Spectrum Investment under Uncertainty: A Behavioral Economics Perspective
SSpectrum Investment under Uncertainty:A Behavioral Economics Perspective
Junlin Yu,
Student Member, IEEE , Man Hon Cheung, and Jianwei Huang,
Fellow, IEEE
Abstract —In this paper, we study a virtual wireless oper-ator’s spectrum investment problem under spectrum supplyuncertainty. To obtain enough spectrum resources to meet itscustomer demands, the virtual operator can either sense forthe temporarily unused spectrum in a licensed band, or leasespectrum from a spectrum owner. Sensing is usually cheaperthan leasing, but the amount of available spectrum obtained bysensing is uncertain due to the primary users’ activities in thelicensed band. Previous studies on spectrum investment problemsmainly considered the expected profit maximization problem of arisk-neutral operator based on the expected utility theory (EUT).In reality, however, an operator’s decision is influenced by notonly the consideration of expected profit maximization, but alsothe level of its risk preference. To capture this tradeoff betweenthese two considerations, we analyze the operator’s optimaldecision problem using the prospect theory from behavioraleconomics, which includes EUT as a special case. The sensingand leasing optimal problem under prospect theory is non-convexand challenging to solve. Nevertheless, by exploiting the unimodalstructure of the problem, we are able to compute the uniqueglobal optimal solution. We show that comparing to an EUToperator, both the risk-averse and risk-seeking operator achievea smaller expected profit. On the other hand, a risk-averseoperator can guarantee a larger minimum possible profit, whilea risk-seeking operator can achieve a larger maximum possibleprofit. Furthermore, the tradeoff between the expected profitand the minimum possible profit for a risk-averse operator isbetter when the sensing cost increases, while the tradeoff betweenthe expected profit and the maximum possible profit for a risk-seeking operator is better when the sensing cost decreases.
Index Terms —Prospect theory, expected utility theory, spec-trum trading, spectrum sensing.
I. I
NTRODUCTION
A. Background and Motivation T HE business model of virtual operator has achieved sig-nificant success worldwide in recent years [2]. Accordingto a recent market report published by Transparency MarketResearch, the global virtual operator market is expected toexpand at an annual rate of . % and reach a value ofUS$ 75.25 billions by 2023 [3]. As a virtual operator (e.g.,Consumer Cellular in the US [4]) does not own any licensed Manuscript received May 1, 2016; revised August 5, 2016; accepted August29, 2016. This work is supported by the General Research Fund (CUHK14202814) established under the University Grant Committee of the HongKong Special Administrative Region, China. Part of this paper was presentedin [1].J. Yu, M. H. Cheung, and J. Huang are with the Department of InformationEngineering, the Chinese University of Hong Kong, Hong Kong, China;Emails: { yj112, mhcheung, jwhuang } @ie.cuhk.edu.hk. Here we focus on a “virtual operator” in the wireless industry, which isalso referred to as the “mobile virtual network operator (MVNO)” in theliterature. spectrum, it needs to acquire spectrum from a spectrum owner (e.g., AT&T) in order to provide services to its customers. Asvirtual operators often rely on leased network infrastructureinstead of building and maintaining their own infrastructure,their investment and operational costs are usually lower thanthe spectrum owners. Such an advantage often enables themto provide cheaper and more flexible data plans to theircustomers, hence reaching niche markets that are underservedby the spectrum owners [5].Motivated by the recent development of cognitive radiotechnology and dynamic spectrum sharing, a virtual operatorcan acquire spectrum in two different ways: spectrum sensingand spectrum leasing. With spectrum sensing [6], [7], a virtualoperator detects the temporarily unused spectrum in a licensedband, and uses it to provide services to its customers as longas such operator does not cause any harmful interferencesto the primary (licensed) customers of the spectrum owner.With spectrum leasing [8], [9], a spectrum owner explicitlyallows the virtual operator to operate over a given licensedband during a given period of time with a leasing fee. In thispaper, we will consider a hybrid spectrum investment schemeinvolving both approaches.The key feature of the problem is the uncertainty ofspectrum acquisition through spectrum sensing, as the virtualoperator does not know the activity levels of the primarycustomers beforehand. When facing uncertainty, most priorstudies of spectrum investment applied the expected utility the-ory (EUT) to compute the operator’s optimal decisions (e.g.,[10]–[13]). In these models, a (virtual) operator optimizesthe decisions to maximize its expected profit. Such an EUTmodel, however, does not fully capture the rather complicateddecision process obtained in our daily life, and hence mayhave a poor predication power [14]. Alternatively, the Nobel-prize-winning prospect theory (PT) (e.g., [14]–[16]), whichestablishes a more general model than the EUT, provides apsychologically more accurate description of the decision-making under uncertainty. PT incorporates three main factorsin the modeling: (1) Impact of reference point : A decisionmaker evaluates an option based on the potential gains or To better understand PT, consider the following two lottery settings.Lottery A1: 50% to win $200, and 50% to win $0; Lottery A2: 100% to win$100. Experimental results [14] showed that most people prefer Lottery A2to A1. The result reflects that people are risk averse, which we will discuss inmore details later in this section. Next we further consider another two lotterysettings. Lottery B1: 1% to win $99, and 99% to loss $1; Lottery B2: 100%to win $0. Experimental results [14] showed that most people prefer LotteryB1 to B2. The result reflects that people will have a subjective probabilitydistortion of small probability events, which we will also introduce later inthis section. a r X i v : . [ c s . N I] A ug losses with respect to a reference point, and the choice ofthe reference point significantly affects the valuation. (2) The s-shaped asymmetrical value function : A decision maker oftenexperiences a diminishing marginal utility when evaluating again, and a diminishing marginal disutility when evaluatinga loss. Furthermore, it often prefers avoiding losses thanachieving gains. As a result, the value function is s-shapedand asymmetrical: concave in the gain regime, convex in theloss regime, and steeper for losses than for gains (see Fig.2(a) in Section III-D for a concrete example). (3) Probabilitydistortion : A decision maker tends to overreact to small prob-ability events, but underreact to medium and large probabilityevents (see Fig. 2(b) in Section III-D for a concrete example).This characteristic is useful in explaining behaviors relatedto lottery and insurance [14], where people usually purchaselottery and insurance at prices higher than the expected returns.As PT fits better into the reality than EUT based on manyempirical studies, researchers and practitioners have appliedPT in many areas, such as understanding the behavior ofinvestment agents in finance [17] and the effort and wagelevels of workers and firms in labor markets [20]. However,there isn’t any existing work of using PT to understand thespectrum investment behaviors in today’s wireless market. B. Key Results and Contributions
In this paper, we study the spectrum sensing and leasingdecisions of a virtual operator under sensing uncertainty , andformulate it as a two-stage sequential optimization problem.In Stage I, the virtual operator determines the optimal amountof licensed spectrum to sense . Due to the stochastic nature ofthe primary licensed customers’ traffic, the amount of availablespectrum obtained through sensing is a random variable. If thespectrum obtained through sensing is not sufficient to satisfyits customers’ demand, the virtual operator will lease someadditional spectrum from the spectrum owner in Stage II.Under this sensing uncertainty , we can model the decisionmaking of a risk-neutral operator, who aims to maximize itsexpected profit by EUT. However, in reality, a decision makeris rarely risk-neutral. Besides aiming to achieve a high ex-pected profit, it is often affected by its own risk preference. Tobe more specific, a risk-seeking decision maker is aggressiveand wants to achieve a high profit even with a high risk,while a risk-averse decision maker is conservative and wantsto guarantee a satisfactory level of minimum possible profit. Inorder to capture this tradeoff between the expected profit andrisk preference, we apply the PT to study the optimal sensingand leasing decisions. It leads to a non-convex optimizationproblem, which is very challenging to solve. Nevertheless,by exploiting the unimodal structure of the problem, we canobtain the global optimal solution analytically.Our key contributions are summarized as follows: We note that prospect theory is not a theory only about “irrationalbehaviors”. Instead, it is about how decision makers decide the tradeoffbetween the maximum/minimum possible profit and the expected profit (inour context), which applies to professionals as well. In fact, there have beenvarious studies (e.g., [18], [19]) that focus on the professionals’ decisionsproblems based on prospect theory in areas such as finance and politics. • Behavioral economics modeling of virtual operator’s in-vestment decision under uncertainty : We model a virtualoperator’s investment decisions under sensing uncertaintyusing PT, which captures the tradeoff between the ex-pected profit maximization and risk preference. We char-acterize the feature of a risk-averse operator (who is mostconcerned of potential losses) and a risk-seeking operator(who is most concerned of potential gains). • Characterization of the unique optimal solution of thenon-convex decision problem : Despite the non-convexityof the spectrum sensing problem, we characterize theuniqueness of the optimal solution and compute it nu-merically. We further evaluate how different behavioralcharacteristics (i.e., reference point, probability distortion,and s-shaped valuation) affect this optimal solution. • Engineering insights based on comparison between EUTand PT : We show that a risk-averse operator can achieve abetter tradeoff between the expected profit and minimumpossible profit in the high sensing cost scenario than thelow sensing cost scenario. The result for the risk-seekingoperator is exactly the opposite.Next we will review the literature in Section II. In Sec-tion III, we introduce the spectrum investment model andformulate the sequential optimization problem. In Section IV,we compute the global optimal solution of the non-convexoptimization problem, and discuss various engineering insightsderived from such a solution. In Section V, we illustratethe impact of probability distortion and reference point byconsidering the special case of binary sensing outcomes. InSection VI, we provide simulation results to evaluate thesensitivity of the optimal decision with respect to severalmodel parameters. We conclude the paper in Section VII.II. L
ITERATURE R EVIEW
A. Expected Profit Maximization in Spectrum Investment Us-ing Expected Utility Theory
Spectrum investment problem under uncertainty has beenstudied extensively through expected profit maximization us-ing EUT (e.g., [10]–[13]). Kasbekar and Sarkar in [10] con-sidered a spectrum auction problem under the uncertainty ofthe number of secondary customers. Gao et al. in [11] studiedthe spectrum contract between a primary spectrum owner andthe secondary customers considering the uncertainty of thecustomer types. In [12], Jin et al. presented an insurance-based spectrum trading problem between a primary spectrumowner and the secondary customers, where the uncertainty alsocomes from the types of the customers. Duan et al. in [13]considered the spectrum investment of a virtual operator underthe spectrum sensing uncertainty. In all the above studies basedon EUT, a decision maker aims to maximize the weightedaverage of its utilities under different outcomes, which doesnot fully capture the realistic human decision behaviors exam-ined in several well known psychological studies in the pastfew decades [14]–[16]. Thus, in this paper we apply the moregeneral PT, which takes into account both the expected payoffand risk preference in the human decision making.
B. Resource Allocation in Communication Networks andSmart Grids Using Prospect Theory
The study of resource allocation in communication networksand smart grids based on PT only emerged recently. Thefirst paper is due to Li et al. in [21], which compared theequilibrium strategies of a two-user random access game underEUT and PT. Yang et al. in [22] considered the impact of end-user decision-making on wireless resource pricing, when thereis an uncertainty in the quality of service (QoS) guaranteesrelying on PT. Several other recent studies applied PT to studythe decision making in smart grid systems. In [23], Wang etal. formulated a non-cooperative game among consumers in anenergy exchange system. They applied PT to explicitly accountfor the users’ subjective perceptions of their expected utilities.Xiao et al. in [24] studied the static energy exchange gameamong microgrids that are connected to a backup power plant.They analyzed the Nash equilibria under various scenariosbased on PT, and evaluated the impact of user’s objectiveweight on the equilibrium of the game. To reflect the factthat realistic decision making is different from expected profitmaximization, the studies in [21]–[24] considered a linearvalue function with the probability distortion. However, tomodel the impact of risk preference on a decision maker, usingonly a linear value function is not comprehensive enough.In fact, based on the psychological studies in [14]–[16], thethree characteristics of PT (i.e., reference point, s-shaped valuefunction, and probability distortion) together determine the riskpreference of a decision maker. Our paper is the first one thatconsiders all three characteristics of PT for a more accurateand comprehensive understanding of the optimal decisionproblem. III. S
YSTEM M ODEL
A. Spectrum Sensing and Leasing Tradeoff
We consider a cognitive radio network with a spectrumowner and a virtual operator. From the empirical data in[25], [26], it is possible for the spectrum owner to estimatethe spectrum utilization at a particular location accuratelybased on past measurements. In this paper, we assume thatsuch estimations are accurate, so that the spectrum ownercan divide its licensed spectrum into the primary band and secondary band according to the spectrum utilization. Thespectrum owner uses the primary band to serve its primarycustomers (PCs), while reserves the secondary band to meetthe potential leasing requests from the virtual operator. For thevirtual operator, it can either try to sense the idle spectrumin the primary band (as PCs’ activities are stochastic duringthe time period of interest), or lease the spectrum from thesecondary band.As a more concrete (hypothetical) example, we considerthe spectrum trading between Consumer Cellular and AT&T(shown in Fig. 1). AT&T is a spectrum owner, who provideswireless services to its PCs. However, it cannot fully utilize itsspectrum in some rural areas, so it will divide its spectrum intothe primary band (channel 1-8) and secondary band (channel9-16) at those under-utilized locations. Consumer Cellularwants to provide spectrum services to its own customers, but
Primary Band Secondary Band 𝐵 𝑠 𝐵 𝑙
1 2 3 4 5 6 7 8 AT&T (Spectrum Owner) Consumer Cellular (Virtual Operator) Channels obtained through sensing Channels obtained through leasing Idle Occupied 9 10 11 12 13 14 15 16 9 10 11 4 7 Stage I Stage II
Fig. 1. An Example of Consumer Cellular and AT&T. it does not own any spectrum. As a result, Consumer Cellularsenses for spectrum holes (not used by the PCs) in the primaryband (channel 3-8) without explicit payment to AT&T, and canalso choose to lease spectrum in the secondary band (channel9-11) with explicit payment to AT&T.From the virtual operator’s point of view, sensing is oftena cheaper way to obtain spectrum than leasing, because theenergy and time overhead involved in sensing is often muchlower than the explicit cost of spectrum leasing [13]. However,the available amount of spectrum obtained through sensing isuncertain due to the spectrum owner PCs’ stochastic activitiesover time. We would like to understand the virtual operator’soptimal spectrum investment decisions in every time slot thatstrike the best tradeoff between the cost and the risk . B. Two-Stage Decision Model
We formulate the virtual operator’s spectrum investmentproblem as a two-stage sequential optimization problem ineach time slot.In Stage I (i.e., the sensing stage), the virtual operatordetermines its sensing decision B s (measured in Hz). Forsimplicity, we assume a linear sensing cost c s per unit ofsensed bandwidth, which represents the time and energyoverhead for sensing [27]. Due to the stochastic nature ofPCs’ traffic, only a fraction α ∈ [0 , of the sensed spectrumis temporarily available and can be utilized by the virtualoperator’s own customers. Hence, the virtual operator obtainsa bandwidth of B s α at the end of the stage. In other words, a We choose the length of time slot such that the primary customers’activities remain unchanged within a time slot [13]. For simplicity, we assume perfect sensing in this paper. For imperfectsensing [28], it involves an additional level of uncertainty, which is challengingto consider due to the framing effect [14] in behavioral economics. large α corresponds to a high sensing realization , and a small α corresponds to a low sensing realization . As an examplein Fig. 1, Consumer Cellular senses six channels in Stage I(i.e., channel 3-8), and only channels 4 and 7 are available.Hence, α = 1 / in this case. We assume that the virtualoperator knows the distribution of α through historical sensingresults . Notice that in the sensing stage, the virtual operatorhas uncertainty of sensing realization, and its optimal decisionwill be influenced by balancing the expected profit and its riskpreference on the sensing uncertainty. We will discuss the PTmodeling on this aspect in more details in Section III-D.In Stage II (i.e., the leasing stage), the virtual operatordetermines the leasing decision B l (measured in Hz) afterknowing the available amount of spectrum through sensing, B s α . We consider a linear leasing cost c l , which is determinedthrough negotiation between the virtual operator and spectrumowner and is considered to be a fixed parameter in our model.As an example in Fig. 1, after Consumer Cellular acquires twochannels from sensing (i.e., channels 4 and 7), it further leasesthree more channels in Stage II (i.e., channel 9-11). As thereis no uncertainty involved in Stage II, there is no differencebetween EUT and PT modeling in terms of results . C. Virtual Operator’s Profit
When serving its customers, the virtual operator can obtaina revenue of π per unit of sold spectrum. We assume that theprice π is exogenously given and cannot be changed by thevirtual operator, due to the intensive market competition [35].Under a fixed usage-based price π , we assume that the virtualoperator’s secondary customers’ maximum spectrum (band-width) demand is D . However, the demand may not be fullysatisfied if the virtual operator does not have enough spectrumobtained through sensing and leasing discussed before. Hence,the profit of the virtual operator is R ( B s , B l , α ) = π min { D, B l + B s α } − ( B s c s + B l c l ) , (1)where the revenue (first term on the right hand side) dependson the minimum of the demand D and the spectrum supply B l + B s α , and the cost (second term on the right handside) depends on both the sensing decision B s and leasingdecision B l . As α is a random variable before making thesensing decision B s , we will incorporate the operator’s riskpreference towards such uncertainty through the modelingbased on prospect theory. D. Prospect Theory Modeling in Sensing Decision
To model the virtual operator’s decision under spectrumsensing uncertainty, we consider the following three key In practice, it is difficult for the operator to obtain the exact distribution of α . However, according to [29]–[32], the operator can estimate the distributionof α through learning based on the updated historical sensing results [33],[34]. In fact, as long as λ = β = α = 1 , choosing a non-zero value of R p willjust induce a constant shift of the EUT utilities, without affecting the optimaldecision under EUT. The reference point affects the analysis of PT, and ouranalysis shows the impact of reference point in Section V-B. We do not assume any specific relationship between price π and demand D in this paper. Please refer to [35] for some further discussions along thisline. −2 −1 0 1 2−3−2−1012 x v ( x ) GainLoss λ = 1, β = γ = 0.3 λ = 1, β = γ = 1 λ = 2, β = 0.3, γ = 0.6 λ = 2, β = γ = 0.6 (a) v ( x ) p w ( p ) µ =0.4 µ =0.6 µ =0.8 µ =1 (b) w ( p ) Fig. 2. The s-shaped asymmetrical value function v ( x ) and the probabilitydistortion function w ( p ) in PT. features of PT: reference point R p , s-shaped value function v ( x ) , and probability distortion function w ( p ) .First, the choice of reference point R p will significantlyaffect the evaluation of profit R ( B s , B l , α ) in PT. Morespecifically, we define the net gain as x = R ( B s , B l , α ) − R p . (2)The reference point is a benchmark to evaluate the payoff,where x ≥ means a gain, while x < means a loss. Thevirtual operator will have different decision mechanisms (to beexplained in the next paragraph) when dealing with a gain ora loss in PT. A higher reference point means that the operatorexpects a higher profit (at the benchmark), which implies theoperator is more risk-seeking.Second, as shown in Fig. 2(a), the value function v ( x ) isconcave for a positive argument (gain), and is convex fora negative argument (loss). Moreover, the impact of loss isusually larger than the gain of the same absolute value. Acommon choice of value function [15], [16], [36] is v ( x ) = (cid:40) x β , if x ≥ , − λ ( − x ) γ , if x < , (3)where λ > , < β < , and < γ < . The parameter λ is the loss penalty parameter, where a larger λ indicatesthat the virtual operator is more concerned of loss, and henceis more risk-averse . The parameters β and γ are the riskparameters, where the value function of the gain part is moreconcave (i.e., the virtual operator is more risk-averse ) when β approaches zero, and the value function of the loss part is moreconvex (i.e., the virtual operator is more risk-seeking ) when γ approaches zero. The impact of β and γ can be interpretedby the risk-seeking behavior in loss and risk-averse behaviorin gain. As an example, a gambler will be more addicted intothe gambling when it loses money, and will be less willing tocontinue when he wins money.We note that EUT is a special case if we choose λ = 1 and γ = β = 1 . The result under the case γ = β has beendiscussed in the conference version of this work in [1], and wefocus on the more complicated case of β < γ in this paper , The analysis can be readily extended to the case of β > γ , although thereare no additional new insights in that case. Hence we omit the discussion of β > γ here.
TABLE IK EY N OTATIONS
Symbols Physical Meanings B s (decision variable) Sensing amount B l (decision variable) Leasing amount c s Sensing cost per unit c l Leasing cost per unit π Price per unit D Secondary users’ demand α Sensing realization factor β Risk parameter for gain γ Risk parameter for loss λ Loss penalty parameter µ Probability distortion parameter which models the scenario that the marginal utility in gain isdiminishing faster than the marginal disutility in loss [37].
Assumption 1.
The risk parameter for gain is less than therisk parameter for loss (i.e., β < γ ). Third, as shown in Fig. 2(b), the probability distortionfunction w ( p ) models the fact that virtual operator overweighsa small probability event and underweighs a large probabilityevent. A common choice of probability distortion function(e.g., [15], [16], [36]) is w ( p ) = exp ( − ( − ln p ) µ ) , < µ ≤ , (4)where p is the objective probability of high sensing realization,and w ( p ) is the virtual operator’s corresponding subjective probability. Here the probability distortion parameter µ re-veals how a virtual operator’s subjective evaluation distortsthe objective probability, where a smaller µ means a largerdistortion.The key notations of the virtual operator’s profit maximiza-tion problem are listed in Table I. In the next section, wewill study the virtual operator’s optimal sensing and leasingdecisions that maximize its profit .IV. S OLVING THE T WO - STAGE O PTIMIZATION P ROBLEM
In this section, we use backward induction [38] to solvethe two-stage sequential optimization problem. In SectionIV-A, we derive the operator’s optimal leasing decision inStage II. In Section IV-B, we obtain the operator’s optimalsensing decision in Stage I under sensing uncertainty by thePT model . A. Optimal Leasing Decision in Stage II
In Stage II, given a fixed value of sensing bandwidth B s determined in Stage I, the operator’s leasing optimizationproblem is max B l ≥ R ( B s , B l , α ) = π min { D, ( B l + B s α ) }− ( B s c s + B l c l ) . (5)In our analysis, we make the following assumptions. We will simply use “operator” to denote “virtual operator” for the restof the paper. It should be noted that the models under EUT and PT share the sameStage II (i.e., the leasing stage), because uncertainty only appears in Stage I(i.e., the sensing stage).
Assumption 2. (a) The sensing cost is less than the leasingcost (i.e., c s < c l ); (b) The leasing cost is less than theoperators usage-based price (i.e., c l < π ). Both parts of Assumption 2 allow us to focus on the non-trivial case in our analysis. When c s ≥ c l , we can show that theoperators’ optimal decision of the two-stage process is always B ∗ s = 0 and B ∗ l = D for any type of risk preferences, becauseleasing is both cheaper and risk-free. When c l ≥ π , we canshow that the operator will always choose B ∗ l = 0 for anytype of risk preferences. Hence, we can focus on Assumption2 without loss of any generality. Under Assumption 2, we canshow that the optimal leasing decision that solves Problem (5)is B ∗ l = max { D − B s α, } , (6)which is the difference between the total demand D and theavailable spectrum through sensing, B s α . If B s α exceeds D ,then B ∗ l = 0 . Under (6), the operator’s profit in (1) can bewritten as a function of B s and α : R ( B s , B ∗ l , α ) = πD − B s c s − max { D − B s α, } c l . (7) B. Optimal Sensing Decision in Stage I
We assume that the sensing realization factor α follows adiscrete distribution with I possible outcomes , hence has afinite number of sensing realization outcomes [28], [39]. Thecorresponding probability mass function is denoted as p ( α i ) (cid:44) P ( α = α i ) = p i , i ∈ I = { , ..., I } . (8)Without loss of generality, we assume that α i < α j if i < j .By substituting (2), (3), (4) and (8) into (7), we obtain the expected utility of the operator under PT as U ( B s ) = I (cid:88) i =1 v [ R ( B s , B ∗ l , α i ) − R p ] w ( p ( α i ))= I (cid:88) i =1 v [ πD − ( B s c s +max { D − B s α i , } c l ) − R p ] w ( p ( α i )) . (9)The operator’s spectrum sensing optimization problem inStage I with sensing uncertainty is max B s ≥ U ( B s ) . (10)
1) Optimal Sensing Decision under a Risk-free ReferencePoint R p = D ( π − c l ) : Problem (10) is a non-convex optimiza-tion problem, because it involves the s-shaped value functionin (3). Hence, it is challenging to analytically characterizethe closed-form optimal solution. However, we can show thatthere exists a unique global optimal solution by exploiting thespecial unimodal structure of the problem.In problem (10), a common choice of reference point is therisk-free profit . For example, in finance, investors naturally Since the spectrum is usually divided into a finite number of channelsin practical systems, it is reasonable to consider a discrete distribution of α . Mathematically, when we choose the number of possible realizations I to be large, then the discrete distribution can well approximate a continuousdistribution. A complete analysis for an arbitrary reference point is challengingbecause of the non-convexity in Problem (10). In Section V, we will furtherlook into the impact of reference point in a simplified model with binarysensing results.
TABLE IIO
PTIMAL S ENSING AND L EASING D ECISION UNDER PT Condition Optimal Sensing Decision B ∗ s Optimal Leasing Decision B ∗ l D ≤ M ˆ ı +1 B ∗ s = Dα ˆ ı +1 B ∗ l = max { , D − Dαα ˆ ı +1 } M j < D < H j for j = ˆ ı + 1 , ..., I − B ∗ s = g − j (0) B ∗ l = max { , D − αg − j (0) } H j ≤ D < M j +1 for j = ˆ ı + 1 , ..., I − B ∗ s = Dα j +1 B ∗ l = max { , D − Dαα j +1 } D ≥ M I B ∗ s = M I α I B ∗ l = D − M I αα I choose the risk-free return as a benchmark to evaluate theirinvestment performances [36]. Here, we choose the maximumprofit that the operator can achieve without sensing (hence arisk free choice) as the reference point. This corresponds to theoperator leasing a bandwidth B l = D and choosing B s = 0 ,which leads to the profit of R p = D ( π − c l ) . (11)Substituting (11) into (10), we solve problem (10) andsummarize the key result in Table II. To understand TableII, we first define some notations. We define ˆ ı ∈ I as theunique index that satisfies both the constraints α ˆ ı ≤ c s c l and α ˆ ı +1 > c s c l . We define H j (for j = ˆ ı + 1 , ..., I − ) (see (17)in Appendix A) and M j (for j = ˆ ı +1 , ..., I ) (see (15) and (16)in Appendix A) in Table II as decision indicators. The decisionindicators are increasing functions of the leasing cost c l andthe risk parameter for gain β , and decreasing functions of thesensing cost c s , the risk parameter for loss γ , and the losspenalty parameter λ . In other words, when the sensing cost c s is higher or when the operator is more risk-averse (e.g., with alarger λ , a smaller β , or a larger γ ), the decision indicators H j and M j decrease. The function g j ( B s ) (for j = ˆ ı +1 , ..., I − )(see (27) in Appendix A) is a decreasing function in B s . Theorem 1.
Under Assumptions 1 and 2, the optimal sensingdecision B ∗ s for problem (10) and the optimal leasing decision B ∗ l for problem (5) are summarized in Table II. By using the unimodal structure of Problem (10), we showthat there is at most one inner local maximum point, and hencethe global optimum is either at the local maximum point orat the boundaries [40]. For the detailed proof of Theorem1, please refer to Appendix A. As shown in Table II, theoperator’s optimal sensing and leasing decisions depend on thedecision indicators H j and M j . Notice that M j < H j < M j +1 for every j , and all H j (for j = ˆ ı + 1 , ..., I − ) and M j (for j = ˆ ı + 1 , ..., I ) are increasing in c l and β , and are decreasingin c s , γ , and λ . For a more risk-averse operator (with larger λ , smaller β , and larger γ ), it has smaller decision indicators H j and M j (which refers to a lower row in Table II), henceit leads to a smaller B ∗ s .To better illustrate the insights behind Table II, we furthercharacterize the impact of sensing cost c s and the risk param-eters λ , β and γ on B ∗ s in the following corollaries. Corollary 1.
The optimal sensing decision B ∗ s for problem(10) is decreasing in the loss penalty parameter λ in (3) andthe risk parameter for loss γ in (3). Both the case α ≥ c s c l and the case α I ≤ c s c l are trivial, and thediscussion under these two cases are covered in the discussions under thecase α < c s c l < α I . The proof of Corollary 1 is given in Appendix B. Corollary1 indicates that if an operator is more risk-averse (with a largerloss penalty parameter λ or a larger risk parameter for loss γ ), it will sense less, as it prefers avoiding loss due to a lowsensing realization to achieving gain due to a high sensingrealization. Corollary 2.
When the demand is larger than the I th decisionindicator (i.e., D > M I ), the optimal sensing decision B ∗ s forproblem (10) is increasing in the risk parameter for gain β in(3). The proof of Corollary 2 is given in Appendix C. Corollary2 indicates that for the case of a large demand (i.e.,
D > M I ),a more risk-seeking operator (with a larger risk parameter forgain β ) will sense more to achieve a larger gain.However, in the small demand case (i.e., D ≤ M I ), as longas the total available spectrum from sensing can satisfy thedemand (i.e., B s α ≥ D ), a larger sensing decision B s leadsto a constant revenue Dπ but a larger total sensing cost B s c s ,hence a less profit. Therefore, the optimal sensing decision B ∗ s is not always increasing with β when D ≤ M I . Corollary 3.
The optimal sensing decision B ∗ s for problem(10) is decreasing in the sensing cost c s . The proof of Corollary 3 is given in Appendix D. Corollary3 indicates that the operator is more willing to sense when thesensing cost decreases.For comparison, we also consider the operator’s optimalsensing and leasing decisions under the EUT model. Thedetailed discussion is given in Appendix E.V. S
PECIAL C ASE : O
PTIMAL S ENSING D ECISION WITH B INARY O UTCOMES
In Section IV, we have focused on the discussion of theimpact of the parameters of the s-shaped value function. Inthis section, we consider a special case with binary sensingoutcomes (i.e., I = 2 ), to further illustrate the impact ofreference point and probability distortion on the sensing de-cision. More specifically, the binary results are α = 0 and α = 1 with probabilities p and p as in (8) respectively,where p + p = 1 .A practical motivation of this case is the spectrum utilizationat those areas such as the subway stations or overpasses, wherethe traffic patterns of PCs follow a very high peak-valley ratio[25], [26]. In these areas, there is hardly any spectrum leftfor the virtual operator when the traffic is at its peak, butthe situation completely changes during the off-peak hours.Another example is the spectrum used by a rotating radarsystem [41], where the radar antenna gain and radiation patternseen by the operator vary in a very high peak-valley ratio TABLE IIIO
PTIMAL S ENSING D ECISION UNDER DIFFERENT µ Condition Optimal Sensing Decision B ∗ sc l c s ≤ (cid:16) w ( p ) w ( p ) (cid:17) β + 1 B ∗ s = 0 c l c s > (cid:16) w ( p ) w ( p ) (cid:17) β + 1 B ∗ s = D with the rotation of main beam. Hence, there are periods oftime where the sensing realization is very high, and otherperiods of time when the sensing realization is very low. Tobetter illustrate the impact of reference point and probabilitydistortion, we consider the case β = γ in this section tosimplify the analysis . Hence, by plugging I = 2 into (9),we can obtain the utility function: U ( B s ) = w ( p ) v ( πD − B s c s − Dc l − R p )+ w ( p ) v ( πD − B s c s − max { , D − B s } c l − R p ) . (12) A. Impact of Probability Distortion
First, we discuss the impact of probability distortion onthe operator’s optimal sensing decision B ∗ s . To compute theoptimal sensing decision, we consider the same reference point R p = D ( π − c l ) as in Section IV-B, and obtain the followingtheorem. Theorem 2.
For the case of binary outcomes, under Assump-tions 1 and 2, the optimal sensing decision B ∗ s for problem(10) is summarized in Table III. The proof of Theorem 2 is given in Appendix F. The resultsin Table III depend on the ratio w ( p ) /w ( p ) . The operatorwill sense B ∗ s = D when the leasing and sensing cost ratio c l /c s exceeds the sensing threshold [ w ( p ) /w ( p )] β +1 , andwill sense B ∗ s = 0 otherwise. Since p + p = 1 , accordingto the probability distortion effect in PT, the larger probabilityis underweighed and the smaller probability is overweighed.From Table III, for the case p < p , p is overweightedand p is underweighted in PT (i.e., w ( p ) /w ( p ) > p /p ),where the operator underestimates the chance of having a highsensing realization. Thus, the PT operator is more risk-averseand will sense less than or equal to an EUT operator becauseit has a higher sensing threshold. On the other hand, when p > p , p is overweighted (i.e., w ( p ) /w ( p ) < p /p ),where the operator overestimates the chance of a high sensingrealization. Thus, the PT operator is more risk-seeking andwill sense larger than or equal to that of an EUT operatorbecause the PT operator has a lower sensing threshold. B. Impact of Reference Point
Next, we discuss the impact of reference point R p on theoperator’s optimal sensing decision B ∗ s . A high reference point R p indicates that the operator has a high expectation on theprofit, and it is more likely to experience a loss since theoutcome is often less than its expectation. On the other hand,a low reference point R p indicates that the operator has a lowexpectation, and it is more likely to experience a gain since We do not obtain additional new insights on reference point and proba-bility distortion in the case β (cid:54) = γ , hence we omit the discussion of β (cid:54) = γ here. TABLE IVO PTIMAL S ENSING D ECISION B ∗ s UNDER R HIGH p = D ( π − c s ) AND R LOW p = D ( π − c l − c s ) Condition B ∗ s under R high p B ∗ s under R low p (i.e., Risk-Seeking) (i.e., Risk-Averse) c l c s < w ( p ) w ( p ) B ∗ s = U (cid:48) − RPH (0) B ∗ s = 0 c l c s ≥ w ( p ) w ( p ) B ∗ s = D B ∗ s = U (cid:48) − RPL (0) the outcome is often beyond its expectation. As we will see inthe following, whether an outcome is considered as a loss or again can significantly affect the operator’s subjective valuationof the outcome, and hence its sensing decision.To better illustrate the impact of reference point, we focuson two choices: (i) A high reference point R high p = D ( π − c s ) , which reflects the operator’s expectation of realizing allthe sensing spectrum D . (ii) the low reference point R low p = D ( π − c l − c s ) , which reflects the operator’s expectation ofnot realizing any of the sensing spectrum D . In other words,the same outcome is more likely to be considered as a lossunder the high reference point than under the low referencepoint.From (12), we obtain the utility under the high referencepoint as U RPH ( B s ) = − λw ( p ) ( B s c s + Dc l − Dc s ) β − λw ( p ) [ D ( c l − c s ) − B s ( c l − c s )] β , (13)and the utility under the low reference point as U RPL ( B s )= w ( p )( Dc s − B s c s ) β + w ( p ) [ B s ( c l − c s )+ Dc s ] β . (14)By studying the first order derivatives under the two ref-erence points U (cid:48) RP H ( B s ) and U (cid:48) RP L ( B s ) , we can computethe optimal sensing decision that solves problem (10) in thefollowing theorem. Theorem 3.
For the case of binary outcomes, under Assump-tions 1 and 2, the optimal sensing decision B ∗ s for problem(10) under different reference points R highp = D ( π − c s ) and R lowp = D ( π − c l − c s ) are summarized in Table IV. The proof of Theorem 3 is given in Appendix G. Theorem3 indicates that B ∗ s under R high p is always larger than B ∗ s under R low p . This means that an operator with R high p is more willing tosense compared to an operator with R low p . When an operatorhas a high expectation (due to a high reference point), it ismore likely to encounter losses than gains under uncertainty.Since the operator’s valuation function v ( x − R p ) is convexin the loss region, it will sense more in order to gain more inthe case of high sensing realization (i.e., α = 1 ). In contrast,when an operator has a low expectation (due to a low referencepoint), it is more likely to encounter gains than losses. Sincethe operator’s valuation function v ( x − R p ) is concave interms of gains, it will sense less, in order to avoid the riskof low sensing realization (i.e., α = 0 ). To summarize, anoperator who expects a higher profit is more risk-seeking, andan operator who expects a lower profit is more risk-averse.VI. P ERFORMANCE E VALUATIONS
In this section, we illustrate the operator’s optimal sensingdecision and the corresponding expected profit under different
Loss Averse Parameter λ R i sk A v e r s e P a r a m e t e r f o r Lo ss γ (a) β = 0 . , R p = ( π − c l ) D Loss Averse Parameter λ R i sk A v e r s e P a r a m e t e r f o r Lo ss γ (b) β = 1 , R p = ( π − c l ) D Loss Averse Parameter λ R i sk A v e r s e P a r a m e t e r f o r Lo ss γ (c) β = 1 , R p = ( π − c s ) D Fig. 3. Optimal sensing decision B ∗ s versus γ and λ for different β and R p . Other parameters are c l = 5 , c s = 2 , and D = 10 . (The color representsoptimal sensing decision B ∗ s .) system parameters. The key insights under the PT modelinginclude: (a) A risk-averse operator will sense less and leasemore, which leads to a smaller profit with a lower risk of loss;while a risk-seeking operator will sense more and lease less,which leads to a larger profit with a higher risk of loss. (b) Risk preference changes with the probability of high sensingrealization. When the probability of high sensing realizationchanges from very high to very low, the operator changesfrom risk-averse to risk-seeking. (c) Both risk-averse and risk-seeking operators face a tradeoff between satisfying their riskpreferences and maximizing expected profit. A risk-averse operator achieves a better tradeoff in a high sensing costscenario than in a low sensing cost scenario, while a risk-seeking operator achieves a better tradeoff in a low sensingcost scenario than in a high sensing cost scenario.
A. Evaluation of the Optimal Sensing Decision
We first evaluate the impact of the three characteristics ofPT on the operator’s optimal decision.
1) Impact of s-shaped Value Function:
First, we illustratethe operator’s optimal sensing decision under different param-eters of s-shaped value function (i.e., γ , β , and λ ), assumingreference point R p = ( π − c l ) D and linear probabilitydistortion (i.e., µ = 1 ). We compare the optimal sensingdecision and the corresponding expected profit with the EUTbenchmark, where λ = β = γ = 1 .First, in Fig. 3(a) and Fig. 3(b), we study the optimal sensingdecision B ∗ s against the loss penalty parameter λ and the riskaverse parameter for loss γ . To illustrate the impact of β , weset β = 0 . in Fig. 3(a) and β = 1 in Fig. 3(b). The othersystem parameters are fixed at c l = 5 , c s = 2 , and D = 10 .We observe the behaviors of both risk-averse and risk-seekingoperators in Fig. 3(a) and Fig. 3(b). The upper right parts ofthe figures correspond to the risk-averse operators, and thelower left parts of the figures are risk-seeking operators. TheEUT benchmark corresponds to the case of λ = β = γ = 1 (i.e., upper left corners of Fig. 3(b) and Fig. 3(c)). We observethat risk-averse operators sense less and risk-seeking operatorssense more. Impact of γ on B ∗ s : We can see that for fixed β and λ ,the operator senses more when γ decreases (see y-axis in Fig.3(a) and Fig. 3(b)), as stated in Corollary 1. The intuition isthat when γ decreases from 1 (hence the operator is morerisk-seeking than an EUT operator), the operator experiences less marginal disutility from loss. In order to win a potentiallylarge gain, the operator is more willing to take risk and sensemore. An example to illustrate this impact is that a gambler,who has already lost a lot, cares less of losing an additional$ than a gambler who just starts to gamble. Impact of λ on B ∗ s : As stated in Corollary 1, for fixed γ and β , the operator senses less when λ increases (see the x-axis in Fig. 3(a) and Fig. 3(b)). The intuition is that when λ islarger, the penalty of loss to the operator is larger (hence theoperator is more risk-averse than an EUT operator). In orderto avoid a potential loss, the operator will sense less. Impact of β on B ∗ s : By comparing Fig. 3(a) and Fig. 3(b),we can observe that the operator senses more when β increasesfor fixed λ and γ , which verifies Corollary 2. The intuition isthat when β decreases from 1 (hence the operator is more risk-averse than an EUT operator), the operator experiences lessmarginal utility from the same gain. Hence, the operator willsense less to achieve a certain gain, rather than taking risk fora very large gain. An example to illustrate this impact is thata rich man is less willing to earn an additional $ than a poorman if doing so requires a fixed amount of effort.
2) Impact of Reference Point:
In Fig. 3(b), we have consid-ered the reference point of R p = D ( π − c l ) . Next, we furtherillustrate the operator’s optimal sensing decision under anotherhigh reference point of R p = D ( π − c s ) . As stated in SectionV-B, whether an outcome is considered a loss and gain willsignificantly affect the operator’s subjective valuation of theoutcome, hence will affect its sensing decision.In Fig. 3(c), we consider the case of R p = D ( π − c s ) ,which reflects the operator’s expectation of realizing all of thesensing spectrum D . Hence the same outcome is more likelyto be considered as a loss under R p = D ( π − c s ) than under R p = D ( π − c l ) . We plot the optimal sensing decision of therisk-seeking and risk-averse operators for different values of λ and γ . The other system parameters are the same as those inFig. 3(b). By comparing Fig. 3(b) with Fig. 3(c), we observethat the operator with a low reference point R p = D ( π − c l ) senses less. The intuition is that when an operator has a lowreference point, it has a low expectation, so it is more likelyto encounter gains than losses. Due to the concavity of itsvaluation function v ( x − R p ) in gain, the operator will becomemore risk-averse and will sense less to avoid the risk of lowsensing realization.
3) Impact of Probability Distortion:
Then, we illustrate theoperator’s optimal sensing decision under different probability
Probability of Low Sensing Realization p S en s i ng T h r e s ho l d r µ =0.4 µ =0.6 µ =0.8 µ =1 Fig. 4. Sensing threshold r versus probability of low sensing realization p with different probability distortion parameter µ for β = γ = λ = 1 , and R p = D ( π − c l ) .
40 45 50 55 60283440
Maximum Possible Profit E x pe c t ed P r o f i t γ =0 γ =0 γ =0 γ =1 (EUT) c s = 2.2c s = 2.6c s = 3
20 25 30 35 40404244464850
Minimum Possible Profit E x pe c t ed P r o f i t β =0 β =1 β =1 β =1 (EUT)(EUT)(EUT) c s = 1.2c s = 1.6c s = 1.9 Fig. 5. (a) Tradeoff between optimal expected profit and maximum possibleprofit for β = 1 under different c s and γ . (b) Tradeoff between optimalexpected profit and minimum possible profit for γ = 1 under different c s and β . Other parameters are R p = ( π − c l ) D , c l = 4 , π = 8 , and D = 10 . distortion parameters µ for the case of binary sensing out-comes (i.e., I = 2 ). We compare the result with the non-distorted benchmark, where µ = 1 . The PT operator is risk-seeking when the probability of low sensing realization p ishigh, and it is risk-averse when p is low.We notice that the operator’s decisions can be characterizedby a threshold r related to the leasing and sensing cost ratio c l c s .When c l c s ≥ r , meaning that leasing is expensive, the operatorwill choose to sense for all the demand D . Otherwise, when c l c s < r , the operator will sense less than the demand D andlease for part of the demand . Hence, a larger r means thatthe operator is less willing to sense.In Fig. 4, we plot the sensing threshold r against p fordifferent values of µ , where we assume β = γ = λ = 1 , and R p = D ( π − c l ) . We can see that the threshold r decreasesin µ when p < . , and increases in µ when p > . .As a smaller µ means that the operator will overweigh thelow probability more, it becomes more risk-averse when p issmall. Similarly, since a smaller µ means that the operator willunderweigh the high probability more, it is more risk-seekingwhen p is large. B. Expected Profit and Risk Preference Tradeoff
We then evaluate the tradeoff between the expected profitand risk preference of an operator. A risk-seeking operator is Notice that the threshold r is different under different scenarios. Forexample, with R p = D ( π − c l ) for binary sensing outcomes, we have r =[ w ( p ) /w ( p )] β + 1 from Theorem 2. On the other hand, with R p = D ( π − c s ) for binary sensing outcomes, we have r = 1 + w ( p ) /w ( p ) from Theorem 3. aggressive and mainly interested in earning a high maximumprofit, while a risk-averse operator is conservative and mainlyinterested in guaranteeing a high minimum profit. Given thesystem parameters (i.e., c s , c l , π , and D ) and risk perferenceparameters (i.e., λ , β , and γ ), we let B ∗ s and B ∗ l be theoptimal sensing and leasing decisions discussed in SectionIV. The optimal expected profit E α [ R ( B ∗ s , B ∗ l , α )] is the aver-aged profit over different sensing realizations. The maximumpossible profit R ( B ∗ s , B ∗ l , is the profit with α = 1 , andthe minimum possible profit R ( B ∗ s , B ∗ l , is the profit with α = 0 . Tradeoff of a risk-seeking operator:
In Fig. 5(a), we plotthe tradeoff between the expected profit and the maximumpossible profit for β = 1 under different c s and γ . Since anEUT operator ( γ = 1 ) makes decision only by maximizingexpected profit, we can see from Fig. 5(a) that it can achievethe highest expected profit. On the other hand, a PT operatormakes decision by taking into account both the expected profitand its risk preference. More specifically, although a risk-seeking PT operator (i.e., γ < ) achieves a lower expectedprofit comparing to an EUT operator, it can earn a highermaximum possible profit than an EUT operator. Notice inFig. 5(a), when the operator is very risk-seeking ( γ → ), theexpected profit and maximum possible profit both decrease.This is because the operator can achieve the maximum possibleprofit when the sensed spectrum is fully realized. However,when the sensing decision B ∗ s is larger than demand D (hencethe maximum realized spectrum is larger than D), being morerisk-seeking (which leads to a larger sensing decision B ∗ s )will not lead to a larger maximum possible profit, but willonly lead to a larger probability of achieving that maximumpossible profit. This explains the “bending” in the figure.From Fig. 5(a), we also observe that the tradeoff varieswith sensing cost c s . We can see that under the three casesof c s , an EUT operator has the same expected profit andmaximum possible profit, because it has the same optimalsensing decision ( B ∗ s = 0 ). However, a risk-seeking operatorwill have a smaller loss in expected profit but a larger gainin maximum possible profit than an EUT operator ( γ = 1 )when the sensing cost c s decreases. In other words, a risk-seeking operator achieves a better tradeoff when the sensingcost decreases. Tradeoff of a risk-averse operator:
In Fig. 5(b), we plotthe tradeoff between optimal expected profit and minimumpossible profit for γ = 1 under different c s and β . We can seefrom Fig. 5(b) that a risk-averse operator ( β < ) achievesa lower expected profit comparing to an EUT operator ( β =1 ), but guarantees a higher minimum possible profit than anEUT operator. For example, an extremely risk-averse operator( β → ) has a similar expected profit and minimum possibleprofit under any sensing cost c s , because its optimal sensingdecision B ∗ s is always close to zero. We can also observe thata risk-averse operator will have a smaller loss in expectedprofit but a larger gain in minimum possible profit than anEUT operator when the sensing cost c s increases. In otherwords, a risk-averse operator achieves a better tradeoff whenthe sensing cost increases. VII. C
ONCLUSIONS AND F UTURE W ORK
In this paper, we considered a spectrum investment problemwith sensing uncertainty, where an operator decides its spec-trum sensing and leasing decisions by considering both ex-pected profit and its risk preference based on prospect theory.This is the first paper that studied the optimal decisions basedon all three characteristics of prospect theory in the wirelesscommunication literature, and compared and contrasted thesedecisions with those under the more widely used expectedutility theory. Our results suggested that a risk-averse operatorcan achieve a large expected profit while guaranteeing asatisfactory level of minimum possible profit when the sensingcost is high. On the other hand, a risk-seeking operator canachieve both a large expected profit and maximum possibleprofit when the sensing cost is low.This study demonstrated that a more realistic modelingbased on prospect theory is important in understanding the op-erator’s decisions in the wireless industry. On the other hand,this study is only a small first step, as we have only consideredthe operator’s decision in a single time slot. Regarding thefuture work, we will consider a more general problem withdecisions to be made in multiple time slots. In such a model,the operator’s reference point may change over time, and thestudy of dynamic reference point is a recent active researchfield in prospect theory [17], [42]. We will also conduct asurvey to evaluate different people’s risk preferences.R
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IEEE Transactions on Automatic Control , vol. 49, no. 3,pp. 349–360, Mar. 2004. A PPENDIX
A. Proof of Theorem 1
In this proof, we divide the feasible range of B s into threeintervals, (cid:104) , Dα I (cid:105) , (cid:104) Dα I , Dc l c s (cid:105) , and (cid:16) Dc l c s , ∞ (cid:17) , and analyze theoptimal decision B ∗ s in each interval. By this division, in theinterval (cid:104) , Dα I (cid:105) , B s α i ≤ D for all i ∈ I , so that we do notneed to consider the possibility of sensing realization exceedsthe demand. In the interval ( Dc l c s , ∞ ) , we have Dc l < B s c s ,which means that the total cost of sensing B s c s will be largerthan the cost of leasing only Dc l . Hence the optimal solutionwill not be in this range. The above reason leads to the divisionof three intervals.We will use the following notations in the proof: M I (cid:44) I (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β ˆ ı (cid:80) i =1 λγ ( c s − c l α i ) γ (cid:16) α I (cid:17) γ − β w ( p i ) γ − β , (15) M j (cid:44) j (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β − βc s ( c l α j − c s ) β − I (cid:80) i = j +1 w ( p i ) ˆ ı (cid:80) i =1 λγ ( c s − c l α i ) γ (cid:16) α j (cid:17) γ − β w ( p i ) γ − β ,j = ˆ ı +1 , ˆ ı +2 , ..., I − , (16)and H j (cid:44) j (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β − βc s ( c l α j +1 − c s ) β − I (cid:80) i = j +1 w ( p i ) ˆ ı (cid:80) i =1 λγ ( c s − c l α i ) γ (cid:16) α j +1 (cid:17) γ − β w ( p i ) γ − β ,j = ˆ ı +1 , ˆ ı +2 , ..., I − . (17)Next, we analyze the optimal decision B ∗ s within eachinterval. Case I: B s ∈ (cid:104) , Dα I (cid:105) . We first compute the optimaldecision B ∗ s in this interval using the first order condition.In this case, B s α i ≤ D for all i ∈ I , so the optimal leasingdecision B ∗ l = D − B s α i ≥ . From (7), the revenue is R ( B s , B ∗ l , α i ) = ( π − c l ) D − B s c s + B s c l α i . (18)Since α i follows a discrete distribution in [0 , , we can plug(18) into (9), and get (19) by taking the proper expectation. U ( B s ) = I (cid:88) i =ˆ ı +1 ( B s c l α i − B s c s ) β w ( p i ) − ˆ ı (cid:88) i =1 λ ( B s c s − B s c l α i ) γ w ( p i )= I (cid:88) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) B βs − ˆ ı (cid:88) i =1 λ ( c s − c l α i ) γ w ( p i ) B γs . (19) We consider the first order derivative of (19) with respect to B s : U (cid:48) ( B s )= I (cid:88) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) βB β − s − ˆ ı (cid:88) i =1 λγ ( c s − c l α i ) γ w ( p i ) B γ − s = B β − s (cid:34) I (cid:88) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β − ˆ ı (cid:88) i =1 λγ ( c s − c l α i ) γ w ( p i ) B γ − βs (cid:35) . (20)We can obtain U (cid:48) ( B s ) > ⇔ ˆ ı (cid:88) i =1 − λγ ( − c l α i + c s ) γ w ( p i ) B γ − βs + I (cid:88) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β > ⇔ B s < (cid:34) (cid:80) Ii =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β (cid:80) ˆ ıi =1 λγ ( c s − c l α i ) γ w ( p i ) (cid:35) γ − β = M I α I , (21)where M I is defined in (15). Since the second order derivative U (cid:48)(cid:48) ( B s ) < , we know that U (cid:48) ( B s ) in (20) decreases in B s .Hence, we only need to compare the right boundary Dα I andthe critical point M I α I . From (21), we have B ∗ s = M I α I , if M I α I < Dα I ,Dα I , if M I α I ≥ Dα I . (22)Intuitively, when the leasing cost c l is high (and hence every H j in (17) is large), the utility increases when the operatorsenses more (and leases less). Case II : B s ∈ (cid:104) Dα I , Dc l c s (cid:105) . We compute the optimal decision B ∗ s in this interval by capturing the unimodal structure. In thiscase, profit R ( B s , B ∗ l , α ) can be represented as a piecewisefunction as follows. R ( B s , B ∗ l , α ) = πD − B s c s , if α > DB s ,B s c l α − B s c s , if < α < DB s . (23)We substitute (23) into (9), and the utility function becomes U ( B s ) = ˆ ı (cid:88) i =1 − λ [ B s ( c s − c l α i )] γ w ( p i )+ j (cid:88) i =ˆ ı +1 [ B s ( c l α i − c s )] β w ( p i )+ I (cid:88) i = j +1 ( Dc l − B s c s ) β w ( p i ) . (24)The value of j in (24) varies when B s belongs to differentsub-intervals. The utility function U ( B s ) in (24) is a contin-uous function in the whole interval, and is differentiable ineach of the I − ˆ ı sub-intervals, Dα j +1 ≤ B s ≤ Dα j , j = ˆ ı + 1 ,..., I − , and Dα ˆ ı +1 ≤ B s ≤ Dc l c s . Although the utility functionis not globally differentiable, we can evaluate the derivativeof each sub-interval, and find the optimal point for each ofthe I − ˆ ı sub-intervals. Then we can find the optimal solutionof the whole interval by comparing the optimal points in the I − ˆ ı sub-intervals. We conclude the optimal sensing decision B ∗ s,j in theinterval Dα j +1 ≤ B s ≤ Dα j , j = ˆ ı + 1 , ..., I − in Proposition1. Proposition 1.
The maximum utility value U (cid:0) B ∗ s,j (cid:1) in eachsub-interval among Dα j +1 ≤ B s ≤ Dα j , j = ˆ ı + 1 , ..., I − isachieved at B ∗ s,j = Dα j , if D ≤ M j ,g − j (0) , if M j < D < H j ,Dα j +1 , if H j ≤ D < M j +1 . (25) Proof:
We prove Proposition 1 by capturing the unimodalstructure of (24) in each sub-interval Dα j +1 ≤ B s ≤ Dα j . Fora unimodal problem, the optimal point is either at the uniquelocal maximum point or the boundaries. For Dα j +1 ≤ B s ≤ Dα j ,we consider the first order derivative of (24) with respect to B s : U (cid:48) ( B s )= ˆ ı (cid:88) i =1 − λ ( c s − c l α i ) γ w ( p i ) γB γ − s + j (cid:88) i =ˆ ı +1 β ( c l α i − c s ) β w ( p i ) B β − s − c s I (cid:88) i = j +1 β ( Dc l − B s c s ) β − w ( p i )= j (cid:88) i =ˆ ı +1 β ( c l α i − c s ) β w ( p i ) B β − s (cid:34) ˆ ı (cid:80) i =1 − λ ( c s − c l α i ) γ B γ − βsj (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β +1 − c s I (cid:80) i = j +1 (cid:16) Dc l B s − c s (cid:17) β − w ( p i ) j (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) (cid:35) , (26)where we define g j ( B s ) (cid:44) ˆ ı (cid:80) i =1 − λ ( c s − c l α i ) γ B γ − βsj (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β +1 − c s (cid:16) Dc l B s − c s (cid:17) β − I (cid:80) i = j +1 w ( p i ) j (cid:80) i =ˆ ı +1 ( c l α i − c s ) β w ( p i ) , j = ˆ ı +1 , ˆ ı +2 , ..., I − . (27)With (27), we can rewrite U (cid:48) ( B s ) as U (cid:48) ( B s ) = j (cid:88) i =ˆ ı +1 β ( c l α i − c s ) β w ( p i ) B β − s g j ( B s ) , (28)where (cid:80) ji =ˆ ı +1 β ( c l α i − c s ) β w ( p i ) B β − s > . The function g j ( B s ) in (27) is a strictly decreasing function of B s , whichmeans the first order derivative U (cid:48) ( B s ) in (28) will only bezero at most once, thus there is at most one local maximumpoint . If this point is local minimum, then there is no local maximum point,and the maximum point in this sub-interval is at the boundaries.
We then consider the two boundary points: B s = Dα j +1 and B s = Dα j . When B s = Dα j +1 , we have: g j (cid:18) Dα j +1 (cid:19) = (cid:80) ˆ ıi =1 − λ ( − c l α i + c s ) γ (cid:16) Dα j +1 (cid:17) γ − β γw ( p i ) (cid:80) ji =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β +1 − c s (cid:80) Ii = j +1 ( c l α j +1 − c s ) β − w ( p i ) (cid:80) ji =ˆ ı +1 ( c l α i − c s ) β w ( p i ) . (29)We can obtain g j (cid:18) Dα j +1 (cid:19) > ⇔ D < H j . (30) When B s = Dα j , we have: g j (cid:18) Dα j (cid:19) = (cid:80) ˆ ıi =1 − λ ( − c l α i + c s ) γ (cid:16) Dα j (cid:17) γ − β γw ( p i ) (cid:80) ji =ˆ ı +1 ( c l α i − c s ) β w ( p i ) β + 1 − c s (cid:80) Ii = j +1 ( c l α j − c s ) β − w ( p i ) (cid:80) ji =ˆ ı +1 ( c l α i − c s ) β w ( p i ) . (31)We can obtain g j (cid:18) Dα j (cid:19) > ⇔ D < M j . (32)We can see that g j ( B s ) and U (cid:48) ( B s ) in (28) have the samesign within the interval (cid:104) Dα j +1 , Dα j (cid:105) . From (30), we have that U (cid:48) (cid:16) Dα j +1 (cid:17) ≥ when D ≥ H j and U (cid:48) (cid:16) Dα j +1 (cid:17) < when D < H j . From (32), we have that U (cid:48) (cid:16) Dα j (cid:17) ≥ when D ≥ M j and U (cid:48) (cid:16) Dα j (cid:17) < when D < M j . Thus U (cid:48) ( B s ) canbe either negative within the entire interval (cid:104) Dα j +1 , Dα j (cid:105) , orpositive within the entire interval, or first positive and thennegative within that interval based on the value of demandparameters H j and M j .To sum up, the optimal solution B ∗ s,j in each sub-interval B s ∈ (cid:104) Dα j +1 , Dα j (cid:105) , j = ˆ ı + 1 , ..., I − depends on the valueof demand parameters H j and M j as in (25).Then we are going to study the cases at the boundaries ofthe interval (cid:104) Dα I , Dc l c s (cid:105) . The case of left boundary ( B s = Dα I ) isincluded in Proposition 1. When B s > Dα ˆ ı +1 , we can show theutility U ( B s ) in (24) is decreasing in B s , hence the optimal B ∗ s ≤ Dα ˆ ı +1 .We then study the relation between the adjacent sub-intervals. Since (i) U ( B s ) is continuous, (ii) g j ( B s ) >g j − ( B s ) for all j , and (iii) g j ( B s ) is decreasing in B s forall j , we know in (cid:104) Dα I , Dc l c s (cid:105) , B ∗ s = Dα j , if g j (cid:18) Dα j (cid:19) ≥ and g j − (cid:18) Dα j (cid:19) ≤ ,g − j (0) , if g j (cid:18) Dα j (cid:19) < and g j (cid:18) Dα j +1 (cid:19) > . (33) Based on (30), (32) and (33), we can have the followingsummary. B ∗ s = Dα j , if H j ≤ D ≤ M j +1 ,g − j (0) , if M j < D < H j , (34)where j = ˆ ı, ..., I − . Case III: B s ∈ (cid:16) Dc l c s , ∞ (cid:17) . In this case, Dc l − B s c s < .From (7) and (11), we know R ( B s , B ∗ l , α ) − R p = Dc l − B s c s − max { D − B s α, } < , (35)which means the total cost of sensing B s c s will be larger thanthe cost of only using spectrum leasing Dc l . However, theamount of revenue Dπ from sensing or leasing is the same,since the total demand is limited. Hence it is impossible tochoose the optimal B ∗ s in this range to maximize the utility.By summarizing the analysis of the above three cases, weobtain Table II. B. Proof of Corollary 1
From (26), we obtain h λ ( λ ) (cid:44) ∂U (cid:48) ( B s ) ∂λ = ˆ ı (cid:88) i =1 − γw ( p i ) ( c s − c l α i ) γ B γ − s < , (36)and h γ ( γ ) (cid:44) ∂U (cid:48) ( B s ) ∂γ = ˆ ı (cid:88) i =1 − λw ( p i ) ( c s − c l α i ) γ B γ − s + ˆ ı (cid:88) i =1 − λw ( p i ) γ ln ( c s − c l α i ) ( c s − c l α i ) γ − B γ − s ˆ ı (cid:88) i =1 λw ( p i ) γ ln B s ( c s − c l α i ) γ B γ − s < . (37)If λ > λ , then h λ ( λ ) < h λ ( λ ) for every B s , whichmeans U (cid:48) ( B s ) with λ = λ is less than U (cid:48) ( B s ) with λ = λ for every B s , and U (cid:48) ( B s ) with λ = λ will become zero witha smaller B s . Hence B ∗ s decreases in λ .If γ > γ , then h γ ( γ ) < h γ ( γ ) for every B s , whichmeans U (cid:48) ( B s ) with γ = γ is less than U (cid:48) ( B s ) with γ = γ for every B s , and U (cid:48) ( B s ) with γ = γ will be zero with asmaller B s . Hence B ∗ s decreases in γ when D > M I . C. Proof of Corollary 2
From (21), we obtain h β ( β ) (cid:44) ∂U (cid:48) ( B s ) ∂β = I (cid:88) i =ˆ ı +1 w ( p i ) ( c l α i − c s ) β B β − s − I (cid:88) i =ˆ ı +1 λw ( p i ) β ln ( c l α i − c s ) ( c l α i − c s ) β − B β − s I (cid:88) i =ˆ ı +1 λw ( p i ) β ln B s ( c l α i − c s ) β B β − s < . (38) If β > β , then h β ( β ) < h β ( β ) for every B s , whichmeans U (cid:48) ( B s ) with β = β is less than U (cid:48) ( B s ) with β = β for every B s , and U (cid:48) ( B s ) with β = β will be zero with asmaller B s . Hence B ∗ s decreases in β . D. Proof of Corollary 3
From (26), we obtain h c s ( c s ) (cid:44) ∂U (cid:48) ( B s ) ∂c s = ˆ ı (cid:88) i =1 − λγ w ( p i ) ( c s − c l α i ) γ − B γ − s + c s I (cid:88) i = j +1 B s β ( β − w ( p i ) ( Dc l − B s c s ) β − − I (cid:88) i = j +1 β ( Dc l − B s c s ) β − w ( p i ) − j (cid:88) i =ˆ ı +1 β w ( p i ) ( c l α i − c s ) β − B β − s < . (39)If c s > c s , then h c s (cid:0) c s (cid:1) < h c s (cid:0) c s (cid:1) for every B s , whichmeans U (cid:48) ( B s ) with c s = c s is less than U (cid:48) ( B s ) with c s = c s for every B s , and U (cid:48) ( B s ) with c s = c s will be zero with asmaller B s . Hence B ∗ s decreases in c s . E. The EUT Benchmark in Section IV
For comparison, we also consider the operator’s optimalsensing and leasing decisions under the EUT model. Asmentioned in Section III, EUT model is a special case of thePT model with β = 1 , γ = 1 , λ = 1 , and µ = 1 . Underthe EUT model, we can obtain the solution of problem (10)analytically. Theorem 4.
The optimal sensing decision B ∗ s for problem (10)and the optimal leasing decision B ∗ l for problem (5) underEUT are summarized in Table V.Proof: In the proof, we divide the feasible range of B s into two intervals, (cid:104) , Dα (cid:105) , and (cid:104) Dα , ∞ (cid:17) . Case I: B s ≤ Dα . We first compute the optimal decision B ∗ s in this interval. In this case, B s α is not always largerthan D , thus by (9), we can write the expectation of revenue R ( B s , B ∗ l , α ) with respect to α . In each sub-interval ≤ B s ≤ Dα I and Dα j +1 ≤ B s ≤ α j , j = 1 , ..., I − , we canwrite U ( B s ) = E α [ R ( B s , B ∗ l , α )]=( πD − B s c s ) I (cid:88) i = j +1 p i + j (cid:88) i =1 [( π − c l ) D − B s ( c s − c l α i )] p i = πD − j (cid:88) i =1 p i Dc l + (cid:34)(cid:32) j (cid:88) i =1 α i p i (cid:33) c l − c s (cid:35) B s . (40) In the case ≤ B s ≤ Dα I , the utility function U ( B s ) is equivalent to U ( B s ) in (40) with j = I . TABLE VO
PTIMAL S ENSING D ECISION AND L EASING D ECISION UNDER
EUT
Condition Optimal Sensing Decision B ∗ s Optimal Leasing Decision B ∗ lc l c s ≤ (cid:32) I (cid:80) i =1 p i α i (cid:33) − B ∗ s = 0 B ∗ l = D (cid:32) j +1 (cid:80) i =1 p i α i (cid:33) − < c l c s < (cid:32) j (cid:80) i =1 p i α i (cid:33) − for j = 1 , ..., I − B ∗ s = min { Dα j +1 , Dc l c s } B ∗ l =max { , D − α min { Dα j +1 , Dc l c s }} c l c s ≥ ( α p ) − B ∗ s = Dα B ∗ l = D − α Dα From (40), we know that U ( B s ) is continuous and piece-wise linear in B s . Since j (cid:80) i =1 α i p i c l in (41) is increasing in j , we know that there is at most one local maximum point.Hence, the optimal B ∗ s is either at the local maximum pointor the boundaries, depending on the value of c l c s , B ∗ s = , if c l c s ≤ (cid:32) I (cid:88) i =1 α i p i (cid:33) − .Dα j , if (cid:32) j +1 (cid:88) i =1 α i p i (cid:33) − < c l c s < (cid:32) j (cid:88) i =1 α i p i (cid:33) − , j = 1 , , ..., I − ,Dα , if c l c s ≥ ( α p ) − . (41)From the analysis of Case III in Appendix A, we know thatthe optimal B ∗ s ∈ (cid:104) , Dc l c s (cid:105) . By comparing the value of Dc l c s with the optimal B ∗ s in (40), we obtain the results in the firsttwo rows of Table V. Case II: B s ∈ (cid:104) Dα , ∞ (cid:17) .In this case, B s α ≥ D . From (9), we know U ( B s ) = E [ R ( B s , B ∗ l , α )] = πD − B s c s . (42)Since R ( B s ) is decreasing in B s in this case, the optimalsensing decision B ∗ s = Dα , and the corresponding utility U ( B ∗ s ) = πD − Dα c s Since U ( B s ) is continuous, we combine the optimal utilitiesfrom Case I and Cases II, and obtain the optimal B ∗ s withdifferent values of c l c s as in Table V.The results in Table V also depend on the cost ratio c l c s .When c l c s ≤ (cid:16)(cid:80) Ii =1 p i α i (cid:17) − , the leasing is cheap enoughso that the operator will choose to lease only ( B ∗ s = 0 ).For the case c l c s ≥ ( p α ) − (hence leasing is significantlymore expensive), we have B ∗ s = Dα . The threshold value (cid:16)(cid:80) ji =1 p i α i (cid:17) − is based on the distribution of α , as theexpected “effective cost” of getting one unit of idle spectrumthrough sensing is c s (cid:16)(cid:80) ji =1 p i α i (cid:17) − . F. Proof of Theorem 2
From (12), when B s > D , U ( B s ) is decreasing in B s , so B ∗ s ∈ [0 , D ] . Hence, we obtain U ( B s ) = − λw ( p ) ( B s c s ) β + ( c l − c s ) β B βs w ( p )= (cid:104) w ( p ) ( c l − c s ) β − λw ( p ) c sβ (cid:105) B βs . (43) From (43), we find that U ( B s ) is a monotonic function of B s . Hence, we can find the optimal sensing decision B ∗ s = (cid:40) , if w ( p ) ( c l − c s ) β < λw ( p ) c sβ ,D, if w ( p ) ( c l − c s ) β ≥ λw ( p ) c sβ . (44) G. Proof of Theorem 3
From (13) and (14), when B s > D , both U RP H ( B s ) and U RP L ( B s ) are decreasing in B s , so B ∗ s ∈ [0 , D ] . We proveTheorem 3 by capturing the unimodal structure of (13) and(14). For a unimodal problem, the optimal point is either atthe unique local maximum point or the boundaries. We firstcompute the first order derivatives of U RP H and U RP L withrespect to B s : ∂U RPH ( B s ) ∂B s = − λc s w ( p ) β [ B s c s + D ( c l − c s )] β − + λw ( p ) β ( c l − c s ) β [( D − B s )] β − , (45)and ∂U RPL ( B s ) ∂B s = − c s β ( − B s c s + Dc s ) β − w ( p )+ β ( c l − c s ) [ B s ( c l − c s )+ Dc s ] β − w ( p ) . (46)Since the second order derivatives ∂ U RPH ( B s ) ∂B s > and ∂ U RPL ( B s ) ∂B s < , the function ∂U RPH ∂B s is a strictly increasingfunction of B s , and the function ∂U RPL ( B s ) ∂B s is a strictlydecreasing function of B s , which means ∂U RPH ( B s ) ∂B s and ∂U RPL ( B s ) ∂B s will only be zero at most once, thus at most onelocal maximum point for both U RP H ( B s ) and U RP L ( B s ) .We then consider the two boundary points (a) B s = 0 + (cid:15) and (b) B s = D − (cid:15) , with (cid:15) being a small positive numberapproaching zero (i.e., (cid:15) → ), to see if the optimal point isat the local maximum point or at the boundaries.(a) When B s = 0 + (cid:15) , we have: lim (cid:15) → U (cid:48) RPH ( (cid:15) ) = βλ [ D ( c l − c s )] β − [ w ( p )( c l − c s ) − w ( p ) c s ] , (47)and lim (cid:15) → U (cid:48) RPL ( (cid:15) ) = βDc β − s [ − w ( p ) c s + ( c l − c s ) w ( p )] . (48)(b) When B s = D − (cid:15) , we have: lim (cid:15) → U (cid:48) RP H ( D − (cid:15) ) = ∞ , (49) and lim (cid:15) → U (cid:48) RP L ( D − (cid:15) ) = −∞ . (50)We can obtain lim (cid:15) → U (cid:48) RP H ( (cid:15) ) < ⇔ − w ( p ) c s + ( c l − c s ) w ( p ) < ⇔ lim (cid:15) → U RP L ( (cid:15) ) < . (51)Since U (cid:48) RP H ( (cid:15) ) and U (cid:48) RP L ( (cid:15) ) have the same sign from (47)and (48), either U (cid:48) RP H ( B s ) is first negative then positiveand U (cid:48) RP L ( B s ) is all negative within the range (0 , D ) , or U (cid:48) RP H ( B s ) is all positive and U (cid:48) RP L ( B s ) is first positive thennegative within the range (0 , D ) .Since U ( B s ) is continuous in B s ∈ [0 , D ] , the optimalsolution B ∗ s under the two reference points depends on thevalue of c l and c s as in Table IV. Junlin Yu (S’14) is working towards his Ph.D.degree in the Department of Information Engi-neering at the Chinese University of Hong Kong.His research interests include behavioral economicalstudies in wireless communication networks, andoptimization in mobile data trading. He is a studentmember of IEEE.
Man Hon Cheung received the B.Eng. and M.Phil.degrees in Information Engineering from the Chi-nese University of Hong Kong (CUHK) in 2005 and2007, respectively, and the Ph.D. degree in Electricaland Computer Engineering from the University ofBritish Columbia (UBC) in 2012. Currently, he is apostdoctoral fellow in the Department of Informa-tion Engineering in CUHK. He received the IEEEStudent Travel Grant for attending
IEEE ICC 2009 .He was awarded the Graduate Student InternationalResearch Mobility Award by UBC, and the GlobalScholarship Programme for Research Excellence by CUHK. He serves as aTechnical Program Committee member in
IEEE ICC , Globecom , and
WCNC .His research interests include the design and analysis of wireless networkprotocols using optimization theory, game theory, and dynamic programming,with current focus on mobile data offloading, mobile crowd sensing, andnetwork economics.
Jianwei Huang (F’16) is an Associate Professor andDirector of the Network Communications and Eco-nomics Lab (ncel.ie.cuhk.edu.hk), in the Departmentof Information Engineering at the Chinese Univer-sity of Hong Kong. He received the Ph.D. degreefrom Northwestern University in 2005. He is theco-recipient of 8 Best Paper Awards, including IEEEMarconi Prize Paper Award in Wireless Communica-tions in 2011. He has co-authored five books:
Wire-less Network Pricing , Monotonic Optimization inCommunication and Networking Systems , CognitiveMobile Virtual Network Operator Games , Social Cognitive Radio Networks ,and