lead to a (not necessarily thesame amount of) decrease in the other’s utility in thematching pair. Therefore, we face a two-sided marketwith transferrable utility, where each user’s utility (orpreference) in a given matching pair is not fixed, butdepends on the resource exchange with the matched user(i.e., the access time reward and relay power).

Based on the above discussion, we address the follow-ing matching problem for the proposed market:

Question 1 (Relay-Assignment): What kind of matchingpairs will emerge from users’ strategic interactions?

Question 2 (Resource-Exchange): What is the detailedresource exchange (i.e., the access time and relay power)for each pair of matched PU and SU?

4.2 Market Equilibrium – Stable MatchingNow we consider the above matching problem froma game-theoretical analysis. Note that game theory isapplicable to our model, as PUs and SUs are rationaland self-interested, and always want to maximize theirown utilities. Furthermore, one user’s decision will affectothers’ utilities and decisions. To solve the matchingproblem, it would be useful to understand what kindof matching will be “broken” (and thus unstable).

Definition 3 (Unstable Matching): A matching will bebroken if one of the following conditions is satisfied:

(a) There exist a PU and an SU who are not matchedto each other but both prefer to be;

(b) There exists a PU or an SU with a negative utilityin the existing matching.

5. Since each binary matching is associated with a set of resource ex-changes (each for a pair of matched users), we will use the terminology“matching” to denote both (i) the binary matching between PUs andSUs, and (ii) the resource exchange for each pair of matched users.

GM,n ,Gn,MGM,1 ,G1,M

GM,N ,GN,M

G1(M)

Gn(M)

Gn(1)

GN(1)

GN(M)

G1,1 ,G1,1(2nd)G1,n ,Gn,1G1,N ,GN,1

G1(1)

Relay channel information

PU channel information

SU channel information

PU-s

ide SU

-side

. . .

. . .

. . .

. . .

. . .

. . .

1

m

M N

n

1G1(m)

Gn(m)

GN(m)

Gm,n ,Gn,m

Gm,N ,GN,m

Gm,1 ,G1,m

Incomplete IncormationComplete Information

G1

Gm

GM

Fig. 4. Illustration of complete information scenario andincomplete information scenario.

In (a), the PU and the SU would discard their currentpartners and match to each other. In (b), the PU or SUwould remain single rather than stick to the currentpartner. Thus, a broken matching is “unstable”.

As long as a matching is not broken, we call it a stablematching, or equivalently, a market equilibrium.

Definition 4 (Market Equilibrium): A market equilibri-um is a stable matching, where no user can improveits utility via unilateral deviation (i.e., choosing anotherpartner or changing resource exchange details).

Based on the above definitions, we can rewrite thematching problem (given in Questions 1 and 2) into thefollowing equivalent question: what market equilibrium(stable matching) will emerge in the two-sided matchingmarket? Notice that there are many important factors af-fecting the market equilibrium realization, one of whichis: which side of the market has the market power topropose cooperation offers to the other side [18]. Inthis paper, we consider a PU-proposal market, wherePUs have the authority to propose offers.6 That is, eachPU decides which SU to select and what time rewardand relay power request to/from that SU, and eachSU passively accepts or rejects the received offer. Thestudy of a PU-proposal market is motivated by the factthat PUs usually have more market power than SUsin cooperative spectrum sharing, since their number islimited and they own the scarce spectrum licenses.

4.3 Network InformationIn addition to who are the proposers, the informationscenario also affects the equilibrium. That is, whichequilibrium will eventually appear also depends on howmuch network information are known to PUs. In ourmodel, the key network information mainly include7

(a) Gm: the direct channel gain of each PU m;

6. Note that the analysis for the PU-proposal market can be directlyextended to the SU-proposal market, where SUs propose offers toPUs, and PUs simply accept or reject the received offers.

7. Notice that we view the cooperative transmission time Tm of eachPU m and the power sensitivity Cn of each SU n as public information.This is because these parameters are usually pre-defined and keepunchanged in a long time, given the types of PUs or SUs.



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TABLE 2Information Scenarios and Associated Market Equilibria.

Information Scenario Each PU m’s Knowledge Equilibrium Property Section No.Complete {Gm, Gn(m), Gm,n, Gn,m}n∈N ,m∈M PU-Optimal-EQ Pareto-optimal for PUs Section 6

Incomplete {Gm, Gn(m), Gm,n, Gn,m}n∈N PU-Robust-EQ Guaranteed for PUs Section 7

(b) Gn(m): the direct channel gain of each SU n on eachPU m’s spectrum band;

(c) Gm,n and Gn,m: the relay channel gains betweeneach pair of PU m and SU n.

For convenience, we illustrate the network information(between PU m and SU n) in Figure 3, where the labelin each link denotes the associated channel gain.

In this work, we consider two different informationscenarios: complete information and incomplete informa-tion, depending on how much network information thePUs know. Specifically, in the complete information sce-nario, each PU m is assumed to know the whole networkinformation, i.e., {Gm, Gn(m), Gm,n, Gn,m}n∈N ,m∈M. Inthe incomplete information scenario, each PU m is as-sumed to know its local network information only, i.e.,{Gm, Gn(m), Gm,n, Gn,m}n∈N , but not those of other PUs.Note that the incomplete information scenario is morepractical, as a PU can use reference signal strength (RSS)to detect the channel gains to all potential SUs, yet it ishard for him to estimate other PUs’ channel gains. Forconvenience, we summarize these information scenariosin Figure 4.8

In what follows, we will characterize the sufficientand necessary conditions for the market equilibrium(of the PU-proposal market) in Section 6, and studywhich equilibria will emerge in complete and incompleteinformation scenarios in Section 7. For convenience, wesummarize these information scenarios and the associ-ated market equilibria in Table 2.

5 A SIMPLIFIED MODEL

Before studying the market equilibrium in the generalmodel, we first consider a simplified model where theaccess time reward offered by each PU (in Phase III) andthe relay power contributed by each SU (in Phase II)are fixed. Thus, we only need to consider the binarymatching between PUs and SUs (i.e., the relay assign-ment problem defined in Question 1).9

Note that we use this simple model to illustrate somekey properties of the proposed matching market, whichis essential for understanding the general model in Sec-tions 6 and 7. Moreover, this simplified model is usefulfor the network scenario with inflexible relay protocolsor static physical layer specifications, where secondaryaccess time and relay power cannot be freely adjusted.

8. We further study a strongly incomplete information scenario in theonline technical report [29], where each PU m does not even knoweach SU n’s channel gain, i.e., Gn(m). The equilibrium analysis forsuch a stronger incomplete information scenario is similar.

9. This model is similar to the two-sided market model in [18], [?],and our results in this section also refer to those in [18], [?].

5.1 Users’ Preferences

With fixed time reward and relay power, each user’sutility for each particular partner on the other sideis fixed. Thus, the model degenerates to a two-sidedmatching problem without utility transferring (similar tothe Marriage model considered in [18]).

Let tm denote the (fixed) spectrum access time reward-ed by PU m for any SU collaborating with him, andpn the (fixed) relaying power offered by SU n for anyPU collaborating with him. The preferences of a PU mcan be represented by an ordered list in the decreasingpreference, P (m), on the SUs,10

P (m) = {n1, n2, ..., np}, (9)

subject to (i) Πmni

(tm, pni) ≥ Πmnj

(tm, pnj ) if i < j,and (ii) Πm

n (tm, pn) ≥ 0 for any SU n ∈ P (m), andΠmn′(tm, pn′) < 0 for any SU n′ /∈ P (m). Here Πm

n (·) is thePU m’s utility defined in (4). The first condition meansthat PU m prefers an SU who provides him a higherutility, and the second condition means that PU m neverselects an SU who brings him a negative utility.

Similarly, the preference list of an SU n on the PUs canbe represented by

Q(n) = {m1,m2, ...,mq}, (10)

subject to (i) ∆min (tmi

, pn) ≥ ∆mjn (tmj

, pn) if i < j, and(ii) ∆m

n (tm, pn) ≥ 0 if m ∈ Q(n), and ∆m′

n (tm′ , pn) < 0m′ /∈ Q(n). Here ∆m

n (·) is the SU n’s utility in (8).

5.2 An Example and Its Equilibrium

We provide an example to illustrate the equilibria underthe fixed preferences. Recall that a binary matching be-tween PUs and SUs is represented by a correspondenceµ(·), where µm = n (or equivalently, µn = m) means PUm is matched to SU n, and µm = ∅ or µn = ∅ meansPU m or SU n remains single (i.e., not engaged in thecooperative spectrum sharing).

Example 1: Consider a two-sided market with 3 PUs(m1,m2,m3) and 3 SUs (n1, n2, n3) with preferences:

PUs’ preferences SUs’ preferencesP (m1) n1, n2, n3 Q(n1) m2, m3, m1

P (m2) n2, n3, n1 Q(n2) m3, m1, m2

P (m3) n3, n1, n2 Q(n3) m1, m2, m3

According to the definition of equilibrium in Defini-tion 4, we can easily obtain the following three equilibriaµa, µb, and µc for the above example:

10. The number of elements in set P (m) could be smaller than N ,that is, PU m is only interested in a subset of SUs.



7

Equilibria µa(·) µb(·) µc(·)PUs m1 m2 m3 m1 m2 m3 m1 m2 m3

SUs n1 n2 n3 n2 n3 n1 n3 n1 n2

Let us check the feasibility of the above equilibriausing the definition of market equilibrium. Take theequilibrium µb as an example, where PU m1 is matchedto SU n2, PU m2 is matched to SU n3, and PU m3 ismatched to SU n1. We can easily find that PU m1 prefersSU n1 than its current partner SU n2 (as Πm1

n2< Πm1

n1),

but SU n1 has no incentive to discard its current partnerPU m3 and pair with PU m1 (as ∆m3

n1> ∆m1

n1). Similarly,

PU m2 prefers SU n2 than its current partner SU n3, butSU n2 has no incentive to discard its current partner PUm1 and pair with PU m2. PU m3 prefers SU n3 than itscurrent partner SU n1, but SU n1 has no incentive todiscard its current partner PU m2 and pair with PU m3.Thus, there does not exist a pair of PU and SU who arenot matched to each other but both prefer to be. �

5.3 Common & Conflicting Interests on EquilibriumBy Example 1, we can find that there exists a uniquePareto-optimal equilibrium for PUs: µa, where every PUachieves a no worse utility than in any other equilibrium.Similarly, there exists a unique Pareto-optimal equilibri-um for SUs: µc, where every SU achieves a no worseutility than in any other equilibrium. Next we show thatthis observation is generally applicable.

Proposition 1: There always exists a unique Pareto-optimal equilibrium for PUs, where every PU achievesits maximum utility among all equilibria. Similarly, therealways exists a unique Pareto-optimal equilibrium forSUs, where every SU achieves its maximum utility a-mong all equilibria.

We further show that for any two equilibria, theequilibrium that is better for all users on one side isalways worse for all users on the other side.

Proposition 2: For any equilibria µ and µ′, all PUsprefer µ to µ′, if and only if all SUs prefer µ′ to µ.

Proposition 1 and 2 show that users on the same sidehave a common interest and users on the opposite sideshave conflicting interests, regarding the set of equilibria.This observation was first discovered by Gale and Shap-ley in [18], wherein readers can find the detailed proof.

5.4 Equilibrium under Complete InformationNow we study which equilibrium will actually emerge(in the PU-proposal market) under complete informa-tion. In this case, any SU’s misrepresentation (of itspreference list) is not allowed, as PUs know the wholenetwork information and thus know the complete pref-erence lists of SUs.

We first introduce the “deferred acceptance” (DAC) algo-rithm in [18], and then show in Proposition 3 the DACalgorithm converges to the Pareto-optimal equilibriumfor PUs (PU-Optimal-EQ).

Proposition 3: The DAC algorithm converges thePareto-optimal equilibrium for PUs, i.e., PU-Optimal-EQ.

Algorithm 1 DAC1: while (at least one PU is rejected in previous round) do {2: Each PU proposes to the first SU in its preference list;3: Each SU rejects all proposals but the most preferred;4: Each PU updates the preference list by removing the SU

who just rejected him; }

Notice that every PU can achieve a better utility underthe PU-Optimal-EQ than under any other equilibrium.That is, there does not exist another equilibrium, inwhich at least one PU achieves a higher utility than in thePU-Optimal-EQ. Thus, all PUs are willing to follow theproposing rule in the DAC algorithm in order to achievethe PU-Optimal-EQ. This implies that the PU-Optimal-EQis the only equilibrium that will emerge under completeinformation.

5.5 Equilibrium under Incomplete InformationNow let us consider the incomplete information scenari-o, where PUs know their local network information only,and thus cannot know the complete preference lists ofSUs. Therefore, it is possible for SUs to misrepresent theirpreference lists to seek more utilities. We show this bythe following example.

Example 2: Consider the same model in Example 1,except that here each PU does not know SUs’ preferencelists. Suppose SU n1 misrepresents its preferences byQ(n1) = {m2,m3}. Then it is easy to check that the DACalgorithm will lead to the equilibrium µb. Furthermore,if SU n1 misrepresents its preferences by Q(n1) = {m2},then the DAC algorithm will lead to the Pareto-optimalequilibrium µc for SUs. In both misrepresentations, SUn1 achieves a higher utility than truth-telling. �

By the above example, we can see that differentmisrepresentation behaviors of SUs may lead to differ-ent market equilibria. Unfortunately, characterizing allmisrepresentation behaviors of SUs in the incompleteinformation scenario is an NP-hard problem. Hence, it isdifficult to characterize which specific equilibrium willactually emerge in the incomplete information scenario,due to the uncertainty of SU misrepresentation. To thisend, we study the worst-case equilibrium for PUs, andcharacterize a robust equilibrium for PUs (PU-Robust-EQ), which gives every PU a guaranteed utility underany possible misrepresentations of SUs.

Specifically, the robust equilibrium PU-Robust-EQ canbe achieved by a “reversed deferred acceptance” (RDAC)algorithm as in [18], which is same as the DAC algorithmexcept that it reverses the roles of PUs and SUs inproposing offers. That is, in the RDAC algorithm, eachSU proposes to its most preferred PU among those whohave not yet rejected him, and each PU accepts the mostpreferred proposal and rejects the others. The details ofthe RDAC algorithm are skipped due to space limit. Bythe symmetry between two algorithms, the RDAC algo-rithm converges to the Pareto-optimal equilibrium forSUs, which, by Lemma 2, is the worst-case equilibrium



8

for PUs. Therefore, every PU is guaranteed to achieve autility no worse than that in this worst-case equilibrium,under any possible misrepresentations of SUs. In thissense, we refer to this worst-case equilibrium for PUs asthe robust equilibrium for PUs.

6 MARKET EQUILIBRIUM ANALYSIS

Now we characterize the market equilibrium in a generalmodel where the access time reward and the relay powerare not fixed. In this case, each user’s utility (with a par-ticular partner at the other side) is no longer fixed, butchanges with the resource exchange (i.e., the access timereward and relay power) between them. Accordingly,the preference list of each user depends on its currentresource exchanges with users at the other side. Thus, weneed to consider not only the binary matching betweenPUs and SUs (i.e., the relay assignment problem definedin Question 1), but also the resource exchange betweeneach pair of matched users (i.e., the resource exchangeproblem defined Question 2).

In what follows, we will start by defining the utilitytransfer function, which captures the interaction withineach matching pair. Then we characterize the necessaryand sufficient conditions for market equilibrium. Wewill show that there are multiple (infinite number) ofmarket equilibria. In the next section, we will furtherstudy which specific equilibrium will actually emerge indifferent information scenarios.

6.1 Utility Transfer Function - UTFSuppose that PU m is matched to SU n. We first studyhow their utilities change with the access time rewardand the relay power.

Recall in the PU-proposal market, PUs propose theoffers and SUs decide whether to accept or not. Let δmndenote the SU n’s highest utility from all other PUs’offers (except PU m’s) and from remaining single (notmatched to any PU), called reservation utility of SU n(towards PU m).11 Then, to attract SU n’s cooperation,PU m has to offer SU n a utility no less than δmn . Thus,the PU m’s maximum utility is given by

Πm∗n = max

{pmn ,tmn }Πmn (pmn , t

mn )

s.t. ∆mn (pmn , t

mn ) ≥ δmn ,

(11)

where Πmn (pmn , t

mn ) is PU m’s utility defined in (4),

∆mn (pmn , t

mn ) is SU n’s utility defined in (8). Notice that

the constraint must be tight at the optimality. This im-plies that the variable tmn (or pmn ) can be rewritten as afunction of pmn (or tmn ), and thus the problem (11) can betransformed into a single-variable optimization problem.Hence, problem (11) can be solved efficiently using manyone-dimension exhaustive search methods. We denotethe computational complexity of solving (11) as ρ.

11. Note that an SU n’s reservation utility is PU-dependent, i.e., δm1n

may be different from δm2n , as it is the SU’s highest utility from all but

one of the PUs.

TABLE 3Important Notations in Section 6

fmn (·) PU m’s maximal utility given the matched SU’s utility;gmn (·) SU n’s maximal utility given the matched PU’s utility;µn The PU matched to SU n;µm The SU matched to PU m;δn The SU n’s achieved utility in a given matching;δ̄n The SU n’s highest achievable utility in a matching;δn The SU n’s lowest acceptable utility in a matching;

It is easy to check that the PU m’s maximal utility Πm∗n

given by (11) is strictly decreasing in SU n’s reservationutility δmn . Thus, we can write Πm∗

n as a decreasingfunction of δmn :

Πm∗n , fmn (δmn ).

We refer to fmn (·) as the Utility Transfer Function (UTF).Furthermore, we denote

gmn (·) , fmn (−1)(·)as the Inverse Utility Transfer Function (IUTF), i.e.,πmn = fmn (δmn ) if and only if δmn = gmn (πmn ). Note thatfmn (δmn ) denotes the PU m’s maximal utility when givingSU n a utility δmn , and gmn (πmn ) denotes the SU n’s maximalutility when giving PU m a utility πmn . Figure 5 in thesimulations illustrates the UTF.

To facilitate the understanding of the later analysis, welist some important notations in Table 3.

6.2 Sufficient & Necessary ConditionsNow we study the sufficient and necessary conditionsfor an equilibrium. The optimization problem (11) showsthat the PU m’s maximal utility depends on SU n’s utilityδmn , so does the optimal access time tmn and relay powerpmn . Thus, the resource exchange problem in Question 2is equivalent to the following utility division problem:what is the utility for each SU in a matching? This meansthat we can rewrite a matching as

{(µn, δn), ∀n ∈ N}, (12)

where µn denotes the PU matched to SU n, and δndenotes the SU n’s utility in the given matching.12 Ob-viously, in such a matching, the utility of PU µn (i.e.,the PU matched to SU n) is fµn

n (δn), which is PU µn’smaximum achievable utility computed by (11). Note thatthe utility is zero for an unmatched SU n (with µn = ∅)or PU m (with m 6= µn,∀n ∈ N ).

In what follows, we will present the lower-bound andupper-bound of each δn (for the matched PU µn andSU n), such that none of PUs and SUs has an incentiveto break the current matching. To achieve this, for eachpair of matched PU µn and SU n: (i) PU µn (or SU n) hasno incentive to break the current matching by remainingsingle, or by pairing with another SU (or PU), and (ii)all PUs other than µn (or all SUs other than n) have noincentives to discard their respective partners and pairwith SU n (or PU µn).

12. Note we omit the superscript µn in δµnn for simplicity of writing.



9

According to Definition 4, we first have the followingnecessary conditions for an equilibrium.

Lemma 1: If {(µn, δn),∀n} is an equilibrium, then thefollowing conditions hold: for each SU n ∈ N ,

(IR) fµnn (δn) ≥ 0;

(IC) fµnn (δn) ≥ fµn

k (δk), ∀k 6= n,where µn denotes the PU matched to SU n.

The first condition is generally referred to as IndividualRationality (IR), and the second condition is generallyreferred to as Incentive Compatibility (IC). The proof fol-lows directly from the definition of equilibrium: If the IRcondition is violated by a PU µn, then PU µn will discardits partner (SU n) and remain single; If the IC conditionis violated by a PU µn, i.e., there exists another SU k 6= nsuch that fµn

n (δn) < fµn

k (δk), then PU µn can get a betterutility by paring up with SU k instead.13

The IC and IR conditions for a PU µn ensure that PUµn has no incentive to change its decision. We furtherintroduce the Competitive Compatibility (CC) condition forPU µn, which ensures that no other PUs has the incentiveto compete with a PU µn (for SU n), that is, no otherPU m can achieve a higher utility on SU n than on itscurrent partner SU µm. Formally, we have the followingnecessary conditions for an equilibrium, where the prooffollows directly from the definition of equilibrium.

Lemma 2: If {(µn, δn),∀n} is an equilibrium, then thefollowing condition holds: for each PU µn ∈M,14

(CC) fmµm(δµm

) ≥ fmn (δn), ∀m 6= µn,where µm denotes the SU matched to PU m.

From the IR condition for a PU µn, we have:

fµnn (δn) ≥ 0⇒ δn ≤ gµn

n (0),

where gµnn (0) = fµn

n(−1)(0) is the maximum achievable

utility of SU n when leaving zero utility to PU µn.From the IC condition for a PU µn, we have:

fµnn (δn) ≥ fµn

k (δk), ∀k 6= n

⇒ δn ≤ gµnn

(fµn

k (δk)), ∀k 6= n

⇒ δn ≤ mink 6=n{gµnn

(fµn

k (δk))},

where gµnn (x) = fµn

n(−1)(x) is maximum achievable

utility of SU n when leaving a utility x to PU µn.From the CC condition for a PU µn, we have:

fmµm(δµm

) ≥ fmn (δn), ∀m 6= µn

⇒ δn ≥ gmn(fmµm

(δµm)), ∀m 6= µn

⇒ δn ≥ maxm6=µn

{gmn(fmµm

(δµm))}.

Based on the above discussions, we can obtain thelowest acceptable utility δn and the highest achievable utilityδn for an SU n in an equilibrium {(µ(n), δn),∀n}, i.e.,

δn , maxm6=µn

{gmn(fmµm

(δµm)), 0}, (13)

13. With a proper offer from PU µn (e.g., giving SU k a utility δk+ε,where ε is any infinitesimal positive number), SU k will also improveits utility through the new pairing and thus will accept.

14. Note that fmµm(·) ≡ 0 if µm = ∅, which implies that the utility

of PU m is always zero when not engaging in any collaboration.

δn , mink 6=n{gµnn

(fµn

k (δk)), gµn

n (0)}. (14)

Intuitively, PU µn cannot offer a utility δn lower thanδn to SU n, otherwise some other PU m 6= µn will havethe incentive to pair up with SU n (by offering SU na utility slightly higher than δn). This will make theoriginal matching broken. On the other hand, PU µn willnever offer a utility δn higher than δn to SU n, otherwisePU µn would be better off by pairing up with an otherSU. This will also make the matching broken.

Therefore, we can rewrite the necessary conditions inLemmas 1 and 2 more explicitly as follows: If a matching{(µn, δn), ∀n} is an equilibrium, the condition holds:

δn ≤ δn ≤ δn, ∀n ∈ N . (15)

The left inequality is due to the CC condition, and theright inequality is due to the IC and IR conditions.

The next theorem shows the above conditions are notonly necessary, but also sufficient.

Theorem 1 (Sufficient & Necessary Condition): Amatching {(µn, δn), ∀n} is an equilibrium, if and only if

δn ≤ δn ≤ δn, ∀n ∈ N ,where δn and δn are given by (13) and (14).

Proof: We have already shown the necessity by theabove analysis. Next we prove the sufficiency.

We first show that such a matching does not violateany PU or SU’s IR condition. This is because each SU ngets a utility δn ≥ δn ≥ 0 and each PU m gets a utilityfµnn (δn) ≥ fµn

n (δn) ≥ fµnn (gµn

n (0)) = 0. By (11), we canfurther show that each pair of matched PU µn and SUn have no incentive to change δn, otherwise at least oneuser will lose certain utility (since an increase of one’sutility will lead to a decrease of another’s utility).

We then show that no PU or SU will deviate fromthe current matching by choosing a different partner. Weprove this by contradiction. Suppose that both PU µn(who is currently matched to SU n) and SU k 6= n haveincentives to break the matching by pairing up with eachother. Then there must exist a value δ′k such that (i) δ′k >δk (i.e., SU k achieves a larger utility with PU µn) and (ii)fµn

k (δ′k) > fµnn (δn) (i.e., PU µn achieves a larger utility

with SU k). On the other hand, we have

fµnn (δn) ≥ fµn

k (δk) > fµn

k (δ′k),

where the first inequality follows directly from δn ≤ δn,and the second inequality follows from the assumptionδ′k > δk. This leads to a contradiction.

6.3 Property of EquilibriumTo facilitate the characterization of equilibrium in differ-ent information scenarios (in the next section), we firstshow the following property for equilibrium.

Lemma 3 (Generalized Lattice Theorem): Given anypossible binary matching µ(·), if {(µn, δIn),∀n} and {(µn,δIIn ),∀n} are equilibria, then the matching {(µn, δXn ),∀n}with δXn = min(δIn, δ

IIn ) is also an equilibrium.



10

Proof: To prove {(µn, δXn ), ∀n} is an equilibrium, weonly need to show that δXn satisfies the conditions inTheorem 1. By Theorem 1, we have: for all n ∈ N ,

δin ≤ δin ≤ δi

n, i = I, II.

Without loss of generality, we consider an arbitrarymatching pair, say, PU µn and SU n. Suppose δIn ≤ δIIn .Then δXn = min(δIn, δ

IIn ) = δIn. On one hand,

δXn = δIn ≥ δIn = maxm6=µn

{gmn(fmµm

(δIµm)), 0}

≥ maxm6=µn

{gmn(fmµm

(δXµm)), 0}, δXn .

The second line follows because gmn(fmµm

(·))

is an increas-ing function, and δXµm

≤ δIµm,∀µm. On the other hand,

δXn = min{δIn, δ

IIn

}≤ min

{δI

n, δII

n

}= min

k 6=n

{gµnn

(fµn

k (δIk)), gµn

n

(fµn

k (δIIk )), gµnn (0)

}= min

k 6=n

{gµnn

(fµn

k (min(δIk, δIIk ))

), gµnn (0)

}, δ

X

n .

The last line follows because h(·) , gµnn

(fµn

k (·))

is anincreasing function, thus min(h(x), h(y)) = h(min(x, y)).

Lemma 3 can be seen as a generalization of the LatticeTheorem proposed by Kunth [25] for the basic two-sidedmatching (without utility transferring). By repeatedlyapplying Lemma 3 on any two equilibria, we will reachthe unique optimal equilibrium for PUs.

Theorem 2 (Optimality): There exists a Pareto-optimalequilibrium {(µn, δ∗n),∀n} for PUs, where every PUachieves its maximum utility among all equilibria. Inaddition, {(µn, δ∗n),∀n} satisfies:15

δ∗n = δ∗n, ∀n ∈ N .Proof: The existence and uniqueness of PU-optimal

equilibrium can be easily proved by iteratively applyingLemma 3 on any two equilibria.

Next we prove that the unique PU-optimal equilibri-um {(µn, δ∗n),∀n} satisfies the conditions: δ∗n = δ∗n,∀n.Suppose it does not satisfy the above conditions. Then,there must exist one or multiple n such that δ∗n > δ∗n,and δ∗n can be further reduced (which implies that PUn’s utility can be further increased) without affecting theIC and IR conditions for equilibrium. Obviously, thisviolates the optimality of {(µn, δ∗n),∀n}.

Theorem 2 not only shows the existence of the Pareto-optimal equilibrium for PUs, but also suggests a methodto calculate the Pareto-optimal equilibrium for PUs.Specifically, given any feasible matching µ(·), we canderive each SU’s utility (and the associated time rewardand relay power) under the Pareto-optimal equilibriumby jointly solving the following function set:

δn = maxm6=µn

{gmn(fmµm

(δµm)), 0}, ∀n ∈ N . (16)

15. Similarly, there always exists a Pareto-optimal equilibrium{(µn, δ∗n), ∀n} for SUs, where every SU achieves its maximum utilityamong all equilibria. In addition, {(µn, δ∗n), ∀n} satisfies: δ∗n = δ

∗n, ∀n.

Obviously, if there is a unique solution for (16), it mustbe the unique Pareto-optimal equilibrium. If there is nosolution for (16), the matching µ(·) is not feasible.

However, directly solving the above function set is dif-ficult. Consider a single equation in the function set (say,δn). To solve this equation, we first need to compute theutilities of all PUs other than µn on their own matchingpairs, i.e., fmµm

(δµm), ∀m 6= µn. Then, we need to compute

the SU n’s maximum achievable utility when givingeach PU m 6= µn a utility fmµm

(δµm), i.e., gmn(fmµm

(δµm)),

∀m 6= µn. Finally, the SU n’s utility δn is the maximum ofthe above computed maximum achievable utilities andzero. Obviously, for this single equation, we need torepeatedly solve the optimization problem (11) 2·(M−1)times. Furthermore, jointly solving N equations makesthe problem even more complicated. This motivates usto study low complexity algorithms to realize differentmarket equilibria in the next section.

7 MARKET EQUILIBRIUM REALIZATION

In the previous section, we characterized the necessaryand sufficient conditions of market equilibrium. In thissection, we will study which equilibrium will actuallyemerge in different information scenarios.

Specifically, we consider two typical information sce-narios: complete and incomplete information, dependingon how much network information the PUs know (seeSection 4.3 for details). Notice that in the PU-proposalmarket, PUs propose offers and SUs respond by rejectingor accepting offers. Under complete information, eachPU knows the whole network information, and canprecisely predict the SUs’ preferences. Thus, SUs cannotmis-represent their preferences, and have to respond inthe truthful manner. That is, each SU will always acceptthe best offer that brings him the maximum positiveutility. Under incomplete information, however, each PUknows only its local network information, and cannotpredict the SUs’ preferences with other PUs. Thus, SUscan potentially mis-represent their preferences, and re-spond in a non-truthful manner. That is, each SU mayreject the best offer that brings him the maximum posi-tive utility, as long as such a mis-representation can bringhim a better utility.

7.1 Equilibrium under Complete InformationUnder complete information, each SU has to respond ina truthful manner, and accept the best offer that bringshim the maximum positive utility. In this sense, we canview the SUs’ (truthful) responses as natural reactions.Therefore, the matching problem can be viewed as anM -player game with complete information, where gameplayers are PUs, whose objectives are to maximize theirown utilities. By (11), each PU’s maximal utility strictlydecreases with the utility of the matched SU. Thus, givenall other PUs’ offers (strategies), the best strategy for aparticular PU µn (i.e., the PU matched to SU n) is tooffer the lowest acceptable utility to its matched SU n,



11

Algorithm 2 G-DAC1: Initialization2: Each PU m ∈M set δmn = 0, ∀n = 1, ..., N ;3: do {4: Each PU m proposes to n if fmn (δmn ) ≥ maxk{fmk (δmk ), 0};5: Each SU n accepts the PU offering him highest utility;6: Each PU m updates δmn = δmn + ε if rejected by SU n; }7: while (at least one δmn is changed)

i.e., δn = δn. This essentially lead to the Pareto-optimalequilibrium for PUs (PU-Optimal-EQ).

Next we show how to compute the PU-Optimal-EQin the complete information scenario. Notice that whenµ(·) is given, the PU-Optimal-EQ can be computed byjointly solving (16), which is complicated as mentionedpreviously. Moreover, an exhaustive search of all feasibleµ(·) requires going through a total of M ! possibilities,which has an exponential-time complexity. This makesthe solving process mathematically intractable.

To this end, we introduce a “Generalized Deferred Ac-ceptance” (G-DAC) algorithm in Algorithm 2, which con-verges to the PU-Optimal-EQ in polynomial-time com-plexity. Specifically, in the G-DAC algorithm, each PU mmaintains a vector ∆m = (δm1 , ..., δ

mN ), which records the

utilities it is willing to offer to each SU. The basic ideaof the G-DAC algorithm is: (i) in each round, each PUm proposes to the best SU based on ∆m, and each SUn accepts the best proposal and rejects others; and (ii)if PU m is rejected by SU n in a round, it increases theoffer δn by a small value ε in the following rounds.

The G-DAC algorithm generalizes the DAP algorithm[18] in the following aspect. In the DAP algorithm, whena PU is rejected by an SU, it will propose to another SUin later rounds (as the SU who has rejected its offer willnever accept its offer in the future); while in the G-DACalgorithm, the PU will continue to increase the utility tothe SU (attempting to attract the SU), until the offer isaccepted by the SU or the PU can achieve a higher utilityby proposing an offer to another SU.16 Hence, it bringsan additional degree of freedom for PUs making theirdecisions. Namely, each PU decides not only the targetSU that it is going to propose to, but also the utility thatit is going to transfer to the SU.

Lemma 4: The G-DAC algorithm converges to thePareto-optimal equilibrium for PUs (PU-Optimal-EQ), ifthe stepsize ε is small engouth.

To prove the above lemma, we can first show that theconvergence state of G-DAC satisfies the conditions inTheorem 1, and is therefore an equilibrium. Then wecan prove the optimality by showing that the G-DACwill stop at the first equilibrium it achieves, which isexactly the Pareto-optimal equilibrium for PUs.

16. Intuitively, the G-DAC algorithm is equivalent to an English-Auction [26] with SUs as auctioneers and PUs as bidders. The auctionworks as follows: (i) each SU asks a “price” (the required utility), andgradually increases the price if multiple PUs bid to him, and (ii) eachPU submits bid to the best SU, according to the maximum utilities thatit can achieve from SUs at the current price level of all SUs.

Algorithm 3 G-RDAC1: Initialization2: Each SU n sets πmn = 0,∀m = 1, ...,M ;3: do {4: Each SU n proposes m if gmn (πmn ) ≥ maxk{gkn(πkn), 0};5: Each PU m accepts the SU offering him highest utility;6: Each SU n updates πmn = πmn +ε if rejected by PU m; }7: while (at least one πmn is changed)

Next we discuss the communication overhead andcomputational complexity of the G-DAC algorithm. Ineach round, each PU m needs to send one requestmessage to a particular SU, and each SU n needs to sendback one acceptance message to a particular PU. Thus,the communication overhead is linear in the number ofPU-SU pairs (and hence is low). In terms of computation,each PU m needs to solve the optimization problem (11)at most once with complexity ρ in each round (except thefirst round where the PU needs to solve the optimizationproblem (11) N times with complexity Nρ), since at mostone price in the vector ∆m = (δm1 , ..., δ

mN ) is changed in

each round. Furthermore, each PU needs to find the max-imal utility among N utilities, and each SU needs to findthe best offer among at most M proposals. Thus, the totalcomputational complexity is O(T · (Mρ+MN +NM)),where T =

∑n∈N δ

∗n

ε is the maximum possible iterations,and ρ is the complexity of solving problem (11).

7.2 Equilibrium under Incomplete Information

Under incomplete information, each SU can respond ina non-truthful manner, and may reject the best offer thatbrings him the maximum positive utility, as long as sucha mis-representation can bring a better utility for the SU.This can be seen from the following example.

Example 3: Suppose PU µn proposes the lowest accept-able utility to its matched SU n (as in the completeinformation scenario). The SU n can simply reject thisoffer (which is its best offer for SU n according to thedefinition of the lowest acceptable utility), such that PUµn increases the utility offered to SU n. �

By the above example, we can see that different mis-representation behaviors of SUs (e.g., determining whento accept a PU’s offer, and when to reject) may leadto different market equilibria. Unfortunately, due to thecontinuity of utility, there are infinite number of mis-representation behaviors of SUs. Thus, it is impossibleto characterize all possible mis-representation behaviorsof SUs. To this end, we will focus on characterizing arobust equilibrium for PUs (PU-Robust-EQ), which pro-vides every PU a guaranteed utility under any possiblemisrepresentations of SUs.

Similarly, the robust equilibrium for PUs (PU-Robust-EQ) can be achieved by a “Generalized Reversed DeferredAcceptance” (G-RDAC) algorithm in Algorithm 3, whichis same as the G-DAC Algorithm 2 except that it reversesthe roles of PUs and SUs in proposing offers.



12

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

SU n’ Utility - δ n

PU

m’Maxim

alUtility

-Π

m∗

n

UTF - fmn1(.)

UTF - fmn2(.)

1 PU {m} and 2 SUs {n1, n2}

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

SU n’ Utility - δ n

PU

m’Maxim

alUtility

-Π

m∗

n

UTF - fm1n (.)

UTF - fm2n (.)

2 PUs {m1, m2} and 1 SU {n}

Fig. 5. Utility Division under Equilibrium. Left: 1 PU and 2 SUs; Right: 2 PUs and 1 SU.

Since the G-DAC algorithm and G-RDAC algorithmare symmetric, we can easily show that the G-RDACalgorithm converges to the Pareto-optimal equilibriumfor SUs, which is also the worst-case equilibrium for PUs(due to the conflict interests of PUs and SUs regardingequilibrium). The convergence of the G-RDAC can beshown in a similar way as that of the G-DAC givenin Lemma 4 for the G-DAC. Therefore, every PU isguaranteed to achieve a utility no worse than that in thisworst-case equilibrium, under any possible misrepresen-tations of SUs. In this sense, we refer to this worst-caseequilibrium for PUs as the robust equilibrium for PUs.

7.3 Equilibrium under Arbitrary Information

It is important to note that the PU-Robust-EQ guaranteesevery PU’s lowest achievable utility among all equilibria,while the PU-Optimal-EQ ensures every PU’s highestachievable utility among all equilibria. This implies thatunder any possible equilibrium (in any information sce-nario), every PU’s utility must be bounded by its utilitiesunder the PU-Robust-EQ and the PU-Optimal-EQ. Ournumerical studies in the next section further show that ina typical market where the number of PUs and SUs aredifferent, the gap between the PU-Robust-EQ and thePU-Optimal-EQ is quite small. Hence, the PU-Robust-EQ or PU-Optimal-EQ can be viewed as an effectiveapproximation to understand all equilibria.

8 SIMULATION RESULTS

To highlight the performance gain achieved from coop-erative spectrum sharing, we focus on such a networkscenario in simulations, where PUs’ direct channel gainsare much smaller than SUs’ relay channel gains. This mayhappen when PUs’ direct links are highly attenuated dueto shadow fading or obstacles. This is very intuitive: onlywhen an SU can provide a large enough benefit to aPU, the PU has the incentive to cooperating with theSU, which is a prerequisite of the cooperative spectrumsharing studied in this paper.

We distribute PUs’ and SUs’ transceiver pairs in asquare area of size 1500 × 1500m2, and set the distanceof each PU’s transceiver pair close to 1000m and thedistance of each SU’s transceiver pair close to 400m

(which are the typical transmission ranges between mo-bile phones and cellular base stations). Unless otherwisestated, the following default network setting will beused: (i) Cn = Tm = 1, (ii) σ2 = −105dBm, and (iii) thechannel gains are chosen based on the free-space pathloss model in [27].

8.1 Utility Division under Equilibrium

We first illustrate the utility division (or equivalently, re-source exchange) under market equilibria to provide anillustrative impression on the market equilibrium.

Figure 5 illustrates the utility transfer functions (UTF)and the associated equilibrium utility division in an arti-ficial network with (i) one PU {m} and two SUs {n1, n2}(left figure) and (ii) two PUs {m1,m2} and one SU {n}(right figure). Recall that the UTF function representshow a PU’s maximal utility changes with the utility ofthe matched SU. In the left figure, PU m is matched to SUn1 under the market equilibrium. In the right figure, SUn is matched to PU m1 under the market equilibrium. Inboth figures, the equilibrium utility division among thematched PU and SU is illustrated by the black bold line(between the circle and the square). More specifically,the circle denotes the utility division under the Pareto-optimal equilibrium for PUs (PU-Optimal-EQ), and thesquare denotes the utility division under the Robustequilibrium for PUs (PU-Robust-EQ). More detailed ex-planations for these equilibria are given below.

(1) PU-Optimal-EQ: In the left figure, the PU-Optimal-EQ can be achieved by offering the lowest acceptableutility (i.e., 0 in this example) to SU n1. Obviously, thePU achieves its maximum utility under PU-Optimal-EQamong all equilibria. In the right figure, the PU-Optimal-EQ can be achieved when PU m1 offers the lowestacceptable utility (i.e., 1.7 in this example) to SU n. Notethat due to the competition among PUs, PU m1 cannotoffer a utility lower than 1.7 to SU n, otherwise, PU m2

can offer a utility of 1.7 to SU n such that SU n accepts theoffer of PU m2. Similarly, PU m1 achieves its maximumutility under the PU-Optimal-EQ among all equilibria.17

(2) PU-Robust-EQ: In the left figure, the PU-Robust-EQ can be achieved by offering the highest achievable

17. Note PU m2 achieves a zero utility under the PU-Optimal-EQ.



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1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

Number of SUs - N

PUs’

TotalUtility

PU−Optimal−EQPU−Robust−EQ

N > M : Gap is small.

N = M = 2: Gap is large.

N < M : Gap is small.

1 2 3 4 5 6 7 8 9 100

1

2

3

Number of SUs - N

PUs’

TotalUtility

PU−Optimal−EQPU−Robust−EQ

N = M = 4: Gap is large.

N > M : Gap is small.

N < M :Gap is small.

Fig. 6. Total PUs’ utility vs Number of SUs N under Equilibrium. Left: 2 PUs; Right: 4 PUs.

utility (i.e., 0.5 in this example) to SU n1. Note that dueto the competition among SUs, SU n1 cannot requestinga utility higher than 0.5 from the PU (which will leavethe PU a utility lower than 0.8), otherwise, SU n2 cansuccessfully pair with the PU by requesting a utilityof zero (which will leave the PU a utility of 0.8). Thisimplies that the PU is guaranteed to obtain a utility of0.8 (i.e., that under PU-Robust-EQ). In the right figure,the PU-Robust-EQ can be achieved when PU m1 offersthe highest achievable utility (i.e., 2.2 in this example) toSU n. Obviously, the PU-Robust-EQ provides PU m1 aguaranteed utility of zero.

8.2 Performances EvaluationNow we evaluate the PUs’ performance under differentmarket equilibria. Note that the SUs’ performance can beestimated directly from the conflicting interests betweenPUs and SUs, that is, an equilibrium that is better for allPUs must be worse for all SUs. In the following, eachresult is the average of 1000 simulations with randomlygenerated network topologies.

Figure 6 illusrates the PUs’ total utilities vs the num-ber of SUs N under different market equilibria, wherethe number of PUs is M = 2 in the left figure, andM = 4 in the right figure. The red curve denotes the totalPUs’ utility under the PU-Optimal-EQ, and the blue dashcurve denotes the total PUs’ utility under the PU-Robust-EQ. Obviously, the total PUs’ utility under any otherequilibrium is bounded by these two curves.

From Figure 6, we can see that the PUs’ total utilityunder equilibrium increases with the number of SUs N(as the competition among SUs increases with N ), anddecreases with the number of PUs M (as the competitionamong PUs increases with M ). Moreover, when M < N ,PUs can extract most of the generated utility (as thecompetition among SUs is more intense), and whenM > N , SUs can extract most of the generated utility (asthe competition among PUs is more intense). We furtherhave the following observations.

(1) In the general case where the numbers of PUs andSUs are different (i.e., M 6= N ), the utility gap betweenPU-Optimal-EQ and PU-Robust-EQ is quite small (e.g.,less than 10% in both figures). This implies that the infor-mation scenario (i.e., the amount of network informationthat PUs know) has a small impact on the equilibrium.

(2) In the special case where the numbers of PUs andSUs are identical (i.e., M = N ), the utility gap betweenPU-Optimal-EQ and PU-Robust-EQ is large. This impliesthat the information scenario has a significant impact onthe equilibrium.

Intuitively, when N 6= M , the imbalance of competi-tion among PUs and among SUs reduces the effect ofinformation incompleteness. The reason is as follows. Ifthe number of SUs is larger than that of PUs (N > M ),the intensified competition among SUs pushes SUs toreveal their preferences close to the true values. If thenumber of PUs is larger than that of SUs (M > N ), theintensified competition among PUs already drives mostof the generated welfare to the SU-side even in the worst-case equilibrium for SUs, and thus the SUs’ performancegain in other equilibria is small. When M = N , however,none of the two sides has a dominated competition levelin the market, thus the effect of information incomplete-ness becomes significant.

9 CONCLUSIONIn this paper, we study the cooperative spectrum shar-ing between multiple PUs and multiple SUs using thematching theory. We formulate the problem as a two-sided matching market, and characterize the marketequilibrium systematically. We further study which e-quilibrium will actually emerge in both complete andincomplete information scenarios. Our study can facili-tate the design of large networks, in which players can besplit into two different types, interacting with each other.The interaction between players is not restricted withinthe cooperative spectrum sharing studied in this work,but can be quite general (e.g., spectrum trading). Thereare some possible directions to extend the results in thispaper. An interesting direction is to consider a moregeneral cooperative spectrum sharing protocol whereeach PU can cooperate with multiple SUs to relay itstraffic, and each SU can cooperates with multiple PUs toaccess their spectrum. In this case, it is no longer a one-to-one matching, but a multiple-to-multiple matchingbetween PUs and SUs.

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Lin Gao (S’08-M’10) is an Associate Professorin the School of Electronic and Information En-gineering, Harbin Institute of Technology (HIT)Shenzhen Graduate School. He received theM.S. and Ph.D. degrees in Electronic Engineer-ing from Shanghai Jiao Tong University (China)in 2006 and 2010, respectively. He worked asa Postdoc Research Fellow at the Chinese U-niversity of Hong Kong from 2010 to 2015. Hisresearch interests are in the area of networkeconomics and games, with applications in wire-

less communications, networks, and internet of things.

Lingjie Duan (S’09-M’12) is an Assistant Pro-fessor with Engineering Systems and Design,Singapore University of Technology and Design(SUTD), Singapore. He received the Ph.D. de-gree from the Chinese University of Hong Kong,Hong Kong, in 2012. In 2011, he was a VisitingScholar University of California at Berkeley, CA,USA. He is an Editor of IEEE CommunicationsSurveys and Tutorials, and is a SWAT memberin the Editorial Board of IEEE Transactions onVehicular Technology. He is also a Guest Editor

of IEEE Wireless Communications magazine for a special issue aboutgreen networking and computing for 5G. His research interests includenetwork economics and game theory, cognitive communications andcooperative networking, energy harvesting wireless communications,and network security. Recently, he served as the Program Co-Chairof the IEEE INFOCOM 2014 GCCCN Workshop, ICCS 2014 specialsession on Economic Theory and Communication Networks, the Wire-less Communication Systems Symposium of the IEEE ICCC 2015, theGCNC Symposium of the IEEE ICNC 2016, and the IEEE INFOCOM2016 GSNC Workshop. He also served as a technical program com-mittee (TPC) member of many leading conferences in communicationsand networking (e.g., ACM MobiHoc, IEEE SECON, ICC, GLOBECOM,WCNC, and NetEcon). He was the recipient of the 10th IEEE ComSocAsia-Pacific Outstanding Young Researcher Award in 2015, Hong KongYoung Scientist Award (Finalist in Engineering Science track) in 2014,and CUHK Global Scholarship for Research Excellence in 2011.

Jianwei Huang (S’01-M’06-SM’11-F’16) is anAssociate Professor and Director of the Net-work Communications and Economics Lab (n-cel.ie.cuhk.edu.hk), in the Department of Infor-mation Engineering at the Chinese University ofHong Kong. He received the Ph.D. degree fromNorthwestern University in 2005, and workedas a Postdoc Research Associate in Prince-ton University during 2005-2007. He is the co-recipient of 8 international Best Paper Awards,including IEEE Marconi Prize Paper Award in

Wireless Communications in 2011. He has co-authored four books:”Wireless Network Pricing,” ”Monotonic Optimization in Communicationand Networking Systems,” ”Cognitive Mobile Virtual Network OperatorGames, and ”Social Cognitive Radio Networks”. He has served as anAssociate Editor of IEEE Transactions on Cognitive Communicationsand Networking, IEEE Transactions on Wireless Communications, andIEEE Journal on Selected Areas in Communications - Cognitive RadioSeries. He is the Vice Chair of IEEE ComSoc Cognitive NetworkTechnical Committee and the Past Chair of IEEE ComSoc MultimediaCommunications Technical Committee. He is a Fellow of IEEE and aDistinguished Lecturer of IEEE Communications Society.

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