On multi-layer game-theoretical modelling of spectrummarkets and cognitive radio networks

FORTH-ICS TR 423, AUGUST 2011

Georgios Fortetsanakis1,2, Maria Papadopouli1,2, Luc Rey-Bellet5, Markos Katsoulakis3,4

1Department of Computer Science, University of Crete2Institute of Computer Science, Foundation for Research and Technology - Hellas

3Department of Applied Mathematics, University of Crete4Institute of Applied and Computational Mathematics,Foundation for Research and Technology - Hellas

5Department of Mathematics & Statistics, University of Massachusetts

ABSTRACTThis paper presents a powerful innovative multi-layer game the-

oretical modelling framework for the evolution of spectrum mar-

kets. It integrates models of the channel, client and network op-

erator (provider) paradigms, wireless infrastructures, types of in-

teraction, and price adaptation. This modelling framework incor-

porates an extensive set of parameters that allow the modelling of

various complex interactions of CRNs entities and business-driven

cases in multiple spatio-temporal scales in a realistic manner. It

also provides a modular simulation environment that implements

this framework to enable researchers to instantiate various mod-

els and perform comparative assessments of spectrum-sharing and

spectrum-provision mechanisms. To address several practical and

systems-based issues, we proposed two novel mechanisms: a price

adaptation algorithm for providers and the u-map, a user-centric

community-based mechanism that enables clients to record and

upload their feedback about their QoE during a call in a shared

location-based database. To illustrate how the proposed mod-

elling framework and simulation platform can be used, this paper

analyzes the evolution of a cellular-based market that uses the

proposed price adaptation algorithm. In the context of this mar-

ket, we also evaluated the u-map and show how it can improve the

network operator selection process. Finally, we discuss how this

framework and simulation platform can be extended to analyze

various spectrum markets and cognitive radio networks (CRNs).

1. INTRODUCTIONCognitive radio networks (CRNs), an emerging dis-

ruptive technology, aims to improve spectrum utiliza-tion, enabling dynamic spectrum use. This researchfocuses on the design of a complete multi-layer mod-elling framework of CRNs, incorporating both systemsand business aspects using statistical mechanics, gametheory and economics. Its main components include thechannel, network operators/providers, wireless networkinfrastructure and access, and clients.

A distinct characteristic in modelling the evolution ofsuch markets is the multiple time and spatial scales inwhich the various parameters, interactions, and decisionmaking mechanisms are manifested and/or need to bestudied. For example, the rate at which a network oper-ator changes its prices for its spectral resources is oftensmaller than the rate at which clients (e.g., primary orsecondary devices) demand for spectral resources or se-lect a provider. The modelling and simulation of suchmarkets in fine-detail (i.e., microscopic-level) results inan extremely large number of events, generated by theirentities and their (often complex) interactions. From amodelling and computational point of view however, itcan be very expensive to keep track of all the details andnot amenable to analysis. As shown in statistical me-chanics, it is actually unnecessary to keep track of alldetails, when mesoscopic or macroscopic phenomena,which involve a very large number of clients, need to beanalyzed. This observation motivated us to propose aninnovative, powerful multi-layer modelling frameworkand a modular simulation environment that implementsthis framework to enable researchers to instantiate var-ious models and perform comparative assessments ofspectrum-sharing or spectrum-provision mechanisms invarious spatio-temporal scales. The framework allowsresearchers to capture different networks, types of in-teractions, negotiation strategies, price-setting mecha-nisms, provider selection approaches, information avail-ability and trust among entities at the appropriate levelof detail.

Most of the proposed models for CRNs focus on a spe-cific sub-problem/aspect of CRNs omitting their inher-ent features. Specifically, they can be classified into twocategories, namely, the microscopic- and the macroscopic-level ones. The microscopic-level approaches considerinteractions among secondary devices at a very fine spa-

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tial level, mostly assuming a limited number of primarydevices due to the high computational complexity (e.g.,[1, 4, 12, 17, 5, 19, 18]). On the other hand, the macro-scopic approaches focus on the revenue of providers,considering only an average (over large temporal orspatial scales) behaviour of secondary devices (e.g., [8,7, 10, 11]). Unlike these approaches, this frameworkmodels the interactions of the entities at several spatialscales, from large metropolitan areas to small neigh-borhoods (e.g., within the coverage of a wireless accesspoint), enabling the instantiation of various parametersat different time granularities. Moreover, it provides theset of models and mathematical transformations thatallow to scale up or down a modelling environmentin order to analyze a certain phenomenon in mesoscopicor microscopic level. At the microscopic level of theframework, the various entities are modeled in fine tem-poral and spatial detail. On the other hand, the meso-scopic level exhibits various aggregations. In a coarse-graining procedure that results to a loss of informationin a controlled and hierarchical fashion [9], the individ-ual entities (e.g., clients) are replaced by local aver-ages over space and/or time. For example, clients aremodeled as a population with certain attributes, com-puted as spatial averages of the characteristics of theindividual users of that population. To the best of ourknowledge, it is the only modelling framework that at-tempts to incorporate such an extensive set of parame-ters that allow the modelling of various complex inter-actions of CRNs entities and business-driven cases inmultiple spatio-temporal scales in a realistic manner.

Providers and clients (i.e., agents that sell/subleaseor buy spectral resources, respectively) are modeled in agame-theoretical framework. Updating mechanisms arechosen to model their learning and decision mechanismsand can be instantiated by a rich set of strategy updat-ing rules (e.g., best-response, perturbed best-response,mutation, imitation). The choice of utilities, spatialstructure and (stochastic) updating mechanisms definecompletely a multi-agent evolutionary game.

Two important aspects of this framework are theamount of information available to agents and the typeof their interactions. This paper makes several contribu-tions both in systems and modelling; First, it presentsthis innovative multi-layer game-theoretical frameworkand discusses the theoretical issues and properties (inSection 2). Then, it describes a simulation platformthat implements this framework and can instantiatedifferent spectrum and cognitive radio markets (in Sec-tion 3). As spectrum and cognitive radio markets evolve,various mechanisms that improve the wireless accesshave been proposed. We envision a novel system thatenables clients to upload their feedback/measurementsabout providers and their quality-of-experience (QoE)of their access in a shared map-based database (super-

imposing their reports on the region of the map thatcorresponds to their location). Based on this feedbackand statistics on these measurements, in a grass-root,user-centric fashion, a QoE/coverage map can be built.We call this system u-map. Clients may use statis-tics provided in this map to select a network opera-tor. Network operators could potentially employ suchstatistics to improve their network (capacity planning).Furthermore, they can use the collected statistics (e.g.,about user characteristics, requirements and popula-tions), to instantiate the market in the modelling frame-work/simulation platform and analyze its evolution. Toillustrate how the proposed framework and simulationplatform can be used, we analyzed the evolution of themost typical spectrum provision paradigm, a cellular-based market (in Section 3). In the context of this mar-ket, addressing practical and systems-based issues, weproposed different access paradigms (e.g., subscribersand rechargeable card users) and a novel price adap-tation algorithm. Section 4 analyzes the evolution ofthis market under different client populations and in-formation availability (e.g., u-map) scenarios. Finally,Section 5 summarizes our main findings and future workplans.

2. MODELLING FRAMEWORKThe main entities of the modelling framework are the

clients, providers, and channels. Each provider owns aspectrum band that consists of a number of discreetchannels and offers them for an appropriate remunera-tion. A client requests for a channel from a provider.

Consider a set of clients U and a set of providers P .Each provider (e.g., p P ) maintains an infrastructureof base stations BSs (set Bp). The set of channels (of allproviders) in the spectrum is indicated by Y , while theset of available transmission power levels is T . Denoteas B the set of all BSs (of all providers) (B = pPBp),and BYp = {(x, y) Bp Y : channel y is allocated toBS x} the set of BS and channel pairs that belong tothe provider p P . Assume that BY = pP BYp.

Each client u is characterized by two constraints, namely,the price tolerance threshold Cmax(u), i.e., the maximalprice that the client u accepts to pay, and the targettransmission rate (Rtar(u)) i.e., the minimal transmis-sion rate that the client u aims to achieve. When thesetwo constraints cannot be satisfied by any provider, theclient chooses to remain disconnected. BSs and clientshave a physical location in a region A R2. The func-tion L : U B A R2 maps each client and each BSto their corresponding locations.

The set of strategies of a client u is Su = BY T{e}.We represent the strategy of a client u U as a triplet(u) = (b(u), c(u), (u)), where (b(u), c(u)) BYare the BS and channel, respectively, to which the clientu connects, while (u) is the value of transmission

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Table 1: Main modelling parameters

Notation DescriptionA region with client/BS distributionL mapping of client/BS to location in AU set of clientsB set of BSsP set of providersY set of channelsT set of trx power levels

Providers(p) strategy of provider pBp BSs belonging to provider p PBYp

{(x, y) Bp Y : channel yis allocated to BS x}

BY pPBYpSp set of strategies

F (p;, ) utility function of provider pClients

(u) strategy of client ub(u) BS to which u connects (u) transmission power of uc(u) channel to which u connectsI(u; ) observed interference when u selects (u)R(u; ) achievable transmission rateC(u; , ) financial cost of selecting (u)Cmax(u) price tolerance of uRtar(u) target transmission rate of uSu set of strategies of u

F (u;, ) utility function of client u

power that this client invests. Moreover, the strategy erepresented by the triplet (0, 0, 0) indicates the discon-nection state. The set of strategies of a provider p isSp = {sp,1, sp,2 ..., sp,Mp}, where sp,i is the ith choiceof price of the provider p.

The state of the system 1 corresponds to the positionsof all clients and BSs (given by the function L) and thestrategies of all providers and clients = ((p))pP ,where (p) Sp for each p P , and = ((u))uU ,where (u) Su for each u U .

Each provider seeks to optimize its revenue by in-creasing both market share and price. The utility func-tion F (p;, ) for a provider p P , given strategies and for providers and clients, respectively, is given by

F (p;, ) =uU

(x,y)BYp

(b(u) x, c(u) y) (u)(p)+uU

(x,y)/BYp

(b(u)x, c(u)y)(Cmax (u)(p))

1Note that the ;, (or ;,n) indicates the state of thesystem.

where Cmax is the average maximum price that clientscan tolerate, i.e., Cmax = |U |1

uU Cmax(u), and

(, ) is a function (Kroneckers delta) that returns 1when both arguments are zeros, otherwise the value 0.The first term in the utility F (p;, ) expresses the ob-jective of a provider to charge the maximal price to itscustomers, while the second term expresses the objec-tive to increase the number of its clients. To penal-ize an aggressive increase of the transmission power,the providers adopt a pricing scheme that charges theclients proportionally to the transmission power theyinvest.

Each client u U has two competing objectives,namely, to maximize the achievable transmission rateR(u; ), and to minimize the financial cost C(u; , ) toachieve this transmission rate. The transmission rateR(u; ) is computed based on the Shannon capacitytheorem, although more sophisticated models that takeinto consideration the modulation schemes can also beincorporated easily [13]:

R(u; ) = B log2(

1 + (u)G(L(u), L(b(u)))

I(u; )

),

where B is the channel width and G(x, x) is the channelgain between a transmitter and a receiver at positionsx and x, respectively. The channel gain has the formG(x, x) = PL(d0)(x x2/d0)nX, with a path lossexponent n, the channel gain PL(d0) at the referencedistance d0, and the shadowing parameter X, a log-normal random variable of zero mean. The amount ofinterference power that the client u observes when itconnects to the BS b(u) at the channel c(u) is givenby the following formula:

I(u; ) =

{u:c(u)=c(u)} (u

)G(L(u), L(b(u))) (1)

The financial cost is given by C(u; , ) = (u)(p(b(u))),where the mapping p(b(u)) returns the provider towhich the BS b(u) belongs. Note that the utility func-tion can be extended to incorporate more user pref-erence aspects. For example, if QoE evaluations areavailable (e.g., via the u-map, as described earlier), theobjective of a client can be described as a weighted2

sum of the transmission rate, the financial cost, andthe QoE indicator, as follows:

Z(u; , ) = a1R(u; ) a2C(u; , ) + a3q((u)) (2)

To complete the definition of the payoff function of aclient, we must also take into consideration the afore-mentioned client constraints. The utility function of a

2The role of the weights is to perform a prioritization andunit transformations among the various objectives.

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client u U , F (u;, ) can now be defined as follows:

F (u;, ) =

Z(u; , ) if (u) 6= e,R(u; ) Rtar(u),

C(u; , ) Cmax(u),

l 1 if (u) = e

2 otherwisewhere l = inf

u,,Z(u; , ) and 1, 2 are positive numbers

with 2 l 1. Note that this utility function as-sumes that the client aims to remain always connectedas long as its criteria (on price and target rate thresh-olds) are satisfied. Only if any of the thresholds arenot satisfied, the client chooses to remain disconnected(2 l 1). Observe that in a more general case, aclient may prefer to stay disconnected, e.g., under theabsence of traffic demand or limited battery. The util-ity function can be easily generalized to consider suchcases by introducing additional constraints.

Our models are agent-based, in the sense that theyemploy detailed properties and preferences of all clientsand providers in an individual basis. Throughout thepaper, the word agent is used to indicate a client ora provider. This approach makes very easy the integra-tion and enrichment of these models with, as many asdesired, additional individual client or provider char-acteristics by augmenting their strategy set with newvariables and defining new corresponding utility func-tions.

Dynamical updatingOnce the utilities of the agents have been specified, wehave defined a multi-agent game in the sense of tra-ditional game theory [16]. We will follow further theevolutionary game theoretic approach [14] and com-plement the game by specifying dynamical (stochas-tic) rules with which providers and clients update theirstrategies. The updating mechanisms are chosen tomodel the learning and decisions mechanisms of agents.In the context of evolutionary games, one assumes thatthe updating rates are based essentially only on the util-ity functions. In particular, in traditional game theory,one often assumes that every agent knows the strate-gies of all the others to compute its own utility func-tion. This is a strong assumption which is very usefulin modelling but unrealistic in practice. We will furtherdiscuss and relax this assumption later on.

(a) Updating times The updating can occur at dis-crete times intervals t which, in general, may dependon the type of agents. We consider here instead con-tinuous time dynamics: each agent is equipped with itsown clock, i.e. an exponential random variable with arate which determines the time at which it will updateits strategy. The memoryless property of the exponen-tial random variable ensures that this gives rise to a

continuous time Markov process.(b) Updating rates A change in the strategies of

providers is indicated by the transition p,,where p, is the same as with the difference thatprovider p has changed its strategy from (p) to .

p,(p) ={(p) if p 6= p Sp if p = p

One can further restrict the range of the values that can take in each update, for instance, in a neighborhoodof the current value (p), in order to model an aversionto abrupt changes in the pricing. Similarly, one canincorporate more elaborate schedules in the selection ofthe new price , depending on the modelling needs.

A change in the strategies of clients is indicated bythe u,J , where u,J is the same as withthe difference that the client u has switched from thestrategy (u) to J .

u,J(u) =

{(u) if u 6= uJ = (Jb, Jc, J ) Su if u = u

In the context of cognitive radios, it is natural to restrictthe values of the coordinates of BS Jb of the new con-figuration J = (Jb, Jc, J ) to be in the neighborhoodof the current location of a client (e.g., L(u)). Onespecifies updating rates cp(,

p,; ) for providers andcu(,

u,J ;) for clients. In most examples consideredhere, the transition rates between two states depend onthe payoff of both states through an appropriate func-tion (e.g., gp for providers and gu for clients)

cp(, p,; ) = gp(F (p;, ), F (p;

p,, )) (3)

cu(, u,J ;) = gu(F (u;, ), F (u;,

u,J)), (4)

where the functional forms gp and gu (e.g., Gibbs sam-pler, Metropolis) correspond to the strategy updatingrules that are defined below. The generator of thecontinuous-time Markov process has then the form

Lf(, ) = prLprf(, ) + clLclf(, )where the rates pr and cl specify the time scales atwhich clients and providers update their strategies and

Lprf(, ) =pP

Sp

cp(, p,; ) [F (p;p,, ) F (p;, )]

Lclf(, ) =uU

JSu

cu(, u,J ;)

[F (u;, u,J) F (u;, )]

This can also be easily generalized for other types ofupdating mechanisms. 3

3A generator is a compact way to write the Kinetic MonteCarlo (KMC) algorithm that implements the Markov pro-cess based on an efficient calculation of the waiting timesbetween updatings and the transition probabilities at theupdatings.

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(c) Updating rules We consider here a number ofuseful rules, but other choices are possible, see e.g. [15].

1. Best-response dynamics This dynamics is thenatural choice to model rational agents: each agent eval-uates the utility functions for all the accessible strate-gies and chooses the one with the highest possible pay-off. If K strategies are accessible with payoffs F1, FKthen upon updating it chooses the strategy argmaxFiwith a suitable tie-breaking rule. For example, in ourmodel, a client would search for all available BSs andchannels in its neighborhood, and all possible transmis-sion power levels, and choose the more advantageousone (including the disconnection). This may lead tonontrivial optimization problems.

2. Generalized Gibbs sampler This dynamicsmodels an agent who tends to act in a rational mannerbut makes mistakes which are encoded in a param-eter 0. If an agent has K strategies with payoffsF1, FK , then upon updating, it will choose strategyj with probability eFj//

Ki=1 e

Fi/. Note that as tends to 0 the Gibbs sampler dynamics rule tends tothe best response dynamics.

3. Generalized Metropolis dynamics The Metropo-lis rule is well adapted to model rational agents whichhave a very large number of strategies to choose from.Rather than evaluating all the possible strategies, theagent picks one strategy at random, say strategy j, andcompares it to its current strategy, say strategy i. Up-dating to strategy j occurs with probability max{1,e(FjFi)/}, that is the agent always updates to bet-ter strategies but makes mistakes with an exponentiallysmall probability. As tends to 0, the Metropolis dy-namics tends to a dynamics where agents pick a newstrategy, and then update it, if it corresponds to ahigher payoff.

4. Imitation dynamics The agent picks one of itspeers and imitates it, if the peer strategy is more favor-able than its own. In the context of spatial games, onespecifies a probability to choose any given one of its peer(depending for example of the distance between agents),and the agent updates with a rate max {0, Fj Fi}, ifits current strategy is i and its peer strategy is j.

5. Mutation dynamics Often it is natural to add acertain drift (in the biological sense) to the strategiesof agents. In that case, the agent if it has currentlystrategy i, upon updating, keeps its current strategywith probability 1 and drifts to a randomly chosenstrategy with probability .

The different updating mechanisms can be easily com-bined in a flexible way: clients and providers may havedifferent updating rules from each other; a given agentmay update by combining various behaviors, e.g. imi-tation and mutation, or imitation and best response; ordifferent groups of clients can have different updatingmechanism.

Incomplete informationThe previous dynamics consider a well-defined modelof the payoff of each provider, which is used to per-form the price adaptation. As discussed earlier, thisis a strong and unrealistic assumption, which we willrelax. Specifically, we propose a novel price determina-tion/adaptation algorithm which assumes that providersknow only their own prices and the prices of their com-petitors and measure their own revenue. That is, weassume that providers do not a priori know the util-ity that corresponds to a price they may offer in themarket. Let us introduce now this new dynamics:

6. Polynomial-based approximation of payoffA provider performs a novel price adaptation algorithmbased on a second-degree concave polynomial approxi-mation of the payoff function and estimates its param-eters based on its own history of the game evolution.This approximation is simple yet appropriate to cap-ture the mathematical properties of the payoff functionof a provider. Specifically, each provider keeps track ofthe last combinations of prices that have been offeredas well as the corresponding values of its revenue. It pe-riodically fits the polynomial to the recently collecteddata by solving a least-square problem with the addi-tional constraint that the polynomial is concave, formu-lated as a semi-definite program. The price is adaptedby moving towards the direction of the partial deriva-tive of the polynomial that corresponds to that specificprovider and with a certain step. Providers adapt theirprices at time instances generated via a stochastic pro-cess (for example Poisson distribution). Furthermore,notice that in the previous models (e.g., Gibbs sam-pler, metropolis), agents (e.g., clients) choose strategiesat times generated by a Poisson process. In contrast,in this dynamics, agents may consider different distri-butions. In principle, to include such generalized dis-tributions in our modelling framework is possible (e.g.,general waiting times have been used in other queue-ing/network models) but we shall not pursue this fur-ther in the modelling part of this paper. This dynamicshas been fully implemented and evaluated in our sim-ulations (presented in the next sections). It also natu-ral to allow client mobility. This can be implementedfairly easily in our theoretical model and has been im-plemented in the simulations.

Convergence and stabilityThere are few if any general results on the convergenceof the stochastic algorithms at the level of generalityused here. Let us assume that the number of clientsand providers is finite so that the dynamics is given bya finite state Markov chain. Furthermore, if agents up-date strategies in such a way that all of the states ofthe Markov chain can be reached by a sequence of up-dating of the various agents (irreducibility), then the

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Markov chain converges to a unique steady state fromany initial condition (ergodicity of the Markov process).A Markov chain with perturbed best response (gener-alized Metropolis and Gibbs sampler) is naturally ir-reducible, if well-designed. The fact that unfavorablestrategies are chosen (with exponentially small proba-bilities) makes it usually fairly easy to verify the irre-ducibility of the Markov chain. If the dynamics con-tains exclusively imitative mechanisms, then the dy-namics has the intrinsic property that no new currentlyunused strategy is introduced in the system. For ex-ample, imagine that clients (quite unrealistically) up-date their strategies only by imitating their neighbors,then a newly entered provider would never gain any newclients. In this case, one does not expect convergence toa unique steady state, but rather the existence of severalabsorbing states. For example, in our model, theseabsorbing states could correspond to the extinctionof a provider. However any level of rational behavior,or mutation of clients will ensure a stable algorithm. Ifthe updating is based uniquely on best-response mecha-nisms, then intuitively, the Markov chain has very littlestochasticity (most updates occur with probability one).The question of convergence for these systems can beextremely difficult to study, especially if its number ofstates is very large. One should remark further thatin the limit where the number of agents tend to infin-ity, a number of interesting phenomena can occur. Onemay observe various phase transition, for example onemay loose stability and uniqueness of the steady statefor perturbed best-response rules, or in the contrary forimitative dynamics one may regain some form of ergod-icity and observe complex steady states, see. e.g. [2].

Coarse-graining and population gamesThe choice of utilities, spatial structure and (stochas-tic) updating mechanisms define completely a multi-agent evolutionary game. This is an extremely pow-erful tool to model the actions of agents in as muchdetail as wished. However, as mentioned earlier, froma modelling and computational point of view, it can bevery expensive, if at all possible, to keep track of allthe details about strategies of every single agent, andactually unnecessary, if one is interested in macroscopicor mesoscopic quantities which involve averages overmany clients. We shall describe here a reduced modelwhere we keep full track of the strategies of the typicallyvery few providers but coarse-grain the clients, con-structing an evolutionary population game model [14]which describes the time evolution of local populationsrather than individual clients. This coarse-graining pro-cedure corresponds to a loss of information in a con-trolled (hierarchical) fashion, [9], replacing individualagents/quantities by local averages over space.

We make the following assumptions: (a) The popu-

Table 2: Coarse-grained model parameters

Notation DescriptionD(k) Region kn(k) Number of clients in D(k)

U(k, b, i) representative client in D(k)at BS b with trx power i

n(k, b, i)Number of clients in D(k) connected

at BS b with trx power in vector describing client populationsU(k) representative client in region D(k)G(k, b) avg channel gain btwn U(k) & BS b

R(k, b, i; n) avg achievable trx rate of U(k, b, i)I(k, b; n)

avg interference experiencedby U(k, b, ) connected at BS b

F (k, b, i;,n) utility of U(k, b, i)F (p;,n) utility of provider p

lation in the region A is naturally divided in subgroupslocated in the regions D(k) with k = 1, ,K. Forexample, one might imagine that populations are nat-urally divided in neighborhoods with different densitiesor transmission power. We have A = Kk=1D(k) and de-note by n(k) the total number of clients in region D(k).For the sequel, we assume that n(k) is fixed in time,although this could be easily modified. We only keeptrack of the region of a client (e.g., D(k)) and not itsspecific position (L(u)).

(b) Rather than keeping track of the transmissionpower of each client, we divide the transmission powerrange T into two subsets: the low and high transmissionpower ranges (T = Tlow Thigh), and assign two repre-sentative transmission power levels low and high, re-spectively. We assign to each client a label i {0, low, high},where 0 indicates the no transmission. Of course,this could be easily generalized to more than three lev-els, if desired.

(c) We stop keeping track of channels choice of clients,and simply, record the BS b to which clients are con-nected. We define new dynamical variables n(k, b, i) toindicate the number of customers in region D(k) con-nected to BS b and investing a transmission power i,i.e.,

n(k, b, i) =

u :L(u)D(k), (u)Ti

(b(u) b) (5)

for b B and k = 1, ...,K, i {0, low, high}. Wealso set n(k, 0, 0) to denote the number of disconnectedclients in the region D(k).

(d) In order to estimate the interference, recall thateach provider p P is assigned a specific frequencyband that consists of a number of discrete channels andthat frequency bands of distinct providers may be over-lapping. Furthermore each provider allocates channels

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to BSs according to a frequency reuse scheme. The setof channels that are allocated to a specific BS b B ac-cording to this scheme, is denoted as Yb. The numberof channels |Yb Yb |, which are shared by two BSs willbe instrumental to determine the interference.

(e) We replace the channel gain by a suitable average:

G(k, b) =1

n(k)

u:L(u)D(k)

G(L(u), L(b)) .

Coarse-grained payoff By performing various av-erages we now express the payoff exclusively in termsof the vector n describing the various client popula-tions n = (n(k, b, i)),with k {1, ,K}, b B, i {0, low, high} . We approximate the utility F (u;, )by a new function F (k, b, i;,n) which is the utilityof a representative client in region D(k) that connectsto BS b and invests a transmission power i. Assumethat Rtar(k) and Cmax(k) are the average target trans-mission rate and the average price tolerance thresh-old of the population of clients at region D(k), respec-tively. First, we set the achievable transmission rateof a representative client in the region D(k) that isconnected to BS b using transmission power i to be

R(k, b, i; n) = B log2(

1 + iG(k,b)I(k,b;n)

)where

I(k, b; n) =b 6=b

k

j

jG(k, b)

n(k, b, j)|Yb |

|Yb Yb||Yb|

(6)The interference I(k, b; n) (defined in the above Eq.) ex-perienced by a representative client in the region D(k)connected with BS b is derived as follows: As part ofthe coarse-graining, one assumes that a client selects achannel uniformly at random (from the set of channelsof a BS), when it connects to that BS. Let assume aclient of type j in D(k), connected at BS b. It con-nects to an arbitrary channel of BS b with probabilityn(k,b,j)|Yb | . Then jG(k

, b)) is the interference that itcauses to a representative client in D(k) which is con-nected to the same channel at the BS b. The repre-sentative client in D(k) picks the same channel whenit connects to BS b as the first client with probability|YbYb||Yb| .The utility function of a representative client in re-

gion D(k) connected at BS b, with transmission level iis as follows:

F (k, b, i;,n) =

Z(k, b, i;,n) if b 6= 0,R(k, b, i; n) Rtar(k),C(b, i;) Cmax(k)

l 1 if b = 0

2 otherwise

where l = infk,b,i;,n

Z(k, b, i;,n) and

Z(k, b, i;,n) = a1R(k, b, i; n)a2C(b, i;)+a3q(k, b, i).The cost that the representative client pays when con-nected to BS b using transmission power i is C(b, i;) =i(p(b)) and finally the utility function for a provider pconsidering the client populations is defined as following

F (p;,n) =bBp

k

i

n(k, b, i)i(p) (7)

+b /Bp

k

i

n(k, b, i)(Cmax i(p))

Coarse-grained dynamicsWe can easily derive a coarse-grained dynamics for thevariables and n. As in the microscopic case, the defi-nition of dynamics entails two steps: defining an updat-ing mechanism and a corresponding rate. An updatingoccurs at the transition of a client in some region D(k)from one BS b and type i to another BS b and typei with (b, i) 6= (b, i) (recall that b = 0 means dis-connected). Therefore, the transitions are

n 7 n = nl,(bi)(bi) (8)where j {1, ,K}, b, b B, i, i {0, low, high},and

nj,(bi)(bi)(k, b, i) := n(k, b, i) if j 6= kn(k, b, i) 1 if j = k, b = b, i = i

n(k, b, i) + 1 if j = k, b = b, i = i

Since the utilities depend only on the variables andn, then the updating rates can also be expressed onlyin terms of the variables and n as in (3) and (4),based on the general strategy updating rules, such asbest response, Gibbs sampler, and the coarse-grainedpayoffs.

The convergence and stability properties of the coarse-grained stochastic processes are closely related to theones of the original agent-based. Under stochastic rulessuch as Metropolis and Gibbs sampler one obtains aunique stable steady state and the system converges tothe steady state for any initial condition. The samecaveat about imitation and best-response dynamics ap-ply here too.

It is important to observe that the coarse-grainingand population games methodology does not simplyperform an averaging, reducing the initial client pop-ulation size. The population size remains the same.Instead it provides the set of mathematical transfor-mations and models for creating various populations,scaling up certain parameters to reduce the unneces-sary information.

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3. SIMULATION PLATFORMThe modelling framework is parameterized based on

the channel, infrastructure and network topology, typeof users (e.g., requirements, demand, mobility, profile),providers (e.g., access protocol, price adaptation mech-anism), and their interactions, client/provider distribu-tions, mobility, and available information. The simula-tion environment based on this framework is modular,in that, it can instantiate and implement different mod-els for these parameters. For example, the channel canbe modeled using large-scale propagation models (e.g.,path-loss and shadowing) and small-scale models (e.g.,multi-path fading).

Providers This work considers the cellular topolo-gies of two providers, owners of spectral resources, thatoffer wireless access via their BSs to clients in a smallcity. Furthermore, we assume that the providers dividetheir channels into time-frequency slots according to aTDMA scheme.

Channel To simulate the channel quality, we em-ployed the Okumura Hata path-loss model for smallcities. Moreover, the contribution of shadowing (ex-pressed in dB) to the channel gain at the positions ofBSs follows a multivariate Gaussian distribution withmean 0 and covariance matrix defined in Eq. (9).

C(i, j) =

2s if i = j,

2seL(i)L(j)/Xc if i 6= j,

(9)

where s is the standard deviation of shadowing (2.5dB in our simulations), Xc is the correlation distancewithin which the shadowing effects are correlated [6],and L(i), L(j) are the positions of the BSs i and j, re-spectively.

To model the effect of angular correlation of shadow-ing, we represent each BS using six points, locatedon a circle with center the BS position. When a clientcommunicates with a specific BS, the contribution ofshadowing to the channel gain is equal to the value thatcorresponds to the point representing the BS, whose di-rection is the closest to the direction of arrival of thesignal [3].

The interference power at a BS during a time fre-quency slot is computed considering the contributionof all interfering devices at cochannel BSs. Moreover,cochannel BSs of the same provider may not be synchro-nized, resulting in overlapping time frequency slots,and thus, in devices that cause interference during morethan one slots.

Client populations Two types of client populationsare present: the card users and the subscribers. A carduser selects a BS at the start of each call (dynamically),while subscribers choose their provider upon their ar-rival in the region and connect only to BSs of that

provider for the remaining duration of the experiment.A client is characterized by a price-tolerance thresh-

old and a target transmission rate threshold (as de-scribed in Section 2). Based on their preference, clientscan also be distinguished in two categories, namely theprice-preference and transmission rate-preference ones.In rate preference, clients aim to optimize only theirtransmission rate when selecting a BS, given that thisBS can satisfy both the price tolerance and target trans-mission rate thresholds (e.g., a1 = 1, a2 = 0, a3 = 0in Section 2, Eq.2). Clients with price preference aimto minimize the cost of acquiring a time-frequency slot,when selecting a BS, given that this BS can satisfy boththe price-tolerance and target-transmission rate thresh-olds.

Client demand Clients generate requests to con-nect to a BS (i.e., calls). The duration of calls and dis-connection periods are given by appropriate stochasticprocesses.

Client mobility Clients move with pedestrian speed.The position of a client at the end of a disconnectionperiod is chosen randomly from a circular region withcenter the position of the client at the beginning of thedisconnection and radius the maximum possible trav-elled distance during this period. If a client ends upin a position outside the borders of the city it is re-flected back in the city. The simulation platform canbe extended to consider other mobility models (e.g., ve-hicular mobility) and mobility traces.

User-centric QoE map (u-map) The u-map is adata structure that corresponds to a grid-based rep-resentation of a region. Each cell of the grid storesstatistics about the providers and the QoE of calls.In the context of this analysis, we assume that thecell size corresponds to the simulated small city. Atthe end of a call, a card client reports the numberof available time-frequency slots of the closest BS ofeach provider. Subscribers also report the number ofavailable time-frequency slots of the closest BS of theirproviders. This information is uploaded and stored ina centralized database (u-map). Based on this infor-mation, the average spectrum availability i.e., numberof available time-frequency slots of a BS of a provider,averaged over all collected measurements, is computed.Subscribers select the provider with the highest aver-age spectrum availability. In this paper, the main pur-pose of the u-map is to reduce the call blocking prob-ability. In a more general context, different type (e.g.,QoE-based) of measurements can be recorded on the u-map, the cell size of the u-map may vary, and differentBS/provider selection mechanisms can be employed toimprove the QoE of a user.

4. PERFORMANCE EVALUATION

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4.1 Simulation scenariosWireless network infrastructure The simulation

platform considers a small city, represented by a grid of11 Km x 9 Km. Each provider has a cellular networkthat consists of 49 BSs placed on the sites of a triangulargrid, with a distance between two neighboring sites of1.6 Km. Moreover, each provider owns bandwidth of5.6 MHz, that is divided into 28 channels of 0.2 MHzwidth. These channels are allocated to BSs accordingto a frequency reuse scheme with spatial reuse factorsof 4 and 7, for Provider 1 and Provider 2, respectively.The closest BSs at the same frequency band as a givenBS in a topology with a spatial reuse factor of 4 can belocated by moving two steps towards any directionon the grid. On the other hand, in a topology with aspatial reuse factor of 7, by moving two steps towardsany direction, then turning by 60 degrees, and movingone more step, the closest BSs at the same frequencyband as a given BS can be located. This is illustratedin Fig. 1. Each channel is further divided into threetime-frequency slots in a TDMA scheme, resulting in 21time-frequency slots per BS of Provider 1 and 12 slotsper BS of Provider 2. Note that a single time-frequencyslot of a given BS can be offered to only one client. Also,the demand of each client is exactly one slot.

We consider two different BS deployments, namely,the uniform deployment, in which the network of eachprovider covers the entire city, and the non-uniform de-ployment, in which six BSs (out of 49) of provider 2 areremoved. Clients located in the neighborhood of theremoved BSs can buy spectral resources only from theprovider 1. This is an example of a partial monopoly,in the sense that there are regions in which clients haveonly the option of connecting to BSs of a single provider.

Providers use the polynomial-based approximation oftheir payoff function (as described in Section 2) to de-termine the prices for card clients dynamically, at timeinstances generated by a Poisson process with a meanrate 0.002 renewals per minute. We assume that bothproviders offer the same prices to subscribers, whichremain fixed through the entire duration of the experi-ment. This is a reasonable assumption, given the timeduration (30500 minutes or 21 days) and scale of theseexperiments. To penalize an aggressive increase of thetransmission power, the providers adopt a pricing schemethat charges the clients proportionally to the trans-mission power they invest (as described in Section 2).Moreover, the maximum allowable transmission powerthat a client can invest is 2 Watts.

Client populations There are 5000 clients in to-tal (4400 card users and 600 subscribers), distributedaccording to a uniform distribution in the simulated re-gion of this small city. In our experiments, the pricetolerance threshold (in euros per minute) and targettransmission rate (in Mbps) follow a Gaussian distri-

0 2 4 6 8 100

2

4

6

8

Distance (Km)

Dis

tanc

e (K

m)

Cochannel BSs

(a)

0 2 4 6 8 100

2

4

6

8

Distance (Km)

Dis

tanc

e (K

m)

Cochannel BSs

(b)

Figure 1: Closest BSs using the same frequency bandwhen the spatial reuse factor is 4 (left plot) and 7 (rightplot).

bution. Specifically, we simulated client populationswith normal price tolerance (m = 0.15, = 0.0375)and high price tolerance (m = 0.2, = 0.0375). Wealso simulated client populations with a normal targettransmission rate (m = 0.1, = 0.01) and high targettransmission rate (m = 0.2, = 0.01).

Client demand A client generates a sequence ofcall requests. The call duration follows a Pareto dis-tribution (xs = 3.89, a = 4.5) of mean 5min, while thedisconnection period follows a Log-normal distribution(m = 3.22, = 0.37) of mean 27min. We assume thatduring disconnection periods, clients move with pedes-trian speed of maximum value 1 m/sec, while they re-main stationary during calls. Furthermore, during acall, the client remains connected at the same BS forthe entire duration of the call. Fig. 2 shows a snap-shot of the network topology. Specifically, it representsthe BS deployment and all clients with a call at thatparticular time instance.

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

Distance (Km)

Dis

tanc

e (K

m)

Network topology

Provider 1 BSsProvider 2 BSsClients in active periodRegion of interest

Figure 2: A snapshot of the network topology.

Region of interest To avoid the effect of boundaryconditions, each provider takes into consideration onlythe interactions between BSs and clients located in asmall rectangular region, corresponding to the centerof the city (marked as region of interest, the inner

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(a) (b) (c)

(d) (e) (f)

Figure 3: Main results for a cellular-based market. Providers: Revenue (a) and (b). Spectrum utilization (d) and(e). Clients: Percentage of blocked calls (c) and (f). Averages over 10 simulation experiments, each lasting 30500min.

rectangle shown in Fig. 2). The region of interestincludes 9 BSs of each provider. In the non-uniform BSdeployment (partial monopoly), two BSs of the provider2 are removed from the region of interest and four BSsare removed from the remaining area. Only the BSslocated in that region and calls originated from thatregion are considered in the price adaptation algorithm.

Metrics This analysis will evaluate the impact ofclient characteristics and preferences, BS distribution(presence of partial monopoly) on the performance ofproviders and clients.

The performance of a provider is characterized by itsrevenue and spectrum utilization while the performanceof a client is indicated by the percentage of blocked calls.The revenue of a provider corresponds to the averagetotal revenue of all BSs in the region of interest thatbelong to that provider, averaged over all Monte-Carloruns. The spectrum utilization of a BS corresponds tothe average percentage of time frequency slots allocatedto clients. The spectrum utilization of a provider cor-responds to the average utilization of all its BSs in theregion of interest, averaged over all Monte-Carlo runs.The percentage of blocked calls of a client is the ratioof its successful calls over the total number of call re-quests. Our reported results are average statistics overall clients.

We implemented the simulation platform and thismarket in Matlab. 10 Monte Carlo runs were performed

for each scenario (shown in Table 3). Each scenariosimulates a homogeneous client population with respectto preference and thresholds. Specifically, P scenarioscorrespond to a price-preference population, while Rscenarios to a rate preference ones. For each scenario inTable 3, we simulated two client populations, one withprice-preference (P) and another with rate-preference(R) (Fig. 3). Note that in partial monopoly (rnd) sce-narios, subscribers select randomly their provider. Eachrun represents the evolution of the market in the micro-scopic layer and lasts 30500 minutes (including a warmup period of 500 min). Compared to clients, the deci-sion making and updating times of providers occur inlonger time scales. The relatively long duration of oursimulations is required in order to better observe theevolution of providers and their interaction with clientsin this simulated small-city environment. As describedin Section 2, the mesoscopic and macroscopic layers en-able us to assess the evolution of such markets in evenlarger spatial and temporal scales.

Table 3: Description of Scenarios

ScenarioPrice Rate BS u-map

threshold threshold deployment usedNormal normal normal uniform yes

High price tolerance high normal uniform yesHigh target rate normal high uniform yesPartial monopoly normal normal non-uniform yes

Partial monopoly (rnd) normal normal non-uniform no

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4.2 AnalysisIn general, price preference (P) triggers a more in-

tense competition among providers than rate preference(R). This results in relatively lower prices: fewer userswill be blocked due to their price tolerance threshold(Fig. 3 (c)&(f)). Furthermore, in rate preference (R),the revenue is much larger (an order of magnitude) thanin price preference (P), in which the competition be-tween providers forces them to keep their prices rela-tively low (Fig. 3 (a)&(b)). In rate preference, clientstend to buy with a price equal to their maximum pricetolerance threshold (in order to increase their transmis-sion rate), while in price preference, clients are moreconservative (in that they aim at paying the minimumpossible price to achieve the targeted transmission rate).We observed similar trends in scenarios with stationaryusers that are always on call (with a continuous de-mand for access that lasts during the entire simulation).These results have been omitted due to lack of space.

In the case of increased target rate, as expected, theblocking probability also increases (Fig. 3 (c)). In-terestingly, in rate preference, the revenue of providerswill decrease. This is due to the fact that, although inrate preference scenarios, clients invest their maximumtransmission power that satisfies the price threshold inorder to achieve the highest possible data rate, for hightarget rates, fewer clients will achieve their target rate,and therefore, the blocking probability will increase, re-sulting to a smaller revenue and spectrum utilization.

In price preference scenarios, clients select the leastexpensive BS (if any) that satisfies their rate and priceconstraints. As the target rate increases, the price-based selection criterion deteriorates, since a clientwill tend to select more frequently the BS that is clos-est to it (i.e., BS with the best channel quality) thanthe least expensive one (compared to lower target ratescenarios) in order to satisfy the increased data raterequirement. This allows providers to increase theirprices, and thus, their revenue. (Fig. 3 (a)). Note thatas the target rate increases in price preference scenarios,the BS selection mechanism exhibits more similarities asin rate preference scenarios (i.e., clients tend to choosethe BS with the best channel quality). As observed alsoin rate-preference, the blocking probability is increased,which results to smaller spectrum utilization.

In rate preference, as the price threshold increases, wewould expect that the blocking probability decreases,while the spectrum utilization also increases (Fig. 3 (c)and (d)). Interestingly, these changes are small, due tothe inter-dependency of the price tolerance threshold ofclients and price setting mechanism of providers. Theincrease of price tolerance threshold allows providersto increase their prices even further. The increase ofprices is directly reflected on the increased revenue ofproviders. Although the blocking probability and spec-

trum utilization have not changed, the prices are nowhigher.

As a result of the relatively higher prices in rate pref-erence compared to price preference, the blocking prob-abilities are larger. Note that this is true only for cardusers. For subscribers, the prices are the same in bothscenarios and remain fixed for the entire duration of theexperiment. In addition, in price preference, the higherthe price tolerance threshold, the lower the blockingprobability.

Card clients have smaller blocking probability thansubscribers, since on average, a subscriber is furtheraway from the best BS than a card client. This is be-cause a subscriber belongs to a provider, and thus, se-lects a BS from the set of BSs deployed by that provider,while a card client may select a BS from a larger set ofBSs that belong to various providers.

In general, provider 1 has a higher spectrum avail-ability (i.e., larger number of time frequency slots) re-sulting in larger revenue compared to provider 2 andsmaller spectrum utilization. Moreover, this is evenmore prominent in the partial monopoly case, in whichthe difference in the spectrum availability of the twoproviders is increased.

In partial monopoly, unlike the case of rate prefer-ence, in which the revenue increase is not dramatic, inprice preference, the revenue of the monopoly provideris doubled. This is due to the price tolerance thresholdand the competition with the other provider (shown inFig. 3 (b)). Actually, the price-tolerance-threshold isthe dominant factor, given that in rate preference, theimpact of competition is less prominent since clients se-lect the BS with the best channel quality (and not thelowest price). Note that in monopoly scenarios, thereare some regions in which BSs of both providers arepresent, resulting in a competition. In the region ofmonopoly, the price setting mechanism of the provideris constrained by the price tolerance threshold, whilein the remaining regions, by mainly the competitionamong providers. The larger the region of a monopoly,the larger the flexibility for that monopoly (provider)to set its price. The competition between providers inthe other regions and the tendency of the monopolyprovider to increase its price give the opportunity tothe other provider to also increase its price, and thus,its revenue. This is an example of cases where par-tial monopolies provide opportunities to non-monopolyproviders to increase their revenue.

The values of u-map indicate that the spectrum avail-ability of provider 1 exceeds the spectrum availabilityof provider 2. As expected, all subscribers select theprovider 1. To evaluate the impact of the u-map, weemploy a baseline scenario in which subscribers selecta provider randomly (rnd). Compared to the randomselection (rnd), the map-based selection exhibits lower

11

average blocking probability for subscribers, both inrate and price preference (Fig. 3 (f)). Clearly a u-map related system can be beneficial to clients. Po-tentially providers could also take advantage from thereported information about the call arrivals and distri-butions and user price tolerance threshold. For exam-ple, an increased blocking probability in certain areasmay alarm providers for further investigation and bettercapacity planning. In this study, the price adaptationalgorithm of providers does not employ any informationabout clients. It is important to note that the integra-tion of additional knowledge about the population mayfurther improve the performance of the price adaptationmechanism by satisfying the price tolerance threshold ofa larger client population. The u-map can provide theinformation that the more traditional, game-theoreticaldynamics require (e.g., well-defined payoff functions forproviders). Even more importantly, a network operatorcould use information collected from such databases toinstantiate a market in the mesoscopic and macroscopiclayers (as presented in our modelling framework) andanalyze its evolution.

5. CONCLUSIONS AND FUTUREWORKThe paper presented a powerful innovative game the-

oretical framework that models spectrum markets inmultiple spatio-temporal scales. It enables a researcherto control the loss of information and scale up or downthe models of various parameters in order to analyzecertain phenomena or trends in the appropriate spatio-temporal scales. Furthermore, it describes a simulationplatform that allows researchers to instantiate variousparadigms, driven by business cases/scenarios of suchmarkets and analyze their evolution. As an example,it implemented and evaluated a cellular-based marketin the microscopic scale. The modelling framework andsimulation platform can be extended to instantiate dif-ferent network and cooperation paradigms (e.g., meshand cognitive radio networks). For example, a device,e.g., an AP in a mesh network or a primary device ina cognitive radio network, may act at certain times asa provider and at other times as a client, correspond-ing to two agents. In other cases, providers may formcoalitions or bargain with each other or employ jointlypower-control and channel allocation.

The amount of available information and its reliabil-ity are critical parameters. Our proposed price adapta-tion algorithms assumed that providers have no knowl-edge about the client preference and constraints or thepayoffs of other providers. It is a part of our on-goingresearch to compare this approach with the other moretraditional game-theoretical dynamics (also describedin this work) that assume such knowledge. Note thatthe u-map serves as a connecting axis among thevarious types of dynamics enabling the integration of

different types of information in the simulation plat-form. We plan to incorporate the presence of malicious,mis-configured or non-rational entities in our simula-tion platform. For example, malicious/mis-configuredclients may upload erroneous information on the u-map,while non-rational entities can make mistakes (as-pects that have been incorporated in our modelling frame-work).

An important long-term objective of this research isthe incorporation of measurements from a real-life net-work environment in our simulation platform. Specif-ically, we consider the integration of information col-lected from a metropolitan-area wireless network, e.g.,BSs deployment, empirical-based channel models (e.g.,ray-tracing), mobility models, and user traffic traces.This will further enrich the simulation platform by en-abling the cross-validation and analysis of various paradigmsand spectrum markets in even more realistic settings.The proposed modelling framework sets the fundamen-tals for enabling the assessment of the evolution of mar-kets of cognitive radio technologies using game-theoryand micro-economics.

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