On the Optimal Duration of Spectrum Leases in Exclusive License Markets with Stochastic Demand
Gourav Saha, Alhussein A. Abouzeid, Zaheer Khan, Marja Matinmikko-Blue
OOn the Optimal Duration of Spectrum Leases inExclusive License Markets with Stochastic Demand
Gourav Saha, Alhussein A. Abouzeid, Zaheer Khan, and Marja Matinmikko-Blue
Abstract —This paper addresses the following question whichis of interest in designing efficient exclusive-use spectrum licensessold through spectrum auctions. Given a system model inwhich customer demand, revenue, and bids of wireless operatorsare characterized by stochastic processes and an operator isinterested in joining the market only if its expected revenue isabove a threshold and the lease duration is below a threshold,what is the optimal lease duration which maximizes the netcustomer demand served by the wireless operators? Increasingor decreasing lease duration has many competing effects; whileshorter lease duration may increase the efficiency of spectrumallocation, longer lease duration may increase market competitionby incentivizing more operators to enter the market. We formu-late this problem as a two-stage Stackelberg game consistingof the regulator and the wireless operators and design efficientalgorithms to find the Stackelberg equilibrium of the entiregame. These algorithms can also be used to find the Stackelbergequilibrium under some generalizations of our model. Usingthese algorithms, we obtain important numerical results andinsights that characterize how the optimal lease duration varieswith respect to market parameters in order to maximize thespectrum utilization. A few of our numerical results are non-intuitive as they suggest that increasing market competition maynot necessarily improve spectrum utilization. To the best of ourknowledge, this paper presents the first mathematical approachto optimize the lease duration of spectrum licenses.
Index Terms —Spectrum license, spectrum auctions, lease du-ration, spectrum utilization, Stackelberg game, Nash equilibrium
I. I
NTRODUCTION
With the rapid growth of wireless services and devices,wireless data traffic is increasing. Cisco’s forecast [2] showsa 6-fold increase in global data traffic from 2017 to 2022.There is only a finite amount of wireless spectrum that canbe used to support the growing wireless data traffic. Thereare various reports [3], [4] that show that many licensedspectrum channels are underutilized, leading to inefficient useof the spectrum. It is widely accepted that the legacy policyof static spectrum allocations is a major cause of inefficientspectrum utilization [5]. Long-term spectrum leases can alsolead to spectrum hoarding [6]. Long-term spectrum leasesare likely to have higher license fees. Higher license feeslead to a lower number of wireless operators in the marketand can also lead to collusion among wireless operators [7].
This material is based upon work supported by the National Science Foun-dation under grant numbers CNS-2007454 and CNS-1456887, FiDiPro Fellowaward from Business Finland (MOSSAF), Academy of Finland 6GenesisFlagship (grant 318927), and partly funded by Infotech Oulu (2018-2021). Apreliminary version of this work appeared in IEEE International Symposiumon Dynamic Spectrum Access Networks (DySPAN) 2018 [1].G. Saha and A. A. Abouzeid are with the Department of Electrical, Computerand Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180,USA; Email: [email protected], [email protected] Khan and M. Matinmikko-Blue are with Centre for Wireless Commu-nications (CWC), University of Oulu, Finland; Email: zaheer.khan@oulu.fi,Marja.Matinmikko@oulu.fi.
This can reduce market competition and hence may leadto inefficient spectrum utilization. For the recently proposedCitizens Broadband Radio Service band [8], the lease durationof Priority Access Licenses (PAL) is an important topic ofdebate. Since potential wireless operators prefer different leaseduration of PALs, it has changed multiple times over the lastfew years of debate; year in [8], years in [9], years in [10]. In spite of the importance of leaseduration, there is no formal study to optimize lease duration(except for our previous work [1]).In this paper, we present a mathematical model to capturethe effect of lease duration on spectrum utilization whenchannels are allocated for exclusive use . Our model canbe summarized as follows. First , the customer demand, therevenue of an operator and its bids are modeled as statisticallycorrelated stochastic processes.
Second , spectrum utilization isequal to the net customer demand served by the operators overa long time horizon.
Third , the revenue of an operator and itsvaluation of channel is solely dependent on the amount ofcustomer demand it can serve using the channel; the morethe customer demand served by the operator, the higher itsrevenue and valuation of the channel.
Fourth , an operator willjoin the market only if the lease duration is below a thresholdso that it can afford the licensing fees and if the lease durationis such that its expected revenue is above a threshold so thatit can generate return sufficient return on its investments.Based on our system model, we formulate an optimizationproblem whose objective is to maximize spectrum utilization.The optimization problem manifests itself as a two-stageStackelberg game consisting of the regulator and the wirelessoperators. The optimization problem, in essence, has only onescalar decision variable, the lease duration. Shorter lease dura-tion increases the frequency of spectrum auctions. Therefore,there is frequent re-allocation of channels to those operatorswho values it the most. This leads to more efficient allocationof spectrum and hence improves spectrum utilization. On theother hand, longer lease duration, in general, ensures thatthe operators get their desired return on investment. Thisincentivizes more operators to join the market and hencelead to more competition which in turn improves spectrumutilization. However, if the lease duration is too long, someof the operators may not join the market because they cannotafford license fees [10]. These opposing factors suggest thatthe optimization problem should have a non-trivial solution.There have been several active areas of research related tospectrum licenses, such as pricing [11], auction design [12],flexible licensing [13], enforcement [14], etc. But, to the bestof our knowledge, a mathematical treatment of the impact oflease duration of spectrum licenses has been only consideredin [15]. In [15], the authors took a data-driven approach and a r X i v : . [ c s . G T ] F e b oncluded that lease duration has no significant impact onspectrum market competition. But there are several workslike [5], [6], [7] that suggest otherwise and also data-drivenapproaches cannot be generalized, especially for extrapolation.Furthermore, higher market competition may not necessarilyimply higher spectrum utilization as we show in section IV.Other than [15], we found no paper that mathematical studiesthe impact of lease duration even in other synergistic areassuch as electricity markets and cloud computing.However, there are a few works in the spectrum sharingliterature that consider the effect of certain “duration aspects”on the overall performance of the system. The work in [16]considers a market of only two service providers with acommon customer base. Time is divided into intervals. At thebeginning of every interval, an auction is conducted whichredistributes the available bandwidth based on the bids ofindividual service providers. The ratio of the customer demandreaching each service provider is governed by evolutionarygame theory. The authors use simulations to conclude thatshorter allocation interval corresponds to better spectrum uti-lization. In [17], the authors model various factors that asecondary service provider considers when buying spectrumresources from primary service providers. The authors design autility function for the secondary service provider that suggestthat longer contract duration is better. In [18], the primaryuser leases its bandwidth to secondary users for a fraction oftime in exchange for cooperation (relaying). If the fraction oftime is too small, it will not compensate for the overall costof transmission (including relaying), and hence the secondaryusers may not agree to cooperate. For opportunistic spectrumuse, optimal spectrum sensing time is an area which receivedwide attention from the spectrum community [19]. There arefew works in economic journals like [20] that consider theproblem of optimizing contract duration for welfare analysis.The fundamental idea governing these works is a tradeoffbetween opportunity cost and transaction cost. The definitionsof transaction cost and opportunity cost change with themarket setting, like housing property market [21], priorityservice market [22], etc.In Section II-A, we present a system model to study theeffect of lease duration on spectrum utilization. Our modelcaptures important properties of the effect of lease durationand market competition on spectrum utilization and an opera-tor’s expected revenue. We also discuss how some of theseproperties are affected by bidding accuracy, the statisticalcorrelation between an operator’s bid and its revenue. Theseproperties are discussed in Section II-C. Up to our knowledge,this is the first system model that enables the mathematicalanalysis of optimal lease duration. This constitutes the firstcontribution of the paper. The work done in this paper extendsto other system models as long as it satisfies the propertiesdiscussed in Section II-C. Few of these generalizations arehypothesized in Section II-C as well.In Section III-A, we capture the interaction between aregulator and the operators as a two-stage Stackelberg gamewith incomplete information . In the first stage, the regulatorsets the lease duration to optimize spectrum utilization. Inthe second stage (subgame), the operators decide whether to enter the market or not based on the lease duration set bythe regulator in the first stage. Our model admits an uniquesubgame Nash equilibrium (NE) and hence finding the Stack-elberg Equilibrium of the two-stage Stackelberg game reducesto solving the optimization problem of the first stage which hasonly one scalar decision variable, the lease duration. Yet, theoptimization problem is not trivial because it is reminiscentof combinatorial optimization. To elaborate, the debate overlease duration of PALs shows that it may not be possible tochoose a lease duration which interests all the operators. Infact, a lease duration which interests all the operators, even if itexists, may not lead to the optimal spectrum utilization. Hence,in certain sense, we want to find the optimal set of interestedoperators which is a combinatorial optimization problem. Theformulation of the Stackelberg game is the second contributionof the paper.In Section III, we design algorithms to solve the opti-mization problem of the first stage for two scenarios: (i)homogeneous market with complete information, (ii) het-erogeneous market with incomplete information. Since theoptimization problem has a combinatorial nature, the numberof candidate sets of interested operators may be exponentialin the number of operators in the market. The design of an efficient optimization algorithm relies on the result that, withlease duration as the decision variable, the number of candidatesets of interested operators is polynomial in the number ofoperators in the market. Designing an efficient optimizationalgorithm for the first stage game is the third contributionof the paper. The final contribution is the numerical resultspresented in Section IV. We use our optimization algorithmto numerically study the variation of optimal characteristics,i.e., optimal lease duration and optimal value of the objectivefunction, as a function of market parameters. We also studyhow bidding accuracy and incomplete information decreasesspectrum utilization. Few of our numerical results are non-intuitive as they suggest that increasing market competitionmay not necessarily improve spectrum utilization.II. P ROBLEM F ORMULATION
We present our system model in Section II-A and alsointroduce the revenue function and the objective function,which capture the revenue of an operator and spectrum uti-lization, respectively. The expressions of the revenue and theobjective function are derived in Section II-B. The propertiesof the objective and the revenue function are discussed inSection II-C which reveal their practical relevance. We alsodiscuss few generalizations of our system model in Sec-tion II-C. Table I lists frequently used notations while othernotations are standard.
A. System Model
We consider a time slotted model where t ∈ Z + is a timeslot. Let T ∈ Z + denote the lease duration. The word epoch denotes a lease duration. Hence, the time slots correspondingto the c th epoch are t ∈ [( c − T + 1 , cT ] where c ∈ Z + .There are N operators indexed k = 1 , , . . . , N . Let S ⊆{ , , . . . , N } be the set of operators who are interested inentering the market. In our model, the number of interested able IA TABLE OF FREQUENTLY USED NOTATIONS . Notation Description Z + Set of positive integers. (cid:100) x (cid:101) Ceiling Function. T Lease duration. N Number of operators. M Number of channels. x k ( t ) Revenue of the k th operator in t th time slot. Y k ( c, T ) Net revenue of the k th operator in c th epoch if leaseduration is T . (cid:98) Y k ( c, T ) Bid of the k th operator in c th epoch if lease duration is T . µ k , (cid:98) µ k True and estimated mean respectively of the revenue processof the k th operator. σ k , (cid:98) σ k True and estimated standard deviation respectively of therevenue process of the k th operator. a k , (cid:98) a k True and estimated autocorrelation coefficient respectively ofthe revenue process of the k th operator. ρ k , (cid:98) ρ k True and estimated bid correlation coefficient respectively ofthe k th operator. λ k , (cid:98) λ k True and estimated minimum expected revenue (MER) re-quirement respectively of the k th operator. Λ k , (cid:98) Λ k True and estimated maximum lease duration respectivelyabove which the k th operator cannot afford a channel. ξ k , (cid:98) ξ k Tuples representing the true and estimated parameters of the k th operator resp. We have, ξ k = ( µ k , σ k , a k , ρ k , λ k , Λ k ) and (cid:98) ξ k = (cid:16)(cid:98) µ k , (cid:98) σ k , (cid:98) a k , (cid:98) ρ k , (cid:98) λ k , (cid:98) Λ k (cid:17) . S Set of interested operators. s Number of interested operators. We have s = |S| . (cid:101) S Set of interested operators according to the regulator. S Lk Largest set of interested operators who may join the marketaccording to the k th operator. (cid:101) S L Largest set of interested operators according to the regulator. R k ( S , T ) Revenue function of the k th operator. (cid:98) R k ( S , T ) Revenue function of the k th operator as perceived by itself. (cid:101) R k ( S , T ) Revenue function of the k th operator as perceived by theregulator. U ( S , T ) Objective function as a function of set of interested operators S and lease duration T . (cid:101) U ( S , T ) Objective function as perceived by the regulator as a functionof set of interested operators S and lease duration T . U ( T ) Objective function as a function of lease duration T . (cid:101) U ( T ) Objective function as perceived by the regulator as a functionof lease duration T . R ( s, T ) Revenue function of an operator for a market that is homo-geneous in µ k , σ k , a k and ρ k . U ( s, T ) Objective function as a function of number of interestedoperators s and lease duration T . It only applies for a marketthat is homogeneous in µ k , σ k , a k and ρ k . operators, s = |S| , is the measure of market competition [15].There are M channels indexed m = 1 , , . . . , M which areto be allocated to the operators in set S , at the beginning ofevery epoch, for exclusive use. Similar to prior works like [23],[24], these channels are assumed to be identical. Our modelassumes spectrum cap of one, i.e. an operator is allocated atmost one channel in every epoch.Let x k ( t ) denonte the revenue of the k th operator at timeslot t if it is allocated a channel. The revenue of an operatoris if it is not allocated a channel. We model the revenue x k ( t ) as a first order Gaussian Autoregressive (AR) process.Modeling time-series with AR models is a common practice inacademic literature [25], [26]. Federal Communication Com-mission report [9] expresses the need for “periodic, market-based reassignment of channels in response to changes in localconditions and operator needs.” A first order AR process isa simple stochastic process capturing autocorrelation among time series data. We can model fast (slow) “changes in localconditions and operator needs” by setting a lower (higher)autocorrelation among x k ( t ) . Mathematically, x k ( t + 1) = a k x k ( t ) + ε k ( t ) ; ∀ t ≥ (1)where a k ∈ [0 , is the autocorrelation coefficient, ε k ( t ) isan iid Gaussian random process with mean µ εk and standarddeviation σ εk , i.e. ε k ( t ) ∼ N ( µ εk , σ εk ) , ∀ t , and x k (1) is agaussian random variable with mean µ k and standard deviation σ k , i.e. x k (1) ∼ N ( µ k , σ k ) where µ k = µ εk − a k ; σ k = σ εk (cid:112) − a k (2)It can be shown that x k ( t ) is a stationary Gaussian randomprocess [27] with mean µ k and standard deviation σ k , i.e. x k ( t ) ∼ N ( µ k , σ k ) , ∀ t . It should be noted that x k ( t ) ,as given by (1), can become negative for some t . Thishowever is not practical because revenue is always positive.We can reduce the probability of x k ( t ) becoming negativeby setting a low coefficient of variation σ k µ k . Mathematically, P [ x k ( t ) <
0] = (cid:16) erf (cid:16) − µ k √ σ k (cid:17)(cid:17) . So if σ k µ k ≤ . , P [ x k ( t ) < ≤ . . This model is similar to other ap-proaches for modeling non-negative quantities by Gaussianprocesses for ease of analysis, e.g. [28], [29]. Proposition 1.
Let Y k ( c, T ) = cT (cid:80) t =( c − T +1 x k ( t ) be the netrevenue of the k th operator in c th epoch if it is allocateda channel and the lease duration is T . Then, Y k ( c, T ) is agaussian random variable with mean, (cid:101) µ k ( T ) , and standarddeviation deviation, (cid:101) σ k ( T ) , where (cid:101) µ k ( T ) = µ k T (3) (cid:101) σ k ( T ) = (cid:113) T − a k (cid:0) − a Tk + a k T (cid:1) (1 − a k ) σ k (4) Mathematically, Y k ( c, T ) ∼ N (cid:0)(cid:101) µ k ( T ) , (cid:101) σ k ( T ) (cid:1) ; ∀ c .Proof: Please refer to Appendix A for the proof.Channels are allocated through auctions in every epoch. Thereare M channels to be allocated and the spectrum cap isone. In a given epoch, the regulator allocates the channelsto the wireless operators with the M highest bids in thatepoch. Let (cid:98) Y k ( c, T ) denote the bid of the k th operator in c th epoch if the lease duration is T . Our model assumestruthful spectrum auctions. Therefore, the bid of an operatoris equal to its valuation of a channel. An operator’s valuationof a channel is equal to its revenue in an epoch if it isallocated the channel. Our model does not account for thestrategic value of a spectrum as discussed in [30] which arisesbecause “markets are not fully competitive, and there is valuein controlling access to that market.” To this end we have, (cid:98) Y k ( c, T ) = Y k ( c, T ) . But this is true only if during bidding,an operator exactly knows the true net revenue it will earn inthat epoch. In reality, the bid (cid:98) Y k ( c, T ) is only has an estimateof the true revenue Y k ( c, T ) . In our model, (cid:98) Y k ( c, T ) and Y k ( c, T ) assumes the following joint probability distribution, (cid:20) Y k ( c, T ) (cid:98) Y k ( c, T ) (cid:21) ∼ N (cid:18)(cid:20)(cid:101) µ k ( T ) (cid:101) µ k ( T ) (cid:21) , (cid:20) (cid:101) σ k ( T ) ρ k (cid:101) σ k ( T ) ρ k (cid:101) σ k ( T ) (cid:101) σ k ( T ) (cid:21)(cid:19) ; ∀ c (5)here ρ k ∈ [0 , is the correlation coefficient between bid (cid:98) Y k ( c, T ) and true revenue Y k ( c, T ) . A higher ρ k implies ahigher accuracy of bidding estimate. Also note that in (5), thebid (cid:98) Y k ( c, T ) has the same marginal distribution as Y k ( c, T ) , (cid:98) Y k ( c, T ) ∼ N (cid:0)(cid:101) µ k ( T ) , (cid:101) σ k ( T ) (cid:1) ; ∀ c (6)In our model, an operator generates revenue solely byserving customer demand. Let d k ( t ) denonte the customerdemand served by the k th operator at time slot t if it isallocated a channel. An operator’s revenue in time slot t is x k ( t ) = p k ( t ) d k ( t ) . In a competitive market, the price p k ( t ) charged by an operator to serve a unit of customerdemand cannot vary significantly with operators. Otherwise,the operator may suffer a significant loss of its market share.Hence, our model assumes that p k ( t ) = p ( t ) , ∀ k . Similarresults are shown in [31]. In fact, the famous Bertrand andCournot competition models [32], [33] suggest that with twoor more operators, the market reaches perfect competition andall operators sell at the same price. So we have, x k ( t ) = p ( t ) d k ( t ) . This implies that, in our model, an operator whois generating more revenue is also utilizing the spectrum betteras it is serving more customer demand. The results presentedin this paper may not hold if p k ( t ) varies significantly amongoperators, e.g. when operators have strategic valuation ofspectrum because such markets are not competitive.An operator has to invest in infrastructure development toenter the market and further invest to lease a channel. Sincethe cost of leasing a channel generally increases with leaseduration, some operators cannot afford to lease a channel ifthe lease duration is too high [10]. This is captured in ourmodel using Λ k , the maximum lease duration above whichthe k th operator cannot afford a channel. In order to get returnon infrastructure development cost and the cost of leasing achannel, the k th operator wants to make a minimum expectedrevenue (MER) λ k in an epoch. The k th operator is interestedin entering the market iff the lease duration is less than Λ k and its expected revenue in an epoch is greater than λ k .If k ∈ S , then R k ( S , T ) is the revenue function of the k th operator and it denotes its expected revenue in an epochif the set of interested operators is S and the lease durationis T . In our model, the objective function is U ( S , T ) whichis proportional to the spectrum utilization when the set ofinterested operators is S and the lease duration is T . We deriveexpressions for R k ( S , T ) and U ( S , T ) in the next section. B. Analytical Expressions of Revenue and Objective Function
We start by introducing few notions and notations which arerequired for the derivation of revenue and objective function.Let w mc denote the index of the operator who is allocatedthe m th channel in c th epoch. Without any loss of generalitylet us assume that w mc is decided by the following rule: m th channel is allocated to the operator having the m th highestvalue of (cid:98) Y k ( c, T ) in the c th epoch. The number of channelsbeing allocated is (cid:102) M = min ( M, s ) . The revenue function R k ( S , T ) can be expressed as R k ( S , T ) = (cid:102) M (cid:88) m =1 E [ Y k ( c, T ) | w mc = k ] P [ w mc = k ] + 0 · P (cid:34) (cid:102) M (cid:84) m =1 w mc (cid:54) = k (cid:35) (7) = (cid:102) M (cid:88) m =1 E [ Y k (1 , T ) | w m = k ] P [ w m = k ] (8)In (7), P [ w mc = k ] is the probability that the k th operator isallocated the m th channel in the c th epoch in which case its netexpected revenue is E [ Y k ( c, T ) | w mc = k ] . P (cid:34) (cid:102) M (cid:84) m =1 w mc (cid:54) = k (cid:35) is the probability that the k th operator is not allocated achannel in the c th epoch in which case its revenue is .In (7), w mc is dependent on the random variable (cid:98) Y k ( c, T ) .This shows that Y k ( c, T ) and (cid:98) Y k ( c, T ) are the only randomvariables in (7). Hence, the expectation in (7) is over Y k ( c, T ) and (cid:98) Y k ( c, T ) . Based on Proposition 1 and (6), statisticalproperties of Y k ( c, T ) and (cid:98) Y k ( c, T ) are not dependent onepoch c . Therefore, the expectation in (7) and hence therevenue function does not depend on c . This means that we cansimply substitute c = 1 to get (8). Please note that R k ( S , T ) as given by (8) is a function of µ k , σ k , a k and ρ k because thestatistical properties of Y k ( c, T ) and (cid:98) Y k ( c, T ) if governed by(5) which in turn depends on µ k , σ k , a k and ρ k . Equation 8 isenough for the remaining discussion in the paper. However, wehave derived a more explicit equation to calculate R k ( S , T ) in Appendix B. This equation involves numerical integration.In this paper, we consider a scenario where the regulatorwants to maximize the expected spectrum utilization. Asdiscussed in Section II-A, in our model, an operator who gen-erates more revenue also utilizes the spectrum better. There-fore, maximizing expected spectrum utilization is equivalentto maximizing the net expected revenue V in optimizationhorizon T (cid:29) T . Assume that T is a multiple of T , i.e T = CT where C ∈ Z + . We have, V = C (cid:80) c =1 E (cid:34) (cid:102) M (cid:80) m =1 Y w mc ( c, T ) (cid:35) (9)In (9), E (cid:34) (cid:102) M (cid:80) m =1 Y w mc ( c, T ) (cid:35) denotes the net expected rev-enue in the c th epoch over all the (cid:102) M = min ( M, s ) allocatedchannels. Similar to (7), the expectation in (9) is over ran-dom variables Y k ( c, T ) and (cid:98) Y k ( c, T ) and hence the term E (cid:34) (cid:102) M (cid:80) m =1 Y w mc ( c, T ) (cid:35) is not dependent on epoch c . In otherwords, the net expected revenue is equal in all epochs. Hence,(9) can be simplified to V = CE (cid:34) (cid:102) M (cid:80) m =1 Y w m (1 , T ) (cid:35) = T T E (cid:34) (cid:102) M (cid:80) m =1 Y w m (1 , T ) (cid:35) (10)Maximizing V in (10) is equivalent to maximizing E (cid:34) (cid:102) M (cid:80) m =1 Y wm (1 ,T ) (cid:35) T . This holds even if T is not a multiple of T provided T (cid:29) T . Finally, the regulator wants to maximize U ( S , T ) = E (cid:34) (cid:102) M (cid:80) m =1 Y w m (1 , T ) (cid:35) T (11) (cid:102) M (cid:80) m =1 (cid:80) k ∈S E [ Y k (1 , T ) | w m = k ] P [ w m = k ] T (12) = 1 T (cid:88) k ∈S R k ( S , T ) (13)Equation 12 is obtained by first applying linearity of ex-pectation and then applying law of iterated expectation overall possible w m in (11). To obtain (13), we change the orderof summation and then observe that R k ( S , T ) is equal to (cid:102) M (cid:80) m =1 E [ Y k (1 , T ) | w m = k ] P [ w m = k ] (refer to (8)).We end this section by defining two new notations. Let R ( s, T ) and U ( s, T ) denote the revenue and the objectivefunction respectively if the market is homogeneous in µ k , σ k , a k and ρ k , i.e. µ k = µ , σ k = σ , a k = a and ρ k = ρ , ∀ k .In such a market, the revenue and the objective functiononly depend on the number of interested operators. Also,the revenue function is the same for all the operators. Theformula for R ( s, T ) is derived in Appendix C. The objectivefunction U ( s, T ) is obtained by substituting R k ( S , T ) = R ( s, T ) , ∀ k in (13) which yields, U ( s, T ) = sT R ( s, T ) (14) C. Properties of the Revenue and the Objective Function
In this section, we discuss few properties of the revenueand the objective function that are crucial for formulating andsolving the Stackelberg game in Section III-A and III. We alsodiscuss a few generalizations of the system model proposedin Section II-A under which these properties of the revenueand the objective function should remain valid. R k ( S , T ) and U ( S , T ) have the following properties. Property 1. R k ( S , T ) is unimodal in T with a maximum. Property 2. R k ( S , T ) is monotonic decreasing in S , i.e. R k ( S , T ) ≥ R k ( S (cid:83) { a } , T ) where a / ∈ S . Property 3. U ( S , T ) is monotonic increasing in T or it isunimodal in T with a minimum. Consider an operator indexed a , where a / ∈ S , whose bidcorrelation coefficient is ρ a . Property 4. As ρ a → , U ( S (cid:83) { a } , T ) ≥ U ( S , T ) . As ρ a decreases, U ( S (cid:83) { a } , T ) decreases. Figure 1 is a pictorial representation of these properties.According to Property 1, the revenue function is unimodal in T with a maximum. This is shown in Figure 1.a where boththe blue and the green curves first increase with T and thenstart decreasing after a certain value of lease duration. This canbe qualitatively explained as follows. The increase of revenuefunction with T simply happens because an operator can earnmore revenue if it has the channel for a longer duration.However, the decrease in revenue function with T is non-intuitive . This can be explained as follows. The coefficient ofvariation of bids (cid:98) Y k ( c, T ) is (cid:101) σ k ( T ) (cid:101) µ k ( T ) (refer to (6), (3) and (4)). (a) (c)(b) (d)(e) NOTPOSSIBLE (f) (g)
Figure 1. (a) A typical trend of R k ( S , T ) with respect to T and S in aheterogeneous market. (b) A typical trend of R ( s, T ) with respect to T and s in a homogeneous market. (c, d, and e) A typical trend of U ( S , T ) withrespect to T in a homogeneous market (c) and in a heterogeneous marketwith high ρ k (c), mid-range ρ k (d) and low ρ k (e). (f) A trend of U ( S , T ) which is not possible in any market scenario. (g) A typical trend of U ( S , T ) with respect to S . The black and the blue curves corresponds to objectivefunction, U ( { , , } , T ) , if bid correlation coefficient of operator is ρ l and ρ h respectively where ρ h > ρ l . As T increases, coefficient of variation of (cid:98) Y k ( c, T ) , (cid:101) σ k ( T ) (cid:101) µ k ( T ) ,tends to zero and hence (cid:98) Y k ( c, T ) → (cid:101) µ k ( T ) = µ k T . In ourmodel, those operators with high bids (cid:98) Y k ( c, T ) are allocatedchannels in the c th epoch. Since (cid:98) Y k ( c, T ) is approximatelyequal to µ k T for large T , operators with low µ k are notlikely to be allocated channels as T increases. This leads toa decrease in their revenue function. This result shows thatnot all the operators would prefer a long lease duration. Fora heterogeneous market, we could only verify Property 1 nu-merically. But for a homogeneous market, we could rigorouslyprove that revenue function R ( s, T ) is monotonic increasingin T (special case of unimodal function with maximum atinfinity). This is shown in Figure 1.b where both the blueand the green curves increase as T increases. Please refer toAppendix D for the proof.According to Property 2, the revenue function decreasesas the set of interested operators S grows. This is shown inFigure 1.a and 1.b where the blue curve is below the greencurve. As S grows, an operator competes with more operatorsto get a channel. Hence, the probability of an operator beingallocated a channel decreases which in turn decreases itsrevenue function. We have verified Property 2 numerically.According to Property 3, the objective function U ( S , T ) isunimodal in T with a minimum or it is monotonic increasingin T . This happens because of two competing causes. First ,as lease duration increases, the regulator is allocating thechannels to the operators with the best spectrum utilization lessoften. This reduces the efficiency of spectrum allocation andhence spectrum utilization decreases.
Second , due to biddinginaccuracy an operator may place a lower bid and hence notallocated a channel even though it has a high net revenuein an epoch (and hence a higher spectrum utilization). Thechances of such inefficient spectrum allocation increases aslease duration decreases because spectrum auctions occursmore frequently and hence more the chances of such erroneousspectrum allocation. As bidding coefficient ρ k decreases, thebidding inaccuracy increases making the second cause moredominating than the first one. Therefore, the objective functions monotonic non-increasing (special case of unimodal func-tion with minimum at infinity) in T if ρ k is high (Figure 1.c),unimodal in T with a minimum if ρ k is mid-range (Figure1.d) and monotonic non-decreasing in T if ρ k is low (Figure1.e). However, the objective function will never be unimodalin T with a maximum (Figure 1.f). For a heterogeneousmarket, we could only verify Property 3 numerically. But for ahomogeneous market, we could rigorously prove that objectivefunction U ( s, T ) is monotonic decreasing in T (Figure 1.c)for any value of ρ k . Please refer to Appendix D for the proof.Property 4 discusses the effect of a set of interestedoperators on objective function and how it changes withbid correlation coefficient. According to Property 4, as ρ a → , U ( S (cid:83) { a } , T ) ≥ U ( S , T ) ; but as ρ a decreases, U ( S (cid:83) { a } , T ) decreases as well. According to our model,market competition increases as the set of interested oper-ators grows from S to S (cid:83) { a } . Qualitatively, property 4states that increasing market competition may have two out-comes: (a) Increase in spectrum utilization if bid correlationis high: as ρ a → , there is a value of ρ a above which U ( S (cid:83) { a } , T ) ≥ U ( S , T ) implying that the increase incompetition due to addition of operator a lead to an increasein spectrum utilization. This is shown in Figure 1.g wherethe blue curve is above the green curve. (b) Decrease inspectrum utilization if bid correlation is low: as ρ a decreases,the bid of operator a is not a good estimate of its truenet revenue which in turn is proportional to operator a ’sspectrum utilization. Therefore, with decrease in ρ a , thereis a higher probability of erroneous spectrum allocation tooperator a when its spectrum utilization is low. In other words,spectrum allocation becomes less efficient with decrease in ρ a which in turn decreases objective function U ( S (cid:83) { a } , T ) .In some cases, the decrease in spectrum utilization may belarge enough that U ( S (cid:83) { a } , T ) < U ( S , T ) . This is non-intuitive as it suggests that increasing market competitionmay not necessarily improve spectrum utilization. This isshown in Figure 1.g where the black curve is below thegreen curve. For a heterogeneous market, we could only verifyProperty 4 numerically. But for a homogeneous market, wecould rigorously prove that objective function U ( s, T ) ismonotonic increasing in s for any value of ρ k . Please referto Appendix D for the proof.We end this section by stressing that the results in thesubsequent sections remain valid as long as Properties 1-3hold. We believe that these properties are robust and holdunder various generalizations of our system model. Two suchgeneralizations are as follows. First , we can relax our systemmodel such that an operator can be allocated more than onechannel.
Second , we can generalize the revenue process givenby (1) to other stochastic processes. An interesting generaliza-tion is to consider cross-correlation among operators’ revenueprocesses. Such generalizations may not guarantee closed-form expressions of the revenue function, in which case, itcan be estimated using
Monte-Carlo simulations .III. S
TACKELBERG G AME F ORMULATION AND S OLUTION
In this section, we formulate the problem as a Stackel-berg game in in Section III-A. We solve Stage-2 of the Stackelberg game and use the result to formulate Stage-1of the Stackelberg game as an optimization problem OP .We then design algorithms to solve OP for two marketsettings: (a) homogeneous market with complete informationin Section III-B (Proposition 3), and (b) heterogeneous marketwith incomplete information in Section III-C (Algorithm 1). A. Stackelberg game formulation
In this section, we formulate the optimization problem fromthe regulator’s and the operators’ perspective. The optimizationproblem manifests itself in the form of a two-stage Stackelberggame. In Stage-1, the regulator sets the lease duration to max-imize spectrum utilization. The payoff of an operator dependson the lease duration. In Stage-2, the operators decide whetherto join the market or not depending on the decision whichmaximizes its payoff. To make decisions the regulator needsinformation about the operators and the operators need infor-mation about other operators in the market. The k th operatorcan be completely characterized by six parameters which canbe represented using the tuple ξ k = ( µ k , σ k , a k , ρ k , λ k , Λ k ) .Only the k th operator knows the true value of these six pa-rameters. To model incomplete information games , we assumethe regulator and other operators in the market only has apoint estimate of the k th operator’s parameters [34]. Let theestimate be (cid:98) ξ k = (cid:16)(cid:98) µ k , (cid:98) σ k , (cid:98) a k , (cid:98) ρ k , (cid:98) λ k , (cid:98) Λ k (cid:17) . Please note thatfor simplicity, we have assumed that the entire market hasone common estimate of k th operator’s parameters. This canbe easily generalized where the regulator and the operatorshave different estimates of k th operator’s parameters.Stackelberg equilibrium of a Stackelberg game can be foundusing backward induction [35]. To apply backward induction,we start with Stage-2 and analyze the operators’ decision strat-egy given a lease duration. Then we solve for Stage-1 wherethe regulator decides the lease duration knowing the possibleresponse of the operators to the lease duration. Consider Stage-2 of the game, also referred as the subgame. The outcome ofthis process is the function S ( T ) which characterizes the setof interested operators as a function of lease duration T . Anoperators decision to enter the market depends on its revenuefunction R k ( S , T ) . R k ( S , T ) is the true revenue functionof the k th operator. To compute R k ( S , T ) , the k th operatorneeds to know ξ j of all the operators in the market whichit does not. Let (cid:98) R k ( S , T ) denotes the revenue function ofthe k th operator as perceived by the k th operator due toincomplete information scenario. To compute (cid:98) R k ( S , T ) , the k th operator uses its true parameters ξ k and estimates (cid:98) ξ j ,where j (cid:54) = k , of other operator’s parameters. If T > Λ k ,then the k th operator will definitely not enter the market. If T ≤ Λ k , then the payoff of the k th operator is π k ( X ) = (cid:98) R k ( X , T ) − λ k (15)if it enters the market where X is the set of operators whodecided to enter the market and k ∈ X . Payoff of the k th operator is if it does not enter the market. With (15) as thepayoff function, the subgame can have multiple pure strategyNash equilibria. For example, consider a complete informationgame which is a subset of incomplete information game with (cid:98) ξ k = ξ k ; ∀ k . There is M = 1 channel and N = 2 operators.arameters ξ k of these operators are µ k = µ , λ k = λ , Λ k = ∞ ; k = 1 , . If the lease duration T = λµ , then (cid:98) R k ( { k } , T ) = µT = λ ; k = 1 , and (cid:98) R k ( { , } , T ) < µT = λ ; k = 1 , (due to Property 2). Hence, there are two pure strategy Nashequilibria; Operator enters the market while Operator doesnot and vice-versa.If the subgame has multiple pure strategy Nash equilibria,then Stage-1 will have multiple optimal solutions for leaseduration each corresponding to one of the NE of Stage-1. Inother words, for a given set of market parameters, there canbe multiple optimal lease durations. In order to simplify theanalysis, we consider a setting where the subgame has a uniquepure strategy NE. In one such setting, an operator is interestedin maximizing its minimum payoff. The obtained NE is called Max-Min NE and has been considered in previous works like[36]. According to Property 2, payoff decreases as the set X increases. So the minimum payoff corresponds to the largest X . The largest X is composed of those operators who may join the market. The k th operator will definitely not join themarket if either T > Λ k or µ k T < λ k . The inequality T > Λ k is obvious. To appreciate the inequality µ k T < λ k , note thatthe maximum payoff of the k th operator is µ k T − λ k whichhappens when it is alone in the market, i.e. X = { k } and hence (cid:98) R k ( { k } , T ) = µ k T . This is due to Property 2. If µ k T < λ k ,then the payoff of the k th operator is negative if it enters themarket, irrespective of the decision of other operators. Hence,its dominant strategy is not to enter the market. To conclude,the k th operator may join the market if and only if T ≤ Λ k and µ k T ≥ λ k . But the k th operator does not know the truevalue of µ j , λ j and Λ j if j (cid:54) = k . Therefore, the largest setof interested operators who may join the market, according tothe k th operator, for lease duration T is S Lk ( T ) = (cid:110) j : j = k , T ≤ Λ j , µ j T ≥ λ j or j (cid:54) = k , T ≤ (cid:98) Λ j , (cid:98) µ j T ≥ (cid:98) λ j (cid:111) (16)To this end we can conclude that according to the k th operator, its minimum payoff is (cid:98) R k (cid:0) S Lk ( T ) , T (cid:1) − λ k . Thisleads to the following proposition. Proposition 2.
The subgame has a unique Max-Min Nashequilibrium which is given by the set of interested operators S ( T ) = (cid:110) k : T ≤ Λ k , (cid:98) R k (cid:0) S Lk ( T ) , T (cid:1) ≥ λ k (cid:111) (17)Equation 17 is the solution (true) of the subgame. With slightabuse of notation let U ( T ) = U ( S ( T ) , T ) (18) U ( T ) is the true objective function because it depends onthe true solution of the subgame, S ( T ) , and also becausecomputation of U ( S , T ) is based only on the true parameters ξ j . In Stage-1, the regulator wants to maximize U ( T ) . Hence,to calculate U ( T ) , the regulator needs to know ξ j of allthe operators. But the regulator does not know ξ j of any operator; it only knows the estimates (cid:98) ξ j . Let (cid:101) R k ( S , T ) and (cid:101) U ( S , T ) = T (cid:80) k ∈ S (cid:101) R k ( S, T ) denote the revenue function ofthe k th operator and the objective function as perceived by Figure 2. A typical plot of objective function (true) U ( T ) , perceived objectivefunction (cid:101) U ( T ) , number of interested operators (true) s ( T ) = |S ( T ) | , andthe perceived number of interested operators (cid:101) s ( T ) = (cid:12)(cid:12)(cid:12) (cid:101) S ( T ) (cid:12)(cid:12)(cid:12) . For this plot, N = 10 , M = 2 and the true parameters ξ k are chosen uniformly at random.The estimated parameters (cid:98) ξ k are chosen uniformly at random such that theyare within ± of the true parameters. the regulator respectively. Unlike (cid:98) R k ( S , T ) , (cid:101) R k ( S , T ) iscalculated only based on estimates (cid:98) ξ j . The largest set ofinterested operators as perceived by the regulator, (cid:101) S L ( T ) , andthe perceived set of interested operators, (cid:101) S ( T ) , is given by (cid:101) S L ( T ) = (cid:110) k : T ≤ (cid:98) Λ k , (cid:98) µ k T ≥ (cid:98) λ k (cid:111) (19) (cid:101) S ( T ) = (cid:110) k : T ≤ (cid:98) Λ k , (cid:101) R k (cid:16) (cid:101) S L ( T ) , T (cid:17) ≥ (cid:98) λ k (cid:111) (20)Note that (cid:101) S ( T ) ⊆ (cid:101) S L ( T ) because (cid:101) R k (cid:16) (cid:101) S L ( T ) , T (cid:17) ≤ (cid:101) R k ( { k } , T ) = (cid:98) µ k T (Property 2). With slight abuse ofnotation let (cid:101) U ( T ) = (cid:101) U (cid:16) (cid:101) S ( T ) , T (cid:17) be the perceived objec-tive function. In Stage-1, the regulator solves the followingoptimization problem OP (cid:26) max T ∈ Z + (cid:101) U ( T ) = (cid:101) U (cid:16) (cid:101) S ( T ) , T (cid:17) The regulator chooses lease duration T to maximize theperceived objective function (cid:101) U ( T ) . The perceived objectivefunction and the objective function (true) U ( T ) may not beequal as shown in Figure 2. Therefore, a lease duration whichmaximizes the perceived objective function may not maximizethe objective function. This leads to sub-optimal spectrumutilization due to incomplete information games.The perceived set of interested operators, (cid:101) S ( T ) , can be implicitly controlled by choosing a suitable T . A significantpart of solving OP is to find an (cid:101) S ( T ) that maximizes (cid:101) U ( T ) . The number of combinations of (cid:101) S ( T ) can be expo-nential in N . Therefore, OP is reminiscent of combinatorialoptimization which makes it difficult to solve even thoughit is a scalar optimization problem in T . A typical plotof objective function (cid:101) U ( T ) and the perceived number ofinterested operators (cid:101) s ( T ) = (cid:12)(cid:12)(cid:12) (cid:101) S ( T ) (cid:12)(cid:12)(cid:12) is shown in Figure 2(black curve). The discontinuous and non-smooth nature of (cid:101) U ( T ) is another reason why it is difficult to solve OP .Figure 2 also shows that the optimal lease duration is non-trivial . If the lease duration is too low, MER of manyoperators are not satisfied (Property 1) and hence (cid:101) S ( T ) issmall. Therefore, (cid:101) U ( T ) is low (high) according to Property 4if bid correlation if high (low). If the lease duration is too highand bid correlation is high (low), (cid:101) U ( T ) is low (high) for oneof the two reasons. First , due to Property 3. This is becausethe objective function is monotonic decreasing (increasing) inlease duration if bid correlation is high (low).
Second , (cid:101) S ( T ) ismall either because the operators cannot afford a channel withlong lease duration or because the revenue function decreasesfor a higher value of lease duration as suggested by Property 1.Hence, the optimal lease duration is neither too high nor toolow as shown in Figure 2. B. Stackelberg game solution: Homogeneous Market withComplete Information
For complete information games, (cid:98) ξ k = ξ k ; ∀ k , and forhomogeneous market, ξ k = ξ ; ∀ k . Let, ξ = ( µ, σ, a, ρ, λ, Λ) .Since (cid:98) ξ k = ξ k , the revenue function of the k th operator asperceived by the k th operator, (cid:98) R k ( S , T ) , and as perceivedby the regulator , (cid:101) R k ( S , T ) , are equal to the revenue function(true) R k ( S , T ) . Similarly, the perceived objective function, (cid:101) U ( T ) , is equal to the objective function (true), U ( T ) . Also,since the market is homogeneous in ξ k , the revenue is samefor all the operators, i.e. R k ( S , T ) = R ( s, T ) ; ∀ k . The fol-lowing proposition can be used to calculate the optimal leaseduration, T ∗ , and the optimal value of the objective function, U ∗ , for a homogeneous market with complete information. Proposition 3.
Let θ be the solution to R ( N, θ ) = λ and (cid:100)·(cid:101) be the ceiling function. If (cid:100) θ (cid:101) ≤ Λ , then T ∗ = (cid:100) θ (cid:101) and U ∗ = N (cid:100) θ (cid:101) R ( N, (cid:100) θ (cid:101) ) . However, if (cid:100) θ (cid:101) > Λ , then U ∗ = 0 and T ∗ can be set to any value.Proof: Please refer to Appendix E for the proof.Intuitively, Proposition 3 can be understood as follows. Ina homogeneous market, either all or none of the operators areinterested in joining the market. If none of the operators areinterested in joining the market, then the objective functionis zero which is trivial. Hence, to have U ∗ > , T ∗ shouldbe such that all the operators are interested in joining themarket. If all the N operators join the market, then the revenuefunction of an operator is R ( N, T ∗ ) . Also, for operatorsto be interested in joining the market, T ∗ must also satisfy R ( N, T ∗ ) ≥ λ and T ∗ ≤ Λ (refer to (20)). As discussed inSection II-C, the revenue function of a homogeneous marketis monotonic increasing in lease duration. Hence, the solutionto R ( N, T ∗ ) ≥ λ is T ∗ ≥ (cid:100) θ (cid:101) where θ if the solution to R ( N, θ ) = λ . The ceiling function (cid:100)·(cid:101) is needed becausewe consider a time slotted model. To this end we concludethat, U ∗ > if and only if T ∗ satisfies (cid:100) θ (cid:101) ≤ T ∗ ≤ Λ .If (cid:100) θ (cid:101) > Λ , then there does not exists a T ∗ that satisfies (cid:100) θ (cid:101) ≤ T ∗ ≤ Λ . Hence, U ∗ = 0 and T ∗ can be set to any value.If (cid:100) θ (cid:101) ≤ Λ , then T ∗ = (cid:100) θ (cid:101) maximizes the objective functionbecause for a homogeneous market, objective function ismonotonic decreasing in lease duration (refer to Section II-C).For T ∗ = (cid:100) θ (cid:101) , all the operators are interested in joining themarket. Hence, according to (14), U ∗ = N (cid:100) θ (cid:101) R ( N, (cid:100) θ (cid:101) ) . Thiscompletes the explanation of Proposition 3.Finally, to solve OP for a homogeneous market withcomplete information, we have to compute θ . Since, R ( N, θ ) is monotonic increasing in θ , the equation R ( N, θ ) = λ canbe solved using binary search or Newton-Raphson method.
C. Stackelberg game solution: Heterogeneous Market withIncomplete Information
Algorithm 1:
Optimization algorithm to solve OP fora heterogeneous market with incomplete information Input: N , M , (cid:98) ξ k ; k = 1 , . . . , N Output: (cid:101) T ∗ , (cid:101) U ∗ , (cid:101) S ∗ Initialize an empty list Q L whose elements are orderedpairs ( T, k ) where T denotes lease duration and k denotes operator index for k ← to N do Append (cid:16)(cid:108) (cid:98) λ k (cid:98) µ k (cid:109) , k (cid:17) and (cid:16)(cid:98) Λ k + 1 , k (cid:17) onto Q L Sort Q L in ascending order of lease duration Set (cid:101) U ∗ = 0 and X L = ∅ for i ← to (cid:12)(cid:12) Q L (cid:12)(cid:12) do if Q Li .k ∈ X Li − then set X Li = X Li − − (cid:8) Q Li .k (cid:9) ; else set X Li = X Li − (cid:83) (cid:8) Q Li .k (cid:9) if Q Li .T < Q Li +1 .T or i = (cid:12)(cid:12) Q L (cid:12)(cid:12) then X Li is one of the sets in F L . Set θ Li = Q Li .T and Θ Li = Q Li +1 .T − Initialize an empty list Q whose elements are or-dered pairs ( T, k ) where T denotes lease durationand k denotes operator index for k in X Li do Find γ k and Γ k , the minimum and the maximumlease duration resp. in the interval (cid:2) θ Li , Θ Li (cid:3) s.t. (cid:101) R k (cid:0) X Li , T (cid:1) ≥ (cid:98) λ k ; ∀ T ∈ [ γ k , Γ k ] if γ k and Γ k exists then Append ( γ k , k ) onto Q if Γ k < Θ Li then append (Γ k + 1 , k ) onto Q Sort Q in ascending order of lease duration Set X = ∅ for j ← to | Q | do if Q j .k ∈ X j − then set X j = X j − − { Q j .k } ; else set X j = X j − (cid:83) { Q j .k } if ( Q j .T < Q j +1 .T or j = | Q | ) then X j is one of the sets in F . Set θ j = Q j .T and Θ j = Q j +1 .T − if (cid:101) U ( X j , θ j ) > (cid:101) U ∗ then Set (cid:101) T ∗ = θ j , (cid:101) U ∗ = (cid:101) U ( X j , θ j ) , (cid:101) S ∗ = X j if (cid:101) U ( X j , Θ j ) > (cid:101) U ∗ then Set (cid:101) T ∗ = Θ j , (cid:101) U ∗ = (cid:101) U ( X j , Θ j ) , (cid:101) S ∗ = X j Algorithm 1 is a psuedocode to solve OP for a hetero-geneous market with incomplete information. Proposition 3which solves OP for a homogeneous market with completeinformation is a special case of Algorithm 1. The maindifficulty about solving OP is the change in (cid:101) S ( T ) withchange in T . This leads to discontinuities in the objectivefunction of OP . Algorithm 1 solves this issue by dividingthe entire positive real axis which represents the lease durationinto intervals such that (cid:101) S ( T ) does not change within theseintervals. The optimal lease duration within these intervals willlie in its boundaries because of Property 3. Finally, the optimallease duration can be found by comparing the maximum leaseduration within each of these intervals. In the rest of thissection, we will device an efficient approach to find theseintervals. We will approach this in steps. First , we will convert P into a combinatorial optimization problem OP . Bydoing so, we formalize the idea discussed in this paragraph. Second , we will discuss how to divide the entire positive realaxis into intervals such that (cid:101) S L ( T ) does not change withinthese intervals. This is required because (cid:101) S ( T ) is a functionof (cid:101) S L ( T ) (refer to (20)). Therefore, to find the intervalscorresponding to (cid:101) S ( T ) , we have to first find the intervalscorresponding to (cid:101) S L ( T ) . The process of finding the intervalscorresponding to (cid:101) S L ( T ) will be exemplified using Example1, Example 2 and Figure 3. Finally, we discuss how to findthe intervals corresponding to (cid:101) S ( T ) which is very similar tofinding the intervals corresponding to (cid:101) S L ( T ) .Let F be a family of sets containing all possible per-ceived sets of interested operators. Mathematically, F = (cid:110) S : ( ∃ T ∈ Z + ) (cid:104) (cid:101) S ( T ) = S (cid:105)(cid:111) . For a given S ∈ F , therecan be several values of T satisfying (cid:101) S ( T ) = S . Let T m (cid:0) S (cid:1) and T M (cid:0) S (cid:1) denote the minimum and the maximum T respectively satisfying (cid:101) S ( T ) = S . According to Property 3,either T m (cid:0) S (cid:1) or T M (cid:0) S (cid:1) maximizes OP if (cid:101) S ( T ) = S .Based on this discussion, OP is equivalent to the followingoptimization problem OP (cid:26) max S∈F (cid:101) U (cid:0) S (cid:1) = max (cid:16) (cid:101) U (cid:0) S , T m (cid:0) S (cid:1)(cid:1) , (cid:101) U (cid:0) S , T M (cid:0) S (cid:1)(cid:1)(cid:17) OP is a combinatorial optimization problem in S . Let (cid:101) T ∗ be the optimal solution of OP . If (cid:101) S ∗ is an optimalsolution of OP , then (cid:101) T ∗ is either T m (cid:16) (cid:101) S ∗ (cid:17) or T M (cid:16) (cid:101) S ∗ (cid:17) ,whichever maximizes (cid:101) U (cid:16) (cid:101) S ∗ , T (cid:17) . In Algorithm 1, we find (cid:101) S ∗ (and hence (cid:101) T ∗ ) by iterating over all S ∈ F to find the onewhich maximizes (cid:101) U (cid:0) S (cid:1) . To do so, we need a constructivemethod to find all the sets in F . In the rest of the section, wediscuss the steps involved in finding all the sets in F and thecorresponding line number of Algorithm 1 which implementsthat step. One of the outputs of Algorithm 1 is (cid:101) U ∗ , the optimalvalue of the perceived objective function. But the value of theobjective function (true) corresponding to optimal solution of OP , (cid:101) T ∗ , is given by (18) and is equal to U ∗ = U (cid:16) S (cid:16) (cid:101) T ∗ (cid:17) , (cid:101) T ∗ (cid:17) (21)In order to find all the sets in F , we have to first find allthe sets in F L = (cid:110) S : ( ∃ T ∈ Z + ) (cid:104) (cid:101) S L ( T ) = S (cid:105)(cid:111) . F L is afamily of sets containing all possible largest sets of interestedoperators as perceived by the regulator. According to (19), k ∈ (cid:101) S L ( T ) if and only if T ≥ (cid:108) (cid:98) λ k (cid:98) µ k (cid:109) and T < (cid:98) Λ k + 1 .The ceiling function (cid:100)·(cid:101) is needed because we consider atime slotted model. Consider the ordered pairs (cid:16)(cid:108) (cid:98) λ k (cid:98) µ k (cid:109) , k (cid:17) and (cid:16)(cid:98) Λ k + 1 , k (cid:17) where the first element is lease duration and thesecond element is operator index. A list Q L contains suchordered pairs corresponding to all the N operators ( line 1-3 ).The size of Q L is (cid:12)(cid:12) Q L (cid:12)(cid:12) = 2 N as there are ordered pairscorresponding to each of the N operators. Let Q Li be the i th element of Q L . We use the dot ( · ) operator to access the leaseduration and the operator index of the elements of Q L . Inother words, Q Li .T and Q Li .k denote the lease duration and the operator index respectively corresponding to ordered pair Q Li . All the sets in F L can be found using the following steps:(A1) Sort Q L in ascending order of lease duration ( line 4 ).Traverse the sorted list Q L from i = 1 to (cid:12)(cid:12) Q L (cid:12)(cid:12) andrepeat steps (A2) to (A4) in every iteration ( line 6 ). Let X Li be the largest set of interested operators as perceivedby the regulator which is obtained in the i th iteration.Set X L = ∅ ( line 5 ). Let i be the current iteration.(A2) If the operator with index Q Li .k is not in set X Li − , add Q Li .k to X Li − to get X Li . Else if the operator with index Q Li .k is in set X Li − , remove Q Li .k from X Li − to get X Li .This is implemented in line 7 .(A3) If Q Li .T < Q Li +1 .T or i = (cid:12)(cid:12) Q L (cid:12)(cid:12) , then the obtained X Li in step (A2) is one of the sets in F L . (cid:101) S L ( T ) = X Li forall T ∈ (cid:2) Q Li .T , Q Li +1 .T − (cid:3) .(A4) If Q Li .T = Q Li +1 .T , then the obtained X Li in step (A2)is not one of the sets in F L . This is because operators Q Li .k and Q Li +1 .k corresponding to ordered pairs Q Li and Q Li +1 respectively must update X Li − simultaneouslyas both these ordered pairs have the same lease duration.The if statement in line 8 implements steps (A3) and (A4).If Q Li .T = Q Li +1 .T , then the if statement in line 8 is false and the algorithm simply loops to the next iteration withoutconsidering the obtained X Li in line 7 as one of the sets in F L . The following examples exemplifies the working of steps(A1) to (A4). Example 1.
Consider N = 3 operators with (cid:98) µ k = 1 ; ∀ k , (cid:98) λ k ’sare [175 , , and (cid:98) Λ k ’s are [300 , , . The sorted list Q L for this example is shown in Figure 3.a. As we traverseFigure 3.a from left to right, (cid:101) S L ( T ) is ∅ if T ≤ , { } if T ∈ [100 , , { , } if T ∈ [175 , , { , , } if T ∈ [200 , , { , } if T ∈ [300 , , { } if T ∈ [450 , and ∅ if T ≥ . Hence, F L consists of the sets { } , { , } , { , , } , { , } , { } , and ∅ . Example 2.
This example demonstrates the importance of step(A4) by considering mutiple ordered pairs with same leaseduration. The setting is the same as Example 1 except that (cid:98) λ k ’s are [200 , , . The sorted list Q L for this exampleis shown in Figure 3.b. As we traverse Figure 3.b from leftto right, (cid:101) S L ( T ) is ∅ if T ≤ , { } if T ∈ [100 , , { , , } if T ∈ [200 , , { , } if T ∈ [300 , , { } if T ∈ [450 , and ∅ if T ≥ . Hence, F L consists ofthe sets { } , { , , } , { , } , { } , and ∅ . Unlike Example 1, { , } / ∈ F L since ordered pairs (200 , and (200 , havethe same lease duration. Figures 3.a and 3.b show that steps (A1) to (A4) dividesthe set of positive integers into contiguous intervals of leaseduration. Each interval has its corresponding X Li . A gen-eral setup is shown in Figure 3.c. Let i , i , . . . , N ,where i < i < · · · < N , denote all the iterations such that Q Li .T < Q Li +1 .T (or i = (cid:12)(cid:12) Q L (cid:12)(cid:12) = 2 N ). Refering to step(A3), F L consists of the sets X Li , X Li , . . . , X L N . Eachof these sets are associated with a corresponding intervalof lease duration. As shown in Figure 3.c, (cid:101) S L ( T ) is equalto X Li in the interval (cid:2) Q Li .T , Q Li +1 .T − (cid:3) , X Li in theinterval (cid:2) Q Li .T , Q Li +1 .T − (cid:3) etc. These intervals are non-overlapping and their union spans the entire set of positive ncreasing Lease DurationDirection of TraversalIncreasing Lease DurationDirection of TraversalIncreasing Lease DurationDirection of TraversalOrdered Pairswith sameLease Duration Ordered Pairswith sameLease Duration Ordered Pairswith sameLease Duration (b)(c)(a) Q L N . T Q Li − Q L Q Li ... ... Q Li Q Li − Q Li +1 ... Q L N Q L N − Q Li b N +1 (300 ,
1) (450 ,
2) (625 ,
99 100 199 200 299 300 449 450 624 625 (100 ,
2) (200 , , ∅ { } { , , } { , } { } ∅ (100 ,
2) (300 , ,
1) (200 ,
3) (450 ,
99 100 174 175 199 200 299 300 449 450 624 625 ∅ { } { , } { , , } { , } { } ∅ (625 , Q L ∅ Q L Q L X Li X L N = ∅X Li Q L . T − Q L i . T Q L i + . T − Q L i . T Q L i + . T − Q L i . T Q L i b N + . T − Figure 3. (a) Figure showing sorted list Q L (in blue) and the sets in F L (ingreen) for Example 1. (b) Figure showing sorted list Q L (in blue) and thesets in F L (in green) for Example 2. (c) A generic pictorial representationshowing the sorted list Q L (in blue), the sets in F L (in green) and the intervalof lease duration corresponding to the sets in F L (in red). integers. We want to design an algorithm to find all the setsin F . F contains all the sets (cid:101) S ( T ) as T varies in the setof positive integers. This problem is equivalent to finding allthe sets (cid:101) S ( T ) as T varies in each one of these intervals. Theequivalence is due to the fact that the union of these intervalsspans the entire set of positive integers.Let (cid:2) θ Li , Θ Li (cid:3) be one such interval where Q Li .T< Q Li +1 .T, θ Li = Q Li .T , and Θ Li = Q Li +1 .T − ( line 9 ). We have (cid:101) S L ( T ) = X Li ; ∀ T ∈ (cid:2) θ Li , Θ Li (cid:3) . In the interval (cid:2) θ Li , Θ Li (cid:3) ,the perceived set of interested operators (cid:101) S ( T ) ⊆ X Li .If k ∈ X Li , then T ≤ (cid:98) Λ k (refer to (19)).Hence, for the interval (cid:2) θ Li , Θ Li (cid:3) , (20) is equivalent to (cid:101) S ( T ) = (cid:110) k ∈ X Li : (cid:101) R k (cid:0) X Li , T (cid:1) ≥ (cid:98) λ k (cid:111) . According to Prop-erty 1, (cid:101) R k (cid:0) X Li , T (cid:1) is unimodal in T . Therefore, the solutionto (cid:101) R k (cid:0) X Li , T (cid:1) ≥ (cid:98) λ k in the interval (cid:2) θ Li , Θ Li (cid:3) is also aninterval [ γ k , Γ k ] . γ k and Γ k are the minimum and the maxi-mum lease duration satisfying θ Li ≤ γ k ≤ Γ k ≤ Θ Li such that (cid:101) R k (cid:0) X Li , T (cid:1) ≥ (cid:98) λ k ; ∀ T ∈ [ γ k , Γ k ] . If T ∈ (cid:2) θ Li , Θ Li (cid:3) , there arethree possible cases:(B1) γ k and Γ k exist and Γ k < Θ Li : One such example isshown in Figure 4.a. In this case, the k th operator is as-sociated with two ordered pairs ( γ k , k ) and (Γ k + 1 , k ) implying that k ∈ (cid:101) S ( T ) if and only if T ≥ γ k and (a) (c)(b) Figure 4. Graph of unimodal function (cid:101) R k (cid:0) X Li , T (cid:1) depicting an examplewhen (a) γ k and Γ k exist and Γ k < Θ Li . (b) γ k and Γ k exist and Γ k = Θ Li .(c) γ k and Γ k do not exist. T < Γ k + 1 .(B2) γ k and Γ k exist and Γ k = Θ Li : One such example isshown in Figure 4.b. In this case, the k th operator isassociated with an ordered pair ( γ k , k ) implying that k ∈ (cid:101) S ( T ) if and only if T ≥ γ k .(B3) γ k and Γ k do not exist: One such example is shown inFigure 4.c. In this case, the k th operator is not associatedwith any ordered pair because k / ∈ (cid:101) S ( T ) for all T .Consider a list Q containing the ordered pairs associatedwith all the operators in X Li . List Q is constructed in lines11-15 . This involves computation of γ k and Γ k in line 12 followed by accounting for cases (B1) to (B3) in lines 13-15 . γ k and Γ k can be computed as follows. First, we find themaximum of (cid:101) R k (cid:0) X Li , T (cid:1) in the interval (cid:2) θ Li , Θ Li (cid:3) time using fibonnaci search [37]. Let (cid:98) Θ be the maxima of (cid:101) R k (cid:0) X Li , T (cid:1) inthe interval (cid:2) θ Li , Θ Li (cid:3) . Second, γ k ( Γ k , resp.) can be found bysolving the equation (cid:101) R k (cid:0) X Li , T (cid:1) = (cid:98) λ k in the interval (cid:104) θ Li , (cid:98) Θ (cid:105) ( (cid:104) (cid:98) Θ , Θ Li (cid:105) , resp.) using binary search . This strategy to compute γ k and Γ k requires O (cid:0) log (cid:0) Θ Li − θ Li (cid:1)(cid:1) computations of (cid:101) R k (cid:0) X Li , T (cid:1) for various values of T .To find all the sets (cid:101) S ( T ) as T varies in the interval (cid:2) θ Li , Θ Li (cid:3) , we simply apply steps (A1) to (A4) to list Q . Thisis implemented in lines 16-20 . Let Q j be the j th element ofthe sorted list E . If Q j .T < Q j +1 .T , then X j is one of the setsin F . We have, (cid:101) S ( T ) = X j ; T ∈ [ θ j , Θ j ] where θ j = Q j .T and Θ j = Q j +1 .T − . Therefore, the objective functionin the interval [ θ j , Θ j ] is (cid:101) U ( X j , T ) which is maximum for T = θ j or T = Θ j (Property 3). We can find the optimallease duration in the interval (cid:2) θ Li , Θ Li (cid:3) by iterating over all X j such that Q j .T < Q j +1 .T . Finally, we can find the optimallease duration (cid:101) T ∗ by repeating the same procedure for all suchintervals (cid:2) θ Li , Θ Li (cid:3) that satisfies Q Li .T < Q Li +1 .T . These stepsare implemented in lines 21-25 .We end this section by discussing the time complexity ofAlgorithm 1 and comparing it with a bruteforce approach tosolve OP . Lines 12 and 22 are the most computationallydemanding steps of Algorithm 1 as it involves numericalintegration to evaluate the revenue function (cid:101) R k ( S , T ) . Allother computations is absorbed (up to a constant factor)by the time taken for evaluating the revenue function. Let (cid:98) Λ L = max ≤ k ≤ N (cid:98) Λ k , the maximum lease duration above whichnone of the operators can afford a channel. roposition 4. Time complexity of Algorithm 1 is O (cid:16) N log (cid:16)(cid:98) Λ L (cid:17) + N (cid:17) .Proof: Please refer to Appendix F for the proof.A bruteforce approach to solve OP involves iterating from T = 1 to (cid:98) Λ L to find the T which maximizes (cid:101) U ( T ) . This isbecause for T > (cid:98) Λ L , (cid:101) S ( T ) = ∅ and hence (cid:101) U ( T ) = 0 . Toevaluate (cid:101) U ( T ) , we need O ( N ) computations of (cid:101) R k ( S , T ) tofind (cid:101) S ( T ) (refer to (17)) and finally (cid:101) U ( T ) (refer to (13)).Therefore, time complexity of the bruteforce approach is O (cid:16) N (cid:98) Λ L (cid:17) . In practice, (cid:98) Λ L is much larger compared to N .Hence, time complexity of bruteforce approach, O (cid:16) N (cid:98) Λ L (cid:17) ,is much larger compared to time complexity of Algorithm 1, O (cid:16) N log (cid:16)(cid:98) Λ L (cid:17) + N (cid:17) .IV. N UMERICAL R ESULTS
In Sections IV-A to IV-B, we use the optimization algo-rithms from Section III to numerically explore the effect oftrue market parameters ξ k on (cid:101) T ∗ , U ∗ = U (cid:16) S (cid:16) (cid:101) T ∗ (cid:17) , (cid:101) T ∗ (cid:17) and s ∗ = (cid:12)(cid:12)(cid:12) S (cid:16) (cid:101) T ∗ (cid:17)(cid:12)(cid:12)(cid:12) for complete information games. Recall that (cid:101) T ∗ is the optimal lease duration corresponding to the perceivedobjective function, U ∗ is the value of the objective function(true) corresponding to (cid:101) T ∗ and s ∗ is the number of interestedoperators corresponding to (cid:101) T ∗ (refer to (21)). For completeinformation games, the perceived objective function, (cid:101) U ( T ) , isequal to the objective function (true), U ( T ) . Hence, (cid:101) T ∗ = T ∗ here T ∗ is the true optimal lease duration corresponding to U ( T ) . In Section IV-D, we discuss how incomplete informa-tion games leads to sub-optimal solutions.One of the market parameters in ξ k is the autocorrelationcoefficient a k . Instead of a k , we use time constant τ k where a k = exp (cid:16) − τ k (cid:17) . A higher time constant implies higherautocorrelation. Throughout this section, number of operators N = 10 and number of channels M = 2 . A. Optimal Trends
The trends of T ∗ and U ∗ as a function of number ofoperators N and parameters µ k , σ k , τ k , and ρ k are discussedin this subsection. Throughout this subsection, Λ k = ∞ ; ∀ k .We consider both homogeneous and heterogeneous market.The default parameters for homogeneous market are µ = 1 , σ = 0 . , τ = 100 , ρ = 0 . and λ = 100 . We vary oneparameter at a time while holding the other parameters attheir default values. We solve for T ∗ and U ∗ as we vary theparameters and plot the result in Figure 5.For heterogeneous markets, we randomly choose the val-ues for the market parameters from an uniform distribution.The default uniform distributions are µ k ∼ U (0 . , . , σ k ∼ U (0 . , . , τ k ∼ U (50 , , ρ k ∼ U (0 . , . ,and λ k ∼ U (50 , ∀ k . Each of these distributions areassociated with a mean and a range, e.g. the mean of µ k is µ = . . = 1 and the range is (1 . − .
8) = 0 . . Therange of these distributions remains the same throughout thissection; only the mean is varied. One of these distributions isvaried at a time while holding the other distributions at their (b)(c)(d)(a) (f)(g)(e) (j)(k)(h) (l)(i) (n)(o)(p)(m) (q)(r)(s)(t) Figure 5. Plots showing T ∗ and U ∗ as a function of mean (a, b, c, and d),standard deviation (e, f, g, and h), time constant (i, j, k, and l), MER (m, n, o,and p), and number of operators (q, r, s, and t). The top two and the bottomtwo rows correspond to homogeneous and heterogeneous markets respectively.For every mean of the market parameters (x-axis) in a heterogeneous market,we averaged T ∗ and U ∗ over instances of market parameters. default value. For every distribution, we generate instancesof market parameters sampled from the five distributions. Wesolve for T ∗ and U ∗ for each of the instances and plotthe result as errorbar graphs in Figure 5. The errobar graphsshow the sample mean and standard deviation of T ∗ and U ∗ .We will now explain the effect of various parameters on T ∗ and U ∗ . These explanations will rely on Properties 1 -4. Also, recall that special cases of Properties 3 and 4 holdsfor homogeneous market, i.e. objective function is monotonicdecreasing in T (Property 3) and monotonic increasing in s (Property 4) for a homogeneous market. Effect of mean:
In a homogeneous market, as µ increases,an operator’s revenue per time slot increases. Therefore, it willtake less time to generate the MER λ . Hence, T ∗ decreasesas shown in Figure 5.b. With decrease in T ∗ , U ∗ increasesaccording to Property 3. This is shown in Figure 5.a. Similartrends hold for heterogeneous market. As the mean of µ k , µ ,increases, the sample mean of U ∗ increases while the samplemean of T ∗ decreases. This is shown in Figures 5.c. and 5.d. Effect of standard deviation:
In a homogeneous market, T ∗ decreases with increase in σ as shown in Figure 5.f. Thiscan be explained as follows. As σ increases, an operator’srevenue fluctuates more around the mean. These fluctuationscan lead to a revenue which is either greater or lower than themean. If an operator is allocated a channel, there is a higherprobability that the revenue is greater than the mean. This isdue to the allocation policy which, in general, ensures thatthe operator who is allocated a channel has a high revenue.This suggests that the revenue function increases with σ . Sincethe revenue function increases, an operator takes less time togenerate its MER. Hence, T ∗ decreases with increase in σ . As T ∗ decreases, U ∗ increases due to Property 3. This is shownin Figure 5.e. For a heterogeneous market, the sample mean of U ∗ increases with increase in mean of σ k , σ . This is shown inFigure 5.g and its is similar to homogeneous market. However,unlike a homogeneous market, the sample mean of T ∗ remainsalmost the same with increase in σ as shown in Figure 5.h.his happens due to a cyclic effect which can be explainedas follows. Similar to homogeneous market, with increase in σ , the revenue function increases. But as the revenue functionincreases, more operators are interested in entering the marketwhich in turn decreases the revenue function (Property 2).These two competing factors negates the impact of σ on therevenue function and hence on T ∗ . Effect of time constant:
Consider the homogeneous marketfirst. Autocorrelation defines the self-similarity of a randomprocess. As autocorrelation increases, an operator with higherrevenue at current time slot will have higher revenue at alater time slot. Therefore, with increase in time constant τ (and hence autocorrelation), the revenue function increases.As the revenue function increases, an operator takes less timeto generate its MER. Hence, T ∗ decreases with increase in τ . As T ∗ decreases, U ∗ increases due to Property 3. Thisis shown in Figure 5.i and 5.j. For a heterogeneous market,the sample mean of U ∗ increases with increase in mean of τ k , τ . This trend is shown in Figure 5.k and its is similarto a homogeneous market. However, unlike a homogeneousmarket, the sample mean of T ∗ increases with increase in τ as shown in Figure 5.l. This is due to the same cyclic effectmentioned while explaining the effect of standard deviation.But in this case, the effect of the increase in number ofinterested operator is more dominant. As a result, the revenuefunction decreases with increase in τ . Since the revenuefunction decreases, the sample mean of T ∗ increases becausean operator takes more time to generate its MER. Effect of bid correlation coefficient:
Consider the homoge-neous market first. As bid correlation coefficient increases, anoperator with a high bid is more likely to generate a higherrevenue. Since channels are allocated to operators with highbids, we can equivalently say that if an operator is allocated achannel, then its revenue increases with increase in bid corre-lation coefficient. Therefore, it will take less time to generatethe MER λ . Hence, T ∗ decreases with increase in ρ . Withdecrease in T ∗ , U ∗ increases according to Property 3. Thisis shown in Figure 5.m and Figure 5.n. For a heterogeneousmarket, the sample mean of U ∗ increases with increase inmean of ρ k , ρ . This is shown in Figure 5.o and its is similarto a homogeneous market. However, unlike a homogeneousmarket, the sample mean of T ∗ increases with increase in ρ as shown in Figure 5.p. This is due to the same cyclic effectmentioned while explaining the effect of time constant. Effect of number of operators:
Consider the homogeneousmarket first. As the number of operators increases, the prob-ability that a given operator is allocated a channel decreases.To compensate for this decrease in probability, an operatorhas to generate more revenue when it is allocated a channelin order to satisfy its MER. Hence, T ∗ increases as shown inFigure 5.r. Now we will explain the effect of N on U ∗ . As N increases, U ∗ increases according to Property 4. However,with increase in N , T ∗ increases which leads to decrease in U ∗ according to Property 3. Because of these two competingfactors, U ∗ first increases and then decreases with increase in N as shown in Figure 5.q. Recall that in our model, the numberof interested operators is a measure of market competition.Then this numerical study shows that too much competition (b)(a) (c)(d) Figure 6. Plots comparing the performance of Algorithm 1 and SUBOP asthe market becomes more heterogeneous in Mean µ k (a, b), MER λ k andmaximum affordable lease duration Λ k (c, d). In (a, b), for each coefficientof variation CV [ µ k ] , s ∗ and ∆ U % have been averaged over instance of µ k . In (c, d), for each pair of coefficient of variations ( CV [ λ k ] , CV [Λ k ]) , s ∗ and ∆ U % have been averaged over instance of λ k and Λ k . may not necessarily improve spectrum utilization.For heterogeneous market, we sampled µ k , σ k , τ k , ρ k and λ k from their default uniform distributions. As N increases,the sample mean of T ∗ and U ∗ increases. This is shown inFigure 5.s and 5.t. These trends are similar to homogeneousmarket. Similar to homogeneous market, we expect the samplemean of U ∗ to start decreasing if N is above a threshold. But,we could not verify the same. This is because as N increases,computing the revenue function for a heterogeneous marketbecomes computationally expensive which in turn makes Al-gorithm 1 computationally expensive. This problem does notexists for homogeneous market because the expression forrevenue function is simpler for homogeneous market. Pleaserefer to the proof of Propositions 5 and equation 60 in theappendix to appreciate the relative complexity of the revenuefunction for a heterogeneous and a homogeneous market. B. Comparison with an intuitive algorithm
In this section, we compare the performance of Algorithm 1with an intuitive, but sub-optimal, algorithm SUBOP whichmaximizes the objective function by setting a lease durationthat satisfies all the N operators in terms of affordability Λ k and MER λ k . Through this comparison we exemplify that asthe market becomes more heterogeneous, it is not optimal tosatisfy all the operators even if it is possible.We start by describing SUBOP. Define S N = { , , . . . , N } .Since SUBOP has to satisfy all the N operators, the k th operator is satisfied if the lease duration satisfies T ≤ Λ k and R k ( S N , T ) ≥ λ k . The solution to R k ( S N , T ) ≥ λ k is a range [ γ k , Γ k ] (refer to line 12 of Algorithm 1). But T ≤ Λ k and hence the range has to be modified as (cid:104)(cid:101) γ k , (cid:101) Γ k (cid:105) where (cid:101) γ k = γ k and (cid:101) Γ k = min (Γ k , Λ k ) . The k th operator isinterested in entering the market iff T ∈ (cid:104)(cid:101) γ k , (cid:101) Γ k (cid:105) . The rangeof lease duration that satisfies all the N operators is (cid:104)(cid:101) θ, (cid:101) Θ (cid:105) where (cid:101) θ = max k (cid:101) γ k and (cid:101) Θ = min k (cid:101) Γ k . If (cid:101) Θ < (cid:101) θ , then there isno lease duration that satisfies all the operators and hence thevalue of the objective function is U ∗ S = 0 where the subscript S implies sub-optimal. If (cid:101) Θ ≥ (cid:101) θ , then either T = (cid:101) θ or T = (cid:101) Θ aximizes the objective function (Property 3) subjected to S = S N . Accordingly, the value of the objective function is U ∗ S = max (cid:16) U (cid:16) S N , (cid:101) θ (cid:17) , U (cid:16) S N , (cid:101) Θ (cid:17)(cid:17) .We first compare the performance of Algorithm 1with SUBOP as the market becomes more heterogeneousin mean µ k . To compare the algorithms, lets define ∆ U % = U ∗ − U ∗ S U ∗ S × U ∗ comparedto U ∗ s . The setup is homogeneous in all market parametersbut µ k . We have σ k = 0 . , τ k = 100 , λ k = 100 , and Λ k = ∞ ; ∀ k . The mean µ k is sampled from a truncatedgaussian distribution with mean , coefficient of variation CV [ µ k ] and the truncation bounds are . and . . As CV [ µ k ] increases, the gaussian distribution spreads out moreand hence there is a wider range of µ k making the marketmore heterogeneous. As shown in Figure 6.a, expected optimalnumber of interested operators s ∗ decreases with increasein CV [ µ k ] . This is because as the market becomes moreheterogeneous in µ k , the revenue function becomes unimodalin nature (Property 1). This suggests that there may not be alease duration that satisfies MER of all the operators. Even ifit is possible to satisfy lease duration of all the operators, suchlease durations may too large which may significantly decreasethe objective function according to Property 3 (assuming bidcorrelation coefficients of the operators are high). It is alsopossible that some of the operators have low bid correlationcoefficient. If they enter the market, objective function functioncan decrease (Property 4). Therefore, it may not be optimalto satisfy those operators whose bid correlation coefficientis low. But since SUBOP tries to satisfy all the operators,its performance compared to Algorithm 1 decreases as themarket becomes more heterogeneous in µ k . This is shown inFigure 6.b. where ∆ U % increases with CV [ µ k ] .Similarly, we compare the performance of Algorithm 1 withSUBOP as the market becomes more heterogeneous in λ k and Λ k . The setup is homogeneous in all market parameters but λ k and Λ k . We have µ k = 1 , σ k = 0 . and τ k = 100 ; ∀ k . λ k is sampled from a truncated gaussian distribution with mean , coefficient of variation CV [ λ k ] and the truncation boundsare and . Λ k is sampled from a truncated gaussiandistribution with mean , coefficient of variation CV [Λ k ] and the truncation bounds are and . As CV [ λ k ] and CV [Λ k ] increases, there is a wider of λ k and Λ k . Asshown in Figure 6.c, expected optimal number of interestedoperators s ∗ decreases with increase in CV [ λ k ] and CV [Λ k ] .This is because as the market becomes more heterogeneous in λ k and Λ k , it is possible that a lease duration that satisfiesMER of one operator is not affordable by another operator.Hence, there may not exist a lease duration that satisfies all theoperator. Even if it is possible to satisfy lease duration of allthe operators, such lease durations may be too large becausefew of the operators have high MER. Setting such a largelease duration may not be optimal according to Property 3. Butsince SUBOP tries to satisfy all the operators, its performancecompared to Algorithm 1 decreases as the market becomesmore heterogeneous in λ k and Λ k . This is shown in Figure 6.d. C. Discontinuity in Optimal Trends
This numerical result deals with complete informationgames. We demonstrate certain interesting discontinuities inoptimal trends as MER of operators changes in a heteroge-neous market. Since we are considering complete informationgames, we have (cid:98) ξ k = ξ k ; ∀ k , and hence S Lk ( T ) is same forall k ’s (refer to (16)). So we have, S Lk ( T ) = S L ( T ) ; ∀ k .Let s L ∗ = (cid:12)(cid:12) S L ( T ∗ ) (cid:12)(cid:12) denote the largest number of inter-ested operators corresponding to optimal lease duration. Thenumerical setup is as follows. The market is homogeneousin all parameters but λ k . We have µ k = µ = 1 ; ∀ k , σ k = σ = 0 . ∀ k , τ k = τ = 100 ; ∀ k and Λ k = ∞ ; ∀ k .MER of the first operators are while the th and the th operator has MER λ ≥ . As our setup is homogeneousin µ k , σ k and τ k , the revenue function of all the operatorsis R ( s, T ) and the objective function is U ( s, T ) (refer toSection II-C). Recall that U ( s, T ) is monotonic decreasing in T (special case of Property 3) and monotonic increasing in s (special case of Property 4) while R ( s, T ) is monotonicincreasing in T (special case of Property 1). Consider if themarket consists of only the first operators. This market iscompletely homogeneous and the optimal lease duration isthe solution to R (8 , T ) = 100 (refer to Section III-A) whichis equal to . Now consider the market with all the operators. The optimal lease duration of this market must beat least because the MER of the first operators must besatisfied for optimality. We explore how U ∗ , T ∗ , s ∗ and s L ∗ vary with λ .As λ varies, there are three discontinuities in U ∗ , T ∗ , s ∗ and s L ∗ as shown in Figure 7. Therefore, we divide our explanationinto three regions. As mentioned in the previous paragraph, T ∗ ≥ . In region G1 , λ ≤ µT ∗ ≤ , implying that the th and the th operator may join the market along with theother operators. Hence, s L ∗ = (cid:12)(cid:12) S L ( T ∗ ) (cid:12)(cid:12) = 10 (refer to(16)). Therefore, the minimum value of the revenue functionis R (10 , T ) which decides whether an operator is interestedin entering the market (refer to Section II-D). There are twopossible candidates for optimal lease duration. First, the leaseduration is T that satisfies R (10 , T ) = 100 . In this case,only the first operators are interested in entering the marketand hence the value of the objective function is U (8 , T ) . Second, the lease duration is T that satisfies R (10 , T ) = λ .Since λ ≥ , all the operators are interested in enteringthe market and hence the value of the objective function is U (10 , T ) . Definitely, T ≥ T because λ ≥ and the R ( s, T ) is monotonic increasing in T . In region G1, T isnot much larger compared to T because λ is very close to . Hence, according to Property 4, U (10 , T ) > U (8 , T ) .Therefore, T ∗ = T , s ∗ = 10 and U ∗ = U (10 , T ) . This isshown in Figure 7.a, 7.b and 7.c. As λ increases, T ∗ = T increases and hence U ∗ = U (10 , T ) decreases (Property 3).In region G2 , λ is much more than and hence T ismuch larger compared to T . Hence, according to Property 3, U (10 , T ) < U (8 , T ) . Therefore, T ∗ = T , s ∗ = 8 and U ∗ = U (8 , T ) . This is shown in Figure 7.a, 7.b and 7.c. As T is a constant, T ∗ and U ∗ are constants in region G2. In region G3 , λ > as shown in Figure 7.a. Since it was not
00 150 200 250 300 3502.562.582.62.622.64100 150 200 250 300 350300350400450500 100 150 200 250 300 3508910100 150 200 250 300 3508910 (c)(a) (d)(b)G1 G1G1G1 G2G2G2G2 G3 G3G3G3
Figure 7. A figure demonstrating the discontinuous nature of U ∗ , T ∗ , s ∗ and s L ∗ as a function of MER λ . optimal to satisfy MER of the th and the th operator inregion G2, it is not optimal to satisfy their MER in region G3.This is because λ in region G3 is greater compared to regionG2. But we must satisfy the MER of first operators. Saythat the lease duration is . It is the least lease duration thatsatisfies MER of the first operators. Also, it does not satisfyMER of the th and the th operator becasuse λ > .Infact, if lease duration is , the dominant strategy of the th and the th operator is not to enter the market and hence s L ∗ = 8 as shown in Figure 7.d. So we conlude that T ∗ = 306 in region G3 and the corresponding U ∗ = U (8 , . .This is shown in Figure 7.a and 7.b.We conclude this section with two critical observations.If the market consisted of only the first operators, then U ∗ =2.61. With the th and the th operator in the market, U ∗ is less than . if λ lies in the interval (111 , . Thisis shown in Figure 7.a. Based on this observation, we canconclude the following. First, too much competition may notnecessarily lead to better spectrum utilization. In region G1,even though all the operators are interested in enteringthe market, U ∗ is less than . if λ > . Second, asa thumb rule, the spectrum utilization is higher if the MERof the operators are low and hence leading to a lower leaseduration or if the MER is high enough that it does not affectthe decision of the other operators to enter the market. Toappreciate the last point, note that in region G2, the first operators enter the market only if their MER is satisfied evenif all operator enters the market. So, even though the th and the th operators did not enter the market, these twooperators affected the decision of the first operators. D. Effect of Incomplete Information
Our final numerical study analyzes how the deviation of theestimated market parameters, (cid:98) ξ k , from the true market parame-ters, ξ k , leads to sub-optimal spectrum utilization. Recall that (cid:101) T ∗ is the optimal lease duration corresponding to estimatedmarket parameters, (cid:98) ξ k . (cid:101) T ∗ is one of the outputs of Algorithm1 of the main paper. Let T ∗ be the optimal lease durationcorresponding to true market parameters, ξ k . In other words, (cid:101) T ∗ = T ∗ when (cid:98) ξ k = ξ k . The value of the objective function(true) corresponding to (cid:101) T ∗ and T ∗ are U (cid:16) S (cid:16) (cid:101) T ∗ (cid:17) , (cid:101) T ∗ (cid:17) and U ( S ( T ∗ ) , T ∗ ) respectively (refer to (21) of the main paper).Let, (b) (g)(c) (d)(a) (e) (f) Figure 8. Plots of δU % , the relative change in the optimal value of theobjective function due to incomplete information, when there is a error in (a)all the parameters in (cid:98) ξ k , (b) (cid:98) µ k , (c) (cid:98) σ k , (d) (cid:98) τ k , (e) (cid:98) ρ k , (f) (cid:98) λ k , and (g) (cid:98) Λ k . δU % = U ( S ( T ∗ ) , T ∗ ) − U (cid:16) S (cid:16) (cid:101) T ∗ (cid:17) , (cid:101) T ∗ (cid:17) U ( S ( T ∗ ) , T ∗ ) · δU % is the relative change in the optimal value of theobjective function (spectrum utilization) due to incompleteinformation. We randomly choose true market parameters fromuniform distributions: µ k ∼ U (0 . , . , σ k ∼ U (0 . , . , τ k ∼ U (150 , , ρ k ∼ U (0 . , . , λ k ∼ U (50 , , and Λ k ∼ U (500 , ∀ k . Consider the true parameter µ k .The estimated parameter (cid:98) µ k is choosen uniformly at randomin the interval (cid:104) µ k − δµ %100 · µ k , µ k + δµ %100 · µ k (cid:105) where δµ % is the error window associated with µ k . Similarly, we haveerror windows δσ % , δτ % , δρ % , δλ % and δ Λ% associatedwith σ k , τ k , ρ k , λ k and Λ k respectively. In Figure 8, we ploterrorbar graphs to study how δU % vary with error window.In Figure 8.a, all the six error windows are varied while inthe remaining six plots, one of the error windows is variedwhile the remaining five error windows are set to zero. Foreach value of error window, we sample over randominstances of true and estimated market parameters to generatethe errorbar graphs.As expected, the sample mean of δU % increases witherror window. Comparing Figure 8.a with the remaining sixgraphs, we can conclude that most of the reduction in spectrumutilization happens due to error in µ k . Finally, we point to anon-intuitive result which can be observed by zooming intoFigure 8. There are some cases where δU % is less thanzero. In other words, it is possible that in an incompleteinformation scenario, spectrum utilization is more than com-plete information scenario. This can happen when an operatormakes an erroneous decision of joining the market (not joiningthe market) due to error in (cid:98) ξ k . This can improve spectrumutilization if the true bid correlation coefficient is high (low).V. C ONCLUSION
The duration of a spectrum lease is a critical parameterthat influences the efficiency of spectrum utilization. The maincontribution of this paper is a mathematical model that isused to find the lease duration which maximizes spectrumutilization. This model captures the effects of lease durationon spectrum utilization for a market where an operators’revenue is a measure of its spectrum utilization. Based on theystem model, we formulate a Stackelberg game with leaseduration as one of the decision variables. We also designalgorithms to find the Stackelberg equilibrium and hence findthe optimal lease duration. Using these algorithms, we findseveral numerical trends that show how lease duration shouldchange with respect to various market parameters in order tomaximize spectrum utilization.There are several possible avenues for extending this work,including: (a) Generalization of our system model to capturethe transaction costs associated with re-allocation of channels.(b) Including variance in our system model to capture riskaversion of the operators. (c) Second price auctions to capturethe variable market-dependent price of a spectrum lease.A
PPENDIX AP ROOF OF P ROPOSITION Y k ( c, T ) = cT (cid:80) t =( c − T +1 x k ( t ) . Since x k ( t ) isa stationary process, the pdf of Y k ( c, T ) is same for all epochs.Therefore, we can simply derive the pdf of Y k (1 , T ) = T (cid:80) t =1 x k ( t ) . Now, Y k (1 , T ) is gaussian because x k ( t ) is gaussianand sum of gaussian random variable is always gaussian. Since Y k (1 , T ) is gaussian, its pdf is completely characterized byits mean (cid:101) µ k ( T ) and standard deviation (cid:101) σ k ( T ) . Based on thisdiscussion we can conclude that Y k ( c, T ) ∼ N (cid:0)(cid:101) µ k ( T ) , (cid:101) σ k ( T ) (cid:1) ; ∀ c (22)Now all we have to do is to find expressions for (cid:101) µ k ( T ) and (cid:101) σ k ( T ) . We have, (cid:101) µ k ( T ) = E (cid:34) T (cid:88) t =1 x k ( t ) (cid:35) = T (cid:88) t =1 E [ x k ( t )] = µ k T (23)For a first order AR process as governed by (1), x k ( t ) canbe expressed as x k ( t ) = a tk x k (0) + t − (cid:88) v =0 a t − − vk ε k ( v ) (24)Equation 24 can be easily proved using mathematical in-duction. Also, T (cid:88) t =1 x k ( t ) = T (cid:88) t =1 (cid:32) a tk x k (0) + t − (cid:88) v =0 a t − − vk ε k ( v ) (cid:33) (25) = x k (0) T (cid:88) t =1 a tk + T (cid:88) t =1 t − (cid:88) v =0 a t − − vk ε k ( v )= x k (0) T (cid:88) t =1 a tk + T − (cid:88) v =0 T (cid:88) t = v +1 a t − − vk ε k ( v ) (26) = x k (0) a k − a T +1 k − a k + T − (cid:88) v =0 − a T − vk − a k ε k ( v ) (27)Equation 25 is obtained using (24). Equation 26 is ob-tained by changing the order of summation. Now, (cid:101) σ k ( T ) = (cid:115) Var (cid:20) T (cid:80) t =1 x k ( t ) (cid:21) where, Var (cid:34) T (cid:88) t =1 x k ( t ) (cid:35) = Var (cid:34) x k (0) a k − a T +1 k − a k + T − (cid:88) v =0 − a T − vk − a k ε i ( v ) (cid:35) (28) = (cid:32) a k − a T +1 k − a k (cid:33) Var [ x k (0)]+ T − (cid:88) v =0 (cid:32) − a T − vk − a k (cid:33) Var [ ε k ( v )] (29) = (cid:32) a k − a T +1 k − a k (cid:33) σ k + T − (cid:88) v =0 (cid:32) − a T − vk − a k (cid:33) ( σ εk ) (30) = T − a k (cid:0) − a Tk + a k T (cid:1) (1 − a k ) · ( σ εk ) − a k = T − a k (cid:0) − a Tk + a k T (cid:1) (1 − a k ) σ k (31)So we have, (cid:101) σ k ( T ) = (cid:113) T − a k (cid:0) − a Tk + a k T (cid:1) (1 − a k ) σ k (32)Equation 28 is obtained from (27). Equation 30 holdsbecause ε k ( v ) are independent random variables. Equations30 and 31 follows from the definition of σ εi and σ i as givenby (2) and the paragraph before it. Finally, (23) and (32) aresame as (3) and (4) respectively. This completes the proof.A PPENDIX BR EVENUE F UNCTION FOR H ETEROGENEOUS M ARKET
In this section, we will derive an expression for the revenuefunction R k ( S , T ) for a heterogeneous market. Let’s redefinefew notations from the main paper for the sake of continuity.Recall the following from the main paper. Y k ( c, T ) is the netrevenue of the k th operator in c th epoch if lease duration is T . (cid:98) Y k ( c, T ) is the bid of the k th operator in c th epoch if leaseduration is T . For notational simplicity, lets denote Y k ( c, T ) by Y k and (cid:98) Y k ( c, T ) by (cid:98) Y k . According to equation 5, the jointprobability distribution of (cid:98) Y k and Y k is (cid:20) Y k (cid:98) Y k (cid:21) ∼ N (cid:18)(cid:20)(cid:101) µ k ( T ) (cid:101) µ k ( T ) (cid:21) , (cid:20) (cid:101) σ k ( T ) ρ k (cid:101) σ k ( T ) ρ k (cid:101) σ k ( T ) (cid:101) σ k ( T ) (cid:21)(cid:19) ; ∀ c (33)According to equation 6, the marginal distribution as (cid:98) Y k is (cid:98) Y k ∼ N (cid:0)(cid:101) µ k ( T ) , (cid:101) σ k ( T ) (cid:1) ; ∀ c (34) Proposition 5.
Say that in every epoch, one channel isallocated to each of the (cid:102) M = min ( M, s ) operators havingthe (cid:102) M highest sum of revenue in that epoch. Let S k = S −{ k } and C ( A, a ) denote all the possible combinations of size a − from set A . Let the function f k (cid:16) Y, (cid:98) Y , T (cid:17) denote theprobability density function (pdf) corresponding to the jointdistribution of bid (cid:98) Y and true revenue Y of the k th operatoras given by (33). Similarly, let the function F k (cid:16) (cid:98) Y , T (cid:17) denotethe cumulative distribution function (cdf) corresponding to thearginal distribution of bid (cid:98) Y of the k th operator as givenby (34). Then the revenue function of the k th operator is R k ( S , T ) = ∞ (cid:90) −∞ ∞ (cid:90) −∞ Y · G k (cid:16) (cid:98) Y , S , T (cid:17) · f k (cid:16) Y, (cid:98) Y , T (cid:17) dY d (cid:98) Y (35) where, G k (cid:16) (cid:98) Y , S , T (cid:17) = (cid:102) M (cid:88) m =1 (cid:88) W ∈C ( S k ,m ) H k (cid:16) (cid:98) Y , T, W (cid:17) (36) H k (cid:16) (cid:98) Y , T, W (cid:17) = (cid:89) j ∈ W (cid:16) − F j (cid:16) (cid:98) Y , T (cid:17)(cid:17) (cid:89) j ∈S k − W F j (cid:16) (cid:98) Y , T (cid:17) (37)
Proof.
Consider the term E [ Y k (1 , T ) | w m = k ] P [ w m = k ] of (8). We have, E [ Y k (1 , T ) | w m = k ] P [ w m = k ]= (cid:88) Y Y P [ Y k = Y | w m = k ] P [ w m = k ]= (cid:88) (cid:98) Y ,Y
Y P (cid:104) Y k = Y, (cid:98) Y k = (cid:98) Y | w m = k (cid:105) P [ w m = k ] (38) = (cid:88) (cid:98) Y ,Y
Y P (cid:104) w m = k | Y k = Y, (cid:98) Y k = (cid:98) Y (cid:105) P [ Y k = Y, (cid:98) Y k = (cid:98) Y (cid:105) (39) = (cid:88) (cid:98) Y ,Y
Y P (cid:104) w m = k | (cid:98) Y k = (cid:98) Y (cid:105) P (cid:104) Y k = Y, (cid:98) Y k = (cid:98) Y (cid:105) (40)Equation 38 is obtained by marginalizing over bid (cid:98) Y k , (39)is obtained using Bayes’ Theorem and (40) is true becausegiven (cid:98) Y k , w m is independent of Y k (spectrum allocation de-pends on operators’ bid in an epoch and not on its net revenuein an epoch). (cid:102) M = min ( M , s ) channels are allocated inevery epoch. As mentioned in the main paper (first paragraphof section II-B), w m is the operator who has the m th highestvalue of (cid:98) Y k where m is the channel index. Let (cid:84) denote thelogical AND operator. Then the event w m = k is equivalentto (cid:92) j ∈ W (cid:98) Y j ≥ (cid:98) Y k , (cid:92) j ∈S k − W (cid:98) Y j ≤ (cid:98) Y k for some W ∈ C ( S k , m ) . Hence, the term P [ w m = k | Y k = Y ] of (40) can be written as P (cid:104) w m = k | (cid:98) Y k = (cid:98) Y (cid:105) = (cid:88) W ∈C ( S k ,m ) P (cid:92) j ∈ W (cid:98) Y j ≥ (cid:98) Y , (cid:92) j ∈S k − W (cid:98) Y j ≤ (cid:98) Y = (cid:80) W ∈C ( S k ,m ) (cid:32) (cid:81) j ∈ W P (cid:104) (cid:98) Y j ≥ (cid:98) Y (cid:105) · (cid:81) j ∈S k − W P (cid:104) (cid:98) Y j ≤ (cid:98) Y (cid:105)(cid:33) (41)Equation 41 holds because the bids of any two operatorsare not correlated and are hence independent . Using (8), (40)and (41) we get, R k ( S , T ) = (cid:88) (cid:98) Y ,Y Y ·G k (cid:16) (cid:98) Y , S , T (cid:17) · P (cid:104) Y k = Y, (cid:98) Y k = (cid:98) Y (cid:105) (42) where, G k (cid:16) (cid:98) Y , S , T (cid:17) = (cid:102) M (cid:88) m =1 (cid:88) W ∈C ( S k ,m ) H k (cid:16) (cid:98) Y , T, W (cid:17) (43) H k (cid:16) (cid:98) Y , T, W (cid:17) = (cid:89) j ∈ W P (cid:104) (cid:98) Y j ≥ (cid:98) Y (cid:105) · (cid:89) j ∈S k − W P (cid:104) (cid:98) Y j ≤ (cid:98) Y (cid:105) (44)The joint probability distribution of Y k and (cid:98) Y k is governedby (33). Let f k (cid:16) Y, (cid:98) Y , T (cid:17) denote the corresponding jointprobability density function of Y k and (cid:98) Y k . The marginal prob-ability distribution of (cid:98) Y k is governed by (34). Let F k (cid:16) (cid:98) Y , T (cid:17) denote the corresponding cumulative distribution function of (cid:98) Y k . Therefore, P (cid:104) Y k = Y, (cid:98) Y k = (cid:98) Y (cid:105) = f k (cid:16) Y, (cid:98) Y , T (cid:17) dY d (cid:98) YP (cid:104) (cid:98) Y k ≥ (cid:98) Y (cid:105) = (cid:16) − F k (cid:16) (cid:98) Y , T (cid:17)(cid:17) P (cid:104) (cid:98) Y k ≤ (cid:98) Y (cid:105) = F k (cid:16) (cid:98) Y , T (cid:17) which when substituted in (42) and (44) yields (35)-(37). Thiscompletes the proof. A
PPENDIX CR EVENUE F UNCTION FOR H OMOGENEOUS M ARKET
We want to derive a simplified expression of revenuefunction for a market that is homogeneous in µ k , σ k , a k and ρ k , i.e. µ k = µ , σ k = σ , a k = a and ρ k = ρ , ∀ k .In a homogeneous market, (cid:101) µ k ( T ) and (cid:101) σ k ( T ) in (3) and (4)respectively, are same for all the operators. We have, (cid:101) µ ( T ) = µT (45) (cid:101) σ ( T ) = (cid:112) T − a (2 − a T + aT )(1 − a ) σ (46)In a homogeneous market, we can drop the subscript k from f k (cid:16) Y, (cid:98) Y , T (cid:17) and G k (cid:16) (cid:98) Y , S , T (cid:17) in (35) because f k (cid:16) Y, (cid:98) Y , T (cid:17) and G k (cid:16) (cid:98) Y , S , T (cid:17) is same for all the operators. Let s = |S| , (cid:102) M = min ( M, s ) and (cid:18) s − m − (cid:19) = ( s − m − s − m )! . Then thesimplified revenue function is R ( s, T ) = ∞ (cid:90) −∞ ∞ (cid:90) −∞ Y ·G (cid:16) (cid:98) Y , s, T (cid:17) · f (cid:16) Y, (cid:98) Y , T (cid:17) dY d (cid:98) Y where(47) G (cid:16) (cid:98) Y , s, T (cid:17) = (cid:102) M (cid:88) m =1 (cid:18) s − m − (cid:19) (cid:16) − F (cid:16) (cid:98) Y , T (cid:17)(cid:17) m − F (cid:16) (cid:98) Y , T (cid:17) s − m (48)Note that revenue function in a homogeneous market isdependent on the number of interested operators s and not onthe set of interested operators S . f (cid:16) Y, (cid:98) Y , T (cid:17) and F (cid:16) (cid:98) Y , T (cid:17) are the pdf and cdf respectively of the normal distributiongiven by (33). We have, (cid:16) Y, (cid:98) Y , T (cid:17) = exp − (cid:34) Y − (cid:101) µ ( T ) (cid:101) σ ( T ) (cid:98) Y − (cid:101) µ ( T ) (cid:101) σ ( T ) (cid:35) T (cid:20) ρρ (cid:21) − (cid:34) Y − (cid:101) µ ( T ) (cid:101) σ ( T ) (cid:98) Y − (cid:101) µ ( T ) (cid:101) σ ( T ) (cid:35) π (cid:112) − ρ (cid:101) σ ( T ) (49) F (cid:16) (cid:98) Y , T (cid:17) = 12 (cid:32) erf (cid:32) (cid:98) Y − (cid:101) µ ( T ) √ (cid:101) σ ( T ) (cid:33)(cid:33) (50)Substituting Y = (cid:101) σ ( T ) y + (cid:101) µ ( T ) and (cid:98) Y = (cid:101) σ ( T ) (cid:98) y + (cid:101) µ ( T ) in (47) we get, R ( s, T ) = α ( ρ, s ) (cid:101) µ ( T ) + β ( ρ, s ) (cid:101) σ ( T ) where (51) α ( ρ, s ) = ∞ (cid:90) −∞ ∞ (cid:90) −∞ G ( (cid:98) y, s ) h ( y, (cid:98) y, ρ ) dy d (cid:98) y (52) β ( ρ, s ) = ∞ (cid:90) −∞ ∞ (cid:90) −∞ yG ( (cid:98) y, s ) h ( y, (cid:98) y, ρ ) dy d (cid:98) y (53) G ( (cid:98) y, s ) = (cid:102) M (cid:88) m =1 (cid:18) s − m − (cid:19) (1 − H ( (cid:98) y )) m − H ( (cid:98) y ) s − m (54) h ( y, (cid:98) y, ρ ) = exp (cid:32) − (cid:20) y (cid:98) y (cid:21) T (cid:20) ρρ (cid:21) − (cid:20) y (cid:98) y (cid:21)(cid:33) π (cid:112) − ρ (55) H ( (cid:98) y ) = 12 (cid:18) erf (cid:18) (cid:98) y √ (cid:19)(cid:19) (56)We want to simplify α ( ρ, s ) in (52) further. We have, α ( ρ, s ) = ∞ (cid:90) −∞ G ( (cid:98) y, s ) (cid:32) exp (cid:0) − (cid:98) y (cid:1) √ π (cid:33) d (cid:98) y (57) = ∞ (cid:90) −∞ G ( (cid:98) y, s ) d (cid:0) H ( (cid:98) y ) (cid:1) (58)Equation 57 is obtained by re-writting (52)as α ( ρ, s ) = ∞ (cid:82) −∞ G ( (cid:98) y, s ) (cid:32) ∞ (cid:82) −∞ h ( y, (cid:98) y, ρ ) dy (cid:33) d (cid:98) y andthen observing that the inner integral is equal to √ π exp (cid:0) − (cid:98) y (cid:1) . Equation 58 follows from (57) because H ( (cid:98) y ) = ∞ (cid:82) −∞ √ π exp (cid:0) − (cid:98) y (cid:1) d (cid:98) y . So we have, α ( ρ, s ) = ∞ (cid:90) −∞ (cid:102) M (cid:88) m =1 (cid:18) s − m − (cid:19) (cid:0) − H (cid:1) m − H s − m dH (59)By using Binomial Expansion of (cid:0) − H (cid:1) m − followed bysome algebraic manipulation, we can show that the RHS of(59) is equal to (cid:102) Ms . So the final simplified form of the revenuefunction for homogeneous market is R ( s, T ) = (cid:102) Ms (cid:101) µ ( T ) + β ( ρ, s ) (cid:101) σ ( T ) (60)where (cid:101) µ ( T ) , (cid:101) σ ( T ) and β ( ρ, s ) are given by (45), (46) and(53) respectively. A PPENDIX DP ROOF OF THE P ROPERTIES OF THE R EVENUE AND THE O BJECTIVE F UNCTION
A. Proof that R ( s, T ) is monotonic increasing in T We want to prove that revenue function in a homogeneousmarket is monotonic increasing in T . Revenue function ina homogeneous market is given by (60). So proving that R ( s, T ) is monotonic increasing in T is same as proving that (cid:101) µ ( T ) and (cid:101) σ ( T ) are monotonic increasing in T . (cid:101) µ ( T ) and (cid:101) σ ( T ) are given by (45) and (46) respectively. It is obviousthat (cid:101) µ ( T ) = µT is monotonic increasing in T . To prove that (cid:101) σ ( T ) is monotonic increasing in T , it is enough to show that ( T + 1) − a (cid:0) − a T +1 + a ( T + 1) (cid:1) ≥ T − a (cid:0) − a T + aT (cid:1) (61)Inequality 61 simplifies to a ≥ a T +1 . Since a ∈ [0 , and T ≥ , a T +1 ≤ a . Therefore, proving a ≥ a T +1 issame as proving a ≥ a or equivalently ≥ a (2 a − .For a ∈ [0 , , (2 a − ≤ and hence ≥ a (2 a − . Thiscompletes the proof. B. Proof that U ( s, T ) is monotonic decreasing in T We want to prove that objective function in a homogeneousmarket is monotonic decreasing in T . Objective function in ahomogeneous market is given by (14). We have, U ( s, T )= sT R ( s, T )= sT (cid:32) (cid:102) Ms (cid:101) µ ( T ) + β ( ρ, s ) (cid:101) σ ( T ) (cid:33) (62) = µ (cid:102) M + sβ ( ρ, s ) σ (1 − a ) (cid:112) T − a (2 − a T + aT ) T (63)Equation 62 is obtained by substituting R ( s, T ) from (60).Equation 63 is obtained by substituting (cid:101) µ ( T ) and (cid:101) σ ( T ) from (45) and (46) respectively. In (63), √ T − a (2 − a T + aT ) T is monotonic decreasing in T . Hence, U ( s, T ) is monotonicdecreasing in T . This completes the proof. C. Proof that U ( s, T ) is monotonic increasing in s The proof is divided into two steps. In the first step, weshow that U ( s, T ) in (63) can be equivalently expressed as U ( s, T ) = µ (cid:102) M + sβ (1 , s ) ( ρσ )(1 − a ) (cid:112) T − a (2 − a T + aT ) T (64)Qualitatively, (64) shows that a homogeneous market withstandard deviation σ and bid correlation coefficient ρ havethe same objective function as a homogeneous market withstandard deviation ρσ and bid correlation coefficient . If bidcorrelation coefficient is , it represents a degenerate casewhere the net revenue of the k th operator in epoch c for leaseduration T , Y k ( c, T ) , is equal to the bid of the k th operatorn epoch c for lease duration T , (cid:98) Y k ( c, T ) . Mathematically, (cid:98) Y k ( c, T ) = Y k ( c, T ) ; ∀ k, c, T . In our second step, we provethat if (cid:98) Y k ( c, T ) = Y k ( c, T ) ; ∀ k, c, T , then U ( s, T ) is mono-tonic increasing in s .The first step is equivalent to showing β ( ρ, s ) = ρβ (1 , s ) . β ( ρ, s ) is given by (53). Equation 53 can be equivalentlywritten as β ( ρ, s ) = ∞ (cid:90) −∞ G ( (cid:98) y, s ) ∞ (cid:90) −∞ yh ( y, (cid:98) y, ρ ) dy d (cid:98) y (65)where G ( (cid:98) y, s ) and h ( y, (cid:98) y ) is given by (54) and (55) respec-tively. We will now evaluate the integral, ∞ (cid:82) −∞ yh ( y, (cid:98) y ) dy ,of (65). Lets consider the term (cid:20) y (cid:98) y (cid:21) T (cid:20) ρρ (cid:21) − (cid:20) y (cid:98) y (cid:21) in (55).Using simple algebra , we can show that (cid:20) y (cid:98) y (cid:21) T (cid:20) ρρ (cid:21) − (cid:20) y (cid:98) y (cid:21) = ( y − ρ (cid:98) y ) − ρ + (cid:98) y (66)Substituting (66) in (55) and then substituting the resulting h ( y, (cid:98) y ) in the integral ∞ (cid:82) −∞ yh ( y, (cid:98) y ) dy we have, ∞ (cid:90) −∞ yh ( y, (cid:98) y, ρ ) dy = h ( (cid:98) y ) ∞ (cid:90) −∞ y √ π (cid:112) − ρ exp − (cid:32) y − ρ (cid:98) y (cid:112) − ρ (cid:33) dy (67)where h ( (cid:98) y ) = √ π exp (cid:16) − (cid:98) y (cid:17) . The integral in (67) is theexpectation of a gaussian random variable Y whose mean andstandard deviation is ρ (cid:98) y and (cid:112) − ρ respectively. Hence,integral in (67) is simply equal to ρ (cid:98) y . To this end we have, ∞ (cid:90) −∞ yh ( y, (cid:98) y, ρ ) dy = h ( (cid:98) y ) · ( ρ (cid:98) y ) (68)Substituting (68) in (65) we get, β ( ρ, s ) = ρ ∞ (cid:90) −∞ (cid:98) yG ( (cid:98) y, s ) h ( (cid:98) y ) d (cid:98) y = ρ · ∞ (cid:90) −∞ (cid:98) yG ( (cid:98) y, s ) h ( (cid:98) y ) d (cid:98) y = ρβ (1 , s ) This concludes the proof of the first step. In our second step,we have to prove that if (cid:98) Y k ( c, T ) = Y k ( c, T ) ; ∀ k, c, T , then U ( s, T ) is monotonic increasing in s . Rather than provingthis for a homogeneous market where Y k ( c, T ) is governedby Proposition 1, we take a more general approach. Weprove the following: In a heterogeneous market, U ( S , T ) ≤ U ( S (cid:83) { a } , T ) where a / ∈ S , for any random variable Y k ( c, T ) if (cid:98) Y k ( c, T ) = Y k ( c, T ) ; ∀ k, c, T .Refering to (9) and (10), we can say that the objectivefunction U ( S , T ) is the expectation of X ( S , T ) = 1 T C (cid:88) c =1 (cid:102) M (cid:88) m =1 Y w mc ( c, T ) (69) Recall that w mc is the index of the operator who is allo-cated the m th channel in c th epoch. w mc is decided basedon operators bids (cid:98) Y k ( c, T ) which in our case is equal to Y k ( c, T ) . Since w mc is among the set of interested opera-tors S , X ( S , T ) is a function of S . If we can prove that X ( S , T ) ≤ X ( S (cid:83) { a } , T ) for all revenue process x k ( t ) ,it directly implies that U ( S , T ) ≤ U ( S (cid:83) { a } , T ) . This isbecause if there are two random variables Z and Z suchthat Z ≤ Z ; ∀ Z , Z , then E [ Z ] ≤ E [ Z ] .Consider a list Y S consisting of Y k ( c, T ) for all the opera-tors in set S . Since the channels are allocated to the operatorshaving the (cid:102) M highest Y k ( c, T ) , the term (cid:102) M (cid:80) m =1 Y w mc ( c, T ) of(69) is equal to the sum of the (cid:102) M highest values in list Y S .Similarly, consider a list Y S (cid:83) { a } which is same as Y S butwith an additional value Y a ( c, T ) . Definitely, the sum of the (cid:102) M highest values is greater for list Y S (cid:83) { a } than list Y S . Thisis because the values in list Y S is a subset of the values in list Y S (cid:83) { a } . Therefore, the term (cid:102) M (cid:80) m =1 Y w mc ( c, T ) of (69) is greaterfor S (cid:83) { a } than S . This is true for all c ’s and for any revenueprocess x k ( t ) . This proves that X ( S , T ) ≤ X ( S (cid:83) { a } , T ) for any revenue process x k ( t ) . This concludes the proof.A PPENDIX EP ROOF OF P ROPOSITION (cid:101) U ( T ) , of OP is (cid:101) U ( T ) = (cid:40) NT R ( N, T ) ; (cid:100) θ (cid:101) ≤ T ≤ Λ0 ; o.w. (70)where θ is the solution to R ( N, θ ) = λ . We will nowexplain (70). The largest set of interested operators accordingto the regulator, (cid:101) S L ( T ) , in a homogeneous market, is equalto { , , . . . , N } if the lease duration T ≤ Λ and µT ≥ λ (refer to (19)). Otherwise, (cid:101) S L ( T ) = ∅ . If (cid:101) S L ( T ) = ∅ , then (cid:101) S ( T ) = ∅ . If (cid:101) S L ( T ) = { , , . . . , N } , the minimum revenuean operator earns in an epoch is R ( N, T ) . Hence, accordingto (23), the perceived set of interested operators is (cid:101) S ( T ) = (cid:40) { , , . . . , N } ; T ≤ Λ , R ( N, T ) ≥ λ ∅ ; o.w. (71)As discussed in Section II-C, in a homogeneous market, R ( N, T ) is monotonic increasing in T . Therefore, the least T satisfying the constraint R ( N, T ) ≥ λ is (cid:100) θ (cid:101) where θ is the solution to R ( N, θ ) = λ . The ceiling function (cid:100)·(cid:101) isneeded because we consider a time slotted model. For any T ≥ (cid:100) θ (cid:101) , we have R ( N, T ) ≥ λ . Therefore, the condition T ≤ Λ , R ( N, T ) ≥ λ is equivalent to (cid:100) θ (cid:101) ≤ T ≤ Λ andhence (71) is same as (cid:101) S ( T ) = (cid:40) { , , . . . , N } ; (cid:100) θ (cid:101) ≤ T ≤ Λ ∅ ; o.w. (72)ccording to (17), the perceived objective function forhomogeneous market is (cid:101) U ( T ) = (cid:101) s ( T ) T R ( (cid:101) s ( T ) , T ) (73)where (cid:101) s ( T ) = (cid:12)(cid:12)(cid:12) (cid:101) S ( T ) (cid:12)(cid:12)(cid:12) . Finally, (70) can be obtained directlyby using (72) and (73). This completes the explanation of(70). According to (70), if (cid:100) θ (cid:101) > Λ , then there exists no T such that (cid:100) θ (cid:101) ≤ T ≤ Λ . Hence, (cid:101) U ( T ) = 0 ; ∀ T . Therefore,optimal lease duration T ∗ can be set to any value and theoptimal value of objective function U ∗ = 0 . If (cid:100) θ (cid:101) ≤ Λ ,then (cid:101) U ( T ) = NT R ( N, T ) in the range (cid:100) θ (cid:101) ≤ T ≤ Λ . Asdiscussed in Section II-C, the objective function is monotonicdecreasing in a homogeneous market. Hence, T ∗ = (cid:100) θ (cid:101) maximizes (cid:101) U ( T ) = NT R ( N, T ) in the range (cid:100) θ (cid:101) ≤ T ≤ Λ and accordingly U ∗ = N (cid:100) θ (cid:101) R ( N, (cid:100) θ (cid:101) ) .A PPENDIX FP ROOF OF P ROPOSITION γ k and Γ k in line 12 of Algorithm 1 is computated O (cid:0) N (cid:1) times. This simply follows from the observation that the forloop in line 6 executes (cid:12)(cid:12) Q L (cid:12)(cid:12) = 2 N times and the for loopin line 12 executes (cid:12)(cid:12) X Li (cid:12)(cid:12) < N times. As discussed before,each computation of γ k and Γ k involves O (cid:0) log (cid:0) Θ Li − θ Li (cid:1)(cid:1) computations of (cid:101) R k ( S , T ) . Since the maximum value of leaseduration in the list Q L is (cid:98) Λ L , we have Θ Li ≤ (cid:98) Λ L ; ∀ i .This suggests that (cid:101) R k ( S , T ) is computed O (cid:16) N log (cid:16)(cid:98) Λ L (cid:17)(cid:17) times in line 12. (cid:101) U ( X j , θ j ) and (cid:101) U ( X j , Θ j ) in lines 22and 24 respectively are also computed O (cid:0) N (cid:1) times whichcan be explained as follows. Corresponding to every operatorin X Li (line 11), there are at most ordered pairs. Hence, | Q | ≤ (cid:12)(cid:12) X Li (cid:12)(cid:12) ≤ N . Since the for loop in line 6 executes (cid:12)(cid:12) Q L (cid:12)(cid:12) = 2 N times and the for loop in line 18 executes | Q | ≤ N times, (cid:101) U ( X j , θ j ) and (cid:101) U ( X j , Θ j ) in lines 22 and 24respectively are computed O (cid:0) N (cid:1) times. Each computationof (cid:101) U ( X j , θ j ) and (cid:101) U ( X j , Θ j ) involves O ( N ) computationsof (cid:101) R k ( S , T ) (refer to (16)). This suggests that (cid:101) R k ( S , T ) iscomputed O (cid:0) N (cid:1) times in lines 22 and 24. Finally, timecomplexity of Algorithm 1 is O (cid:16) N log (cid:16)(cid:98) Λ L (cid:17) + N (cid:17) .R EFERENCES[1] G. Saha, A. A. Abouzeid, and M. Matinmikko-Blue, “Optimal durationof spectrum lease: A mathematical approach,” in .IEEE, 2018, pp. 1–10.[2] V. Cisco, “Cisco visual networking index: Forecast and trends, 2017–2022,”
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Gourav Saha received a B.E. degree from AnnaUniversity, Chennai, India, in 2012, M.S.from IndianInstitute of Technology Madras, India, in 2015, andPh.D. from Rensselaer Polytechnic Institute, Troy,New York, in 2020, all in electrical engineering. Heis currently a postdoctoral scholar in the Depart-ment of Electrical and Computer Engineering. Hisresearch experience includes control systems, onlinealgorithms, game theory, and economics of wirelessspectrum sharing market. His current research in-volves various stochastic control problems related tomillimeter wave communication and cyber-physical systems.
Alhussein A. Abouzeid received the B.S. degreewith honors from Cairo University, Cairo, Egypt,in 1993, and the M.S. and Ph.D. degrees fromUniversity of Washington, Seattle, in 1999 and 2001,respectively, all in electrical engineering. He heldappointments with Alcatel Telecom (1994-1997),AlliedSignal (1999), and Hughes Research Labs(2000). Since 2001 he has been with the Electrical,Computer and Systems Engineering Department atRensselaer Polytechnic Institute where he is now aProfessor. From 2008 to 2010 he served as ProgramDirector in the Computer and Network Systems Division of the U.S. NationalScience Foundation (NSF). He was the founding director of WiFiUS, aninternational NSF-funded US-Finland virtual institute on wireless systemsresearch. He received the Faculty Early Career Development Award (CA-REER) from NSF in 2006, and the Finnish Distinguished Professor Fellowaward from Tekes (now Business Finland) in 2014-2019. He has served as anAssociate Editor for several IEEE and Elsevier journals, and on the organizingand technical committees of several IEEE/ACM conferences.
Zaheer Khan received the M.Sc. degree in electricalengineering from the University of Borås, Sweden,in 2007, and the PhD degree in electrical engineeringfrom the University of Oulu, Finland, in 2011. Hewas a Research Fellow/Principal Investigator withthe University of Oulu from 2011 to 2016. He helda tenure-track Assistant Professor position with theUniversity of Liverpool, U.K., from 2016 to 2017.He is currently working as an Adjunct Professorat the University of Oulu. His research interestsinclude design and implementation of advancedsignal processing, and data analytics algorithms for wireless networks onreconfigurable System-on-Chip (SoC) platforms, application of game theoryto model distributed wireless networks, use of machine learning for industrialasset monitoring and proactive network resource allocation solutions, andwireless signal design. arja Matinmikko-Bluearja Matinmikko-Blue