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ContrAuction: An Integrated Contract and AuctionDesign for Dynamic Spectrum Sharing

Lin Gao, Jianwei Huang, Ying-Ju Chen, and Biying Shou

Abstract—Designing mechanisms with proper economic incen-tives is essential for the success of dynamic spectrum sharing, andmarket-driven secondary spectrum trading is one effective way toachieve this goal. In this paper, we consider secondary spectrumtrading between one seller (i.e., the primary spectrum owner, PO)and multiple buyers (i.e., the secondary users, SUs) in a hybridspectrum market with both guaranteed contract buyers (futuremarket) and spot purchasing buyers (spot market). We focus on thePO’s profit maximization with stochastic network information, andformulate the PO’s expected profit maximization problem, wherethe optimal solution serves as a policy guiding the allocation ofevery spectrum under every possible information realization. Westudy systematically the optimal solutions under both informationsymmetry and information asymmetry, depending on whether thePO can observe the SUs’ realized information. Under informationsymmetry, we show that the optimal solution (benchmark)maximizes both the PO’s expected profit (optimality) and thesocial welfare (efficiency). Under information asymmetry, anincentive-compatible mechanism is necessary for eliciting theSUs’ realized information. We propose the ContrAuction, anintegrated contract and auction design, where the PO actsas virtual bidders (in addition to the role of an auctioneer)on behalf of the guaranteed contracts. We derive the optimalContrAuction under the constraint of efficiency, and characterizethe PO’s expected profit loss (compared to that under informationsymmetry) induced by the information rent for the SUs.

I. INTRODUCTION

With the explosive development of wireless services andnetworks, spectrum is becoming more congested and scarce.Dynamic spectrum sharing has been recently viewed as anovel approach to increase spectrum efficiency and allevi-ate spectrum scarcity. The key idea is to enable unlicensedsecondary users (SUs) to access the spectrum licensed toprimary spectrum owners (POs) opportunistically [1], [2]. Thelong-term successful implementation of dynamic spectrumsharing requires many innovations in technology, economics,and policy. In particular, it is essential to design a sharingmechanism that offers enough incentives for POs to open theirlicensed spectrum for secondary sharing.

Market-driven secondary spectrum trading is a promisingparadigm to address the incentive issue in dynamic spectrum

Lin Gao and Jianwei Huang (corresponding author) are with Departmentof Information Engineering, The Chinese University of Hong Kong, HK, E-mail: {lgao, jwhuang}@ie.cuhk.edu.hk. Ying-Ju Chen is with Departmentof Industrial Engineering and Operations Research, University of California,Berkeley, Berkeley, California 94720, E-mail: chen@ieor.berkeley.edu. BiyingShou is with Department of Management Sciences, City University of HongKong, HK, Email: biying.shou@cityu.edu.hk.

This work is supported by the General Research Funds (Project Number9041458, 412509, 412710 and 412511) established under the University GrantCommittee of the Hong Kong Special Administrative Region, China, and theCityU Project (Project Number 7008116) established by the City Universityof Hong Kong, China.

PO

t

Time (Slot)

Spectrum (Channel)

Busy Channel Idle Channel

Spot MarketFuture Market

2

4

c

a db31

Fig. 1. An illustration of the hybrid spectrum market.

sharing, where POs temporarily lease the un-utilized (idle)spectrums to unlicensed SUs to obtain additional profit. In thispaper, we consider the short-term secondary spectrum trading(e.g., on a slot-by-slot basis) between a single PO (monopolyseller) and multiple SUs (buyers) in a hybrid spectrum marketwith both the future market and the spot market. In the futuremarket, each SU reaches an agreement, called a guaranteedcontract, directly with the PO. The contract pre-specifies somekey elements in trading, e.g., the price and the demand in afuture period. In the spot market, the SUs compete openlywith each other for spectrums (e.g., by an auction), and eachspectrum is traded in a real-time and open manner. Figure1 illustrates a hybrid spectrum market with contract users{1,2,3,4} and spot market users {a,b,c,d}, where two idlespectrums at time slot t are allocated to the contract user SU-3and the spot market user SU-a, respectively.

One significant advantage of such a hybrid spectrum marketis its flexibility in achieving desirable Quality of Service (QoS)differentiations. Specifically, the future market insures SUsagainst uncertainties of the future supply, and insures POsagainst uncertainties of the future demand; the spot marketallows SUs to compete for specific spectrums when they needresource, and permits fine grained resource allocation based onthe SU’s real-time demand and preference. Thus, an SU withelastic services (e.g., file transferring that is delay tolerant)may be more interested in spot trading to achieve the bestresource-price tradeoff, while an SU with inelastic services(e.g., Netflix video streaming) requiring a minimum data ratemay prefer the certainty of contract in the future market.

In this paper, we focus on the PO’s profit maximization: howshould the PO allocate his idle spectrums in a future periodamong the guaranteed contract users and the spot marketusers (and charge) to maximize his overall profit? Since the POusually cannot obtain the complete and deterministic networkinformation in advance, we formulate the PO’s expected profitmaximization problem based on the stochastic information.The optimal solution defines a policy, which determines theallocation of every spectrum with every possible information

978-1-4673-3140-1/12/$31.00 ©2012 IEEE

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realization. We study the optimal solutions under informationsymmetry and information asymmetry systematically.

The main contributions of this paper are as follows:• New modeling and solution technique: As far as we know,

this is the first paper tackling secondary spectrum tradingwith the coexistence of future and spot markets.

• Multiple information scenarios: We study the optimalallocations in two stochastic information scenarios: infor-mation symmetry and asymmetry, depending on whetherthe PO can obtain the SUs’ realized information.

• Optimal Mechanism under information symmetry: Wederive the optimal solution analytically, where each spec-trum is allocated to the most “profitable” SU directly(called perfect price discrimination).

• Optimal Mechanism under information asymmetry: Wepropose the ContrAuction, an integrated contract andauction design, where the PO acts as virtual bidders (inaddition to the role of an auctioneer) on behalf of theguaranteed contracts. We derive the optimal ContrAuctionunder the constraint of efficiency.

• Performance analysis: Under information symmetry, theoptimal solution (benchmark) maximizes both the PO’sexpected profit (optimality) and the social welfare (effi-ciency). Under information asymmetry, the optimal Con-trAuction suffers certain profit loss (15% on average inour simulations) under the constraint of efficiency.

The rest of this paper is organized as follows. In SectionII, we review the related literature. In Sections III and IV,we provide the system model and problem formulation. InSections V and VI, we propose the optimal solutions under in-formation symmetry and asymmetry, respectively. We presentthe simulations in Section VII and conclude in Section VIII.

II. RELATED WORK

Recent years have witnessed a growing body of literatureon the economic analysis (in particular the incentive issue) ofdynamic spectrum sharing [1], [2]. Market-driven secondaryspectrum trading is an effective way to address the incentiveissue [3], [4]. The literature on secondary spectrum tradingoften considers pricing, contract and auction.

Pricing and contract are generally adopted by a futuremarket, wherein the buyers enter certain agreements with thesellers beforehand, specifying the price, quality and demand.Pricing is often used when the seller knows precisely thevalue of the resource being sold (information symmetry) [5]–[7]. Contract is effective when the seller only knows limitedinformation about the buyers’ valuation (information asym-metry) [8]–[10]. In contrast to pricing and contract, auction isgenerally adopted by a spot market, where the buyers competeopenly with each other for spectrums. Auction is particularlysuitable for the information asymmetry scenario, where theseller does not know the value of the resource being soldwhereas the buyers know [11]–[16].

However, the above works on secondary spectrum tradingconsider only the pure spectrum market (i.e., either in the spotmarket or in the future market), while we consider a hybridspectrum market with both the spot market and the future

TABLE IA SUMMARY OF SPECTRUM TRADING LITERATURE

Market Type Related WorkFuture Market Pricing: [5]–[7], Contract: [8]–[10]Spot Market Auction: [11]–[16]

Hybrid Market This paper

market. As far as we know, this is the first work tackling thesecondary spectrum trading with the coexistence of the spotmarket and the future market. For clarity, we summarize thekey literature of secondary spectrum trading in Table I.

III. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Description

We consider a network with one PO and multiple SUs.The PO owns a set of licensed spectrums, and serves thelicensed holders with a slotted transmission protocol (e.g.,GSM, WCDMA, and LTE). That is, the total transmissiontime is divided into fixed-time intervals, called time slots, andthe licensed holders access the PO’s spectrums according toa synchronous slot structure. Depending on the activities oflicensed holders, there may be some idle spectrums at eachtime slot, which can be temporally used by the SUs.

We consider short-term secondary spectrum trading, wherethe PO leases the idle spectrums to the SUs on a slot-by-slot basis. Thus, the basic unit of resource for trading is “aparticular spectrum at a particular time slot”, called a spectrumfor short. The total number of idle spectrums in a certainperiod (say T time slots) is defined as the PO’s supply, denotedby S = {1, 2, ..., S}, where S =

∑Tt=1 St and St is the

number of idle spectrums at time slot t. In Figure 1, we haveT = 12, S = 20, S1 = 1, S2 = 2 and so on.

We assume that each idle spectrum can only be used byone SU, that is, spectrum spatial reuse is not considered inthis work. We further assume St = 1 for each time slot t,and thus a spectrum is equivalent to a slot. Note that thisassumption does not affect the generality of our derivation, aswe can view the multiple idle spectrums in the same time slotas a set of sequentially emerging spectrums.

B. Future Market and Spot MarketWe consider a hybrid spectrum market consisting of the

future market and the spot market.1) Future Market: In the future market, each SUs enters

into an agreement, called a future contract or guaranteedcontract, directly with the PO. A contract pre-specifies notonly the SU’s payment and demand for spectrums in a givenperiod, but also the PO’s penalty when violating the contract.

2) Spot Market: In the spot market, each idle spectrum istraded in a real-time and on-demand manner. That is, an SUinitiates a purchasing request only when needed, and all SUsrequesting the same spectrum compete openly (e.g., throughan auction) with each other for the spectrum. The spectrumis delivered immediately to the winner at a real-time marketprice, which not only depends on the SUs’ valuations, but alsoon the market’s supply-demand relationship.

To summarize, the future market insures SUs (PO) againstuncertainties in the future supply (demand), while the spotmarket allows SUs to compete for specific spectrums based

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on their real-time requirements. Thus, a hybrid market ismore flexible in achieving desirable QoS differentiations. Weassume each SU is only associated with one trading scheme.1

Let N = {1, ..., N} and M = {1, ...,M} denote the sets ofSUs in the future market and spot market, respectively.

A guaranteed contract, in principle, could be quite com-plicated. In order to provide an intuitive expression and gainmeaningful insights, we focus on a basic contract form, whichconsists of (i) the SU’s payment and demand in one period(T time slots), and (ii) the PO’s penalty when violating thecontract. Thus, we can write the contract of SU n ∈ N as

Cn , {Bn, Dn, P̂n}, (1)

where Bn and Dn are the SU’s payment and demand, and P̂nis the unit punishing price for every undelivered spectrum.

C. Valuation

The valuation of an SU for a particular spectrum representsthe SU’s benefit from using the spectrum. It depends onboth the spectrum quality, reflecting how efficiently an SUcan utilize the spectrum, and user preference, reflecting howeagerly an SU needs a spectrum.

We consider a general scenario where the SUs’ valuationsrandomly and independently vary with time. For convenience,we introduce the following random variables:• vm: the valuation of spot market user m;• un: the valuation of contract user n;• Θ , (v1, ...,vM ;u1, ...,uN ): valuation vector of SUs.

We denote the realization of vm, un and Θ at time slot t byvm(t), un(t) and θ(t) =

(v1(t), ..., vM (t);u1(t), ..., uN (t)

),

respectively. Since each spectrum t is fully characterized byθ(t), we also refer to θ(t) as the information of spectrum t.

IV. PROBLEM FORMULATION WITH STOCHASTICINFORMATION

We focus on the PO’s profit maximization: how should thePO allocate the idle spectrums among the contract users andthe spot market users (and charge) to maximize his profit?

Due to the randomly varying of information, the PO usu-ally cannot obtain the complete and deterministic networkinformation (i.e., θ(t) for all t) in advance. Therefore, wefocus on the PO’s profit maximization under stochastic net-work information, where the PO knows only the stochasticdistribution of Θ in advance. To tackle this problem, weformulate the PO’s expected profit maximization problem, andthe optimal solution specifies the allocation of every spectrumunder every possible information realization. Formally, theallocation strategy is defined as follows.

Definition 1 (Allocation Strategy). An allocation strategyA(θ) is a mapping from every θ to a probability vector

A(θ) ,(a0(θ), a1(θ), ..., aN (θ)

), ∀θ ∈ Θ, 2

1For the SU running multiple services, we can simply divide the SU intomultiple virtual SUs, each associated with one trading scheme.

2Here we omit the time index t of θ(t), as we are not talking about aparticular spectrum t. Furthermore, we use the same notation Θ to denotethe information space, i.e., the set of all possible information realization θ.

where a0(θ) denotes the allocation probability of a spectrumwith information θ (also called spectrum θ for short) to thespot market,3 and an(θ) to the contract user n, ∀n ∈ N .

The PO’s expected revenue from the spot market is:

E[R0] = S ·∫θ

a0(θ)r0(θ) · fΘ(θ)dθ, (2)

where fΘ(θ) ,∏Mm=1 fvm

(vm)∏Nn=1 fun

(un) is the jointPDF of Θ, and r0(θ) is the maximum revenue that the PO canachieve from selling a spectrum θ to the spot market (whichdepends on whether the PO can observe the SUs’ realizedinformation at each time slot, i.e., information symmetry orasymmetry). Note that we write dv1...dvMdu1...duN as dθand

∫v1...∫vM

∫u1...∫uN

as∫θ

for convenience.The expected number of spectrums for a contract user n is:

E[dn] = S ·∫θ

an(θ) · fΘ(θ)dθ. (3)

The PO’s expected revenue from a contract user n is:E[Rn] = Bn − P(E[dn], Dn), (4)

where P(x, y) is the PO’s penalty given byP(E[dn], Dn) , [Dn − E[dn]]+ · P̂n (5)

Note that the penalty is based on the expected number (ratherthan the actual number) of spectrums for a contract user n.

The PO’s total expected revenue in one period is:

E[R] = E[R0] +

N∑n=1

E[Rn]. (6)

Although there is no explicit relation between a contractuser’s valuation and his actual payment, allocating a contractuser the spectrums with low valuation will decrease the SU’slong-term satisfaction, which may potentially affect the SU’sfuture payment (or the PO’s future revenue). In other words,the PO runs the risk of losing the contract user in future. Wecapture this potential loss by introducing an elastic cost forthe contract user’s long-term satisfaction (loss).

Let cn(θ) denote the cost for a contract user n’s long-termsatisfaction over an (allocated) spectrum θ. The overall costfor contract user n’s long-term satisfactions is

E[Cn] = S ·∫θ

an(θ)cn(θ) · fΘ(θ)dθ. (7)

The PO’s expected profit is a weighted sum of the currentrevenue (6) and the future cost (7):

E[U ] = E[R0] +N∑n=1

E[Rn]−N∑n=1

wn · E[Cn], (8)

where wn is the weight for evaluating the cost for contractuser n’s long-term satisfaction. A larger wn means that thePO cares more about the satisfaction of contract user n.

The PO’s objective is to find the optimal allocation A∗(θ)for any possible realization θ to maximize his profit:

A∗(θ) = argmaxA(·)

(E[R0] +

N∑n=1

E[Rn]−N∑n=1

wnE[Cn]

),

(9)s.t. an(θ) ∈ [0, 1], ∀n ∈ {0} ∪ N ,∀θ ∈ Θ;∑N

n=0 an(θ) ≤ 1, ∀θ ∈ Θ.

Note that solving (9) does not require the deterministic infor-mation of all spectrums, and thus the PO is able to derive theoptimal solution (i.e., the optimal policy) in advance.

3Once allocated to the spot market, we will show that it is always optimalto allocate the spectrum to the user with the highest valuation.

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V. OPTIMAL SOLUTION UNDER INFORMATION SYMMETRY

Under information symmetry, the PO is able to observe therealized information θ(t) at each time slot t (but not beforetime slot t). Thus, when selling a spectrum on the spot market,the PO can extract all the surplus by allocating the spectrumto the highest valuation SU and charging exactly that SU’svaluation. Such a selling mechanism is referred to as theperfect price discrimination.

Obviously, the maximum revenue r0(θ) is given by

r0(θ) = Y 1M (θ) , max

m∈Mvm ,

where Y 1M (θ) (or Y 1

M for short) denote the highest valuationof all spot market users for a spectrum θ. In what follows, wewill derive the optimal policy, i.e., the optimal solution of (9).Due to space limitations, we present all of the detailed proofsin our technical report [20].

A. Necessary Conditions

Suppose A∗(θ) = (a∗0(θ), a∗1(θ), ..., a

∗N (θ)) is the optimal

solution to (9). The following necessary conditions hold.

Lemma 1 (Necessary Condition I). For any information θ ∈Θ, we have:

∑Nn=0 a

∗n(θ) = 1.

Lemma 2 (Necessary Condition II). For any contract usern ∈ N , we have: E[dn] ≤ Dn.

Intuitively, Lemma 1 states that every idle spectrum willbe sold to an SU. This is because the PO can always gaina positive revenue by selling a spectrum on the spot market.Lemma 2 states that none of contract users will get spectrums(on average) more than his demand. This is because there isno bonus for the PO from allocating extra spectrums to an SUbeyond what is specified in the contract.

B. Optimal Solution

By Lemma 1, we can eliminate the decision variable a0(θ)by substituting a0(θ) = 1 −

∑Nn=1 an(θ). By Lemma 2, we

have P(E[dn], Dn) = P̂n (Dn − E[dn]). Thus, the optimiza-tion problem (9) can be rewritten as:

A∗0(θ) = argmaxA0(θ)

(S∫θ

(1−

∑Nn=1 an(θ)

)r0(θ)fΘ(θ)dθ

+∑Nn=1

(Bn − P̂n(Dn − E[dn])

)−∑Nn=1 wn · E[Cn]

)= argmaxA0(θ)

(F + S ·

∑Nn=1

∫θHn(θ)an(θ)fΘ(θ)dθ

)s.t. (i) an(θ) ≥ 0, ∀n ∈ N ,∀θ ∈ Θ;

(ii)∑Nn=1 an(θ) ≤ 1, ∀θ ∈ Θ;

(iii) E[dn] ≤ Dn, ∀n ∈ N ;(10)

where

• A0(θ) = A(θ)/{a0(θ)} =(a1(θ), ..., aN (θ)

);

• F = S ·∫θr0(θ)fΘ(θ)dθ +

∑Nn=1

(Bn − P̂nDn

);

• Hn(θ) = −r0(θ) + P̂n − wncn(θ).In what follows, we first study the dual problem of (10)

using the primal-dual method [19]. Then we determine theoptimal dual variables and optimal primal solution.

1) Primal-Dual Method: We introduce Lagrange multipli-ers (also called dual variables or shadow prices) µn(θ) forconstraint (i), η(θ) for constraint (ii), and λn for constraint(iii). The Lagrangian can be written as L ,

∫θL(θ)·fΘ(θ)dθ,

where L(θ), called Sub-Lagrangian, is given byL(θ) = F + S ·

∑Nn=1Hn(θ)an(θ) +

∑Nn=1 µn(θ)an(θ)

+ η(θ)(1−

∑Nn=1 an(θ)

)+∑Nn=1 λn

(Dn − S · an(θ)

).

We define the PO’s marginal profit (with respect to theallocation probability of contract user n) as the first partialderivative of L(θ) over allocation an(θ). Formally,

Definition 2 (Marginal Profit).L(n)(θ) , ∂L(θ)

∂an(θ)= S ·Hn(θ) + µn(θ)− η(θ)− S · λn.

From Definition 2, we can see that the marginal utilityL(n)(θ) is independent of A0(θ). By Euler-Lagrange con-ditions for optimality [19], the optimal solution A∗0(θ) mustoccur at the boundaries, i.e.,

a∗n(θ) =

0, L(n)(θ) < 0

1, L(n)(θ) > 0

δ ∈ [0, 1], L(n)(θ) = 0

(11)

From (11), the optimal solution is determined by the dualvariables µn(θ), η(θ), and λn. By the primal-dual method,every dual variable must be nonnegative and can be non-zeroonly when the associated constraint is tight, i.e., the followingdual constraints must hold [19]:

(D.1) µ∗n(θ) ≥ 0, a∗n(θ) ≥ 0,µ∗n(θ)a

∗n(θ) = 0, ∀n ∈ N , θ ∈ Θ;

(D.2) η∗(θ) ≥ 0, 1−∑Nn=1 a

∗n(θ) ≥ 0,

η∗(θ)(1−

∑Nn=1 a

∗n(θ)

)= 0, ∀θ ∈ Θ;

(D.3) λ∗n ≥ 0, Dn − S∫θa∗n(θ)fΘ(θ)dθ ≥ 0,

λ∗n(Dn − S

∫θa∗n(θ)fΘ(θ)dθ

)= 0, ∀n ∈ N .

By the duality principle, the primal problem (10) is equiva-lent to the problem of finding a set dual variables µ∗n(θ), η

∗(θ),and λ∗n, such that all dual constraints are satisfied.

2) Optimal Shadow Prices and Optimal Solution: Now westudy the optimal shadow prices (dual variables) and optimalprimal solution. For convenience, we introduce the mantlemarginal profit and core marginal profit as variations of themarginal profit in Definition 2. Formally,

Definition 3 (Mantle Marginal Profit).J (n)1 (θ) , S ·Hn(θ)− η(θ)− S · λn = L(n)(θ)− µn(θ).

Definition 4 (Core Marginal Profit).J (n)2 (θ) , S ·Hn(θ)− S · λn = L(n)(θ)− µn(θ) + η(θ).

We start by showing the necessary conditions for µ∗n(θ).

Lemma 3 (Feasible Range of µ∗n(θ)). The optimal shadowprices µ∗n(θ) must satisfy the following conditions:(a) µ∗n(θ) = 0, if J (n)

1 (θ) ≥ 0;(b) µ∗n(θ) ∈ [0, |J (n)

1 (θ)|], if J (n)1 (θ) < 0.

It is easy to see that L(n)(θ) = J (n)1 (θ)+µn(θ). Lemma 3

implies that (a) J (n)1 (θ) ≥ 0 if and only if (iff) L(n)(θ) ≥ 0,

and (b) J (n)1 (θ) < 0 iff L(n)(θ) ≤ 0. That is, µ∗n(θ) never

changes the sign of marginal profit L(n)(θ), which impliesthat µ∗n(θ) has no impact on the optimal solution by (11).

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Intuitively, this is because the optimal solution given by (11)never violates the first constraint of primal problem (10).

Let K1(θ) , maxn∈N J (n)2 (θ) denote the highest core

marginal profit, and K2(θ) , maxn∈N/n1J (n)2 (θ) the second

highest, where n1 , argmaxn∈N J (n)2 (θ). Next we show the

necessary conditions for the optimal η∗(θ).

Lemma 4 (Feasible Range of η∗(θ)). The optimal shadowprices η∗(θ) must satisfy the conditions:(a) η∗(θ) ∈ [max(0,K2(θ)), K1(θ)], if K1(θ) > 0,(b) η∗(θ) = 0, if K1(θ) ≤ 0.

It is easy to see that J (n)1 (θ) = J (n)

2 (θ)− η∗(θ),∀n. Thatis, the shadow price η∗(θ) can be viewed as an identicalreduction of each contract user’s marginal profit at point θ.Lemma 4 suggests that a feasible η∗(θ) lies between K1(θ)and max(0,K2(θ)), with which there is at most one SU(say n) with positive J (n)

1 (θ), and therefore L(n)(θ) ≥ 0 byLemma 3 and a∗n(θ) = 1 by (11). Thus the user couplingconstraint (ii) in the primal problem (10) will be satisfied.

By Lemmas 3 and 4, the optimal solution is to allocate eachspectrum θ to the contract user with the highest and positivecore marginal profit J (n)

2 (θ). Let n∗ , {n | J (n)2 (θ) =

K1(θ)} denote the set of contract users with the highest coremarginal profit. Formally, the optimal solution is given by thefollowing lemma.

Lemma 5 (Optimal Solution). For ∀θ ∈ Θ, we have a∗n(θ) =0 if n /∈ n∗(θ); and if n ∈ n∗(θ), then(a) a∗n(θ) ∈ [0, 1] such that

∑n∈n∗ a∗n(θ) = 1, if K1(θ) > 0;

(b) a∗n(θ) ∈ [0, 1] such that∑n∈n∗ a∗n(θ) ≤ 1, if K1(θ) = 0;

(c) a∗n(θ) = 0, if K1(θ) < 0;

Lemma 5 states that only the contract users with the highestpositive J (n)

2 (θ) may win a spectrum θ. If J (n)2 (θ) is smaller

than 0 for all contract users (i.e., K1(θ) < 0), the spectrumis allocated to the spot market (or more strictly, the highestvaluation user in the spot market).

When SUs have continuous and heterogeneous valuations,the range of θ such that J (n)

2 (θ) = 0 (which may lead to case(b) in Lemma 5) or multiple contract users having the sameJ (n)2 (θ) (which may lead to |n∗| > 1) has a zero size support,

and therefore can be ignored. Thus, the optimal allocationgiven by Lemma 5 is equivalent to:a∗n(θ) = 1 iff J (n)

2 (θ) > 0 & J (n)2 (θ) > max

i6=nJ (i)2 (θ). (12)

Let Θ+n ,

{θ|J (n)

2 (θ) > 0&J (n)2 (θ) > maxi6=n J (i)

2 (θ)}

denote the set of spectrums allocated to a contract user n.Obviously, Θ+

n is determined by the shadow price vectorΛ∗ , (λ∗1, ..., λ

∗N ), and thus we can write Θ+

n as a functionof Λ∗, denoted by Θ+

n (Λ∗). The optimal a∗n(θ) can be

equivalently written as: a∗n(θ) = 1 for all θ ∈ Θ+n (Λ

∗),and a∗n(θ) = 0 for all θ /∈ Θ+

n (Λ∗). Thus, the expected

number of spectrums to contract user n can be written asE[dn] = S

∫θ∈Θ+

n (Λ∗)fΘ(θ)dθ.

According to (D.3), the optimal shadow price λ∗n is either0 if the associated constraint is not tight, or otherwise a non-negative value makes the associated constraint tight. Formally,

Lemma 6 (Optimal Shadow Price λ∗n). The optimal shadowprice λ∗n,∀n ∈ N , is given by:

λ∗n = max{0, argλn

S ·∫θ∈Θ+

n (Λ∗−n,λn)

fΘ(θ)dθ = Dn

}.

Substituting the optimal shadow price λ∗n into the coremarginal profit J (n)

2 (θ), we can derive the optimal primalallocation an(θ)

∗ of any spectrum θ according to Lemma 5or (12). Intuitively, the shadow price λ∗n can be viewed as avertical shift for the core marginal profit J (n)

2 (θ) of contractuser n, such that E[dn] meets the demand Dn.

C. Efficiency

An allocation is efficient if it maximizes the social welfare,which is defined as the overall welfare of all SUs.

1) Spot market user’s welfare: For each SU m in thespot market, the welfare is directly defined by his maximumwillingness-to-pay (i.e., his valuation vm).

2) Future market user’s welfare: For each SU n in thefuture market, the welfare consists of three parts: (i) a fixedvalue B̃n related to his willingness to pay, (ii) an elastic costw̃n ·E[Cn] used to evaluate his satisfaction in the long run, and(iii) a potential welfare loss P̃n ·(Dn−E[dn]) if the demandednumber of spectrums is not satisfied.

Given any allocation A(θ), the expected social welfare canbe formally written as:

E[W ] = E[W0] +

N∑n=1

E[Wn], (13)

where E[W0] = S·∫θa0(θ)Y

1M (θ)·fΘ(θ)dθ is the total welfare

from the spot market, and E[Wn] = B̃n − w̃n · E[Cn] − P̃n ·(Dn − E[dn]) is the welfare from contract user n.

To concentrate on the difference induced by different infor-mation scenarios, we introduce the following assumptions:(a) w̃n = wn,∀n, i.e., the PO and the contract user n have

the same evaluating weight for the contract user n’s long-term satisfaction loss (cost);

(b) P̃n = P̂n,∀n, i.e., the PO’s penalty is totally used tocompensate the contract user’s welfare loss induced bythe violation of the contract.

With above assumptions, the social welfare (13) is equivalentto the PO’s expected profit (8) up to a constant, which impliesthat the PO’s profit maximizing solution maximizes the socialwelfare as well.

Lemma 7 (Efficiency). Suppose w̃n = wn and P̃n = P̂n. Theoptimal solution for PO’s profit maximization, i.e., Lemmas3-6, also maximizes the social welfare.

VI. OPTIMAL SOLUTION UNDER INFORMATIONASYMMETRY

Under information asymmetry, the PO cannot observe therealized SUs’ private information at each time slot, and thusan incentive-compatible mechanism is necessary to elicit theSUs’ private information. Without loss of generality, we applyan VCG-based auction (e.g., a second-price auction), which isprovably optimal [18], as the underlying mechanism.

6

A. ContrAuction Mechanism

The first question is: how to involve the contract usersinto the auction mechanism?4 To address this, we propose anintegrated contract and auction design, called ContrAuction,wherein the PO acts as virtual bidders on behalf of theguaranteed contracts. That is, the PO now plays two typesof roles: a seller by offering spectrums on the market, and Nbidding agents by bidding on behalf of the contracts.

The second question for the PO is: how to determine theoptimal bid for each contract? In what follows, we will derivethe optimal bidding rules under the constraint of efficiency.

Note that although we consider information asymmetry, weassume that the PO can obtain the realized information ofcontract users, while cannot obtain those of spot market users.This assumption is motivated by the fact that the contract usershave no incentive to hide their private information, since theirpayments are independent of their valuations.

B. Optimal ContrAuction under Efficiency

By Lemma 7, an efficient mechanism achieves the sameallocation as with information symmetry. Inspired by Lemma5, we propose the following bidding rule for each contractuser n (the PO bids on behalf of the contract user n):

bn(θ) , P̂n − wncn(θ)− λ∗n = xn(θ)− λ∗n, (14)where λ∗n is given by Lemma 6, i.e., the optimal shadow priceunder information symmetry.

As the overall mechanism is a second-price auction, thetruthfulness for spot market users is obvious in ContrAuction.The following two Lemmas show the efficiency and optimalityof the ContrAuction with the bidding rule (14).

Lemma 8 (Efficiency). With the bidding rule (14), the Con-trAuction maximizes the social welfare.

The efficiency can be easily proved by showing that theContrAuction with (14) achieves the same allocation as theoptimal allocation under information symmetry.

Lemma 9 (Optimality). The ContrAuction with (14) maxi-mizes the PO’s profit among all efficient mechanisms.

Any efficient mechanism must achieve the same allocationas that under information symmetry. Thus, we can dividethe whole spectrums (under an efficient mechanism) into twoparts: those allocated to the future market (part I) and those tothe spot market (part II). The optimality can be easily provedby the facts that (i) the profit from part I is the maximumprofit, since it is same as that in information symmetry, and(ii) the profit from part II is also the maximum profit that thePO can achieve in information asymmetry; this can be shownby the revenue equivalent principle [18].

VII. SIMULATION RESULTS

Our simulations show that under the constraint of efficiency,the optimal ContrAuction suffers certain profit loss (15% on

4This question is not trivial, because a contract user usually cares onlyabout whether the contract can be satisfied. Thus, a contract user is usuallynot willing to (or not able to) be involved in the competition of an auction.

average in our simulations) induced by the information rentfor the spot market users. We skip the detailed simulationsdue to space limitations. For details, please refer to [20].

VIII. CONCLUSION

We study the short-term secondary spectrum trading ina hybrid spectrum market with the future market and thespot market, and focus on the PO’s expected profit max-imization problem with stochastic information. We derivethe optimal solutions under both information symmetry andasymmetry. Under information symmetry, the optimal solution(benchmark) maximizes both the PO’s expected profit andthe social welfare. Under information asymmetry, We proposethe ContrAuction, an integrated contract and auction design,and derive the optimal ContrAuction under the constraint ofefficiency. The optimal ContrAuction suffers certain profit loss(15% on average in our simulations) under the constraint ofefficiency.

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