Physical Interference Model Based SpectrumSharing with Generalized Spatial Congestion Games

(Invited Paper)

Richard SouthwellDept. of Information Engineering

The Chinese Univ. of Hong KongE-mail: richardsouthwell254@gmail.com

Jianwei HuangDept. of Information Engineering

The Chinese Univ. of Hong KongE-mail: jwhuang@ie.cuhk.edu.hk

Biying ShouDept. of Management Sciences

City Univ. of Hong KongE-mail: biying.shou@cityu.edu.hk

Abstract— With the rapid development of heterogeneouswireless technologies, the issue of how selfish wireless users canshare spectrum is becoming increasingly relevant. In this paperwe introduce the generalized spatial congestion game (GSCG),and use it to model wireless spectrum sharing over a large area.The idea behind the GSCG is to think of the players as vertices ina weighted graph. The amount of congestion two players causeeach other (when they use the same resource) is determinedby the weight of the edge linking them. The GSCG is moresuitable for modeling spectrum sharing than many previouslyconsidered models, because one can select the edge weightsand payoff functions to correspond with several practical interference models (such as the physical interference model). Wefocus on determining which GSCGs possess pure Nash equilibria(i.e., mutually acceptable resource allocations), and how selfishplayers can organize themselves into pure Nash equilibria.

I. INTRODUCTION

As the number of wireless devices grows rapidly, it isbecoming increasingly important to understand how wirelessusers can share the spectrum. Spectrum allocation has beenstudied extensively from a centralized point of view (e.g.,[1]). The complexity of the problem and the selfish natureof wireless users, however, often make it desirable to achieveefficient and fair spectrum sharing in a distributed fashion(i.e., allowing users to select channels for themselves). Thisis especially true in cognitive radio networks [2], [3], wherethe users are empowered with the ability to scan the spectrumand adapt their channels rapidly. Game theory is ideal formodeling distributed spectrum sharing.

The spectrum sharing problem has an intrinsic spatialelement, because the mutual interferences generated amongusers on the same channel heavily depends on the locations ofthe corresponding transmitters and receivers. The generalizedspatial congestion game (GSCG) model proposed in this paperallows us to use “weights” to reflect how much interferenceone user may cause to an other user. An illustrative exampleis shown in Fig. 1, where vertices represents wireless users(transmitter-receiver pairs) and directed edges represent inter-ference relationships. The edge weights represent the level ofinterference. Notice that two vertices (users) might generatedifferent interferences to each other (e.g., the two edges withweights 4 and 9 between two black nodes 2 and 3), possiblyrepresenting different locations of transmitters/receivers anddifferent transmission power levels. By selecting edge weightsappropriately, we can account for the spatial aspects of spec-

Fig. 1. A GSCG. The users are represented by vertices in a weighted graph.Users suffer interference from other users of the same channel which areconnected to them. For example, user 2 suffers an interference level of 3from user 4 (which is also using the black channel), but user 2 suffers nointerference from user 1 (which is using white channel instead).

trum sharing in an accurate way. This is quite satisfying, asthe spatial aspect of spectrum sharing is less understood thanmany other aspects [4].

Many game theoretic models have been used to studyspectrum sharing (e.g., [5]–[10]), but these generally assumethat the users have complete knowledge about the networkparameters and each other’s information. Congestion gamebased models [11] (including the GSCG model) have theadvantage of modeling network scenarios where the usershave limited information, which is the case in many networks(cognitive radio networks in particular). More precisely, manycongestion game models have the finite improvement property,which means is that if the players keep performing selfishbetter response updates based on local information, then thesystem will eventually reach a pure Nash equilibrium.

In [12]–[14] we introduced spatial congestion games andapplied them to spectrum sharing in wireless networks. Theidea behind these games is to model the users as verticesupon an undirected unweighted graph, which can only causeinterference to their neighbors. This corresponds to the pro-tocol interference model [15], within which a pair of usersare either considered linked (in which case they can causeone another some fixed amount of interference) or not linked(in which case they are considered to be too distant tointerfere with one another). Spatial congestion games serveas a more realistic model for spectrum sharing than classicalcongestion games because they allow for spectrum reuse-

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where distantly spaced individuals can use the same channelwithout interfering. However the protocol inference model(upon which these games are based) is too basic to capturethe real interference relationships between the users. In reality,inference decrease continuously with separation distance, anddepends on transmission power levels. Interference effects canbe captured much more accurately using the SINR basedphysical interference model [15], which is widely used inwireless network modeling.

In this paper we extend the spatial congestion game model,from undirected unweighted graphs to directed weightedgraphs. This non-trivial generalization greatly increases theflexibility of the model. Rather than simply assuming a pairof players n and m are either linked (i.e., they can cause afixed amount of congestion to one another) or not linked (i.e.,are too distant to cause congestion), we allocate each pair ofplayers with a weight Sn,m ≥ 0 which represents how muchcongestion n can cause to m in a continuous way. The amountof congestion a player n incurs is given by the sum of theweights Sm,n of the directed edges which are incoming uponn from other players m who are using the same resource. Thepayoff a player receives is given by a decreasing function oftheir congestion level. Our key results and contributions canbe summarized as follows:

• A Powerful New Model of Spectrum Sharing: After in-troducing the GSCG formally (Section II-A), we explainhow GSCGs can be used to model spectrum sharingbased on the physical interference model (Section II-B).

• Properties of Resource-Homogeneous GSCGs: We char-acterize several properties of resource-homogeneousGSCGs, which are very useful for modeling spectrumsharing (Section III).

• Realistic Simulations of Spectrum Sharing: We performrealistic simulations of spectrum sharing (Section IV),and illustrate the practical implication of our theoreticalresults.

II. THE MODEL

A. GSCG Game Formulation

Let us define a generalized spatial congestion game(GSCG) as a 5-tuple

(N ,R, (Rn)n∈N , (frn)n∈N ,r∈Rn

, S),

where

• N = {1, 2, ..., N} is a finite set of players.• R = {1, 2, ..., R} is a finite set of resources.• Rn ⊆ R is the set of resources available to player n ∈

N .• fr

n is a strictly decreasing continuous function that de-scribes the payoff that player n ∈ N gets when it usesresource r ∈ Rn. The payoff player n gets for usingresource r is fr

n(x) where x is the current congestionlevel of resource r at player n, (as we shall describe).

• S is an n×n matrix of non-negative entries. Here Sn,m

measures the amount of congestion that player n causes

to player m when it uses the same resource. We assume1

Sn,n = 0, ∀n ∈ N .Matrix S captures the spatial information in the game.

Connectivity data from S can be visualized with a directedgraph D(S) = (N , E). The vertex set of this graph is theplayer set N . An edge (n,m) belongs to the directed edge setE if and only if the corresponding edge weight Sn,m > 0. Thegraph D(S) describes which players can cause congestion toone another. Sometimes we refer to a GSCG with a spatialmatrix S as “a GSCG on graph D(S)”. The spatial matrixassociated with the graph shown in Fig. 1 is

S =

⎛⎜⎜⎝

0 7 0 40 0 9 00 4 0 00 3 1 0

⎞⎟⎟⎠ .

A state X = (X1, X2, ..., XN ) ∈ Πn∈NRn is an assign-ment of one2 resource Xn ∈ Rn to each player n ∈ N . Thecongestion level of player n (when the system is in stateX) is

∑m∈N :Xm=Xn

Sm,n or simply∑

m:Xm=XnSm,n.

Graphically, we can think of the congestion level as the sumof the weights Sm,n of all of the edges of D(S) that areincoming upon n, from players m which are using the sameresource. In Fig. 1, the congestion level of the player 2 (whichis using the black resource) is S3,2 + S4,2 = 4 + 3 = 7.

The payoff that a player n gets from using a resourceXn = r (when the game is in state X) is a strictly decreasingcontinuous function of their congestion level, i.e.,

frn

( ∑m∈N :Xm=r

Sm,n

).

In Fig. 1, the payoff of the circled node is f black2 (7).

B. Modeling Wireless Networks Using GSCGs

We start by showing that the GSCG can be used tomodel the SINR-based physical interference model with fixedtransmission power levels. Later we discuss how the GSCGcan be used to model more general scenarios.

Consider a wireless network where each user is a fixedtransmitter-receiver pair. In the physical interference model,the interference received by a user is the summation of thepower received from all other users in the network [15].The maximum achievable transmission rate (according to theShannon capacity) that a user n gets by using channel r isBr log2 (1 + SINR), where Br is the bandwidth of channel rand SINR is the signal-to-interference plus noise ratio,

SINR =hn,nPn

τ0Bi +∑

m:m �=n,Xm=r hm,nPm.

Here τ0 is the thermal noise density, Pm is the transmissionpower of m’s transmitter, and hm,n is the channel gain fromm’s transmitter to n’s receiver.

1We make the assumption for notational convenience. We could relax thisassumption to take into account the congestion a player causes to itself, butit seems to be easier just to modify the payoff functions to achieve this.

2This means our generalized spatial congestion games belongs to the classof singleton congestion games. This is well motivated in practice, for example,a wireless user only has one transceiver and can only access one cellularchannel.

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If each user has a fixed transmission power (which isthe default operation mode in today’s Wi-Fi networks), thenthe spectrum sharing scenario can be modeled as a GSCG(N ,R, (Rn)n∈N , (fr

n)n∈N ,r∈Rn , S) as explained below:• Each player n ∈ N corresponds to a fixed transmitter-

receiver pair.• Each resource r ∈ R corresponds to an orthogonal

channel. When channels have an equal bandwidth (whichis true in Wi-Fi , WiMax, and LTE networks) and thechannels are interleaved (and thus have the same channelconditions for the same user), the system corresponds toa resource-homogenous GSCG.

• Each user n has a user-dependent available channelset Rn ⊆ R. This flexibility is especially useful formodeling cognitive radio networks, where the channelsavailable to a secondary user depend on the activities ofthe licensed users within its vicinity.

• Each player uses exactly one resource/channel at a giventime. This is the case in networks using FDMA technolo-gies.

• Sm,n measures the amount of interference that m causesn when both users are on the same channel. Moreprecisely, Sn,n = 0 and Sm,n = hm,nPm for n �= m.

• Player n’s payoff of using resource r depends on theinterference level x =

∑m:Xm=Xn

Sm,n, and is equal to

frn(x) = Br log2

(1 +

hn,nPn

τ0Br + x

). (1)

When the power levels of the users are equal, and distincttransmitter-receiver pairs are distantly spaced (relative to thedistance between individual transmitters and their receivers),the assumption that the interference relationship between usersis symmetric (i.e., hn,m = hm,n) is at least approximatelyvalid. Such cases correspond to GSCGs with symmetricspatial matrices, or “on undirected graphs”.

In Section IV, we will simulate spectrum sharing in realwireless networks using GSCG models. Note that other as-pects of wireless communication can also be modeled by theGSCG. For example, users may have different payoffs anddifferent channel preferences because of different transmissiontechnologies. We can choose player-specific payoff functions(instead of the same Shannon capacity) to model this. Also,we can also use edge weights to reflect different user prioritiesin cognitive radio networks. For example, if n is a primarylicense holder and m is an secondary unlicensed user, thenwe could set Sn,m to be very large to reflect the price thatm must pay (or the punishment which m may receive) forcausing interference to the license holder n.

III. ANALYTIC RESULTS

A. Better Responses, Nash Equilibria, and the Finite Improve-ment Property

Often we assume our system evolves over discrete timeslots, with no more than one player updating its resourcechoice during any given time slot. This assumption is oftenused in congestion games, it is realistic when the time scaleis so small that simultaneous updating becomes unlikely.

When we have a system in state X , and then a single playern ∈ N changes their resource choice to r ∈ R the systemchanges to a new state (X1, · · · , Xn−1, r,Xn+1, · · · , XN ).We say that such a update is a better response update whenit increases player n’s payoff.

Definition 3.1: The event where a player n ∈ N changesits resource choice from Xn to r ∈ Rn is a better responseupdate if and only if

frn

( ∑m:Xm=i

Sm,n

)> fXn

n

( ∑m:Xm=Xn

Sm,n

).

A pure Nash equilibrium is a stable resource allocation, fromwhich no player has any incentive to deviate unilaterally.

Definition 3.2: A state X is a pure Nash equilibrium ifand only if no player can perform a better response update,i.e., fr

n

(∑m:Xm=r Sm,n

) ≤ fXnn

(∑m:Xm=Xn

Sm,n

), ∀n ∈

N , ∀r ∈ Rn.Definition 3.3: A GSCG has the finite improvement prop-

erty if every sufficiently long sequence of better responseupdates leads to a pure Nash equilibrium.When a game with the finite improvement property evolvesvia asynchronous better response updates, it is guaranteedto reach a pure Nash equilibrium within a finite number oftime slots. Loosely speaking, this means that greedy behavioralways leads to a mutually acceptable resource allocation.

B. Properties of GSCGs

Firstly we should note that there are GSCGs which donot possess pure Nash equilibria. This result was effectivelyproved in [12], where we exhibited a spatial congestion gameon an undirected unweighted graph, which has no pure Nashequilibria. The game has player-specific and resource-specificpayoff functions. One can construct many other GSCGswith no pure Nash equilibria which are played on directedgraphs. For example, the three player GSCG played on adirected triangle, with two resources, and a payoff functionfrn(x) = −x, for each player n and each resource r.

In the following we focus on GSCGs with homogenousresources.

Definition 3.4: A GSCG is resource-homogenous if andonly if f1

n(x) = f2n(x) = ... = fR

n (x), ∀n ∈ N and ∀x (i.e.,all resources appear identical from any particular player n’sperspective).

Resource-homogenous GSCGs can model wireless net-works where each channel provides the same data rate.In most wireless standards (e.g., Wi-Fi, WiMax, LTE, andbluetooth), channels have equal bandwidth. Moreover, inter-leaving techniques can be used to homogenize the quality ofdifferent channels (e.g., the interleaved channelization in IEEE802.16d/e [17]).

We show that no player in a pure Nash equilibrium, of aresource-homogenous GCGWE, will have a congestion levelthat is above their maximum possible congestion level dividedby the number of resources they have available to them. Thisis good news, because it means that if enough resources areavailable then all Nash equilibria will have low levels ofcongestion.

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Theorem 3.1: Suppose we have a resource-homogenousGSCG at a pure Nash equilibrium X . Now the congestionlevel

∑m:Xm=Xn

Sm,n of any player n ∈ N is no larger

than(∑N

m=1 Sm,n

)/|Rn|.

Our next result states that when the resources are ho-mogenous and the spatial relationships between the playersare symmetric, the population will be eventually organizeitself into a pure Nash equilibrium. This combines well withTheorem 3.1, which implies that the resulting equilibria willinvolve relatively low levels of congestion. In many wirelessnetworks the assumptions that channels are homogenousand interference is symmetric between users are often quiteaccurate. In such cases, Theorem 3.2 implies that selfish radiousers can find a way to share the spectrum for themselvesefficiently.

Theorem 3.2: Every resource-homogenous GSCG with asymmetric spatial matrix S has the finite improvement prop-erty.

Full proofs of both theorems can be found in the onlinetechnical report [16]. Asynchronous better response updatingcan be used to reach a pure Nash equilibrium within GSCGswith the finite improvement property. A distributed algorithmto find pure Nash equilibria based upon this fact would beto have each unsatisfied player do a better response updatewith a small (fixed) probability p > 0 every time step. Undersuch randomized dynamics there is a non-zero probabilitythat eventually there will be a long enough sequence ofasynchronous better response updates for a pure Nash equi-librium to be reached. In follows that eventual convergence isguaranteed. More generally, a GSCG may not have pure Nashequilibria, or the finite improvement property. The generalityof the GSCG model suggests that ‘exhaustuve search’ is theonly practcal way to verify that is a generic GSCG does nothave pure Nash equilibria, or the finite improvement property.One may always find a mixed Nash equilibrium 3 of a GSCG(since they are finite games) [18], although the notion ofmixed Nash equilibria may not be appropriate in the contextof wireless networks where switching channels is non-trivial.

IV. SIMULATING WIRELESS NETWORKS

In this section we will use GSCGs to simulate how N = 20selfish radio users (scattered across a square region of lengthL) will share R = 5 homogenous channels. We study how theusers’ ability to share the spectrum is influenced by L. Wesuppose that each player corresponds to a fixed transmitter-receiver pair that wishes to maximize its transmission rate byselecting the best channel. We shall also make the followingassumptions:

• Each of user n transmits at a fixed power level of Pn =100mW.

• Each channel r has a bandwidth of Br = 20MHz, andis available to every user.

• The payoff a user n gets for using a channel Xn = r isequal to its transmission rate, as given by Equation (1).

• We shall use the distance-based physical interferencemodel [15], by writing the channel gain hm,n, from

3A mixed Nash equilibrium is where players can use mixed strategies -which are non-trivial probability distributions over their strategy sets

user m’s transmitter to n’s receiver (see Equation (1))as hm,n = 1/dαm,n, where α is the attenuation factor anddm,n is the distance from m’s transmitter to n’s receiver.

• We will suppose that the attenuation factor α = 4and the spectral noise density τ0 = −174 dBm /Hz =10−17.4 mW /Hz.

• We place each transmitter at a point (chosen uniformlyat random) from our L×L square. Each receiver is uni-formly randomly located within 100m of its transmitter.We insure that no receiver is with 1m of a transmitter(since our distance based SINR model breaks down atsuch close ranges).

For each simulation run, we randomly generate a networkand randomly allocate one of the five channels to each user.Then the network evolves by having one random unsatisfied4

user update its channel to a better response at every time slot.The simulation run stops when a pure Nash equilibrium hasbeen reached (or when some pre-specified large number oftime slots have elapsed). We show the choices of channels ina pure Nash equilibrium of a network under our parameters inFig. 2. Figures 3 and 4 depict the results of further simulations.

Fig. 2. A pure Nash equilibrium in a network where the N = 20 usersare scattered across a square of length L = 200m. Each arrow points froma transmitter (represented by a black router) to its receiver (represented bya gray computer). The link colors represent the channels that users chooseat this Nash equilibrium. The average transmission rate of users at this Nashequilibrium is 101Mbps. Notice how links (users) of the same color naturallyspread out to avoid strong mutual interferences.

The majority of our simulation runs converged to pure Nashequilibria. As Fig. 3 shows, however, convergence time ishighly variable in simulation runs. Figure 3 shows that 20users normally reach a pure Nash equilibrium within 200time slots. In rare cases, however, it can take over 1000time slots for a simulation to converge. This illustrates aninherent difficulty in studying the convergence dynamics ofour systems. When a simulation does not converge at all,it will not have the finite improvement property. It is also

4An unsatisfied user is a user which can increase its payoff by switchingto a different channel.

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Fig. 3. The x-axis gives the length L (in meters) of the square regionwhich the users are scattered over. We performed 10 simulations for eachL ∈ {50m, 100m, ..., 500m}. We ran each simulation for up to 2000 timeslots. The y-axis shows how many time slots it takes a given simulation toconverge. We do not show the points corresponding to simulations which didnot converge. Note that the scale of the y axis is logarithmic.

Fig. 4. The x-axis gives the length L, in meters, of the square region whichthe users are scattered across. We performed 1000 simulations for each valueof L. The y-axis gives the fraction of these simulations which converged toa pure Nash equilibrium within 500 time slots.

possible that the system possesses no Nash equilibria. Theproblem is, each of the systems we are simulating has 520

states. This make it impractical to use a computer to rigorouslyverify whether a given system has the finite improvementproperty or Nash equilibria.

For this reason, we studied how the relative frequency thata given system will converge within 500 time slots, dependsupon the geometry of the network. Figure 4 shows that it iseasier for the users to organize themselves into a pure Nashequilibrium when the area they are spread across is larger.When the users are scattered across a large area, the distancebetween distinct transmitter-receiver pairs will often be muchlarger than the distance between a particular transmitter and itsreceiver. This will cause the interference relationship betweenthe users to be approximately symmetric (in that the distancefrom n’s transmitter to m’s receiver will be approximatelyequal to the distance from m’s transmitter to n’s receiver).This means the system will (approximately) correspond to aresource-homogenous GSCG with a symmetric spatial matrix.In this case, Theorem 3.2 states that the system will havethe finite improvement property, and therefor will eventually

converge to a pure Nash equilibrium.

V. DISCUSSIONS

Our simulation results reveal that the ability of wirelessusers to find a mutually acceptable spectrum allocation de-pends sensitively upon the network topology. GSCGs ondirected graphs with no pure Nash equilibria can easily beconstructed, although many GSCGs on undirected graphs havethe finite improvement property. This suggests that spectrumallocation is easier when the interference relationships be-tween the users are symmetric.

ACKNOWLEDGMENT

This work is supported by the General Research Funds(Project Number CUHK 412710, CUHK 412511 and CityU144209) established under the University Grant Committee ofthe Hong Kong Special Administrative Region, China. Thiswork is also partially supported by grants from City Universityof Hong Kong (7008116 and 7008143).

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