ieee/acm transactions on networking 1 atomic...

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IEEE/ACM TRANSACTIONS ON NETWORKING 1

Atomic Congestion Games on Graphsand Their Applications in Networking

Cem Tekin, Student Member, IEEE, Mingyan Liu, Senior Member, IEEE, Richard Southwell,Jianwei Huang, Senior Member, IEEE, and Sahand Haji Ali Ahmad

Abstract—In this paper, we introduce and analyze the prop-erties of a class of games, the atomic congestion games ongraphs (ACGGs), which is a generalization of the classicalcongestion games. In particular, an ACGG captures the spatial in-formation that is often ignored in a classical congestion game. Thisis useful in many networking problems, e.g., wireless networkswhere interference among the users heavily depends on the spatialinformation. In an ACGG, a player’s payoff for using a resourceis a function of the number of players who interact with it anduse the same resource. Such spatial information can be capturedby a graph. We study fundamental properties of the ACGGs:under what conditions these games possess a pure strategy Nashequilibrium (PNE), or the finite improvement property (FIP),which is sufficient for the existence of a PNE. We show that a PNEmay not exist in general, but that it does exist in many importantspecial cases including tree, loop, or regular bipartite networks.The FIP holds for important special cases including systems withtwo resources or identical payoff functions for each resource.Finally, we present two wireless network applications of ACGGs:power control and channel contention under IEEE 802.11.

Index Terms—Congestion games, convergence, game theory,graph theory, Nash equilibrium, wireless networks.

I. INTRODUCTION

I N THIS paper, we study atomic congestion games ongraphs (ACGGs), which is a generalized form of the

class of noncooperative strategic games known as congestiongames [1], [2]. We analyze the properties of the ACGGs and

Manuscript received November 26, 2010; revised October 14, 2011;accepted December 17, 2011; approved by IEEE/ACM TRANSACTIONSON NETWORKING Editor E. Modiano. This work was supported by theNSF under Grants CNS-0238035 and CIF-0910765, the ARO under GrantW911NF-11-1-0532, the National Basic Research Program of China underGrants 2007CB807900 and 2007CB807901, the National Natural ScienceFoundation of China under Grants 61033001, 61061130540, 61073174, and theGeneral Research Funds (Project no. 412710 and no. 412511) established underthe University Grant Committee of the Hong Kong Special AdministrativeRegion, China. An earlier version of this paper appeared in the InternationalConference on Game Theory for Networks (GameNets), Istanbul, Turkey, May13–15, 2009.C. Tekin and M. Liu are with the Electrical Engineering and Computer Sci-

ence Department, University of Michigan, Ann Arbor, MI 48105 USA (e-mail:[email protected]; [email protected]).R. Southwell was with the Chinese University of Hong Kong, Hong Kong.

He is nowwith the Institute for Interdisciplinary Information Sciences, TsinghuaUniversity, Beijing 100084, China (e-mail: [email protected]).J. Huang is with the Network Communications and Economics Lab, Depart-

ment of Information Engineering, The Chinese University of Hong Kong, HongKong (e-mail: [email protected]).S. H. A. Ahmad was with the Electrical Engineering and Computer Science

Department, University of Michigan, Ann Arbor, MI 48105 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNET.2012.2182779

further discuss their applications in networking such as spec-trum sharing in a multichannel wireless system and channelcontention under IEEE 802.11.In a classical congestion game, multiple players share mul-

tiple resources. A player’s payoff1 for using a particular resourcedepends on the number of players simultaneously using that re-source. A formal description is provided in Section III. The con-gestion game framework is well suited to model resource com-petition where the resulting payoff is a function of the level ofcongestion (number of active/competing players).An improvement step is a move where one player changes

its strategy to increase its payoff. An improvement path is asequence of asynchronous improvement steps. The congestiongame enjoys many appealing properties: It has a pure strategyNash equilibrium (PNE), and any improvement path is finite andwill lead to a PNE. The latter property is also called the finiteimprovement property (FIP): Local greedy updates of selfishplayers collectively optimize a global objective known as thepotential function, and such updates converge in a finite numberof steps regardless of the updating sequence.Due to the above reasons, congestion games are used widely

in modeling networking problems, particularly routing prob-lems (see Section II). In this paper, we introduce a generaliza-tion of this model where players can only affect their neigh-bors according to a graph structure. These generalized gameswill be referred to as ACGGs. In these games, an interactiongraph describes the the congestion relationships between theplayers. A player’s payoff for using a resource is a function ofthe total number of players who are using the same resource andare within its interaction set (i.e., connected to it by edges inthe graph). Therefore, resources are reusable beyond a player’sinteraction set. The original congestion game is now a specialcase of the extended ACGG where the underlying interactiongraph is complete (i.e., every player interacts with every otherplayer).2

Our main motivation behind this generalization comes fromapplications in wireless networks, a key feature of which isspatial reuse: Common spectrum resources may be shared bymultiple players located far apart without causing congestionto each other. This feature cannot be captured by the standard

1One can also consider the cost of using a resource instead of payoff. If wedefine the cost as the inverse of the payoff, then maximizing the payoff is equiv-alent to minimizing the cost. For simplicity of presentation, we will only referto the maximization of payoff in this paper.2In our preliminary work [3], we used the term network congestion games.

However, to better differentiate this class of games from routing games (see e.g.,[4] and [5]) that are also sometimes referred to as network congestion games, wewill use the term atomic congestion games on graphs in this paper. Note thata routing game is essentially a classical congestion game in which a player’sstrategy space consists of a set of feasible routes and each route consists ofmultiple resources (links).

1063-6692/$31.00 © 2012 IEEE


2 IEEE/ACM TRANSACTIONS ON NETWORKING

congestion game, which assumes that all players have an equalimpact on the congestion. Specifically, we consider a systemwhere a user can only access one channel at a time, but canswitch between channels. A user’s principal interest lies in opti-mizing its own performance (e.g., its data rate) by selecting thebest channel for itself. This and similar problems have recentlycaptured increasing interest from the research community, par-ticularly in the context of cognitive radio networks and softwaredefined radio technologies, where devices are expected to havefar greater flexibility in sensing channel availability and movingtheir operating frequencies.In addition to the above, there are other applications of the

ACGGmodel. For instance, it can be used to model competitionamong local businesses, whose locations may be representedby vertices on a graph. Edges connect those within close prox-imity, and resources represent different business ventures. Morediscussion on applications is given in Section VIII. It is alsoworth mentioning that one interpretation of the ACGG is thatit models congestion games with incomplete information, i.e.,all players in reality compete with everyone else, but a playeris only aware of the presence of its neighbors on the interac-tion graph. This is actually the standard justification of graph-ical congestion games; see, e.g., [6]. More on this is discussedin Section II.In subsequent sections, we will examine what properties an

ACGG has. Our main findings are summarized as follows forundirected network graphs and nonincreasing payoff functions.1) The FIP holds in an ACGGwith only two resources. Coun-terexamples exist for three or more resources.

2) The FIP holds in an ACGGwhen all resources are identicalto a player (but may be different to different players).

3) A PNE exists in an ACGG over a tree network, a loop, aregular bipartite network, and when there is a dominatingresource.

4) We also identify counterexamples showing that a PNEdoes not necessarily exist for an ACGG with three re-sources, over a directed graph, or with nonmonotonicpayoff functions.

The organization of the remainder of this paper is as follows.Related work is given in Section II. In Section III, we presenta brief review on the classical congestion game, and formallydefine the class of ACGGs in Section IV. We then derive condi-tions under which an ACGG possesses the FIP in Section V, andunder which a PNE exists in Section VI. We provide negativeresults on existence of a PNE or the FIP in Section VII. We il-lustrate two networking applications of ACGGs in Section VIII,and discuss extensions to our work in Section IX. We concludethe paper in Section X.

II. RELATED WORK

Congestion games have been extensively studied withinthe context of wireline network routing; see, for instance, thecongestion game studied in [7], where each source node seeksthe minimum delay path to a destination node, and the delay ofa link depends on the number of flows going through that link.It has recently been used in wireless network modeling, e.g.,access point selection in Wi-Fi networks [8], [9], resource com-petition in multicamera wireless surveillance networks [10],uplink resource allocation in multichannel wireless access

networks [11], wireless channels with multipacket receptioncapability [12], the spectrum mobility in cognitive radio net-works [13], and the impact of interference set in studyingthe congestion game in wireless mesh networks [14]. In ourrecent work [15], we addressed the user-specific interferenceissue within the traditional congestion game framework byintroducing a concept called resource expansion, where wedefine virtual resources as certain spectral-spatial unit thatallows us to capture pairwise interference. This approach wasshown to be quite effective for user objectives like interferenceminimization. Congestion games played on networks have beenstudied before in [6], where each user has the same linear payofffunction. The authors discuss how these systems can be viewedas congestion games with limited information, where playerscan only observe the actions of their neighbors on a graph. OurACGG model allows player-specific payoff functions of moregeneral forms. In this sense, our model is also a generalizationof that considered in [6].It should be mentioned that game-theoretic approaches have

often been used to devise effective decentralized solutions toa multiagent system. Within the context of wireless commu-nication networks and interference modeling, different classesof games have been studied. An example is the well-knownGaussian interference game [16], [17], in which a player canspread a fixed amount of power arbitrarily across a continuousbandwidth and tries to maximize its total rate in a Gaussianinterference channel over all possible power allocation strate-gies. The Bayesian form of the Gaussian interference game wasstudied in [18] in the case of incomplete information. In ad-dition, a market-based power control mechanism was investi-gated via supermodularity in [19] and using externality in [20].A spectrum sharing similar to the one studied here was inves-tigated in [21] using a mechanism design approach in seekinga globally optimal solution. In our problem, the total power ofa user is not divisible, and it can only use it in one channel ata time. This setup is more appropriate for scenarios where thechannels have been predefined, and the users do not have theability to access multiple channels simultaneously (which is thecase with many existing devices).Another approach to analyzing related networking problems

is the use of evolutionary game theory [22], [23]. Evolutionarygames often assume limited rationality and are applicable typi-cally in the presence of a larger number of users. By contrast, ourapproach applies to any number of users and also works underlimited rationality (better response updating). Furthermore, thebetter response dynamics we consider are simpler and more re-alistic, in the context of wireless networking, than many typesof evolutionary dynamics, e.g., those based on reproduction orimitation.

III. REVIEW OF CONGESTION GAMES

In this section, we provide a brief review on the definition ofcongestion games and their known properties.3 We then intro-duce the ACGG as a generalization.Congestion games [1], [2] are a class of strategic

games given by the tuple , where

3This review along with some of our notations are primarily based on [1], [2],and [24].


TEKIN et al.: ATOMIC CONGESTION GAMES ON GRAPHS AND THEIR APPLICATIONS IN NETWORKING 3

denotes a set of players,a set of resources, the strategy space of player ,and a payoff (or cost) function associated withresource . The payoff (cost) of resource is a function ofthe total number of players using that resource and, in general,is assumed to be nonincreasing (nondecreasing). A player inthis game aims to maximize (minimize) its total payoff (cost),which is the sum total of payoff (cost) over all resources itsstrategy involves. For the rest of the paper, we will only referto payoff maximization.Denoting by a strategy profile, where

, player ’s total payoff is given by

(1)

where is the total number of players using resourceunder the strategy profile , with denoting that playerselects resource under .We can define Rosenthal’s potential function

as

(2)

where the second equality comes from exchanging the twosums, and denotes the number of players who useresource under strategy and whose corresponding indicesdo not exceed (i.e., in the set ).In [1], it is shown that the change in player ’s payoff as a

result of its unilateral move (i.e., all other players’ strategyremain fixed) is exactly the same as the change in the potentialfunction. This implies that the potential function may be viewedas a global objective function. To see this, consider player ,who unilaterally moves from strategy (within the profile

) to strategy (within the profile ). Thechange to the potential function is

The second equality comes from the fact that the number of totalplayers does not change for any resource that is used by bothstrategies and . To see why the first equality is true, set

, in which case this equality is a direct consequence of(2). This is also true for any by noting that theordering of players is arbitrary so any player making a changemay be viewed as the th player.Consider now a sequence of strategy changes made by

players asynchronously, in which each change improves thecorresponding player’s payoff (this is referred to as a sequenceof improvement steps). The potential function improves in

every such change sequence. Since the potential function ofany strategy profile is finite, we have the following result [2].Proposition 1 (FIP): For every congestion game, every se-

quence of asynchronous improvement steps is finite and con-verges to a PNE. Furthermore, this PNE is a local optimumof the potential function , defined as a strategy profile wherechanging one coordinate cannot result in a greater value of .We note that by definition, FIP in any game is sufficient to

guarantee the existence of a PNE for that game, but it is notnecessary. It follows that: 1) the nonexistence of a PNE provesthat the FIP does not hold; and 2) in general, the existence of aPNE does not imply the that the FIP holds. Therefore, the FIPis a much stronger result than the existence of a PNE.Also note that under the above standard definition, the

payoff functions are resource-dependent but player-indepen-dent. This was relaxed in [25], where a player-specific payofffunction was considered. It was shown that in this game,a PNE continues to exist (at least in the “singleton” case whereplayers access one resource at a time), but that the FIP nolonger holds in general.It is not difficult to see why the standard definition of a con-

gestion game does not capture spatial reuse of wireless com-munication. In particular, if we consider channels as resources,then the payoff for using channel when there are si-multaneous players does not reflect reality: The function ingeneral takes a player-specific argument since different playersexperience different levels of interaction even when using thesame resource. This player specificity is also different from thatstudied in [25] mentioned above, where is a player-specificfunction, but takes the same non-player-specific argument . Toanalyze and understand the consequence of this difference, wewould need to extend and generalize the definition of the stan-dard congestion game.

IV. PROBLEM FORMULATION

In this section we formally define our generalized conges-tion game, the ACGG. Specifically, an -player ACGG is givenby , where isthe interaction set of player/user (i.e., players interacting withplayer ), while all other elements maintain the same meaningas in a standard CG. The payoff player receives for using re-source is given by , where

.Our generalization goes in two directions: 1) player ’s payoff

for using resource is a player-specific function, as evidencedby the index in ; and 2) the argument of this functionis also player-specific: It is the number of players interactingwith itself, plus itself. The motivation for making the payofffunctions player-specific is to capture, for example, the fact that,in a wireless system, players with different coding/modulationschemes may obtain different rates from using the same channeleven when facing the same level of interferences.A player’s (total) payoff is the sum of payoffs from all the re-

sources it uses. Note that if a player is allowed to simultaneouslyuse all available resources, then its best strategy is to simply useall of them regardless of other players, provided that is a non-negative function. If all players are allowed such a strategy, thenthe existence of a PNE is trivially true.



In this paper, wewill limit our attention to the case where eachplayer is allowed only one resource at a time, i.e., its strategyspace consists of single resource strategies. In thiscase, the payoff player receives for using a single resource isgiven by , where .It is easy to see that we can equivalently represent this

problem on a directed graph, where a node represents a playerand a directed edge connects node to node if and only if

. The ACGG can now be stated as a graph coloringproblem,4 where each node picks a color and receives a valuedepending on the conflict (number of same-colored neighborsto a node); the goal is to see whether a PNE exists and whethera decentralized selfish scheme leads to a PNE. In this paper,we will limit our attention to the case of undirected graphs,where there is an undirected edge between nodes and ifand only if and . This has the intuitive meaningthat if node interacts with node , the reverse is also true.This symmetry does not always hold in reality, but is often agood approximation and helps us obtain meaningful insight.Another reason for this assumption is that a PNE does notalways exist in a directed graph (as we show in Example 4 viaa counterexample).For simplicity of exposition, in subsequent sections we will

often present the problem in its coloring version and will usethe terms resource, color, and strategy interchangeably. For theremainder of the paper, unless stated otherwise, we shall as-sume that every ACGG we consider has the following prop-erties: 1) players only employ one resource at a given time;2) the payoff functions are player-specific and nonincreasing;and 3) the interaction graph is undirected.

V. FINITE IMPROVEMENT PROPERTY (FIP)

In this section we investigate whether an ACGG always pos-sesses the FIP as in the standard CG. Note that if a game hasthe FIP, it immediately follows that it has a PNE as describedin Section III. Specifically, we show that in the following twocases, an ACGG possesses the FIP: 1) when there are only tworesources to choose from, and 2) when all resources are iden-tical to a player, for all players.

A. Finite Improvement Property for Two Resources

The following theorem shows that FIP holds when eachplayer can only select one of the two available resources.Theorem 1: An ACGG has the FIP when there are only two

resources.Proof: We prove this theorem by a potential function argu-

ment. Consider an ACGGwith two resources, 0 and 1. Considera player . Let denote the number of neighbors thathas on the interaction graph. Define the mapping such thatfor all we have

(3)

Since and are nonincreasing andnondecreasing in , respectively, we have that is nonin-creasing in .

4We will use several colored graphs in our analysis, which may not show aseffectively in a black/white version.

Define a threshold , which can be thought of as the minimalnumber of neighbors using resource 1 that must have, so thatprefers to use resource 0.More precisely, we have the following.1) If , , then let

.2) If , , then let .3) Otherwise, we define to be the minimal value of

such that .To see that is well defined, note that if 1) and 2) are false,we must have and (becauseis nonincreasing). This means that , as described in 3), mustexist. Also note that when condition 3) holds, we have

that implies and im-plies . Consider the function

(4)

where equalsthe number of edges in the interaction graph that link players ofresource 1.Consider a strategy profile , from which some player

makes an improvement by changing its strategy. This improve-ment leads to a new strategy profile such that and

for each . There are only two possible scenarios.Case 1: and , i.e., the player switches

from resource 0 to resource 1. In this case, we can con-clude since there will benew edges in linking players of resource 1. We also have

. From these, it follows that wehave

(5)

Now, we claim that we must also have . To see this,note that in order for user ’s channel switching (i.e., switchingfrom resource 0 to resource 1) to be an improvement step, wemust have , where

. This implies , which meanseither condition 1) or condition 3), listed above, must hold. Ifcondition 1) holds, then . If condition 3)holds, then since and is the minimal valuesuch that , we must have . We havetherefore shown that , and thus proved that

, for this scenario.Case 2: and , i.e., the player switches from

resource 1 to resource 0. In this case,since there will be less edges in linking players ofresource 1. We also have .From these, it follows that we have

(6)

Now, we claim that we must also have . To see this,note that in order for user ’s channel switching (i.e., switchingfrom resource 0 to resource 1) to be an improvement step, wemust have , where

. This implies , which means ei-ther condition 2) or condition 3), listed above, must hold. Ifcondition 2) holds, then . If condition 3)



holds, then since and is minimal such that, we must have that . We have therefore

shown that , and thus proved that ,for this scenario.With the above two cases, we have shown that for every im-

provement step, either the value of the function decreases, orthe value of remains constant but the number of players ofresource 1 decreases. We next show that an improvement loop(where a sequence of improvement steps leads to the same statebeing visited more than once) is impossible. This is done bycontraction.Suppose that a sequence of strategy profiles

form an improvement loop, such that , , and foreach , we have that is obtained bytaking profile and having some player perform an improve-ment step. Now, since never increases during an improvementstep, we must have . At thesame time, since , we have . There-fore, we must have . How-ever, we have just proved that whenever does not decreaseduring an improvement step, the number of players of resource 1must decrease. Hence, this implies profile has less playersof resource 1 than , which contradicts our assumption that

. This contradiction implies that, in fact, an improve-ment loop does not exist.The fact that improvement loops cannot exist means that

every improvement path must be finite (because the set ofdifferent possible profiles is finite, and no profile can be visitedmore than once within an improvement path), so every finiteimprovement path must eventually terminate at a profile fromwhich no further improvement steps can be performed. Sucha terminal profile must be a pure Nash equilibrium, hence oursystem has the finite improvement property.Theorem 1 establishes that when there are only two resources,

the FIP holds, and consequently a PNE exists. Note that theabove proof uses a potential function argument, but technicallythe function is not a “proper” potential function because itdoes not (strictly) decrease with every improvement step. How-ever, the function is a proper potentialfunction, which strictly decreases with every improvement step.Here, is the number of players of resource 1 in profileand is a real chosen to be suitably small (smaller than theamount that decreases by, whenever decreases due to animprovement step).

B. Finite Improvement Property for Identical Resources forEach Player

The next theorem shows the second case in which the FIPholds, when all resources are identical to each player, but dif-ferent players can have different payoff functions. In the con-text of a multichannel wireless system, this can represent thecase where all channels have the same bandwidth and statis-tically similar channel quality to each player (e.g, either withfrequency flat fading or with proper channel interleaving suchas the IEEE 802.16d/e standard [26]), but from player to player,their perceived channel conditions may vary.

Theorem 2: For an ACGG, if for all , , and, we have , then the game has

the FIP.Proof: We prove this theorem by using a potential function

argument. Recall that player ’s total payoff under the strategyprofile is given by , with

, where , and we have suppressed the sub-script since all resources are identical.Now consider the following function defined on the strategy

profile space:

(7)

where the indicator function if is true, and 0 oth-erwise. For a particular strategy profile , this function is thesum of all pairs of players that are connected (neighbors of eachother) and have chosen the same resource under this strategyprofile. Viewed on a graph, this function is the total number ofedges connecting nodes with the same color.We see that every time player improves its payoff by

switching from strategy to and thus reducingto (as is a nonincreasing function), the value of

strictly decreases accordingly.5 Since our potential func-tion (which is equal to the number of edges linking a pair ofplayers of the same resource) takes valuesand decreases with each asynchronous improvement step, ourgame converges to a PNE in quadratic time, when it evolvesvia asynchronous improvement steps. Hence, the game has theFIP.

VI. EXISTENCE OF A PURE STRATEGY NASH EQUILIBRIUM

In this section, we examine what graph properties will guar-antee the existence of a PNE in the absence of the FIP. Specifi-cally, we show that a PNE always exists for ACGGs defined onthe following types of graphs: 1) a tree; 2) a loop; and 3) a reg-ular, bipartite graph with non-player-specific payoff functions.

A. Existence of PNE on a Tree Graph

We show that a PNE exists when the underlying graph isgiven by a tree. We denote by the network (graph) of the-player ACGG . As before, the payoff functions

are nonincreasing, and denotes the number of neighborsof player (excluding ) using strategy .Lemma 1: If every -player ACGG has at least one PNE,

then every -player ACGG formed by connectinga new player to an existing player in a -player networkhas at least one PNE.Remark 1: Note that in this lemma, the network itself

does not have to be a tree. The lemma states that as long as aPNE exists for one class of networks, then by adding one morenode through a single link, a PNE exists in the new network.

5It is easy to see that a nonincreasing functionis an ordinal potential function of this game, as its value improves each

time a player’s individual payoff is improved (which decreases the value of itsargument).



Fig. 1. Adding one more player to the network with a single link.

Proof: By assumption has a PNE denoted by. Suppose is in such a PNE. Now, connect

new player to an arbitrary player in . This is illus-trated in Fig. 1.Let player select its best response strategy

where is defined on the extended network , andit counts the th node’s neighbors using resource under. We now consider three cases depending on ’s strategychange in response to the network expansion from to

.Case 1: . In this case, player selected a re-

source different from ’s, so has no incentive to change itsstrategy in response to the addition of player . In turn,player will remain in as this is its best response, andno other players are affected by this single-link network exten-sion. Thus, the strategy profile is a PNE for thegame .Case 2: , and player ’s best response to

the network expansion remains . That is, even with theadditional interfering neighbor , the best choice for re-mains . In this case, again we reach a PNE for the gamewith the same argument as in Case 1.Case 3: , and player ’s best response to

this network expansion is to move away from strategy . Inthis case, more players may in turn change strategies. Supposewe hold player ’s strategy fixed at . Consider now anew -player ACGG , defined on the original network ,but with the following modified payoff functions for and

:

ifotherwise.

In other words, the game is almost the same as the orig-inal game , the only difference being that the addition ofplayer and its strategy are built into player ’s modi-fied payoff function. By the assumption of Lemma 1, this gamewith players has a PNE, and we denote that by . Supposeis reached in the network with player fixed at

. If we have , then obviously playerhas no incentive to change its strategy because, as far as it isconcerned, its environment has not changed. In turn, no player

in will change its strategy because they are already in a PNEwith player held at . If , then playerhas no incentive to change its strategy because moved awayfrom , which does not decrease player ’s payoff on thisresource, and at the same time its payoff for using any other re-source is no better. Again, is player ’s best response.In either case, strategy profile is a new NE for the game

.Theorem 3: Any ACGG defined over a tree has at least one

PNE.Proof: The proof is easily obtained by noting that any tree

can be constructed by starting from a single node and addingone node (connected through a single link) at a time. Formally,we prove this by induction. Start with a single player indexedby 1. This game has a PNE, in which the player selects

for any payoff functions. Assume that any-player game over a tree with any set of nonincreasing

payoff functions has at least one PNE. Any tree may beconstructed by adding onemore leaf node to some other treeby connecting it to only one of the players in . Lemma 1guarantees that such a formation will result in a game with atleast one PNE.

B. Existence of PNE on a Loop

Theorem 4: Any ACGG defined over a loop network has atleast one PNE.

Proof (Sketch): The complete proof, which is lengthy, canbe found in [27]. Here, we only provide a sketch.We begin this proof by assuming that every player on the loop

always has a unique best response. In the event of a tie wherefor some and , we can impose a unique

best response by assuming that each player has a preferenceorder among colors when the payoffs are the same.6 Note thatthis assumption does not affect the validity of the proof becauserelaxing it only widens the set of PNE a given game on the loophas.Under our assumption, we show that every player can be

associated with a triple of possible bestresponses to different scenarios. The triple has the followingproperties.1) If has no neighbors playing , then ’s best response is

, where .2) If has one neighbor playing , with the other neighbornot playing , then ’s best response is to play .

3) If has one neighbor playing and one neighbor playing, then ’s best response is .

The main idea of the proof is to show the existence of PNEgiven the existence of players with various kinds of triples. Westart by showing that if there exists a player such that

, then a PNE exists. This is done by holding fixed atand letting the other players alter their strategies freely. Sincethe other players are essentially playing on a line graph (whichis a type of tree graph), we use Theorem 3 to construct a strategyconfiguration within which each player in employs theirbest response. We then show that allowing to employ its bestresponse under this configuration constitutes a PNE.

6For example, a player with a color preference of “ ”will pick red if the payoffs of choosing red or blue are the same.



Fig. 2. The cube graph is regular and bipartite. The numbers in the bracketsnear each player represents the resource selected by that player.

Next, we show that if no such player exists (so that, ), a PNE must also exist. This is done by

constructing an algorithm that produces strategy configurationsthat satisfy many of the players around the loop. The algorithmbegins by assigning player 1 a strategy .After this, the algorithm continues to allocate strategiesto in such a way that unless

, in which case . We use this algorithmrepeatedly to demonstrate the existence of PNE under severalcases. The entire set of cases we consider exhausts all possiblegames where , .

C. Existence of PNE on a Regular Bipartite Graph

A graph is regular when all its vertices have the same numberof connections. A graph is bipartite when its vertices can benumbered 1 and 2 (only two numbers) so that no edge connects apair of vertices with the same number.Many well-known graphsare regular and bipartite, including hypercubes and rectangularlattices.Theorem 5: Any ACGG defined over a regular and bipartite

network, with non-player-specific payoff functions, has at leastone PNE.

Proof: As payoff functions are not player-specific, we willsuppress the superscript in the function . Suppose eachvertex has degree (so denotes the number of connectionseach vertex has, e.g., in Fig. 2).Without loss of generality,we order the resources such that the payoff functions satisfy

. If , thenresource 1 dominates, and we can trivially construct an NE byallowing each player to use resource 1.Now consider the case where . Since

our graph is bipartite, we may assign the vertices numbers 1and 2 in such a way that no edge connects a pair of verticeswith the same number. We can think of this numbering as aresource allocation . Under this allocation, each employer of2 will receive payoff (because they have no neighborsemploying 2), whereas they would get ifthey played 1, which is no better. Thus, each employer of 2 isplaying its best response under . In a similar way, the fact that

implies that each employer of 1 isplaying its best response.

D. Existence of PNE for Complete Graphs and a DominantResource

We end this section by stating that an ACGG defined over afully connected graph always has a PNE: ACGG over a com-plete graph simply reduces to the standard CG with player-spe-cific payoff functions. The result has been given in [25].

Fig. 3. PNE counterexample of three resources.

TABLE IBEST-RESPONSE LOOP IN THE THREE-RESOURCE PNE COUNTEREXAMPLE

Theorem 6: Any ACGG defined over a complete graph hasat least one PNE.We also note that regardless of the type of graphs, when-

ever there is a dominant resource , i.e., its payoff function issuch that , where

, for all and all , then a PNE obviouslyexists where all players share the same dominant resource.

VII. COUNTEREXAMPLES

In this section, we present counterexamples on the exis-tence of a PNE and the FIP. Our main result in this section isTheorem 7, which shows that a PNE does not always exist inan ACGG.

A. ACGG With Three Resources

Theorem 7: For an ACGG with three resources, a PNE doesnot always exist. Moreover, the FIP may not hold even when aPNE exists.In Example 1, we give two instances of ACGG that justify

Theorem 7.Example 1: Consider the network topology given in Fig. 3.

Assume that the following set of inequalities holds for de-creasing payoff functions:

According to this set of inequalities, a best-responseloop exists, which is given in Table I. Moreover, one cancheck7 that a PNE does not exist for the payoff vectors

7The checking, unfortunately, can only be done numerically and exhaustivelyto the best of our knowledge.



Fig. 4. FIP counterexample of three resources and non-player-specific payoffs.

given below, whereis the number of neighbors of . They satisfy the above set ofinequalities

We also provide a counterexample where a PNE exists butthere is a best-response loop. Simply let whilekeeping all other payoffs the same. Then, one can easily checkthat is the unique PNE. Since the set of inequali-ties above does not include , they are still satisfied, thus abest-response loop exists.

B. ACGG With Three Resources and Non-Player-SpecificPayoffs

Proposition 2: In an ACGG with non-player-specific pay-offs, the FIP does not always hold even when a PNE exists.Remark 2: Recall that the FIP is sufficient but not necessary

for the existence of a PNE. Both Theorem 7 and Proposition 2are negative results. Theorem 7 implies the FIP does not gener-ally hold in an ACGG with three resources because otherwisea PNE would exist. Proposition 2 is a slightly weaker negativeresult; it says that in the special case with non-player-specificpayoff functions, the FIP does not hold, i.e., there may be im-provement loops. This, however, does not suggest a PNE doesnot exist either in this case, for the latter can exist without theformer. Indeed it remains an intriguing open question whether aPNE always exists in an ACGGwith nonincreasing, non-player-specific payoff functions.In Example 2, we give an instance of ACGG that justifies

Proposition 2.Example 2: Suppose we have three colors to assign, denoted

by , , and . Consider a network topology shown in Fig. 4,where we will primarily focus on nodes , , , and . Inaddition to node , node is also connected to , , andnodes of colors red, green, and blue, respectively. , , ;, , ; and , , are similarly defined and illustrated

in Fig. 4. Note that these sets may not be disjoint, e.g., a singlenode may contribute to both and , and so on.

TABLE IIBEST-RESPONSE LOOP IN THE THREE-RESOURCE FIP COUNTEREXAMPLE

Consider now the sequence of improvement updates shownin Table II involving only nodes , , , and , i.e., withinthis sequence, none of the other nodes change color (note thatthis is possible in an asynchronous improvement path), wherethe notation denotes a color change from to . Attime 0, the initial color assignment is given.We see that this sequence of color changes forms a loop. If

we can show that such a loop is feasible, then we have found acounterexample. For this to be an improvement loop such thateach color change results in an improved payoff, it suffices forthe following sets of conditions to hold. Since we assume allplayers have the same payoff function, we have suppressed thesuperscript in , and the notation “ ” denotes that theimprovement occurs at time

It is straightforward to verify the sufficiency of these conditionsby following a node’s sequence of changes.To complete this counterexample, it remains to show that the

above set of inequalities is feasible given appropriate choicesof , , , and , . There are many suchchoices; one example is , , , , forall . With such a choice, and substituting them intothe earlier set of inequalities and through proper reordering, weobtain the following single chain of inequalities:

(8)

It should be obvious that this chain of inequalities can be easilysatisfied by the right choices of nonincreasing payoff functions.It is easy to see how if we have more than three colors, this loopwill still be an improving loop as long as the above inequalitieshold. This means that for three colors or more, the FIP doesnot hold in general. Note that the updates in this example are



Fig. 5. Counterexample of nonmonotonic payoff functions.

TABLE IIIPNE COUNTEREXAMPLE OF NONMONOTONIC PAYOFFS (FOR EACH MATRIXENTRY , , , FIRST THREE NUMBERS REPRESENT THE PAYOFFS TOPLAYERS 1, 2, 3, RESPECTIVELY, WHILE REPRESENTS THE INDEX OF THEPLAYER WHO CAN IMPROVE ITS PAYOFF BY DEVIATING FROM ITS STRATEGY)

not always best-response updates; they can be better responses,which still result in payoff improvements.Remark 3: The topology of Fig. 4 can easily be made to rep-

resent a tree topology (i.e., any neighbor of has as the onlyneighbor, and so on). Then, by Theorem 3, there exists at leastone PNE of this game. However, we have just shown that theFIP does not hold.

C. ACGG With Nonmonotonic Payoffs or Directed InteractionGraph

We further identify two cases where a PNE does not neces-sarily exist (and thus the FIP does not hold) with non-player-specific payoff functions: when the payoffs are nonmonotonic,and when the graph is directed. These are given in Examples 3and 4, respectively.Example 3: Consider the three-player, two-resource network

given in Fig. 5 with nonmonotonic, non-player-specific payoffsgiven as , , , ,, and . One can easily check that there is no PNEfrom the game matrix corresponding to these payoff functions,given in Table III.Example 4: Consider the three-player, two-resource network

given in Fig. 6. Assume that the payoffs are ,, , , which are non-player-specific andnonincreasing. It is easy to check that PNE does not exist by thegame matrix given in Table IV.

VIII. APPLICATIONS OF ACGG

In this section, we illustrate some applications of the ACGGmodel and give example scenarios that can be modeled by thespecial interaction graphs studied in Section VI.We then discussinmore detail two applications in wireless networks: power con-trol in multichannel CDMA wireless network and IEEE 802.11channel contention. We end this section with some numericalresults on the convergence in an ACGG.As mentioned in the Introduction, in addition to networking

applications, the ACGG can model the following scenario ofbusiness competition, where each vertex in a graph representsa different shop/business premise, two business premises arelinked if they are close to one another, and resources represent

Fig. 6. Counterexample of directed graphs.

TABLE IVPNE COUNTEREXAMPLE OF DIRECTED GRAPH (FOR EACH MATRIX ENTRY ,, (FIRST THREE NUMBERS REPRESENT THE PAYOFFS TO PLAYERS 1, 2, 3,RESPECTIVELY, WHILE REPRESENTS THE INDEX OF THE PLAYER WHO CAN

IMPROVE ITS PAYOFF BY DEVIATING FROM ITS STRATEGY)

business ventures (or average foot traffic per unit area). In a sim-ilar way, the ACGG can be used to model how industrial organi-zations decide which natural resources (e.g., lumber, coal, gold)to harvest within their vicinity. More broadly, the ACGG can beused to model a congestion game with incomplete information.In a congestion game with incomplete information, all playersin reality compete with everyone else, but a player is only awareof the presence of its neighbors on the graph.Within the context of wireless networks, topologies like

tree and loop may correspond to wireless devices deployed ina subway system, mine, or along highways, while a bipartitenetwork may correspond to a scenario where nodes are locatedin two separated areas with transmitter and receiver of the sameuser on different areas with directional antennas. An example ofa regular bipartite graph topology is a wireless sensor networkspaced out regularly to form a two-dimensional grid.

A. Power Control in Multichannel CDMA Wireless Network

In the multichannel CDMA wireless power control problem,the utility/rate player/user gets for using channel is oftentaken to be (see e.g., [28])

where is the channel gain between the transmitter of userand the receiver of user , is the transmit power of user onchannel , is the noise power, and is the spreadinggain. Suppose we adopt the following assumptions: 1) each usercan only choose a single channel to transmit on at full power;2) any user with , for some , does not causeinterference to user ; 3) is (approximately) the same forall users who cause interference to . The latter two as-sumptions validate the graph-based model used here. Then, theresulting power control problem can be modeled as an ACGG.Note that if we assume there are only two channels, then by The-orem 1, this game possesses the FIP, and a PNE can be reachedin a distributed way.

B. IEEE 802.11 Channel Contention

Another application of an ACGG is to analyze the randomaccess scheme under the collision model such as IEEE 802.11.



In this case, only a single user can access a channel at each timeslot, and the reward a user obtains from selecting a channelis the probability of accessing the channel multiplied by therate that channel offers to the user. Suppose we assume, asis commonly done, that users sufficiently far apart from eachother do not interfere with each other, and thus can access thesame channel in the same slot. Let be the probability thatuser accesses channel when the number of ’s neighbors com-peting for channel is , and be the rate of channel seenby user . Then, the resulting channel contention can be mod-eled as an ACGG with user-specific payoff functions given by

.

C. Numerical Results

We now present some numerical results on the expectednumber of asynchronous improvement steps needed to con-verge to a PNE over random graphs, in a context similarto that of channel contention described above. Consider threeresources (or channels) indexed 1, 2, 3, and the following twocases. The first case is the random access scheme with iden-tical channels and nonuser specific payoffs. We assume that thepayoff a user gets from using a channel when there areneighbors using channel is ;this can model a fair share of the channel among all userscontending for the same channel under random access. Thesecond case is the random access scheme with nonidenticalchannels and non-user-specific payoffs, where payoffs of userare given by , .We have shown that in the first case FIP exists, therefore userswill converge to a PNE in finite time by improvement steps. Inthe second case, we have a counterexample showing that betterresponse loops may exist, but we have not shown an instancefor which a PNE does not exist. Thus, we only consider bestresponse updates, and for the second case we also check if theplayers ever enter a best-response loop ( if this everhappens).For each simulation, we consider 20 users randomly placed

on a 10 10 square area. For each random placement of users,we generate the interference graph based on a threshold . Ifthe distance between two users is greater than , we assumethat they do not interfere with each other. We calculate foreach by averaging over 100 random placements of users and100 runs for each random placement of users, where each runstarts with a random strategy profile and the improving user ineach step is randomly selected among all users who can improvetheir payoff by changing their strategies. We plot the averagevalue of for each threshold for both casesin Fig. 7, where the unit of is the number of improvementsteps. The results show that the convergence is fairly fast. In thecase of identical channels, convergence tends to be fast when thenetwork is either sparsely connected (low threshold), or whenthe network is approximately fully connected (high threshold).These observations are intuitively satisfying: When the net-

work is sparsely connected (or even disconnected), a smallernumber of interconnected users leads to a fewer number of up-dates; when the network is near fully connected, the impact ofany update is immediately known to the users, which again leads

Fig. 7. value depending on the threshold .

to fewer number of updates needed. In between these two ex-tremes, when the network is connected but not densely con-nected, an update can potentially impact all users in the network,but this impact may take much longer to propagate through thenetwork. The nonidentical channel case is more complex. How-ever, it does follow the same trend except at very low thresholds.In addition, it generally takes longer to converge in this case thanin the case of identical channels when all other parameters arethe same.We also note for all 10 000 simulations, we did not observe

a single case where the users enter a best response loop. This isconsistent with the difficulty we had in searching for counterex-amples (see Remark 2): In general, the instances in which a PNEor best-response improvement path does not exist for ACGGswith non-user-specific payoffs are very rare.

IX. DISCUSSION

In this section, we discuss the relevance and limitations of theACGGmodel and the results obtained in the context of wirelessapplications and point to directions of further studies.Two results obtained in this paper are of particular interest

in the context of wireless networking, namely Theorems 1 and2. Theorem 1 showed that when users are limited to only twochannels, the finite improvement property holds over arbitrarygraphs with user-specific payoff functions. Theorem 2 showedthat when channels are of equal width and propagation charac-teristics for each user (as is the case when a contiguous blockof bandwidth is evenly sliced into smaller channels), the finiteimprovement property holds. This is true even if the channelsare of different quality to different users, e.g., due to the use ofdifferent modulation schemes. This latter scenario is a very re-alistic one, as this is the case with multiple channels in Wi-Fi,Bluetooth, and so on. The finite improvement property suggeststhat, in such systems, greedy user updates (i.e., using best-replydynamics) will lead, in a finite number of steps, to a PNE, whichis also the local minimizer of the explicit potential function [(7)in this case]. This means that we not only have an easy way ofobtaining a PNE, but also have a sense of the (local) efficiencyof this NE.



In the weaker case where the existence of PNE is established,as shown in Theorems 3–5, players can similarly reach a PNEwith probability one by using the uncoupled dynamics with2-recall given in Theorem 3 of [29].Our primary focus in this paper has been the existence of

PNE. We have not considered other equilibrium concepts suchas mixed strategy Nash equilibrium or correlated equilibriumsince their existence is trivial in the games we study in thispaper. We also have not focused on the performance of an equi-librium. As noted above, when the FIP holds, we can attain aPNE that is also a local optimal solution to the potential func-tion, but there may be multiple PNE that result in different ob-jective function values. In general, the performance of a reachedPNE is not guaranteed. It is also worth noting that although con-vergence to a PNE implies stationary behavior, even when theplayers cycle, there can be cases where the average reward ofall the players during a cycle will be higher than when they playat a PNE. Thus, if the cost of switching is negligible, this typeof cycling behavior may indeed be better than a PNE in terms ofthe objective function. An example of such a situation is givenin [30], where the authors show that a natural learning algorithmcycles, but this results in social welfare higher than that achievedat the unique (mixed) NE. Another interpretation of the cyclicbehavior is that its long-term average can be interpreted as amixed or correlated equilibrium.One limitation of the ACGG model is that it treats all inter-

ference relationships equally, i.e., the underlying network graphis unweighted (see, e.g., the power control problem illustratedin Section VIII-A). In reality, the channel quality perceived bya user depends not only on who else is using the same channeland can potentially interfere, but also its distances to these inter-fering users. One way to address this is to define the congestiongame over a weighted network graph and define the user payoffas a function of the weights on links connecting interfering userswho use the same channel. Analysis along this line will be veryinteresting yet challenging.Throughout our discussion, we have limited our attention to

the case where each user can access one resource/channel ata time. In reality, it is also possible for a user to access mul-tiple channels at a time. As mentioned earlier, if all users canaccess all channels simultaneously and the available transmis-sion power is decoupled across the channels, then the resultingcongestion game is not particularly interesting, as an obviousPNE is where all users use all resources. A more interestingcase is when users are limited to the number of channels theycan access simultaneously. An additional feature may be thatdifferent users have different sets of channels they are allowedto access, i.e., user ’s strategy space , whereis user ’s set of allowed channels. Finally, a user may need tospread the communication resource such as transmission poweramong multiple channels, thus transmitting over one or multiplechannels implies different payoff functions for each channel. Allthese features will make the resulting game much more compli-cated and are subjects of future study.

X. CONCLUSION

In this paper, we considered an extension to the classical con-gestion games by allowing resources to be reused among nonin-

teracting or noninterfering users. The resulting game, the atomiccongestion game on graphs, is applicable in the context of wire-less network where spatial reuse is frequently exploited to in-crease spectrum utilization. We showed that the finite improve-ment property holds when there are only two resources or theresources are identical to each user. We provided a negative re-sult on the existence of a finite improvement path in the generalcase, but showed that a pure strategy NE exists without the FIPif the network can be modeled by a tree, a loop, a regular bipar-tite graph, or with a dominant resource.

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Cem Tekin (S’11) received the B.Sc. degree inelectrical and electronics engineering from theMiddle East Technical University, Ankara, Turkey,in 2008, the M.S.E. degree in electrical engineering:systems and M.S. degree in mathematics from theUniversity of Michigan, Ann Arbor, in 2010 and2011, respectively, and is currently pursuing thePh.D. degree in electrical engineering and computerscience at the University of Michigan.His research interests include online learning algo-

rithms, stochastic optimization, multi-armed banditproblems, multiuser systems, and game theory.

Mingyan Liu (M’00–SM’11) received the B.Sc.degree in electrical engineering from the NanjingUniversity of Aeronautics and Astronautics, Nan-jing, China, in 1995, and the M.Sc. degree in systemsengineering and Ph.D. degree in electrical engi-neering from the University of Maryland, CollegePark, in 1997 and 2000, respectively.She joined the Department of Electrical Engi-

neering and Computer Science (EECS), Universityof Michigan, Ann Arbor, in September 2000, whereshe is currently an Associate Professor. Her research

interests are in optimal resource allocation, performance modeling and analysis,and energy-efficient design of wireless, mobile ad hoc, and sensor networks.Dr. Liu serves on the Editorial Board of the IEEE/ACM TRANSACTIONS ON

NETWORKING, the IEEE TRANSACTIONS ONMOBILE COMPUTING, and the ACMTransactions on Sensor Networks. She is the recipient of the 2002 NSF CA-REER Award, the University of Michigan Elizabeth C. Crosby Research Awardin 2003, and the 2010 EECS Department Outstanding Achievement Award.

Richard Southwell received the B.Sc. degree intheoretical physics and M.Sc. degree in mathematicsfrom the University of York, York, U.K., in 2005and 2006, respectively, and the Ph.D. degree inmathematics from Sheffield University, Sheffield,U.K., in 2009.After receiving the Ph.D. degree, he worked as a

Research Associated with Sheffield University forone year, in the amorphous computing project. Hethen moved to the Chinese University of Hong Kong,Hong Kong, and worked as a Research Assistant

with the Network Communications and Economics Laboratory, within the In-formation Engineering Department. He is currently an Assistant Professor withthe Institute for Interdisciplinary Information Sciences, Tsinghua University,Beijing, China. His research interests include adaptive graph models, gameson graphs, complex systems, and wireless networks.

Jianwei Huang (S’01–M’06–SM’11) received theB.S. degree in electrical engineering from SoutheastUniversity, Nanjing, China, in 2000, and the M.S.and Ph.D. degrees in electrical and computer engi-neering from Northwestern University, Evanston,IL, in 2003 and 2005, respectively.He is an Assistant Professor with the Department

of Information Engineering, The Chinese Universityof Hong Kong, Hong Kong. He worked as a Post-doctoral Research Associate with the Departmentof Electrical Engineering, Princeton University,

Princeton, NJ, from 2005 to 2007.Dr. Huang is on the Editorial Board of the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS and the IEEE JOURNAL ON SELECTED AREASIN COMMUNICATIONS: COGNITIVE RADIO SERIES. He is the recipient of theIEEE Marconi Prize Paper Award in Wireless Communications in 2011, theInternational Conference on Wireless Internet Best Paper Award 2011, theIEEE GLOBECOM Best Paper Award in 2010, the IEEE ComSoc Asia–Pa-cific Outstanding Young Researcher Award in 2009, and the Asia–PacificConference on Communications Best Paper Award in 2009.

Sahand Haji Ali Ahmad was born in Tehran, Iran, in 1980. He received theB.Sc. degree from the Sharif University of Technology, Tehran, Iran, in 2002,the M.Sc. degree from the University of Illinois at Urbana–Champaign in 2005,and the Ph.D. degree from the University of Michigan, Ann Arbor, in 2010, allin electrical engineering.He was winner of the gold medal in Iran’s national mathematical olympiad,

winner of the gold medal in the 39th International Mathematical Olympiad, andalso the recipient of a Vodafone fellowship for research in wireless communi-cation during academic year 2005–2006.

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