1

Cooperative Spectrum Sharing: AContract-based Approach

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

Abstract—Providing economic incentives to all parties involved is essential for the success of dynamic spectrum access. Cooperativespectrum sharing is one effective way to achieve this, where secondary users (SUs) relay traffics for primary users (PUs) in exchangefor dedicated spectrum access time for SUs’ own communications. In this paper, we study the cooperative spectrum sharing underincomplete information, where SUs’ wireless characteristics are private information and not known by a PU. We model the PU-SUinteraction as a labor market using contract theory. In contract theory, the employer generally does not completely know employees’private information before the employment and needs to offers employees a contract under incomplete information. In our problem,the PU and SUs are respectively the employer and employees, and the contract consists of a set of items representing combinationsof spectrum accessing time (i.e., reward) and relaying power (i.e., contribution). We study the optimal contract design for both weaklyand strongly incomplete information scenarios. In the weakly incomplete information scenario, we show that the PU will optimallyhire the most efficient SUs and the PU achieves the same maximum utility as in the complete information benchmark. In the stronglyincomplete information scenario, however, the PU may conservatively hire less efficient SUs as well. We further propose a Decompose-and-Compare (DC) approximate algorithm that achieves a close-to-optimal contract. We further show that the PU’s average utility lossdue to the suboptimal DC algorithm and the strongly incomplete information are relatively small (less than 2% and 1.3%, respectively,in our numerical results with two SU types).

Index Terms—Dynamic spectrum access, spectrum trading, cooperative spectrum sharing, contract theory, game theory.

F

1 INTRODUCTION

W ITH the explosive development of wireless ser-vices and networks, spectrum is becoming more

congested and scarce. Dynamic spectrum access is apromising approach to increase spectrum efficiency andalleviate spectrum scarcity, as it enables unlicensed sec-ondary users (SUs) to dynamically access the spectrumlicensed to primary users (PUs) [2], [3]. The successfulimplementation of dynamic spectrum access requiresmany innovations in technology, economics, and policy.In particular, it is important to design the mechanismsuch that PUs have the incentive to open their licensedspectrum for sharing, and SUs have the incentive toutilize the new spectrum opportunities after consideringpotential costs.

Market-driven spectrum trading is a promisingparadigm to address the incentive issue in dynamicspectrum access [4]. In a market mechanism, PUs tem-porarily sell the spectrum to SUs in exchange of eithera monetary reward or a performance improvement. Aparticular interesting market mechanism is cooperative

• Lingjie Duan is with Engineering Systems and Design Pillar, SingaporeUniversity of Technology and Design (SUTD). Part of the work was donewhen he was with The Chinese University of Hong Kong. Lin Gao andJianwei Huang (corresponding author) are with the Network Communi-cations and Economics Lab, Department of Information Engineering, TheChinese University of Hong Kong, Hong Kong.E-mail: lingjie duan@sutd.edu.sg, {lgao, jwhuang}@ie.cuhk.edu.hk.

Part of the results has appeared in IEEE DySPAN, April 2011 [1]. Thiswork is supported by the General Research Funds (Project Number 412710and 412511) established under the University Grant Committee of the HongKong Special Administrative Region, China. This work is also supported bySUTD Start-Up Research Grant (Project Number: SRG ESD 2012 042).

spectrum sharing, where SUs relay traffics for PUs inorder to get their own share of spectrum. An illustrativeexample of cooperative spectrum sharing is shown inFig. 1. The SUs’ transmitters (ST1 ∼ ST3) act as coopera-tive relays for the PU in Phase I and Phase II (Decodingand Forwarding), and transmit their own data in PhaseIII.

There has been extensive research on the coopera-tive relay networks, often within the scope of physicallayer performance optimization [17]–[25]. Researchershave only recently started to study the incentive issuein cooperative spectrum sharing mechanisms [13]–[16].Prior results all assumed complete network information,where PUs know SUs’ types including information suchas channel conditions, resource constraints, and costs oftransmission. This assumption is often too strong forpractical implementations. In this paper, we study thecooperative spectrum sharing between a single PU andmultiple competitive SUs under incomplete information,where SUs’ types are private information1.

We focus on the PU’s utility maximization under in-complete information: How should the PU employ the SUsas relays to maximize its expected performance improvementwithout knowing the SUs’ exact types? Precisely, which SUsthe PU should employ, what relaying efforts the PU wantto ask from the employed SUs, and what rewards the PUwould like to give to the employed SUs?

To tackle this problem, we propose a contract-basedcooperative spectrum sharing mechanism. Contract the-

1. Consider a cellular network with multiple base stations andmultiple licensed mobile phone users. Each link between one basestation and one licensed user defines one PU.

2

X

X X

X X

X

PT PR

ST1

SR1

ST2

SR2

ST3

SR3

ST4

SR4

ST5

SR5

PT PR

ST1

SR1

ST2

SR2

ST3

SR3

ST4

SR4

ST5

SR5

PT PR

ST1

SR1

ST2

SR2

ST3

SR3

ST4

SR4

ST5

SR5

(Phase I) (Phase II) (Phase III)

T0/2 T0/2 t1 t2 t3 T0/2 T0/2 t1 t2 t3 T0/2 T0/2 t1 t2 t3 time time time

Fig. 1. A single time slot in cooperative spectrum sharing includes three phases. Phase I involves transmission fromprimary user transmitter PT to primary receiver PR and transmitters of the involved SUs (ST1, ST2, and ST3). Phase IIinvolves the transmission of the involved SUs’ transmitters to the primary receiver PR. Phase III involves the dedicatedtransmissions of the involved SUs.

ory is effective in designing incentive compatible mech-anisms in a monopoly market under incomplete infor-mation [30], where the employer needs to sign a con-tract with employees before fully knowing their privateinformation. The key idea is to offer the right contractitems so that all agents have the incentive to truthfullyreveal their private information. In our problem, wecan imagine the network as a labor market. The PUis the employer and offers a contract to the SUs. Thecontact consists of a set of contract items, which arecombinations of spectrum access time (i.e., reward) andrelay power (i.e., contribution). The SUs are employees,and each SU selects the best contract item according to itstype. We want to design an optimal contract that maximizesthe PU’s utility (average data rate) under the incompleteinformation of SUs’ types.

The main contributions of this paper are as follows:

• New modeling and solution technique: As far as weknow, this is the first contract paper tackling cooper-ative spectrum sharing between one PU and multipleSUs under incomplete information.

• Multiple information scenarios (Sections 3 and 4): Wefirst consider the complete information scenario as abenchmark. Then we consider the optimal contrac-t design in two incomplete information scenarios(where the PU does not know the precise type ofevery SU): weakly incomplete information, where thePU knows the percentage of every type, and stronglyincomplete information, where the PU knows onlythe probability distribution of every type.

• Feasibility of contracts (Section 5): Under incompleteinformation, a contract is feasible if and only if it isincentive compatible (IC) and individually rational(IR) for each SU. We characterize the necessary andsufficient conditions for a contract to be IC and IRsystematically.

• Optimality of contracts (Sections 6 and 7): Under

TABLE 1Feasibility conditions and optimality of different

information scenarios

Network Information Feasibility Optimality Section No.

Complete (benchmark) IR Optimal 4

Weakly Incomplete IC & IR Optimal 5, 6

Strongly Incomplete IC & IR Close-to-Optimal 5, 7

weakly incomplete information, we derive the opti-mal contract, which achieves the same maximumPU utility as in the complete information bench-mark. Under strongly incomplete information, wepropose a Decompose-and-Compare algorithm thatachieves a close-to-optimal contract.

• Performance analysis (Section 8): Under strongly in-complete information, the PU’s utility loss is causedby two factors: the suboptimal contract and theincomplete information. We quantify the PU’s av-erage utility loss due to the suboptimal algorith-m (by comparing it with exhaustive searching)and incomplete information (by comparing it withcomplete information) through numerical examples.Both losses turn out to be relatively small, e.g., lessthan 2% and 1.3% for a two-type case, respectively.

The key results and corresponding section numbers inthis paper are summarized in Table 1. We also review therelated literature in Section 2 and conclude in Section 9.

2 RELATED WORK

2.1 Spectrum Trading for Dynamic Spectrum Shar-ing

Recent years have witnessed a growing body of litera-tures on the market-driven spectrum trading [4], whichcan be classified into two types: money-exchange and

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resource-exchange. In the former type, SUs pay PUs inthe form of (virtual) money for the usage of spectrum(e.g., [5]–[12]); in the latter type, SUs provide commu-nication resources (e.g., the power in our model) inexchange for the usage of spectrum (e.g., [13]–[16]).

There has been extensive research on the money-exchange spectrum trading model, often in forms ofpricing (e.g., [5], [6]), auction (e.g., [7]–[10]), and contract(e.g., [11], [12]). Pricing is often used when the sellerknows precisely the value of the resource being sold.Auction becomes a suitable approach when the sellerhas no knowledge about the value of the resource.Contract is more effective in the case where the selleronly knows limited information (e.g., distribution) ofthe buyers’ valuations of the resource. By motivatingthe buyers truthfully reveal their private valuations,the seller can optimally allocate the resource. In [11],Gao et al. proposed a quality-price contract for thespectrum trading in a monopoly spectrum market. In[12], Kalathil et al. focused on 1 PU and 1 SU andproposed a contract-based spectrum sharing mechanismto avoid possible manipulating in auction. Yet all thesemonetary-exchange spectrum trading schemes require atrustworthy and widely acknowledged billing system[33], and such a system is far from industry practice yet.

Money-exchange spectrum trading is most effectivewhen PUs have some temporarily unused spectrum.However, when PUs’ own demands are high or the pri-mary channels’ capacities are low (e.g., due to shadow-ing and deep fading), there will be hardly any resourceleft for sale. Also, such a money-exchange trading re-quires a trustworthy billing system which is still far frompractical implementation. In this case, resource-exchangespectrum trading can be a better choice. Cooperativespectrum sharing is an effective form of resource-exchangespectrum trading [13]–[16], wherein PUs utilize SUs ascooperative relays. Such cooperation can significantlyimprove PUs’ data rate and thus can free up spectrumresources for SUs. Existing cooperative spectrum shar-ing mechanisms are usually based on Stackelberg gameformulations with complete information [13]–[16]. All ofthese works cannot be easily extended to the scenarioof incomplete information. Taking [13] as an example.Under incomplete information, an SU can increase itsutility by misreporting its private information (e.g., itsown channel gain and its cost per unit energy consump-tion). Thus, it is no longer optimal for the PU to stickto the equilibrium of the proposed Stackelberg game.In this case, some incentive compatible mechanism isnecessary for eliciting the SUs’ private information.

In this paper, we consider the cooperative spectrumsharing under incomplete information, and propose acontract-based cooperative spectrum sharing mechanism(which is incentive compatible). As far as we know, thisis the first work considering the cooperative spectrumsharing under incomplete information. Compared tothose works on monetary trading contracts, the resource-exchange contract design is more challenging, as we

TABLE 2A summary of spectrum trading literatures

Network Information Money-Exchange Resource-Exchange

Complete Pricing: [5], [6] Stackelberg: [13]–[16]

IncompleteContract: [11], [12]Auction: [7]–[10]

Contract: This paper

need to consider the PU’s and SUs’ non-transferableutilities (since not everything can be translated to asingle currency) as well as other physical constraints.For example, the resource allocation to one SU will affectother SUs directly, and the physical constraints includetransmission power constraint, spectrum opportunitylimit, and relay number constraint. Thus our contractmodel is more complex to analyze than the pure moneyexchange (e.g., in [12]). We summarize the key literaturesof spectrum trading in Table 2.

2.2 Cooperative RelayThe performance of cooperative spectrum sharing mech-anism greatly depends on the underlying cooperativerelay protocol. There has been extensive research on thecooperative relay protocols, often within the scope ofphysical layer performance optimization [17]–[25]. Theresults in [17]–[19] have shown that the collaboration a-mong mobile users may significantly increase the systemperformance. In [20], Hua et. al. studied the optimizationof power (rate) subject to the rate (power) constraint.In [21], Gazor et. al. studied the problem of relay andprotocol selection using three criteria: rate, energy, andresource. In [22], Hou et. al. proposed a polynomial timealgorithm solving the optimal relay node assignmentproblem. In [23], Rong proposed the optimal structure ofsource and relay matrices for multi-hop MIMO relay sys-tems with QoS constraints. In [24], Sung et. al. proposeda new amplify-and-forward (AF) relay systems using su-perimposed pilot signals. In [25], Viberg et. al. proposeda new technique for self-interference suppression in full-duplex MIMO relays.

In this work, we consider the distributed space-time-coded protocol [26] as an example of the underlyingcooperative scheme. Our contract design can be usedwith other choices of relay protocols as well.

3 SYSTEM MODEL AND PROBLEM FORMULA-TION

We consider a cognitive radio network with a primarylicensed user (PU) and multiple secondary unlicensedusers (SUs) as shown in Fig. 1.2 Each user is a dedicated

2. The results of the single PU case in this paper can be extendedto the multiple PU case. Motivated by the fact that the number ofSUs is generally much larger than that of PUs, we can simply dividethe whole network into multiple sub-networks. Each sub-network stillconsists of one PU and a set of nearby SUs and our approach stillapplies. Note that such division will not cause significant loss of systemperformance.

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transmitter-receiver pair. The PU has the exclusive usageright of the licensed spectrum band, but suffers from thepoor channel condition between its transmitter PT andreceiver PR. We represent M SUs by distinct transmitter-receiver pairs {STk − SRk}Mk=1. Each SU wants to havededicated time to access the licensed band and transmitits own data, and it cannot transmit without the PU’spermission. The PU can employ a subset of or all SUs torelay its traffic; the involved SUs will obtain dedicatedtransmission time for their own data. The interactionbetween the PU and the SUs involves three phases asin Fig. 1: Phases I and II for the cooperative communi-cations with a total fixed length of time T0, and PhaseIII for the SUs’ own transmissions. More specifically,• Phase I: In the first half of the cooperative commu-

nication period (T0/2), the primary transmitter PTbroadcasts its data to the primary receiver PR andthe involved SUs’ transmitters (e.g., ST1, ST2, andST3 in Fig. 1). Note SU 4 and SU 5 are not involvedin this example.

• Phase II: In the remaining half of the cooperativecommunication period (T0/2), the involved SUs’transmitters (STs) decode the data received in PhaseI, and forward to the primary receiver PR simulta-neously using the space-time codes assigned by PUfrom its random codebook. Through proper choiceof space-time codes, SUs’ simultaneous relay signalsdo not interference with each other at the primaryreceiver PR [26], [28].

• Phase III: PU rewards each involved SU with adedicated time allocation for that SU’s own data(e.g., {tk}3k=1 for three involved SUs). SUs accessthe spectrum using TDMA and do not interfere witheach other.3

We assume each involved SU can successfully decodePU’s data in the first phase of cooperative communi-cations. Thus we focus on the relay links between thesecondary transmitters STs and primary receiver PR,which are the performance bottleneck of cooperativecommunications.4 It should be noted that the PU mayadopt a more efficient relay scheme where it can alsotransmit in Phase II, and we will show that our mainresults also extend to that scenario in Appendix G.

The PU and SUs have conflicting objectives in theabove interactions. The PU wants the SUs to relay itstraffic with high power levels, which will increase thePU’s data rate but reduce the SUs’ battery levels. AnSU k wants to obtain a large dedicated transmissiontime tk, which will increase the SU’s own performance

3. Our results also apply to the case where the SUs share thespectrum using FDMA, where a more efficient relay will be rewardedwith a larger bandwidth.

4. This assumption can be relaxed if the PU performs an initialscreening over all SUs as follows. The PU first broadcasts a pilot signalto all SUs. Only those SUs replying correctly can choose to accept thecontract and involve in the cooperative communications later. Noticethat the involved SUs are often the ones whose transmitters are closeto PT, and thus may be not close to PR. This explains why a PT − STchannel is often better than the corresponding ST − PR channel.

TABLE 3Key Notations

pk The received power at PR from SU k (relay);tk The dedicated access time for SU k;Φ The power-time contract, Φ = {(pk, tk),∀k ∈ {1, ...,K}};Rdir The PU’s direct transmission rate;Rtot

PU The PU’s total achievable rate with SUs’ cooperation;UPU The PU’s utility (average transmission rate);prlk The relay power of SU k for PU’s traffic;ptrk The transmission power of SU k for its own traffic;Ck The SU k’s cost for one unit power consumption;Rk The SU k’s own transmission rate;Πk The SU k’s payoff, Πk = tkRk − (tkp

trk + 0.5prlk )Ck;

Πk The SU k’s normalized payoff, also denoted by πk;θk The type of SU k, θk :=

2hSTk,PR(Rk−Ckptrk )

Ck.

but reduce the PU’s utility (time average data rate). InSections 3.1 and 3.2, we will explain in details howthe PU and SUs evaluate the trade-off between relaypowers and time allocations. In Section 3.3, we proposea contract-based framework which brings the PU andSUs together and resolves the conflicts.

We list key notations used in this paper in Table 3. Themeaning of each notation will be explained later.

3.1 Primary User Modeling

In this subsection, we discuss how PU evaluates relaypowers and time allocations.

We first compute the PU’s achievable data rate duringcooperative communications (i.e., Phases I and II inFig. 1). Let us denote the set of involved SUs as N(e.g., N = {1, 2, 3} in Fig. 1). The received power (atthe primary receiver PR) from SU k is pk, and the timeallocation to SU k is tk. Without loss of generality, wenormalize T0 to be 1 in the rest of the paper. Then tk canbe viewed as the percentage of the T0.• In Phase I, PT broadcasts its data, and PR achieves

a data rate (per unit time) of

Rdir = log(1 + SNRPT,PR), (1)

which remains as a constant throughout the anal-ysis. Here SNRa,b denotes the Signal-to-interferenceratio of the signal transmitted from node a to b.

• In Phase II, each involved SU decodes PT’s data(received in Phase I) and forwards to PR.

The PU’s total achievable rate during Phases I and IIis given by [26]:

RtotPU =Rdir

2+

1

2log

(1 +

∑k∈N

SNRSTk,PR

)

=Rdir

2+

1

2log

(1 +

∑k∈N pk

n0

), (2)

where n0 is the noise power, and 1/2 denotes equalpartition of the cooperative communication time intoPhase I and Phase II. We can think (2) as the sum oftransmission rates of two “parallel” channels, one from

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the PT to PR and the other from the set of involved SUs’transmitters to PR.

Based on the above discussion, we next compute thePU’s average data rate during entire time period (i.e.,Phases I, II, and III). The cooperative communicationsonly utilizes 1/(1 +

∑k∈N tk) fraction of the entire time

period. The PU’s objective is to maximize its utility (i.e.,average transmission rate during the entire time period)as follows

UPU =1

1 +∑k∈N tk

RtotPU

=1

1 +∑k∈N tk

(Rdir

2+

1

2log

(1 +

∑k∈N pk

n0

)),

(3)which is decreasing in the total time allocations to SUs(i.e.,

∑k∈N tk), and is increasing in the total received

power from SUs (i.e.,∑k∈N pk).

We want to emphasize that the utility calculation in(3) assumes that the PU involves at least one SU in thecooperative communication. The PU can also choose tohave direct transmissions only in both Phases I and IIand does not interact with the SUs (and thus there isno Phase III). The total rate in this direct transmissiononly approach is Rdir. This means that the PU will onlychoose to use cooperative communications if the utilityin (3) is larger than Rdir. In the rest of the analysis, weassume that Rdir is small such that the PU wants touse cooperative communications. In Section 8, we willfurther explain what will happen when this is not true.

3.2 Secondary User ModelingNext we discuss how SUs evaluate relay powers{pk}k∈N and time allocations {tk}k∈N . We want to em-phasize that the relay power is measured at the PU’s receiver,not at the SUs’ transmitters. We consider a general modelwhere SUs are heterogeneous in three aspects:• SUs have different relay channel gains between

their transmitters and the PU’s receiver (hST,PR). IfSU k wants to reach a received power pk at thePU’s receiver, it needs to transmit with a powerprlk = pk/hSTk,PR.

• SUs achieve different (fixed) rates to their ownreceivers (i.e., data rate Rk over link STk− SRk) withdifferent (fixed) transmission power (i.e., ptrk ).

• SUs have different cost Ck per unit transmissionpower.

Note that the parameters Ck, Rk, prlk , and hSTk,PR are SUk’s private information and are only known to itself.

We define an SU k’s payoff as Πk, which is thedifference between its own transmitted rate (during timeallocation tk in Phase III) and its cost of power consump-tion (during Phase II and its allocated time in Phase III).That is,

Πk = tk ·Rk −(tk · ptrk + 0.5 · prlk

)· Ck

= tk ·Rk −(tk · ptrk + 0.5 · pk

hSTk,PR

)· Ck

(4)

Without loss of generality, we assume that every SU iswilling to transmit if it does not need to relay the PU’straffic, i.e., Rk − ptrk · Ck ≥ 0 for all k ∈ N . Notice that(4) is increasing in time allocation tk, but is decreasingin relay power pk (received at PR).

We can further simplify (4) by multiplying both sidesby 2hSTk,PR/Ck, which leads to the normalized payoff

Πk := Πk2hSTk,PR

Ck=

2hSTk,PR(Rk − Ckptrk )

Cktk − pk. (5)

Such normalization does not affect SUs’ choice amongdifferent relay powers and time allocations. Thus it willnot affect the contract design introduced later.

To facilitate later discussions, we define an SU k’s typeas

θk :=2hSTk,PR(Rk − Ckptrk )

Ck> 0, (6)

which captures all private information of this SU.5 Alarger θk means that the SU’s own transmission is moreefficient (a larger Rk or a smaller ptrk ), or it has a betterchannel condition over relay link STk − PR (a largerchannel gain hSTk,PR), or it has a more efficient batterytechnology (a smaller Ck). With (6), we can simplify SU ’snormalized payoff in (5) as

πk(pk, tk) := Πk = θktk − pk. (7)

which is decreasing in PU’s received power pk andincreasing in SU k’s achieved time tk.

Since each SU is selfish, a type-θk SU wants to chooserelay power and time allocation to maximize its normal-ized payoff in (7). Notice that an SU always has theoption of not helping the PU and thus receiving zerotime allocation and zero payoff (i.e., tk = pk = 0). Thusthe PU needs to provide proper incentives to attract theSUs to help.

3.3 Contract FormulationAfter introducing PU’s utility (3) and SUs’ normalizedpayoffs (7), we are ready to introduce the contract mech-anism that resolves the conflicting objectives between thePU and SUs.

Contract theory studies how economic decision-makers construct contractual arrangements, generally inthe presence of private information [30]. In our case, theSUs’ types are private information. The PU does notknow each SU’s type, and needs to design a contractto attract the SUs to participate in cooperative commu-nications to maximize PU’s utility.

To better understand the contract design in this paper,we can imagine the PU as the employer and SUs asemployees in a labor market. The employer optimizesthe contract, which specifies the relationship between theemployee’s performance (i.e., received relay powers atPU’s receiver) and reward (i.e., time allocations to SUs).

5. From a practical implementation point of view, we need to dis-cretize the users’ types (which are related to users’ channel conditions,data rates, and energy consumption costs) to a finite number of levels(similar to [34], [35]).

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If we denote P as the set of all possible relay powersand T as the set of all possible time allocations, thenthe contract specifies a t ∈ T for every p ∈ P . Eachdistinct power-time association becomes a contract item.Once a contract is given, each SU will choose the contractitem that maximizes its payoff in (7). The PU wants tooptimize the contract items to maximize its utility in (3).

We consider K SU types denoted by set Θ ={θ1, θ2, ..., θK}.6 Without loss of generality, we assumethat θ1 < θ2 < ... < θK . The total number of type-θkSUs is Nk. According to the revelation principle [29], itis enough to consider the class of contract that enablesthe SUs to truthfully reveal their types. Because of this,it is enough to design a contract that consists K contractitems, one for each type. We denote contract item de-signed for type-θk as (pk, tk). The contract can be writtenas Φ = {(pk, tk),∀k ∈ K} where K = {1, 2, ...,K}.7

We consider optimal contract design for three infor-mation scenarios.

• Complete information in Section 4: This is a benchmarkcase, where the PU knows each SU’s type. We willcompute the maximum utility the PU can achievein this case, which serves as an upper-bound ofthe PU’s achievable utility under any informationscenario.

• Weak incomplete information in Section 6: The PU doesnot know each SU’s type, but has knowledge ofthe set of types and the number of each type SUsin the market (i.e., Nk for type-θk SUs). We willshow that the optimal contract in this case achievesthe same maximum PU’s utility as in the completeinformation benchmark.

• Strong incomplete information in Section 7: The PUonly knows the total number of SUs (N ) and thedistribution of each type, but does not know thenumber of each type (Nk). The PU needs to designa contract to maximize its expected utility.

Once a contract is given, the interactions between thePU and SUs follow 4 steps.

1) The PU broadcasts the contract Φ = {(pk, tk),∀k ∈K} to all SUs.

2) After receiving the contract, each SU chooses onecontract item that maximizes its payoff and informsthe PU its choice.

3) After receiving all SUs confirmations, the PU in-forms the involved SUs (i.e., those choosing posi-tive contract items) the space-time codes to use inPhase II and the transmission schedule in Phase

6. Although θk is a continuous random variable depending on therealisation of the channel coefficient, here we assume there are Kdistinct types for simplicity, where any two distinct types can bearbitrary close to each other. A large enough K can well approximatethe continuous case with only a slightly decreased system performance.

7. Note that an SU can always choose not to work for the PU, whichimplies an implicit contract item (p, t) = (0, 0) in the contract (oftennot counted in the total number of contract items).

III.8 Note that the length for transmission time foreach involved SU is specified by the contract itemand the PU can no longer change.

4) The communications start by following three phas-es in Fig. 1.

It is worth noting that our contract design focuses onthe issue of hidden (private) information, that is, it aimsat screening the private type of SUs so as to maximizethe PU’s utility under incomplete information. We donot consider other issues regarding contract design (suchas hidden action or moral hazard), since the contractsfor eliciting private information and for hedging moralhazard are very different and are usually studied inde-pendently. Once a contract has been agreed by the PUand one SU, we assume that neither the PU nor theSU will deviate from the contract. This is reasonablein our model, because the action of each SU can beobserved by the PU through measuring the receivedrelay power from the SU, and the action of the PUcan also be observed by the SU through measuring theduration of the awarded access time. In this case, it isnot difficult to solve the moral hazard or hidden actionproblem by introducing simple penalty mechanisms aspotential threat. For example, the PU can impose a largepenalty on the SU (e.g., not cooperating with the SU inthe future), once observing the SU’s deviation from theagreement, and vice versa.

4 OPTIMAL CONTRACT DESIGN WITH COM-PLETE INFORMATION

In the complete information scenario, the PU knowsprecisely the type of each SU. We will use the maximumPU’s utility achieved in this case as a benchmark to e-valuate the performance of the proposed contracts underincomplete information in Sections 6 and 7. Without lossof generality, we assume that Nk ≥ 1 for all k ∈ K.

As the PU knows each SU’s type, it can monitor andmake sure that each SU accepts only the contract itemdesigned for its type. The PU needs to ensure that theSUs have non-negative payoffs so that they are willingto accept the contract. In other words, the contract needsto satisfy the following individual rationality constraint.

Definition 1: [IR: Individual Rationality] A contractsatisfies the IR constraint if for each type-θk SU, itreceives a non-negative payoff by accepting the contractitem for θk, i.e.,

θktk − pk ≥ 0,∀k ∈ K. (8)

8. The exchange of such information (e.g., space-time codes andother details) between the PU and SUs will introduce additionalcommunication overhead. However, as long as the cooperation lasts fora sufficiently long time (e.g., including many rounds of Phases I-III), theoverhead is relatively small. The overhead may become a major issueif SUs are mobile and their wireless characteristics (e.g., relay channelgains and their own channel gains) change frequently. In that case, onlya few rounds of Phases I-III last before the user types change, and thePU needs to tradeoff the cooperation benefit and the communicationoverhead. Detailed analysis and results are provided in Appendix G.

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We say a contract is optimal if it yields the maximumutility for the PU under the current information scenario.Different information scenarios may lead to differentoptimal contracts.

In the complete information scenario, an optimal con-tract maximizes the PU’s utility as follows

max{(pk,tk)≥0,∀k∈K}

Rdir

2 + 12 log

(1 +

∑k∈KNkpkn0

)1 +

∑k∈KNktk

,

subject to IR Constraints in Eq. (8).

(9)

In this paper, the vector operations are component-wise(e.g, (pk, tk) ≥ 0 means that pk ≥ 0 and tk ≥ 0) unlessspecified otherwise. Then we have the following result.

Lemma 1: In an optimal contract with complete infor-mation, each SU achieves zero payoff by accepting thecorresponding contract item, i.e., tkθk = pk, ∀k ∈ K.

Proof. We prove by contradiction. Suppose that thereexists an optimal contract item (pk, tk) with θktk−pk > 0.Since PU’s utility in (9) is increasing in pk and decreasingin tk, the PU can increase its utility by decreasing tk untilθktk − pk = 0. This contradicts with the assumption that(pk, tk) with θktk−pk > 0 belongs to an optimal contract,and thus completes the proof.

Using Lemma 1, we can replace pk by θktk for each k ∈K and simplify the PU’s utility maximization problem in(9) as

max{tk≥0,∀k∈K}

Rdir

2 + 12 log

(1 +

∑k∈KNkθktk

n0

)1 +

∑k∈KNktk

. (10)

Theorem 1: In an optimal contract with complete in-formation, only the contract item for the highest typeis positive and all other contract items are zero, i.e.,(pK , tK) > 0 and (pk, tk) = 0, ∀k < K.9

Proof. By contradiction, suppose there exists an opti-mal contract item with tk > 0 and k < K (for type-θkSUs). The total time allocation is T ′ =

∑k∈KNktk. Then

the PU’s utility is

U1PU =

1

1 + T ′

(Rdir

2+

1

2log

(1 +

∑k∈K θkNktk

n0

)).

(11)Next we show that given a fixed total time allocationT ′, allocating positive time only to the highest type SUs(i.e., NKtK = T ′) achieves a larger utility for the PU asfollows

U2PU =

1

1 + T ′

(Rdir

2+

1

2log

(1 +

θKNKtKn0

)). (12)

This is because θKNKtK = θKT′ in (12) and∑

k∈K θkNktk < θKT′ in (11), thus (12) is larger than

(11). This contradicts with the optimality of the contract,and thus completes the proof.

Intuitively, the highest type SUs can offer the mosthelp to the PU given total time allocation in Phase III.

9. As NK and tK always appear together in (10), the PU’s optimalutility does not depend on the specific value of NK , and the com-plexity in calculating the optimal time allocations does not dependon NK either. It is also worth noting that when SUs have maximumtransmission power constraints, it may not be optimal for the PU toinvolve the highest type SUs only. For details, please see Appendix A.

Using Theorem 1, the optimization problem in (10) canbe simplified as

maxtK≥0

1

1 +NKtK

(Rdir

2+

1

2log

(1 +

θKNKtKn0

)). (13)

Since NK and tK always appear as a product in (13), wecan redefine the optimization variable as tK = NKtKand rewrite (13) as

maxtK≥0

1

1 + tK

(Rdir

2+

1

2log

(1 +

θK tKn0

)), (14)

This means the PU’s optimal utility does not depend onNK . When NK changes, the optimal time allocation peruser t∗K changes inversely proportional to NK .

At this point, we have successfully simplified thePU’s optimization problem from involving 2K variables{(pk, tk),∀k ∈ K} in (9) to a single variable tK in (14).

Let us denote the objective of the problem (14) asUPU(tK), which is a function of tK but is not concave.Any local optimal solution (denoted as tK) to the prob-lem (14) satisfies

dUPU(tK)

dtK|tK=tK

=

θK(1+tK)

n0+θK tK− (Rdir + log(1 + θK tK

n0))

(1 + tK)2= 0.

(15)We can further compute the second order derivative as

d2UPU(tK)

dt2K|tK=tK

=1

(1 + tK)4

(− θ2K(1 + tK)3

(n0 + θK tK)2

+ 2(1 + tK)

(θK(1 + tK)

n0 + θK tK−(Rdir + log

(1 +

θK tKn0

)))).

(16)By substituting (15) into (16), we can show thatd2UPU(tK)/dt2K is always negative at tK . This meansthat the local optimal solution (without considering thenonnegative constraint) to the problem (14) is uniqueand is actually the global optimal solution t∗K . Thus weconclude that t∗K = max(tK , 0), where tK is the uniquesolution to (15) and can be computed numerically.

5 FEASIBLE CONTRACTS UNDER INCOM-PLETE INFORMATION

In this section, we study the necessary and sufficientconditions for a feasible contract under incomplete in-formation. This will help us derive optimal contracts inSections 6 and 7.

A feasible contract includes K power-time items suchthat any type-θk SU prefers the contract item for its type,i.e., (pk, tk), to any other contract item. Under incom-plete information, a feasible contract must satisfy boththe individual rationality (IR) constraint in Definition 1(introduced in Section 4) and the incentive compatibility(IC) constraint defined next.

Definition 2: [IC: Incentive Compatibility] A contractsatisfies the IC constraint if for each type-θk SU, it prefersto choose the contract item for θk, i.e.,

θktk − pk ≥ θktj − pj ,∀k, j ∈ K. (17)

8

In summary, the PU’s optimization problem is

max{(pk,tk),∀k∈K}

UPU({(pk, tk), k ∈ K}), (18)

subject to θktk − pk ≥ θktj − pj , ∀k, j ∈ K,θktk − pk ≥ 0,∀k ∈ Ktk ≥ 0, pk ≥ 0, ∀k ∈ K.

The first two constraints correspond to IC and IR con-straints, respectively. It is worth noting that with IC andIR constraints, each SU will truthfully reveal its privatetype. Specifically, the IR constraint ensures that a typeθk SU will get a nonnegative payoff by choosing the k-th contract item. The IC constraint ensures that a typeθk SU will get the maximum payoff by choosing the k-th contract item. In other words, the IC constraint willincentivise SUs to truthfully reveal their types to thePU.10

5.1 Sufficient and Necessary Conditions for Feasi-bilityNext we provide several necessary and sufficient con-ditions for the contract feasibility. Let Φ = {(pk, tk),∀k}denote a feasible contract.

Proposition 1: [Necessary Condition 1] For any i, j ∈K, pi > pj if and only if ti > tj .

The proof of Proposition 1 is given in Appendix B. Thisproposition shows that an SU contributing more in termsof received power at the PU receiver should receive moretime allocation, and vice versa. From Proposition 1, wehave the following corollary, saying that the same relaypowers must have the same time allocations, and viceversa.

Corollary 1: For any i, j ∈ K, pi = pj if and only ifti = tj .

The following Proposition 2 shows the second neces-sary condition for contract feasibility.

Proposition 2: [Necessary Condition 2] For any i, j ∈K, if θi > θj , then ti ≥ tj .

The proof of Proposition 2 is given in Appendix C.This proposition shows that a higher type SU shouldbe allocated more transmission time. Combined withProposition 1, we know that a higher type of SU shouldalso contribute more in terms of PU’s received power.

From Propositions 1 and 2, we conclude that for afeasible contract, all power-time combination (contractitems) satisfy

0 ≤ p1 ≤ p2 ≤ ... ≤ pK , 0 ≤ t1 ≤ t2 ≤ ... ≤ tK , (19)

with pk = pk+1 if and only if tk = tk+1.The previous propositions help us obtain Theorem 2

as follows.Theorem 2: [Sufficient and Necessary Conditions for

Contract Feasibility]: For a contract Φ = {(pk, tk),∀k}, itis feasible if and only if all the following three conditionshold:

10. In fact, this contract belongs to the class of screening contract,and the main purpose of a screening contract is to elicit the privateinformation of agents, so as to avoid the false reports of agents.

• Contd.a: 0 ≤ p1 ≤ p2 ≤ ... ≤ pK and 0 ≤ t1 ≤ t2 ≤... ≤ tK ;

• Contd.b: θ1t1 − p1 ≥ 0;• Contd.c: For any k = 2, 3, ...,K,pk−1 + θk−1(tk − tk−1) ≤ pk ≤ pk−1 + θk(tk − tk−1).

(20)The proof of Theorem 2 is given in Appendix D. The

conditions in Theorem 2 are essential to the optimalcontract design under weakly and strongly incompleteinformation in Sections 6 and 7.

6 OPTIMAL CONTRACT DESIGN UNDERWEAKLY INCOMPLETE INFORMATION

In this section, we will look at the weakly incompletescenario where the PU does not know each SU’s type,but knows the set of SUs’ type (i.e., Θ) and the numberof each type (i.e., Nk for any k ∈ K). Without loss of gen-erality, we assume that Nk ≥ 1 for all k ∈ K.11 Differentfrom the complete information case, here the PU cannotdetermine the optimal contract item by Lemma 1 for anSU since the PU does not know the SU’s type. The majordifference is that we need to consider IC constraint here.

A conceptually straightforward approach to derive theoptimal contract is to solve (18) directly. Going throughthis route, however, is very challenging as (18) is a non-convex and involves complicated constraints. Here weadopt a sequential optimization approach instead: wefirst derive the best relay powers {p∗k({tk,∀k}),∀k} givenfixed feasible time allocations {tk,∀k}, then derive thebest time allocations {t∗k,∀k} for the optimal contract,and finally show that there is no gap between the solu-tion {(p∗k, t∗k),∀k} obtained from this sequential approachand the one obtained by directly solving (18).

Proposition 3: Let Φ = {(pk, tk),∀k} be a feasible con-tract with fixed time allocations {tk,∀k : 0 ≤ t1 ≤ ... ≤tK}. The optimal unique relay powers satisfy

p∗1({tk,∀k}) = θ1t1,

p∗k({tk,∀k}) = θ1t1 +

k∑i=2

θi(ti − ti−1),∀k = 2, ...,K.

(21)The proof of this proposition is given in Appendix E.

Using Proposition 3, we can simplify the PU’s optimiza-tion problem in (18) as

max{tk,∀k∈K}

Rdir

2 + 12 log

(1 +

∑k∈KNk(θ1t1+

∑ki=1 θi(ti−ti−1))

n0

)1 +

∑k∈KNktk

,

(22)subject to 0 ≤ t1 ≤ ... ≤ tK .We can further simplify (22) using Theorem 3 below.

Theorem 3: In an optimal contract with weakly incom-plete information, only the contract item for the highestSU type is positive and all other contract items are zero,i.e., (pK , tK) > 0 and (pk, tk) = 0, ∀k < K.

11. Otherwise we can remove type θk from Θ and solve a problemwith K − 1 types.

9

Theorem 3 can be proved in a similar way as The-orem 1 and we skip the proof here due to page limit.Following a similar analysis as in Appendix A, we canshow that it may not be optimal for the PU to involvethe highest type SUs if we impose additional constraints,e.g., SUs’ transmission power constraints. However, evenwithout such constraints, we will later show that the PUunder strongly incomplete information needs to involvemultiple types of SUs to maximize the expected utility.

Using Theorem 3, we can simplify the optimizationproblem in (22) further as

maxtK≥0

1

1 +NKtK

(Rdir

2+

1

2log

(1 +

θKNKtKn0

)). (23)

Notice that (23) under weakly incomplete informationis the same as (13) under complete information. We thusconclude that our sequential optimization approach (firstover {pk,∀k} and then over {tk,∀k}) results in no loss inoptimality, since it achieves the same maximum utilityas in the complete information scenario.

To solve problem (23), we can use an efficient one-dimensional exhaustive search algorithm to find theglobal optimal solution t∗K . We will provide numericalresults in Section 8.

7 OPTIMAL CONTRACT DESIGN UNDERSTRONGLY INCOMPLETE INFORMATION

In this section, we study the strongly incomplete infor-mation scenario, where the PU does not even know thenumber of each type. The PU only knows the total num-ber of SUs N and the probability qk of any SU belongingto type-θk. Obviously qk ∈ [0, 1] and

∑k∈K qk = 1.

Similar to the weakly incomplete information scenario,here the PU needs to consider the IC constraint since itcannot determine the optimal contract item by Lemma 1for an SU as well. The difference from Section 6 is thathere the PU does not know which type is the highesttype among all SUs in the current system and howmany SUs are in the highest type, and thus the simpleapproach of only providing a positive contract item fortype θK as in Theorem 3 may not be optimal. If the PUdoes that and it turns out that NK = 0 in the currentsystem, then there will be no SUs participating in thecooperative communications.

The proper target for the PU is design a contract tomaximize the expected utility subject to the IC and IRconstraints. As the PU knows the total number of SUsN , then the probability density function of the numberof SUs {Nk,∀k} is

Q(n1,...,nK−1)

:= Pr(N1 = n1, ..., NK−1 = nK−1, NK = N −K−1∑i=1

Ni)

=N !

n1!...nK−1!(N −∑K−1i=1 ni)!

qn11 ...q

nK−1

K−1 qN−

∑K−1i=1 ni

K .

(26)

The PU’s optimization problem can be written in (24)subject to the IC and IR constraints.

Similar to Section 6, here we adopt a sequential op-timization approach: we first derive the optimal relaypowers {p∗k({tk,∀k}),∀k} with fixed feasible time allo-cations {tk,∀k}, then derive the optimal time allocations{t∗k,∀k} for the optimal contract. The difference is thatoptimality is no longer guaranteed here as explainedlater.

Proposition 4: Let Φ = {(pk, tk),∀k} be a feasible con-tract with fixed time allocations {tk,∀k : 0 ≤ t1 ≤ ... ≤tK}, then the unique optimal relay powers satisfy

p∗1({tk,∀k}) = θ1t1,

p∗k({tk,∀k}) = θ1t1 +

k∑i=2

θi(ti − ti−1),∀k = 2, ...,K.

(27)

Notice that Proposition 4 under strongly incompleteinformation is actually the same as Proposition 3 underweakly incomplete information. Proposition 4 can beproved in a similar manner as Proposition 3, using thefact that PU’s expected utility is increasing in {pk,∀k}.The proof is omitted here due to space limit. Based onProposition 4, we can simplify the PU’s optimizationproblem in (24) as (25).

Note that (25) is a non-convex optimization problemand all K variables are coupled in the objective function.Furthermore, the number of terms in the objective in-creases exponentially with the number of types K. Thusit is hard to solve efficiently.

Next, we propose a low computation complexityapproximate algorithm, Decompose-and-Compare algo-rithm, to compute a close-to-optimal solution to (25)efficiently. In this algorithm, we will compare K simplecandidate contracts, and pick the one that yields thelargest utility for the PU. There are two key ideas behindthis heuristic algorithm.• One positive contract item per contract: In each of theK individually optimized candidate contracts, thereis only one positive contract item (for one or moretypes). For the contract item in the kth candidatecontract, this positive contract item is offered to SUswith types equal to or larger than type-θk. In oth-er words, optimization of each candidate contractonly involves a scalar optimization, instead of Kvariables as in (25).

• A tradeoff between efficiency and uncertainty: AmongK optimized candidate contracts, we will pick thebest one that achieves the best tradeoff betweenefficiency and uncertainty (i.e., maximizing the PU’sexpected utility). Under strongly incomplete infor-mation, it is not clear which type is the highestamong all SUs existing in the current system. If acandidate contract offers the same positive contractitem for types equal to or larger than θk, then allSUs in these types will choose to accept that contractitem. The corresponding SU’s payoff is increasing in

10

max{(pk,tk),∀k}

N∑n1=0

N−n1∑n2=0

...

N−∑K−2

i=1 ni∑nK−1=0

Q(n1,...,nK−1)

(Rdir

2+ 1

2log

(1 +

∑K−1i=1 nipi+(N−

∑K−1i=1 ni)pK

n0

))1 +

∑K−1i=1 niti + (N −

∑K−1i=1 ni)tK

. (24)

subject to, IC and IR constraints

max{tk,∀k}

N∑n1=0

N−n1∑n2=0

...

N−∑K−2

i=1 ni∑nK−1=0

Q(n1,...,nK−1)

(Rdir

2+ 1

2log

(1 +

∑K−1i=1 nip

∗i ({tk,∀k})+(N−

∑K−1i=1 ni)p

∗K({tk,∀k})

n0

))1 +

∑K−1i=1 niti + (N −

∑K−1i=1 ni)tK

. (25)

subject to, 0 ≤ t1 ≤ ... ≤ tK .

type, i.e., a type θk SU receives zero payoff and atype-θK SU receives the maximum positive payoff.Thus choosing a candidate contract with a thresholdθk too low will give too much payoffs to the SUs(and thus reduce the PU’s expected utility), butchoosing a candidate contract with a threshold toohigh might lead to the undesirable case that noSUs are willing to participate. This requires us toexamine all possibilities (i.e., K candidate contracts)and pick the one with the best performance.

The Decompose-and-Compare algorithm works as fol-lows.

1) Decomposition: Construct K candidate contracts,where for the kth contract,• Offers the same contract item tk > 0 to SUs

with a type equal to or larger than type θk(called critical type), and zero for the SUs belowthe critical type. That is, t1 = t2 = ... = tk−1 = 0and tk = tk+1 = ... = tK .

• Computes the optimal t∗k that maximizes PU’sexpected utility in (25) under the constraints oft1 = t2 = ... = tk−1 = 0 and tk = tk+1 = ... =tK . The corresponding relay power is p∗k = θkt

∗k

as specified in the contract.2) Comparison: Choose the best contract out of the K

candidates to maximize the PU’s expected utility.Intuitively, we can view each candidate contract (cor-

responding to a critical type θk) in the above algorithmas a simple take-it-or-leave contract (with time allocation tkand relay power pk = θktk). Obviously, SUs with typeshigher than or equal to θk will accept the contract, whileSUs with types lower than θk will refuse the contract.The decompose-and-compare algorithm computes all Koptimal take-it-or-leave contracts, and chooses the bestone that maximizes the expected revenue.

Now we analyze the computational complexity of theabove decompose-and-compare algorithm. Without suchan algorithm, we can simply use exhaustive search tooptimally solve the non-convex optimization Problem(25). With such an algorithm, we can still use exhaustivesearch to solve each of K subproblems to derive acandidate contract. To make a fair comparison betweenthe complexities with and without this algorithm, weadopt the basic exhaustive search for complexity mea-surement. Let us denote the possible range of tk as [0,W ].We approximate the continuity of this range through a

proper discretization, i.e., representing all possibilitiesof any tk by T equally spaced values (with the firstand last values equal to 0 and W , respectively). Then,with the optimal contract design, we need to searchover all possible time allocation ranges for K typesjointly (i.e., {tk,∀k ∈ K}) to directly solve Problem (25),and the computation complexity is O(TK). As we usethe decompose-and-compare algorithm, we decomposethe complex Problem (25) into K simple sub-problems,each associated with one critical type. The computationcomplexity for each sub-problem is O(T ), and thus theoverall computation complexity is O(K · T ).

Unlike complete or weakly incomplete information,here the computational complexity increases with thenumber of SU types. This is because that the PU maywant to involve more than one type of SUs in the con-tract to mitigate the uncertainty and avoid risk of havingno relay in a particular network realisation. However,the complexity of this approach will increase with thenumber of types.

In Section 8, we show by numerical results that theproposed Decompose-and-Compare algorithm achievesa performance very close to the optimal solution to (25)in most cases.

8 NUMERICAL RESULTS

8.1 Complete and Weak Incomplete Information S-cenariosAs shown in Sections 4 and 6, the optimal contract is thesame for complete and weakly incomplete informationscenarios. By examining the PU’s optimization in (23)that applies to both scenarios, we have the followingobservations.

Observation 1: The PU’s optimal utility is increasing inthe highest SU type θK and the PU’s direct transmissionrate Rdir.

Figure 2 shows that PU’s utility under the optimalcontract is increasing in θK and Rdir.12 We also plot thedotted baseline that represents the rate Rdir achieved bydirect transmission only. As Rdir increases, the PU hasless incentive to share spectrum with the SUs. When Rdir

is very large, the PU chooses not to use SUs at all, whichcorresponds to No Relay Region in Fig. 2 (where the threecurves with different θK ’s are below the baseline).

12. Without loss of generality, we normalize n0 to be 1 in the rest ofthis paper.

11

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

Direct transmission rate Rdir

PU

’s o

ptim

al u

tility

u* P

U

Relay region

No relay region

θK=10

θK=15

θK=20

Direct transmission

Fig. 2. PU’s optimal utility U∗PU as a function of thehighest type θK and direct transmission rate Rdir. Thedotted 45 degree line divides the figure into two regions:Relay Region and No Relay Region. When the maximumutility achievable under the optimal contract (by using thelargest type θK) falls into the No Relay Region, the PU willchoose direct transmission only.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

The highest SU type θK

PU

’s o

ptim

al to

tal t

ime

allo

catio

n N

Kt K*

Rdir=0

Rdir=0.25

Rdir=0.5Rdir=0.75

Fig. 3. PU’s optimal total time allocation NKt∗K as a

function of the PU’s direct transmission rate Rdir and thehighest type θK .

For the rest of the numerical results, we will onlyexamine the PU’s choice of optimal contract, withoutrestating the need to compare with Rdir and choosedirect transmission only if needed.

Observation 2: Figure 3 shows that the PU’s optimaltotal time allocation to the highest type-θK SUs, NKt∗K ,decreases in PU’s direct transmission rate Rdir. WhenRdir = 0, the PU’s optimal total time allocation is strictlydecreasing in θK ; when Rdir > 0, the total time allocationfirst increases in θK and then decreases in θK .

This can be explained as follows. When Rdir = 0,the PU can only rely on SUs for transmissions and will

always allocate positive transmission time to the highesttype-θK SUs. If we look at the PU’s utility in (23) withRdir = 0, the logarithmic term log(1 + θNKt

∗K) plays a

more important role than the scaling term 11+NKt∗K

inthis case. When θK is small, the PU needs to allocatea large amount time to the SUs to achieve its desirablerate. When θK becomes large, the PU can reach a highrelay rate by allocating less transmission time to the SUs.This explains why we observe a decrease of NKt∗K in θK .Appendix F provides a rigorous proof of Observation 2under Rdir = 0.

When Rdir > 0 (the lower three curves in Fig. 3), thePU has less incentive to allocate transmission time to theSUs, especially when the highest SU type-θK is small. AsθK becomes large, PU is willing to allocate more time inexchange of efficient help from SUs. As θK becomes verylarge, the PU only needs to allocate a small amount oftime to the SUs in order to obtain enough relay help.This explains why the lower three curves in Fig. 3 firstincrease and then decrease in θK .

8.2 Strong Incomplete Information Scenario

Here we show how PU design the contract to max-imize its expected utility. As a performance bench-mark, we first compute the optimal solution to thePU’s expected utility maximization problem in (25)via an K-dimensional exhaustive search. We denotethe corresponding optimal solution as E[UPU]∗. Noticethat E[UPU]∗ is often smaller than the maximum utilityachieved under complete information. The performancegap is due to the strongly incomplete information. Next,we will compare the PU’s expected utility achieved bythe proposed Decompose-and-Compare algorithm withE[UPU]∗.

For illustration purposes, we consider only two typesof SUs: θ1 < θ2. The PU only knows the total number ofSUs N and the probabilities q1 and q2 of two types, withq1 + q2 = 1.

In the Decompose-and-Compare algorithm, we firstconsider two candidate contracts. The first candidatecontract optimizes the same positive contract item t1 =t2 > 0 for both types. The PU’s corresponding maxi-mum expected utility is E[UPU]1−2. The second candidatecontract sets t1 = 0 and optimizes the positive contractitem t2 > 0. The PU’s corresponding maximum expectedutility is E[UPU]2. Then we pick the candidate contractthat leads to a larger PU’s expected utility as the solutionof the Decompose-and-Compare algorithm.

To evaluate the performance of the algorithm, weconsider two different parameter regimes.

8.2.1 Large qN1This means that the probability that all SUs belongto the low type-θ1 is large. This happens when thetotal number of SUs N is small and probability q1 islarge. Figure 4 shows the PU’s expected utility obtained

12

0 1 2 3 40

0.5

1

1.5

2

Direct transmission rate Rdir

PU

’s e

xpec

ted

utili

tyOptimal E[u

PU ] *

Candidate E[uPU

] 1−2

Candidate E[uPU

] 2

Fig. 4. Comparison between PU’s optimal expectedutility using optimal exhaustive search method (E[UPU]∗)and the two candidate contracts of the Decompose-and-Compare algorithm, as a function of the PU’s direct trans-mission rate Rdir. We set q1 = 0.9, N = 2, θ1 = 4, andθ2 = 10.

0 1 2 3 4

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Direct transmission rate Rdir

PU

’s e

xpec

ted

utili

ty

Optimal E[uPU

] *

Candidate E[uPU

] 1−2

Candidate E[uPU

] 2

Fig. 5. Comparison between PU’s optimal expectedutility using optimal exhaustive search method (E[UPU]∗)and the two candidate contracts of the Decompose-and-Compare algorithm, as a function of the PU’s direct trans-mission rate Rdir. We set q1 = 0.5, N = 5, θ1 = 4, andθ2 = 10.

with the two candidate contracts of the Decompose-and-Compare algorithm (E[UPU]1−2 and E[UPU]2)) and theoptimal exhaustive search method (E[UPU]∗), as functionsof PU’s direct transmission rate Rdir. We can see that thecandidate contract that offers the same positive contractitems to both types (i.e., E[UPU]1−2) achieves a close-to-optimal performance with all values of Rdir simulatedhere. This is because very often the PU needs to rely onthe low type-θ1 SUs to relay its traffic.

We also notice that the candidate contract that offers apositive contract item to the high type SU (i.e., E[UPU]2)also achieves a close-to-optimal performance (even larg-

0 2 4 6 8 10 120.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Number of higher type SUs N2

PU

’s o

ptim

al u

tiliti

es

Strongly incomplete informationComplete information

Fig. 6. Comparison between PU’s optimal utilities fordifferent user number realizations under different infor-mation scenarios. We set q1 = 0.5, N = 12, Rdir = 1,θ1 = 10, and θ2 = 20.

er than E[UPU]1−2) when Rdir is large. This is becausethe PU with a large Rdir relies less on the SUs, and willhave more incentive to employ only the high type SUswhen they are available.

8.2.2 Small qN1This means that the probability that at least one SUbelongs to the high type-θ2 (i.e., 1 − qN1 ) is large. Fig-ure 5 shows the PU’s expected utility obtained fromDecompose-and-Compare algorithm and the optimal ex-haustive search method as a function of Rdir. We can seethat E[UPU]2 is always better than E[UPU]1−2 and achievesa close-to-optimal performance under all choices of Rdir.This is because very often the PU can find a high typeθ2 SU to relay its traffic.

Observation 3: The performance of the proposedDecompose-and-Compare algorithm achieves a close-to-optimal performance (i.e., less than 2% according toFig. 4 and Fig. 5) under the strongly incomplete infor-mation.13

Next we study how the strongly incomplete informa-tion reduces PU’s utility comparing with the completeinformation benchmark. First, we note that PU’s contractin the strongly incomplete information scenario does notdepend on N1 and N2, as the PU targets at optimizingthe expected rate and does not know the realizationinformation. However, the actual PU’s utility (not theexpected value) does depend on N1 and N2. Figure 6shows the PU’s utility under different information sce-narios and different user number realizations (i.e., anyrealization of N1 and N2 for a fixed N1 + N2 = N ).Here, qN1 is small and the curve with strongly incom-plete information corresponds to offering t2 > 0 and

13. Due to space limit, we only discuss two-type case here. Similarresults can be obtained for cases of more than two types.

13

t1 = 0 (as in Fig. 5). The (very close to) optimal contractunder strongly incomplete information can be obtainedby using the Decompose-and-Compare algorithm. Theoptimal contract under complete information changes asN1 and N2 change.

Observation 4: Figure 6 shows that the PU’s optimalutility under strongly incomplete information achievesthe maximum value (close to the one under completeinformation) when the realized SU numbers is close tothe expected value (i.e., N1 = N2 = 6 in this example), asthe PU determines the contract for SUs in the expectedsense.

In Fig. 6, the largest performance gap between the twocurves happens when N2 = 0. In this case, the optimalcontract under complete information satisfies t1 > 0 andt2 = 0, as there are no type-θ2 users. However, the PUunder strongly incomplete information chooses t1 = 0and t2 > 0 to maximize the PU’s expected utility. Suchmismatch means that the PU under strongly incompleteinformation has no SUs serving as relays. However,this parameter setting only happens with a very smallprobability qN1 = 0.512 ≈ 2.4× 10−4.

Another useful comparison is the PU’s average util-ity loss due to strongly incomplete information. Wefirst compute the PU’s optimal average utility understrongly incomplete information, which is the solutionof (25) (via a K-dimensional exhaustive search ratherthan the Decompose-and-Compare algorithm). Then wecan compute the PU’s average utility under completeinformation by calculating the weighted sum (weightedby the probability of each parameter (N1, N2)) of the 13values on the upper curve in Fig. 6.14 In this example, theratio is 0.9874. This means that the PU’s average utilityloss due to strongly incomplete information is very small(i.e., less than 1.3%).

9 CONCLUSION AND FUTURE WORK

We study the cooperative spectrum sharing between onePU and multiple SUs, where the SUs’ types are privateinformation. We model the network as a monopolymarket, in which the PU offers the contract and each SUselects the best contract item according to its type. Westudy the optimal contract design for multiple informa-tion scenarios. We first provide the necessary and suffi-cient conditions for feasible contracts under incompleteinformation. For the weakly incomplete information s-cenario, we derive the optimal contract that achievesthe same PU’s utility as in the complete informationbenchmark. For the strongly incomplete information sce-nario, we propose a Decompose-and-Compare algorithmthat achieves a close-to-optimal PU’s expected utility.Both the PU’s average utility loss due to the suboptimalalgorithm and the strongly incomplete information are

14. With complete information, the PU’s optimal utility in a time slotdepends on the number of users in each class and may not be the sameall the time (see N2 = 0 and N2 ≥ 1 in Fig. 6).

small in our numerical example (less than 2% and 1.3%in our numerical results with two SU types).

This work represents an important step towards es-tablishing a general framework of understanding incom-plete information in dynamic spectrum sharing. As thenext step, we will consider more incomplete informationstructures: (1) the PU does not know the distributionof SUs’ types, and (2) each SU knows other SUs’ typesbut PU does not know (a similar setting with a differentapplication has been studied in [27]).

We also want to understand how to design incentivecompatible mechanisms in a market with multiple PUsand multiple SUs. The mechanism proposed in thispaper can be viewed as a sub-optimal solution in themultiple PU scenario. More precisely, we can imaginethe scenario where the number of SUs is much largerthan that of PUs. In this case, we can divide the wholenetwork into multiple sub-networks: each sub-networkconsists of one PU (as a monopolist) and a set of nearbySUs. Our proposed contract approach can work in eachof the sub-network. Of course, the way that we dividesthe network will affect the performance loss of thesub-optimal algorithm. In an optimal trading schemebetween multiple PUs and multiple SUs, a PU needs toconsider not only the information asymmetry (betweenthe PU and SUs), but also the asymmetric informationand potential competition of other PUs. This will makethe optimal contract design much more difficult.

Finally, we plan to study the contact design for con-tinuous SU type distributions where each type θk canbe a continuous random variable depending on therealization of the channel coefficient. Then the contractwould become no longer a discrete vector (with eachelement corresponding to an SU-type), but a continuousfunction of SU types [36].

10 NOTICE FOR APPENDICES OF PROOFS

Most proofs for this paper in Appendices are removedto another separate supplemental file along with thissubmission, according to TMC new requirement on Sup-plemental Material. These appendices can also be foundin our online technical report [32].

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Lingjie Duan (S’09-M’12) is an Assistant Pro-fessor in the Pillar of Engineering Systems andDesign, Singapore University of Technology andDesign. He received Ph.D. degree in Informa-tion Engineering from The Chinese Universityof Hong Kong in 2012. During 2011, he was avisiting scholar in the Department of ElectricalEngineering and Computer Sciences at Univer-sity of California at Berkeley. Prior to that, hereceived B.Eng. degree in Electrical Engineeringfrom Harbin Institute of Technology in 2008. His

research interests are in the area of network economics, game theory,cognitive radios, mobile computing, and small cell technologies. Be-sides, he has served as a technical program committee (TPC) memberfor multiple top-tier conferences (e.g., VTC2011-Fall and 2013-Spring,PIMRC’11, MobiArch’12, WCNC’13, GLOBECOM’13 and ICC’13).

Lin Gao is a Postdoctoral Researcher in theDepartment of Information Engineering at theChinese University of Hong Kong. He receivedthe M.S. and Ph.D. degrees in Electronic En-gineering from Shanghai Jiao Tong University(China) in 2006 and 2010, respectively. Prior tothat, he received the B.E. degree in InformationEngineering from Nanjing University of Postsand Telecommunications (China) in 2002. Hisresearch interests lie in wireless communicationand network technologies, including dynamic

spectrum access, network resource optimization, economic incentive,secondary spectrum market modeling, and game theoretic modeling.

Jianwei Huang (S’01-M’06-SM’11) is an Assis-tant Professor in the Department of InformationEngineering at the Chinese University of HongKong.

He received Ph.D. in Electrical and ComputerEngineering from Northwestern University (E-vanston, IL, USA) in 2005. Dr. Huang currentlyleads the Network Communications and Eco-nomics Lab (ncel.ie.cuhk.edu.hk), with the mainresearch focus on optimization, economics, andgame theoretical analysis of networks. He is the

recipient of the IEEE Marconi Prize Paper Award in Wireless Communi-cations in 2011, the International Conference on Wireless Internet BestPaper Award 2011, the IEEE GLOBECOM Best Paper Award in 2010,Asia-Pacific Conference on Communications Best Paper Award in 2009,and the IEEE ComSoc Asia-Pacific Outstanding Young ResearcherAward in 2009.

Dr. Huang is the Editor of IEEE Journal on Selected Areas inCommunications - Cognitive Radio Series, Editor of IEEE Transactionson Wireless Communication in the area of Wireless Networks andSystems, Guest Editor of IEEE Journal on Selected Areas in Com-munications special issue on ”Economics of Communication Networksand Systems”, Lead Guest Editor of IEEE Journal of Selected Areasin Communications special issue on ”Game Theory in CommunicationSystems”, and Lead Guest Editor of IEEE Communications MagazineFeature Topic on ”Communications Network Economics”. Dr. Huangis the Chair of IEEE ComSoc Multimedia Communications TechnicalCommittee, and Steering Committee Member of IEEE Transactions onMultimedia and IEEE ICME. He has served as TPC Co-Chair of IEEEGLOBECOM Selected Aresa of Communications Symposium (GameTheory for Communications Track) 2013, IEEE WiOpt (InternationalSymposium on Modeling and Optimization in Mobile, Ad Hoc, andWireless Networks) 2012, IEEE GlOBECOM Wireless CommunicationsSymposium 2010, IWCMC (the International Wireless Communication-s and Mobile Computing) Mobile Computing Symposium 2010, andGameNets (the International Conference on Game Theory for Networks)2009.

1

Supplementary Materials of “CooperativeSpectrum Sharing: A Contract-based Approach”

F

APPENDIX ATHE IMPACT OF SUS’ TRANSMISSION POWERCONSTRAINTS

In this appendix, we consider the more general casewhere there are some physical constraints on SUs’ relay-ing power. This constraint can also be imposed by someregulatory body to avoid excessive interferences from theSUs.1 Next we focus on the complete information case,and study how the SUs’ power constraint affects PU’scontract optimal decisions.

Let us denote an SU k’s maximum transmission poweras Pk, which can be different from the other SUs. Thepower constraint can also be written as pk/hSTk,PR ≤ Pk.Similar to Section 4, the PU only needs to consider SUs’IR (participance) constraints and force pk = tkθk for anyk ∈ K (see Lemma 1). Then we can replace pk by θktk foreach k ∈ K, and the PU’s utility maximization problemis

max{tk≥0,∀k}

Rdir

2 + 12 log(1 +

∑k∈KNkθktk

n0)

1 +∑k∈KNktk

,

subject to, tkθk ≤ PkhSTk,PR, ∀k ∈ K. (28)

By solving (28) similarly as in Section 4, we have thefollowing result.

Theorem 1: Let us denote P th as the unique solutiontoθK +NKhSTk,PRP

th

n0 +NKhSTk,PRP th= R+ log

(1 +

NKhSTk,PRPth

n0

).

(29)In an optimal contract with complete information, thePU will only hire the highest SU type (i.e., type-K)if PK ≥ P th,2 otherwise it will also hire some lowertype SUs. In the latter case, a type is involved in thecooperative spectrum sharing only when all its highertypes are involved.

Intuitively, the PU wants to hire the highest type SUssince they can offer the most help to the PU withinthe given total time allocation. But if the number ofthe highest type SUs NK is small or the correspondingpower constraint PK is small, then the highest typeusers’ help alone is not attractive enough to PU. In this

1. For example, the PU may share the spectrum with another PU,and the SUs cannot cause too large interference to that PU.

2. In this case, the optimal contract is the same as that in Section 4.

case, PU needs to involve lower type SUs to maximizethe relay rate.

Under weakly incomplete information, we can similar-ly show that some lower type SUs will be involved in theoptimal contract if the power constraints are tight. Forstrongly complete information, Section 7 already showthat more than one type of users might be involved inthe optimal contract even without power constraints. In-troducing power constraints will not change this result.Due to page limits, we skip the detailed analysis of thesetwo cases.

APPENDIX BPROOF OF PROPOSITION 1We divide the proof into two parts. First, we prove thatif pi > pj , then ti > tj . Due to the IC constraint in (17),we have

θiti − pi ≥ θitj − pj ,

i.e.,θi(ti − tj) ≥ pi − pj .

Since pi > pj , we conclude

θi(ti − tj) ≥ pi − pj > 0,

and thus ti > tj .Next we prove that if ti > tj , then pi > pj . Due to the

IC constraint in (17), we have

θjtj − pj ≥ θjti − pi,which can be transformed to be

pi − pj ≥ θj(ti − tj).Since ti > tj , we conclude

pi − pj ≥ θj(ti − tj) > 0,

and thus pi > pj .

APPENDIX CPROOF OF PROPOSITION 2Proof. We prove by contradiction. Suppose that thereexists ti < tj with θi > θj . Then we have

θitj + θjti > θiti + θjtj . (30)

On the other hand, the feasible contract satisfies the ICconstraints for both type-θi and type-θj SUs, i.e.,

θiti − pi ≥ θitj − pj ,

2

andθjtj − pj ≥ θjti − pi.

By combining last two inequalities, we have

θiti + θjtj ≥ θitj + θjti,

which contradicts with (30). This completes the proof.

APPENDIX DPROOF OF THEOREM 2D.1 Proof of sufficient conditionsWe use mathematical induction to prove this proposi-tion. Let us denote Φ(n) as a subset which contains thefirst n power-time combinations in the complete contractΦ (i.e., Φ(n) = {(pk, tk), |k = 1, ..., n}).

We first show that Φ(1) is feasible. Since there isonly one SU type, the contract is feasible if it satisfiesIR constraint in (8). This is true due to Contd.b inTheorem 2.

We next show that if contract Φ(k) is feasible, then wecan construct the new contract Φ(k + 1) by adding newitem (pk+1, tk+1) and show that the new contract is alsofeasible. To achieve this, we need to show two results:• Result I: the IC and IR constraints for type-θk+1 SUs:{

θk+1tk+1 − pk+1 ≥ θk+1ti − pi, ∀i = 1, ..., k

θk+1tk+1 − pk+1 ≥ 0,(31)

• Result II: for types θ1, ..., θk already contained in thecontract Φ(k), the IC constraints are still satisfiedafter adding the new type θk+1:

θiti − pi ≥ θitk+1 − pk+1,∀i = 1, ..., k. (32)

Note that the new contract Φ(k + 1) will satisfy theIR constraints of all type θ1 to θk SUs as the originalcontract Φ(k) is feasible.

Proof of Result I in (31): First, we prove the IC constraintfor type θk+1. Since contract Φ(k) is feasible, the ICconstraint for a type-θi SU must hold, i.e.,

θkti − pi ≤ θktk − pk,∀i = 1, ..., k.

Also, the right inequality of (20) in Contd.c can betransformed to

pk+1 ≤ pk + θk+1(tk+1 − tk).

By combining the above two inequalities, we have

θkti−pi+pk+1 ≤ θktk+θk+1(tk+1−tk),∀i = 1, ..., k. (33)

Notice that θk+1 > θk and tk ≥ ti in Contd.a. We alsohave

θk+1tk − θk+1ti ≥ θktk − θkti.

By substituting this inequality into (33), we have

θk+1tk+1 − pk+1 ≥ θk+1ti − pi, (34)

which is actually the IC constraint for type θk+1.Next, we show the IR constraint for type θk+1. Since

θk+1 > θi for any i ≤ k, then

θk+1ti − pi > θiti − pi.

Using (34), we also have

θk+1tk+1 − pk+1 ≥ θk+1ti − pi.By combining the last two inequalities, we have

θk+1tk+1 − pk+1 ≥ 0,

which proves the IR constraint.Proof of Result II in (32): Since contract Φ(k) is feasible,

the IC constraint for type θi holds, i.e.,

θitk − pk ≤ θiti − pi,∀i = 1, ..., k.

Also, we can transform the left inequality of (20) inContd.c to

pk + θk(tk+1 − tk) ≤ pk+1.

By combining the above two inequalities, we conclude

θitk + θk(tk+1 − tk) ≤ θiti − pi + pk+1. (35)

Note that θk ≥ θi for any i ≤ k and tk+1 ≥ tk in Contd.a.We also have

θktk+1 − θktk ≥ θitk+1 − θitk.By combining the above two inequalities, we conclude

θitk+1 − pk+1 ≤ θiti − pi,∀i = 1, . . . , k

which is actually the IC constraint for type θi.

D.2 Proof of necessary conditionsIt is easy to check that the sufficient conditions inTheorem 2 are also necessary for a feasible contract.Specifically, Contd.a is the same as necessary conditionssummarized in (19). Contd.b is same as the necessaryIR constraint for the lowest type θ1 in a feasible con-tract. The left inequality of Contd.c can be derived fromthe necessary IC constraint for type θk−1 in a feasiblecontract, and the right inequality of Contd.c can also bederived from the necessary IC constraint for type θk.

APPENDIX EPROOF OF PROPOSITION 3First, it is not difficult to check that the relay powers in(21) satisfy the sufficient conditions of contract feasibilityin Theorem 2. We skip the details here due to the pagelimit. Next we prove the optimality and uniqueness ofthe solutions in (21).

E.1 Proof of optimalityWe first show that the relay powers in (21) maximizethe PU’s utility given fixed time allocations, i.e., {p∗k,∀k}maximize

1

1 +∑k∈KNktk

(Rdir

2+

1

2log

(1 +

∑k∈KNkpk

n0

)).

(36)We prove by contradiction. Suppose that these existsanother feasible relay powers {pk,∀k} which achieves abetter solution than {p∗k,∀k} in (21). Since (36) is increas-ing in total relay power, we must have

∑k∈KNkpk >∑

k∈KNkp∗k. Thus we have at least one relay power

pk > p∗k for one type θk.

3

If k = 1, then p1 > p∗1. Since p∗1 = θ1t1, then p1 > θ1t1.But this violates the IR constraint for type θ1. Then wemust have k > 1.

Since {pk,∀k} is feasible, then {pk,∀k}must satisfy theright inequality of Contd.c in Theorem 2. Thus we have

pk ≤ pk−1 + θk(tk − tk−1).

By substituting θk(tk − tk−1) = p∗k − p∗k−1 as in (21) intothe this inequality, we have

pk−1 > p∗k−1.

Using the above argument repeatedly, we finally obtainthat

p1 > p∗1 = θ1t1,

which violates the IR constraint for type θ1 again.

E.2 Proof of uniquenessWe next prove that the relay powers in (21) is the uniquesolution that maximizes (36). We also prove by contra-diction. Suppose that there exists another {pk,∀k} 6={p∗k,∀k} such that

∑k∈KNkpk =

∑k∈KNkp

∗k in (36).

Then there is at least one relay power pi < p∗i andone relay power pj > p∗j . We can focus on type-θj andpj > p∗j . By using the same argument before, we havep1 > p∗1 = θ1t1. But this violates the IR constraint fortype θ1.

APPENDIX FPROOF OF OBSERVATION 2 FOR Rdir = 0

When Rdir = 0, the PU will always allocate positivetime to the highest type SUs (i.e., t∗K > 0). Its totaltime allocation is decreasing in the highest type θK . PU’sutility in (23) can be written as

UPU(tK) =1

2(1 +NKtK)log(1 + θKNKtK). (37)

Denote the total time allocation to SUs as T ′. We canrewrite (37) as a function of T ′:

UPU(T ′) =1

2(1 + T ′)log(1 + θKT

′),

which can be shown as a concave function of T ′. Sincet∗K > 0, we conclude that the optimal T ′∗ = NKtK∗ > 0and satisfies

dUPU(T ′)

dT ′|T ′=T ′∗=

θK(1+T ′∗)1+θKT ′∗

− log(1 + θKT′∗)

2(1 + T ′∗)2= 0,

i.e.,

F (θK , T′∗) := θK(1+T ′∗)−(1+θKT

′∗) log(1+θKT′∗) = 0.

(38)Since

∂F (θK , T′∗)

∂θK= 1− T ′∗ log(1 + θKT

′∗),

∂F (θK , T′∗)

∂T ′∗= −θK log(1 + θKT

′∗),

thusdT ′∗

dθK= − ∂F/∂θK

∂F/∂T ′∗=

1− T ′∗ log(1 + θKT′∗)

θK log(1 + θKT ′∗). (39)

Since θKT ′∗ > 0, we have

log(1 + θKT′∗)− θKT ′∗ < 0.

By substituting this inequality into (38), we conclude

1− T ′∗ log(1 + θKT′∗) < 0,

and thus dT ′∗

dθKin (39) is negative. Hence, the optimal total

time allocation T ′∗ is decreasing in the highest type θK .

APPENDIX GIMPACT OF COMMUNICATION OVERHEAD IN AMORE EFFICIENT RELAY SCHEME

Recall that the PU needs to follow 4 steps in Section 3.3to communicate with SUs and confirm cooperation rela-tionship in the contract. The communication overheadis not a major issue in the previous analysis, as wehave focused on a sufficiently long-term cooperationbetween the PU and SUs. But when SUs are mobile andchange their locations frequently, the contract needs toadapt according, and one fixed contract may only lastfor several time slots. Then the communication overheadmay become a major issue. In this case, the PU needsto tradeoff the cooperation benefit and the communica-tion overhead. Moreover, we would like to understandwhether our collaboration scheme can be extended tosome more efficient relay scheme. For example, in Fig.1 the PU does not transmit in Phase II and only SUsrelay his traffic. In a more efficient co-operative commu-nication scheme, the PU can also transmit in Phase IItogether with the SUs without any mutual interferences(by properly choosing the orthogonal space-time codes).In this section, we will extend our prior results to sucha more efficient relay scheme and analyze the impact ofcommunication overhead.

Let us denote the total communication time overheadin the 4-step interaction process as ∆, which may belarger or smaller than T0. As we have normalized T0

to be 1, we can treat ∆ as the overhead percentage ofthe T0. The cooperation between the PU and SUs canlast for a long time (e.g., many rounds of Phases I-III) ora short time (e.g., several rounds of Phases I-III), and wedenote the number of cooperation rounds as M . We alsoconsider the case where the PU also transmits in PhaseII, which means that the PU’s average achievable rateduring Phase I and Phase II is at least Rdir even withoutSUs’ relay helps. The PU’s utility now changes from (3)to

UPU =M(Rdir

2 + 12 log

(1 + SNRPT,PR +

∑k∈KNkpkn0

))∆ +M(1 +

∑k∈KNktk)

=

Rdir

2 + 12 log

(1 + SNRPT,PR +

∑k∈KNkpkn0

)∆M + 1 +

∑k∈KNktk

, (40)

which equals Rdir = log(1 + SNRPT,PR) when no SUs areinvolved (i.e., (pk, tk) = 0 for any k ∈ K). Notice that∆/M can be viewed as the average overhead assigned toeach cooperation round, and (40) is decreasing in ∆/M .

4

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

Direct transmission rate Rdir

PU

’s o

ptim

al u

tility

No relay region

Relay region

Δ/M=0Δ/M=0.25Δ/M=0.5Direct transmission

Fig. 7. PU’s optimal utility U∗PU as a function of the averageoverhead ∆/M and direct transmission rate Rdir. Thedotted 45 degree line divides the figure into two regions:Relay Region and No Relay Region. When the maximumutility achievable under the optimal contract (by using thelargest type θK) falls into the “No relay region”, the PU willchoose direct transmission only.

Next we focus on complete information as in Section 4,and the PU’s objective is to maximize (40) subject to IRconstraints in (8). By following the very similar proofs,we can show Lemma 1 and Theorem 1 still hold for thecurrent utility setting, where (pK , tK) = (θKtK , tK) >0 and (pk, tk) = 0 for any k < K. By redefining theoptimization variable as tK = NKtK , we can simplifythe PU’s problem as

maxtK≥0

Rdir

2 + 12 log

(1 + SNRPT,PR + θK tK

n0

)∆M + 1 + tK

, (41)

and this function UPU(tK) is not concave in tK . Any localoptimal solution (denoted as tK) to Problem (41) satisfies

dUPU(tK)

dtK|tK=tK

=θK/n0/2

( ∆M + 1 + tK)(1 + SNRPT,PR + θK tK

n0)

−(Rdir + log(1 + SNRPT,PR + θK tK

n0))/2

( ∆M + 1 + tK)2

= 0.

(42)

We can further compute the second order derivative, andwe can show that

d2UPU(tK)

dt2K|tK=tK

< 0.

This means that the local optimal solution (without con-sidering the nonnegative constraint) to the problem (41)is unique and is actually the global optimal solution t∗K .Thus we conclude that t∗K = max(tK , 0), where tK is theunique solution to (42) and can be computed numericallyas follows.

Observation 1: Figure 7 shows the PU’s optimal utilityU∗PU is decreasing in the average overhead ∆/M andis increasing in his direct transmission rate Rdir. When

0 0.2 0.4 0.6 0.8 1 1.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Direct transmission rate Rdir

PU

’s o

ptim

al ti

me

allo

catio

n N

K t K*

Δ/M=0Δ/M=0.25Δ/M=0.5

Fig. 8. PU’s optimal total time allocation NKt∗K as a func-tion of his direct transmission rate Rdir and the averageoverhead ∆/M .

∆/M or Rdir is large, the PU will choose direct trans-mission only. Using a more efficient relay scheme, PU’stransmitter does not need to remain idle during PhaseII, and the reduced cooperation cost gives the PU moreincentives to use SUs as relays.

In Fig. 7, we plot the dotted baseline that representsthe rate Rdir achieved by the direct transmission only.As Rdir or ∆/M increases, the PU has less incentive toshare spectrum with SUs. When Rdir or ∆/M is large,the relatively decreased cooperation benefit or increasedcooperation cost discourages the PU to use SUs, whicheventually degenerates to the “No relay region” in Fig. 7.Compared to the basic relay scheme in Fig. 1, PU benefitsfrom his direct transmission in Phase II in this moreefficient relay scheme and obtains a larger utility throughthe cooperation with SUs. If the overhead is zero, the PUwill always cooperate with SUs, while with overhead ithas more incentive to cooperate with SUs with a moreefficient relay scheme.

Observation 2: Figure 8 shows that the PU’s optimaltotal time allocation to the highest type-θK SUs, NKt∗K ,decreases in PU’s direct transmission rate Rdir and in-creases in the average overhead ∆/M .