A Controlled Matching Game for WLANsMikael Touati, Rachid El-Azouzi, Marceau Coupechoux, Eitan Altman and Jean-Marc Kelif

(Manuscript received April 30 2016; Revised December 21 2016; Accepted January 24 2017)

Abstract—In multi-rate IEEE 802.11 WLANs, the tra-ditional user association based on the strongest receivedsignal and the well known anomaly of the MAC protocolcan lead to overloaded Access Points (APs), and poor orheterogeneous performance. Our goal is to propose analternative game-theoretic approach for association. Wemodel the joint resource allocation and user association asa matching game with complementarities and peer effectsconsisting of selfish players solely interested in their indi-vidual throughputs. Using recent game-theoretic results wefirst show that various resource sharing protocols actuallyfall in the scope of the set of stability-inducing resourceallocation schemes. The game makes an extensive use ofthe Nash bargaining and some of its related properties thatallow to control the incentives of the players. We show thatthe proposed mechanism can greatly improve the efficiencyof 802.11 with heterogeneous nodes and reduce the negativeimpact of peer effects such as its MAC anomaly. Themechanism can be implemented as a virtual connectivitymanagement layer to achieve efficient APs-user associationswithout modification of the MAC layer.

Index Terms—Game theory, matching games, coalitiongames, stability, control, 802.11.

I. INTRODUCTION

The IEEE 802.11 based wireless local area networks(WLANs) have attained a huge popularity in dense areasas public places, universities and city centers. In suchenvironments, devices have the possibility to use manyAccess Points (APs) and usually a device selects anAP with the highest received Radio Signal StrengthIndicator (best-RSSI association scheme). In this context,the performance of IEEE 802.11 may be penalized bythe so called 802.11 anomaly and by an imbalance inAP loads (congestion). Moreover, some APs may beoverloaded while others are underutilized because of theassociation rule.

In this paper, we consider a fully distributed IEEE802.11 network, in which selfish mobile users andAPs look for the associations maximizing their individ-ual throughputs. The considered distributed associationproblem with selfish users naturally motivates us to adoptmatching game theory. We thus analyze this scenario us-ing matching games and develop a unified analysis of themobile user association and resource allocation problemfor the reduction of the anomaly and for load balancing

Mikael Touati and Jean Marc Kelif are with Orange Lab,France, E-mail: mikael.touati, jeanmarc.kelif@orange.com, RachidEl-Azouzi is with CERI/LIA, University of Avignon, Avignon, France,E-mail: rachid.elazouzi@univ-avignon.fr, Marceau Coupechoux iswith Telecom ParisTech, University Paris-Saclay, France, E-mail:marceau.coupechoux@telecom-paristech.fr, Eitan Altman is withINRIA Sophia Antipolis, France, E-mail: eitan.altman@inria.fr.

in IEEE 802.11 WLANs. In a network characterizedby a state of nature (user locations, channel conditions,physical data rates), composed of a set W of mobileusers and a set F of APs, the user association problemconsists in finding a mapping µ that associates everymobile user to an AP. We call the set formed by an APand its associated mobile users a cell, or a coalition in thegame framework. The set of coalitions induced by µ iscalled a matching or a structure (partition of the playersin coalitions). Once mobile user association has beenperformed, a resource allocation scheme (also called asharing rule in this paper) allocates radio resources of acell to the associated mobile users.

This matching game is characterized by complemen-tarities in the sense that APs have preferences overgroups of mobile users and peer effects in the sensethat mobile users care who their peers are in a celland thus emit preferences also over groups of mobilesusers. Indeed, by definition of DCF implementation ofthe IEEE 802.11 protocol, a users’ throughput does notonly depend on its physical data rate but also on thecoalition size and composition. The following questionsare raised: are there stable associations? Do these associ-ations always exist? Is there unicity? How to reach theseequilibria in a decentralized way? Finally, how to providethe players the incentive to make the system converge toanother association point with interesting properties interms of load balancing?

In this game, some users may remain unassociatedsince every AP has the incentive to associate with asingle user having the best data rate. To counter thisso called unemployment problem and control the set ofthe stable matchings, we design a decentralized threesteps mechanism. In the first step, the APs share theload. In the second step, the matching game is controlledto provide the incentives to enforce the load balancing.The control is based on the notion of Fear of Ruin(FoR) introduced in [6]. In the third step, players playthe controlled matching game with individual payoffsobtained from a Nash Bargaining (NB) sharing rule.Under some assumptions, the NB sharing rule guaranteesthat the set of stable structures is non empty in all statesof nature. A core stable matching is obtained by a decen-tralized algorithm. We propose here a modified versionof the Deferred Acceptance Algorithm (DAA), calledBackward Deferred Acceptance Algorithm (BDAA), formatching games with complementarities and peer effects.Similarly to the DAA, the complexity of the BDAA ispolynomial in the number of proposals.

A. Related Work

IEEE 802.11 (WiFi) anomaly is a well documentedissue in the literature, see e.g. [10], [11], [16]. Severalapproaches for a single cell modify the MAC so as toachieve a time-based fairness [10], [11] or proportionalfairness [16]. Throughput based fairness, time basedfairness and proportional fairness resource allocationschemes are sharing rules that can be obtained from aNash bargaining as we will see later on.

In a multiple cell WLAN, mobile user-AP associationplays a crucial role for improving the network perfor-mance and can be seen as a mean to mitigate the WiFianomaly without modifying the MAC layer. The max-imum RSSI association approach, though very simple,may cause an imbalanced traffic load among APs. Kumaret al. [14] investigate the problem of maximizing thesum of logarithms of the throughputs. Bejerano et al.[19] formulate a mobile user-AP association problemguaranteeing a max-min fair bandwidth allocation formobile user. This problem is shown to be NP-hard andconstant-factor approximation algorithms are proposed.

Arguing for ease of implementation, scalability androbustness, several papers have proposed decentralizedheuristics to solve this issue, see e.g. [13], [22], [24].Reference [13] proposes to enhance the basic RSSIscheme by an estimation of the Signal to Interferenceplus Noise Ratio (SINR) on both the uplink and thedownlink. Bonald et al. in [24] show how performancestrongly depends on the frequency assignment to APsand propose to use both data rate and MAC throughput ina combined metric to select the AP. Several papers haveapproached the problem using game theory based onindividual MAC throughput. Due to the WiFi anomaly,this is not a classical crowding game in the sense thatthe mobile user achieved throughput is not necessarya monotonically decreasing function of the number ofattached devices, as it can be the case in cellular net-works [21], [25]. Compared to proposed decentralizedapproaches, we do not intend to optimize some net-work wide objective function, but rather to study theequilibria resulting from selfish behaviors. Compared toother game-theoretic approaches, we consider a fullydistributed scenario, in which APs are also playersable to accept or reject mobile users. This requires thestudy of the core stability, a notion stronger than theclassical Nash Equilibrium. Moreover, there is a needin understanding the fundamental interactions betweenmobile user association and resource allocation in thepresence of complementarities and peer effects.

In this paper, we tackle the mobile user-AP associ-ation problem using the framework of matching gameswith individual selfish players. This framework providespowerful tools for analyzing the stability of associa-tions resulting from decentralized mechanisms. Match-ing games [7] is a field of game theory that have provedto be successful in explaining achievements and failuresof matching and allocation mechanisms in decentralized

markets. Gale and Shapley published one of the earliestand probably most successful paper on the subject [3]and solved the stable marriage and college admissionsassociation problem with a polynomial algorithm calledDAA.

Some very recent papers in the field of wireless net-works have exploited the theoretical results and practicalmethods of matching games [28], [29], [30], [34], [35],although none has considered the WLAN associationproblem and its related WiFi anomaly. Authors of [28]address the problem of downlink association in wirelesssmall-cell networks with device context awareness. Therelationship between resource allocation and stabilityis not investigated and APs are not allowed to rejectusers. Hamidouche et al. in [29] tackle the problemof video caching in small-cell networks. They proposean algorithm that results in a many-to-many pairwisestable matching. Preferences emitted by servers exhibitcomplementarities between videos and vice versa. Never-theless, the model doesn’t take into account peer effectswithin each group. Reference [30] addresses the problemof uplink user association in heterogeneous wirelessnetworks. Invoking a high complexity, complementaritiesare taken into account by a transfer mechanism thatresults in a Nash-stable matching, a concept weakerthan pairwise stability or core stability. References [34],[35] study cognitive radio networks and do not exhibitcomplementarities or peer effects in the definition ofplayers’ preferences.

B. Contributions

Our contributions can be summarized as follows:• We provide a matching game-theoretic unified ap-proach of mobile user association and resource allocationin IEEE 802.11 WLANs in the presence of complemen-tarities and peer effects. To the best of our knowledge,this is the first game-theoretic modeling of the IEEE802.11 protocol covering such a number of resourceallocation mechanisms proposed in the literature.• We use existing theoretical results to show that if thescheduling and/or the MAC protocol result from a Nashbargaining then there exist stable mobile user associ-ations, whatever the user data rates or locations. Thisresult highlights the importance of the Nash bargainingin wireless networks as a stability inducer.• In order to control the matching game, we designa three steps mechanism, which includes 1) a genericload balancing, 2) a control step, which modifies agents’preferences to provide them the incentive to enforce theresult of the load balancing, 3) a matching game withresource allocation defined as a Nash bargaining and astable matching algorithm. The control step tackles theso called unemployment problem, that would have leftmobile users aside from the association otherwise. Weshow through numerical examples that our mechanismachieves good performance compared to the global op-timum solution. To the best of our knowledge, such a

mechanism is absent from both the game theoretic andwireless networks based on matching games literature.• We show that our BDAA can be efficiently used tofind a stable many-to-one matching in a matching gamewith complementarities and peer-effects. The algorithmhas a polynomial complexity in the number of proposals.

The mechanism has been originally proposed in anextended abstract [33]. Equal sharing has already beenassessed in [32]. BDAA has been originally proposedin a short paper [31]. In this paper, we provide acomplete description and show the mathematical results.We furthermore generalize the mechanism to a genericload balancing scheme and to the Nash bargaining-basedsharing rules (resource allocation schemes). Finally, weshow new numerical results.

The rest of the paper is organized as follows. InSection II, we define the system model. In Section III, weformulate the IEEE 802.11 WLANs resource allocationand decentralized association problem. In Section IV,we show that there exist stable coalition structures undercertain conditions whatever the individual data rates.Section V presents our three steps mechanism. Sec-tion VI shows numerical results. Section VII concludesthe paper and provides perspectives.

II. SYSTEM MODEL

We summarize in Table I the notations used in thispaper. We use both game-theoretic definitions and theirnetworking interpretation. Throughout the paper, they areused indifferently. Let define the set of players (nodes)N of cardinality N as the union of the disjoint setsof mobile users W of cardinality W and APs F ofcardinality F . As in [14], we assume an interference-free model. It is assumed that the AP placement andchannel allocation are such that the interference betweencochannel APs can be ignored. In game-theoretic terms,this implies that there are no externalities. The mobileuser association is a mapping µ that associates everymobile user to an AP and every AP to a subset of mobileusers.

The IEEE 802.11 standard MAC protocol has been setup to enable any node in N to access a common mediumin order to transmit its packets. The physical data ratebetween a transmitter and a receiver depends on theirrespective locations and on the channel conditions. Foreach mobile user i ∈ W , let θif be the (physical) datarate with an AP f where θif ∈ Θ = θ1, . . . , θm,a finite set of finite rates resulting from the finite setof Modulation and Coding Schemes. If i is not withinthe coverage of f then θif = 0. Given an associationµ, let θC = (θwf )(f,w)∈(C∩F)×(C∩W) denote the datarate vector of mobiles users in cell C served by APf . Let nC be the normalized composition vector of C,whose k-th component is the proportion of users in Cwith data rate θk ∈ Θ. Note that an AP is definedin the model as a player with the additional propertyof having maximum data rate on the downlink. Within

each cell, a resource allocation scheme (e.g. induced bythe CSMA/CA MAC protocol) may be formalized asa sharing rule over the resource to be shared in thecell. This resource may be the total cell throughput(as considered in the saturated regime) or the amountof radio resources in time or frequency in the generalcase. More precisely, a sharing rule is a set of functionsD = (Di,C)C∈C,i∈C , where Di,C allocates a part of theresource of C to user i ∈ C. Equal sharing, proportionalfairness, α-fairness are examples of sharing rules.

Assuming the IEEE 802.11 MAC protocol and thesaturated regime, the overall cell resource of cell C isdefined as the total throughput. It is a function of thecomposition vector nC and of the cardinality |C|. Wedenote ri,C the throughput obtained by user i in cell C.From the game theoretic point of view, ri,C is understoodas i’s share of the worth of coalition C denoted v(C).The function v : C → R is called the characteristicfunction of the coalition game and maps any coalitionC ∈ C to its worth v(C). Other MAC protocols andregime can however be modeled by this approach. Forexample time-based fairness proposed in the literature tosolve the WiFi anomaly results from the sharing of thetime resource. In this case, a user i gets a proportionαi,C of the time resource, which induces a throughoutof αi,Cθif , where f is the AP of C. It can be shownthat time-based fairness results in a proportional fairnessin throughputs.

III. MATCHING GAMES FORMULATION

A. Matching Games for Mobile User Association

In this paper, the mobile user association is modeledas a matching game (in the class of coalition games).The matching theory relies on the existence of individualorder relations ii∈N , called preferences, giving theplayer’s ordinal ranking1 of alternative choices. As anexample, w1 f1 [w2, w3] f1 w4 indicates that theAP f1 prefers to be associated to mobile user w4 to anyother mobile user, is indifferent between w2 and w3, andprefers to be associated to mobile user w2 or w3 ratherthan to be associated to w1. Following the notations ofRoth and Sotomayor in [7], let us denote P the set ofpreference lists P = (Pw1

, . . . , PwW , Pf1 , . . . , PfF ).

Definition 1 (Many-to-one bi-partite matching [7]). Amatching µ is a function from the set W ∪ F into theset of all subsets of W ∪F such that:(i) |µ(w)| = 1 for every mobile user w ∈ W and µ(w) =w if µ(w) 6∈ F;(ii) |µ(f)| ≤ qf for every AP f ∈ F (µ(f) = ∅ if f isn’tmatched to any mobile user in W);(iii) µ(w) = f if and only if w is in µ(f).

1In this paper, we use the Individually Rational Coalition Lists(IRCLs) to represent preferences. It can indeed easily be shown thatother representations (additively separable preferences, B-preferences,W-preferences) are not adapted to our problem, see [26] for moredetails.

|set| cardinality of the set set N set of players (mobile users and APs)W set of mobile users F set of Access Points (APs)C set of coalitions (cells) Cf set of coalitions containing AP f ∈ FC coalition (cell) µ matching (AP-mobile user association)Θ set of feasible data rates θwf data rate between w and fri,C throughput of node (user or AP) i in cell C αi,C proportion of resources (time, frequency) of i in cell CD sharing rule (resource allocation scheme) v(C) worth of coalition Csi,C payoff of player i in coalition C ui(.) utility function of player iqf quota of AP f χC fear-of-ruin of coalition Cqf target load of AP f Ω control functionP (i) preferences list of player i over individuals P#(i) preferences list of player i over groups

TABLE INOTATIONS

Condition (i) of the above definition means that amobile user can be associated to at most one AP andthat it is by convention associated to itself if it is notassociated to any AP. Condition (ii) states that an APf cannot be associated to more than qf mobile users.Condition (iii) means that if a mobile user w is associatedto an AP f then the reverse is also true. In this definition,qf ∈ N∗ is called the quota of AP f and it gives themaximum number of mobile users the AP f can beassociated to.

From now on, we focus on many-to-one matchings.In this setting, stability plays the role of equilibriumsolution. In this paper, we particularly have an interestin the pairwise and core stabilities. For more details werefer the reader to the reference book [7]. We say that amatching µ is blocked by a player if this player prefersto be unmatched rather than being matched by µ. Wesay that it is blocked by a pair if there exists a pair ofunmatched players that prefer to be matched together.

Definition 2 (Pairwise sability [7]). A matching µ ispairwise stable if it is not blocked by any player or anypair of players. The set of pairwise stable matchings isdenoted S(P).

Definition 3 (Domination [7]). A matching µ′ dominatesanother matching µ via a coalition C contained inW∪Fif for all mobile users w and APs f in C, (i) if f ′ =µ′(w) then f ′ ∈ C, and if w′ ∈ µ′(f) then w′ ∈ C; and(ii) µ′(w) w µ(w) and µ′(f) f µ(f).

Definition 4 (Weak Domination [7]). A matching µ′

weakly dominates another matching µ via a coalitionC contained in W ∪ F if for all mobile users w andAPs f in C, (i) if f ′ = µ′(w) then f ′ ∈ C, and ifw′ ∈ µ′(f) then w′ ∈ C; and (ii) µ′(w) w µ(w) andµ′(f) f µ(f); and (iii) µ′(w) w µ(w) for some w inC, or µ′(f) f µ(f) for some f in C.

Definition 5 (Cores of the game [7]). The core C(P)(resp. the core defined by weak domination CW (P))of the matching game is the set of matchings that arenot dominated (resp. weakly dominated) by any othermatching.

In the general case, the core of the game C(P)contains CW (P). When the game does not exhibitcomplementarities or peer effects, it is sufficient forits description that the preferences are emitted over

individuals only. In the presence of complementarities orpeer effects, players in the same coalition (i.e. the set ofmobile users matched to the same AP) have an influenceon each others. In such a case, the preferences need tobe emitted over subsets of players and are denoted P#.

In the classical case of matchings with complemen-tarities, the preference lists are of the form P =(Pw1 , . . . , PwW , P

#f1, . . . , P#

fF), i.e., preferences over

groups are emitted only by the APs (see the firms andworkers problem in [7]). Moreover, it may happen thatthe preferences over groups may be responsive to theindividual preferences in the sense that they are alignedwith the individual preferences in the preferences overgroups differing from at most one player. The prefer-ences over groups may also satisfy the substitutabilityproperty. The substitutability of the preferences of aplayer rules out the possibility that this player considersothers as complements.

Definition 6 (Responsive preferences [7]). The pref-erences relation P#(i) of player i over sets playersis responsive to the preferences P (i) over individualplayers if, whenever µ′(i) = µ(i)∪k\l for l in µ(i)and k not in µ(i), then i prefers µ′(i) to µ(i) (underP#(i)) if and only if i prefers k to l (under P (i)).

Before defining the substitutability property of pref-erences, we need to introduce the choice function Chiof a player i. Given any subset C of players, Chi(S)is called the choice set of i in S. It gives the subset ofplayers in C that player i most prefers.

Definition 7 (Substitutable preferences [7]). A playeri’s (i ∈ W ∪F) preferences over sets of players has theproperty of substitutability if, for any set S that containsplayers k and l, if k is in Chi(S) then k is in Chi(S\l).

Considering preference lists of the form P =(Pw1

, . . . , PwW , P#f1, . . . , P#

fF) and assuming either re-

sponsive or substitutable strict preferences, we have theresult that CW (P) equals S(P) [7, p.174]. Any many-to-one matching problem with these properties has anequivalent one-to-one matching problem, which can besolved by considering preferences over individuals only.The set of pairwise stable matching is non-empty.

If the preferences are neither responsive nor substi-tutable, the equality S(P) = CW (P) does not hold ingeneral and the sets of pairwise, weak core and core

stable matchings may be empty. An additional difficultyappears if the preferences over groups have to be consid-ered on the mobile users side, i.e., if we have preferencelists of the form P = (P#

w1, . . . , P#

wW , P#f1, . . . , P#

fF).

Complementarities and peer effect may arise in bothsides of the matching. The user association problemin IEEE 802.11 WLANs falls in this category becausethe performance of any mobile user in a coalition maydepend on the other mobiles in the coalition. To breakthe indifference, we use the following rule: a mobileuser prefers a coalition with AP with the lowest indexand an AP prefers coalitions in lexicographic order ofusers indices.

To see that preferences may not be responsive, con-sider an example with only uplink communications,two APs f1 and f2 and three mobile users w1, w2, w3

such that θ11 = 300 Mbps, θ12 = θ22 = 54 Mbps,θ21 = θ32 = 1 Mbps. Assuming saturated regime andequal packet size, we can show that P#(w1) = f1 f2 w3; f1 w2; f2 w2; f1 w3; f2,which is not responsive. In this example, we also seethat substitutability is not even defined since every choiceset is reduced to a singleton. After the game has beencontrolled according the proposed mechanism, prefer-ences of w1 can be modified as follows: P#(w1) =w3; f1 w2; f2 w2; f1 w3; f2 f1 f2. Considering S = w2, w3; f1, f2, we haveChw1

(S) = w3; f1, while Chw1(S\w3) = w2; f2.

Preferences are thus not substitutable.This general many-to-one matching problem has al-

gorithmically been assessed by Echenique and Yenmenin [17] who propose a fixed-point formulation and analgorithm to enumerate the set of stable matchings. Itis known, that there is no guarantee that this set is nonempty if the individual preferences over groups are not ofa particular form. The problem of complementarities andpeer effects in matchings has been analytically tackledby Pycia in [27]. Nevertheless, no result have beenderived concerning the control of core stable structuresand no matching algorithm with such reduced lists ofpreferences for the mobiles have been derived.

B. Sharing Rules and Matching Game Formulation

We now assume that a player i in a given coalition Cobtains a payoff si,C , which is evaluated (or perceived)by it through a utility function ui : R → R. In thispaper, we assume that functions ui are positive, con-cave (thus log-concave), increasing and differentiable.In such a case, the individual preferences are inducedby the player’s utilities of these payoffs. Note that asutility functions are increasing, user’s preferences areequivalently based on their payoff or their utility. Weextend our model to the framework of finite coalitiongames in characteristic form Γ = (N ; v), where v isa function mapping any coalition to its worth in R+.By definition of the characteristic function v(∅) = 0.In this paper we do not assume a particular form of

the characteristic function v (e.g. super-additivity2). Aneven particular case of coalition games in characteristicform concerns games with an exogenous sharing ruleΓ = (N ; v;EN ;D), where EN is the set of all payoffvectors and D is a sharing rule.

Definition 8 (Sharing Rule). A sharing rule is a col-lection of functions Di,C : R+ → R+, one for eachcoalition C and each of its members i ∈ C, that mapsthe worth v(C) of C into the share of output obtained byplayer i. We denote the sharing rule given by functionsDi,C as D = (Di,C)C∈C,i∈C .

From this definition, the payoff of user i in coalitionC is given by si,C = Di,C v(C) (where is thecomposition) and his utility of this payoff is given byui(si,C). We can now formulate the IEEE 802.11 jointuser association and resource allocation problem as amatching game.

Definition 9 (Resource Allocation and User AssociationGame). Using the above notations, the resource alloca-tion and users association game is defined as a N -playermany-to-one matching game in characteristic form withsharing rule D and rates θ = θwf(w,f)∈W×F : Γ =(W∪F , v,R+N , D,θ). Each pair of players of the form(w, f) ∈ W ∪ F is endowed with a rate θwf from therates space Θ = θ1, . . . , θm. For this game, we definethe set of possible coalitions C:C = f∪J, f ∈ F , J ⊆ W, |J | ≤ qf∪w, w ∈ W.

(1)

Note that for IEEE 802.11 MAC protocol and for thesaturated regime, si,C , ri,C . For other time sharingMAC approaches, si,C , αi,C .

IV. EXISTENCE OF CORE STABLE STRUCTURES

A. Background on Nash Bargaining

The analytical theory of bargaining has mainly beendeveloped on the concept introduced by J.F. Nash in [1]and [2] for the two person game and by Harsanyi in [4]for the N-person game. The bargaining is developed asa cooperative game where the set of acceptable (fea-sible) individual payoffs results from the set of mutualagreements among the players involved. In this paper, weuse the Nash bargaining to model the resource allocationscheme (see Section V).

Let t = (t1, . . . , tN ) be the fixed threat vector, anexogeneous parameter that may or not result from athreat game. The bargaining problem consists in lookingfor a payoff vector (u1, ..., uN ) satisfying five axioms: (i)Strong Efficiency, (ii) Individual Rationality, (iii) ScaleInvariance, (iv) Independence of Irrelevant Alternativesand, (v) Symmetry.We have the following result (see [15] and referencestherein):

2∀C,C, v(C ∪ C′) ≥ v(C) + v(C′) if C ∩ C′ = ∅

Theorem 1. Let the utility functions ui be concave,upper-bounded and defined on X , a convex and com-pact subset of Rn. Let X0 be the set of payoffs s.t.X0 = s ∈ X|∀i, ui(si) ≥ ti and J be the set of userss.t., J = j = 1, . . . , N|∃s ∈ X, s.t. uj(sj) > tj.Assume that ujj∈J are injective. Then there exists aunique Nash bargaining problem as well as a uniqueNash bargaining solution s that verifies uj(sj) > tj ,j ∈ J , and is the unique solution of the problem:

maxs∈X0

∏j∈J

(uj(sj)− tj). (2)

An important result about the Nash bargaining solu-tion is that it achieves a generalized proportional fairnesswhich includes as a special case the well-known andcommonly-used proportional fairness. The proportionalfairness is achieved in the utility space with a null threatvector. Nevertheless, if the players’ utility functions arelinear in their payoffs and the threats vector is null, theproportional fairness is achieved in the payoff space.In other words, the payoff vector induced by the NashBargaining over the coalition throughput is proportionalfair. This makes game-theory and in particular the bar-gaining problem of fundamental importance in networks.In Appendix A in [36], we show that the resourceallocation induced by the MAC protocol of IEEE 802.11can be modeled as the result of a Nash bargaining withpower functions as utilities satisfying the conditions ofTheorem 1.

B. On the Existence of Core Stable Structures

In this section, we show the existence of stablecoalition structures (users-AP association) when pref-erences are obtained under some regularity conditionsover the set of coalitions and some assumptions over themonotonicity of the sharing rules. There exists a stablestructure of coalitions whatever the state of nature θ ifand only if the sharing rules may be formulated as arisingfrom the maximization of the product of increasing,differentiable and strictly log-concave individual utilityfunctions in all coalitions.

Proposition 2 ([27], Corollary 2). If the set of coalitionsC is such that qf ∈ 2, . . . ,W − 1 and F ≥ 2, and ifthe sharing rule D is such that all functions Di,C arestriclty increasing, continuous and limy→+∞Di,C(y) =+∞, then there is a stable coalition structure for eachpreference profile induced by the sharing rule D iff thereexist increasing, differentiable, and strictly log-concavefunctions ui : R+ → R+, i ∈ N , such that ui(0)

u′i(0)= 0

and(Di,C v(C))i∈C = argmax

sC∈BC

∏i∈C

ui(si,C), (3)

where C ∈ C and BC = sC = (si,C)i∈C |∑i∈C si,C ≤

v(C).

According to Proposition 2, there are two conditionsto be satisfied by the coalitions of our resource allocationand user association game: (i) consider scenarios with at

least two APs (which is reasonable when talking aboutload balancing) and (ii) every AP is supposed to be ableto serve at least two users and should not be able to servethe whole set of users. For more details, see AppendixB in [36].

Proposition 2 ensures that there exists a stable coali-tion structure as soon as resource allocation results froma Nash bargaining. The equal sharing resulting fromCSMA/CA MAC protocol in saturated regime, single-flow per device and equal packet length is obtained byconsidering si,C = ri,C and the identity function for ui.The players’ throughputs in the general saturated regimewith multiple flows and heterogenous packet length isobtained by taking si,C = ri,C and utility functions ofthe form ui(si,C) = sαii,C , where the bargaining powerαi ∈ [0, 1] is a function of the average packet lengthof user i (as shown in Appendix A in [36]). In theseexamples, utility functions ui verify the conditions ofProposition 2.

In CSMA/CA under saturated regime, the cellthroughput is increasing with the individual physical datarates and individual throughputs ri,C are sub-additive,i.e., decreasing with the addition of users. Assuming thatthe payoff is the individual throughput, i.e., si,C = ri,C ,then each player has the incentive to form the lowestcardinality coalition with highest composition vector.In this case, the unique stable structure is a one-to-one matching, in which APs are associated to theirbest mobile user. This will further be mentioned in thename of the unemployment problem since it leaves somemobiles users unassociated (unmatched). There is theneed for a control of the players incentives for someequilibrium points with satisfying properties, in terms ofunemployment in the present case. In other words, sincethe players have the incentive to match in a one-to-oneform, one needs to control the underlying cooperativegame so as to provide new incentives for a suitablemany-to-one form as an equilibrium.

V. MECHANISM FOR CONTROLLED MATCHING GAME

In order to tackle the unemployment problem, wepropose in this section a mechanism to control theplayers incentives for coalitions (see Figure 1). Thismechanism is made of three steps. We start by consid-ering for every AP the set of acceptable mobile users,i.e., the mobile users with non zero data rate with thisAP. In the first step (block LB), APs share the loaddefined in number of users. This results in target loadsthat should be enforced by the mechanism. The secondstep (blocks Ω and Φ) is a controlled coalition gamedesigned so as to provide the players the incentives toform coalitions with cardinalities given by the targetloads and reducing heterogeneity (and thus reducing theanomaly in the IEEE 802.11). The third step (blockµ) is a decentralized coalition formation (or matching)algorithm which results in a stable structure induced bythe individual preferences influenced by the controlledcoalitional game.

LB

F

Ω

v

Φ

(ui(.))i∈N

(s∗i,C)i∈C,C∈C

µ MACW q v (u∗i,C)i∈C,C∈C µ (ri,C)i∈C,C∈µ

MECHANISM 802.11 MAC

Fig. 1. Block diagram of the mechanism in the most general form. The APs share the load in the block LB which gives the APs’ targetloads q. The characteristic function v of the original coalition game is controlled in Ω and gives the modified characteristic function v. TheNash bargaining Φ is played in each coalition for the allocation of the worth of the coalition among its members. The players then emit theirpreferences over the coalitions on the basis of their shares and enter a stable matching mechanism in block µ. This block outputs an AP-userassociation µ. Finally, in the block MAC the nodes transmit their packets according to the unmodified IEEE 802.11 MAC protocol.

Our mechanism can be implemented as a virtual con-nectivity management layer on top of the IEEE 802.11MAC protocol. Mobile users and APs form coalitionsbased on the ”virtual rates” provided by this virtuallayer. Once associated, users access the channel usingthe unmodified 802.11 MAC protocol.

A. Load BalancingThe first step of the mechanism is a load balancing.

This step outputs a target load vector of the formq = (q1, . . . , qF ) that defines the size of the coalitionsthe players should be incentivized to form with eachAP. In other words, q gives the number of connectionsthe players should be incentivized to create with eachAP. In numerical implementation (Section VI), userscovered by several APs are equally shared by these APs.Nevertheless, any load balancing scheme can be usedin this mechanism. For example, there are decentralizedschemes that converge to the Nash Bargaining solutionthat is known to achieve a generalized proportional fairallocation [8], [15]. At this stage, there is no incentivefor agents to respect the target loads and if nothing elseis done the unemployment problem remains. The targetloads serve as input for the control step.

B. Controlling coalition gameThe second step of the mechanism is the control of

the coalition game. The control step of the mechanismtackles the problem of the control of the set of stablematchings. We observed that when a coalition game isdefined by a characteristic function and a sharing ruleinducing sub-additive strictly positive individual payoffs(except for coalitions of size one or those containingplayers with zero data rates), the stable structures to beformed are made of coalitions of size two. This stepof the mechanism develops an analytical framework andmethodology for the control of the equilibria by the wayof a control over the players’ incitations for individualstrategies.

Definition 10 (Controller). The controller is any entity(player or other) having the legitimacy and ability tochange the definition of the game (players, payoffs,worths, information, coalitions).

The controller may not be taking part in the game(e.g. the network operator in a wireless network, thegovernment for a firms and workers association problem)or any player of the game with some kind of additionaldecisional abilities. In other words, it may be any entityhaving the ability to create or modify the individualincitations of the players for some strategy and thusthe ability to change the definition of the game. Thesechanges in the definition of the game in view of manipu-lating the players’ equilibria strategies are called controltransformation,

Definition 11 (Control transformation). A control trans-formation Ω is a mapping from the set of coalition gamesin characteristic form in itself.

In the purpose of this paper it is sufficient to re-strict the definition of the control transformations tothe domain of coalition games in characteristic form.In fact, we further assume that the controller cannotarbitrarily move from one game to another withoutconstraints. We assume that he or she can influencethe equilibria by partial changes in the definition of thegame (characteristic function, individual payoffs, ...) butcan neither change the fundamental rules of the game(e.g. the rules of matching games) nor some essentialelements such as the players taking part in the gameor their strategy spaces. If Γ is a coalition game incharacteristic form, then Ω(Γ) is a coalition game incharacteristic form modified by the controller accordingto its (constrained) abilities. The limits of the abilities ofsuch a controller are to be chosen by the game theoristor the designer of the mechanism so as to satisfy thefundamental hypothesis and description of the system heis looking at. As an example of work on the design ofan incitations operator, Auman and Kurz [6] assess theproblem of designing the joint taxation and redistributionscheme in the framework of a political majority-minoritygame. The majority is the controller and the incitationsare induced by a multiplicative tax over the worths ofthe coalitions.

In Appendix C in [36], we give two simple examplesof the mechanism we propose to control the player’sindividual incentives.

We now search for operators modifying the character-istic function v so as to provide players the incentivesto form stable structures with coalitions of sizes q.

An important lever for controlling our matching gameand designing operator Ω is the fear-of-ruin (FoR).Formally, the FoR of user i in coalition C is definedas:

χi(si,C) ,ui(si,C)

u′i(si,C)

. (4)

The FoR of coalition C is obtained as the inverseof the Lagrange multiplier associated to the constraint∑i∈C si,C ≤ v(C) at the optimum of the Nash bargain-

ing optimization problem (3). Two interesting character-istics of the FoR are that (i) in a coalitional game withNash bargaining as sharing rule, the FoR is constant overthe players in a coalition, i.e., χi(si,C) = χC ∀i ∈ Cat the bargaining solution point si,C and (ii) with con-cave increasing utility functions, the individual payoffsincrease in the common FoR [27]. Thus, the playershave the incentives to form coalitions maximizing theirFoR. In terms of control opportunities, this introduces theFoR as a lever to control the set of individual payoff-based incentives for coalitions. As an example, assumetwo coalitions C and C ′ and their FoRs: χC < χC′ .Players in C ∩ C ′ prefer C ′ to C. Changing the valuesof the FoRs to obtain χC > χC′ changes the individualincentives of these players so that they now prefer C toC ′.

Proposition 3. Assume a coalition game Γ = (F ∪W, v, uii∈N ) in characteristic form with the Nashbargaining sharing rule over v(C) for every coalitionC in C. Furthermore assume strictly increasing andconcave utility functions3 ui : R.+ → R+, i ∈ N . Theset of transformations Ω from the set of characteristicfunctions in itself that provide the players the incentivefor some subset C′ of coalitions in C must satisfy:FC′ Ω(v)(C ′) < FC Ω(v)(C) ∀C ′ ∈ C′,∀C ∈ C\C′

(5)

s.t. C ′ ∩ C 6= ∅ and where FC =

(∑i∈C

(u′i

ui

)−1)−1,

where g−1 denotes the inverse function of g.

Proof. See Appendix D in [36] .

In order to derive our last result, we need to definethe concept of single-peaked preferences. Let X =x1, . . . , xn denote a finite set of alternatives, withn ≥ 3.

Definition 12 (Peak of preferences, [23]). A preferencerelation on X is a linear order on X . The peak of apreference relation is the alternative x∗ = peak()such that x∗ x for all x ∈ X\x∗.

Definition 13 (Single-Peaked preferences, [23]). An axisO (noted by >) is a linear order on X . Given twoalternatives xi, xj ∈ X , a preference relation on X

3Such utility functions are bijective and thus injective. Theorem 1applies.

whose peak is x∗, and an axis O, we say that xi andxj are on the same side of the peak of iff one ofthe following two condition is satisfied: (i) xi > x∗ andxj > x∗; (ii) x∗ > xi and x∗ > xj .A preference relation is single-peaked with respect toan axis O if and only if for all xi, xj ∈ X such that xiand xj are on the same side of the peak x∗ of , onehas xi xj if and only if xi is closer to the peak thanxj , that is, if x∗ > xi > xj or xj > xi > x∗.

We use the discrete version of this definition over N+.We immediately obtain the following corollary, whichiteratively uses the condition of Proposition 3 to provideplayers the incentive for coalitions of size qf with APf .

Corollary 1. Assume a coalition game Γ = (F ∪W, v, uii∈N ) in characteristic form with the Nashbargaining sharing rule over the v(C) in every coalitionC ∈ C. Furthermore assume strictly increasing andconcave utility functions ui : R+ → R+, i ∈ N . Theset of transformations Ω from the set of characteristicfunctions in itself that induce single-peaked preferences(peak at qf ) in cardinalities over the set of coalitionswith an AP f ∈ F must satisfy:

maxC∈Cf

s.t.|C|=q

FC Ω(v)(C) < minC∈Cf

s.t.|C|=q+1

FC Ω(v)(C),

∀q ≥ qf (6)and

maxC∈Cf

s.t.|C|=q

FC Ω(v)(C) < minC∈Cf

s.t.|C|=q−1

FC Ω(v)(C),

∀q ≤ qf (7)

where FC =

(∑i∈C

(u′i

ui

)−1)−1.

Proof. See Appendix D in [36] .

Note that the control step can be centralized or de-centralized. In a decentralized implementation, every APapplies the control independently according to its targetload by modifying the worth of the coalitions with thisAP.

C. Access Point Association

The third step of the mechanism is the joint resourceallocation and users association (matching) game wherethe players (APs and mobile users) share the resource inthe coalitions according to a Nash bargaining and thenmatch with each other. The coalition game played hasbeen described in Section III and Section IV. This stepcorresponds to the blocks Φ and µ of the block diagramin Figure 1.

1) Stable Matching Mechanism: We now show thata modified version of the Gale and Shapley’s deferredacceptance algorithm in its college-admission form with

Algorithm 1: Backward Deferred AcceptanceData: For each AP: The set of acceptable (covered) users and

AP-user data rates.For each user: The set of acceptable (covering) APs.Result: A core stable structure S

1 begin2 Step 1: Initialization;3 Step 1.a: All APs and users are marked unengaged.

L(f) = L∗(f) = ∅, ∀f ;4 Step 1.b: Every AP f computes possible coalitions

with its acceptable users, the respective users payoffsand emits its preference list P#(f);

5 Step 1.c: Every AP f transmits to its acceptable usersthe highest payoff they can achieve in coalitionsinvolving f ;

6 Step 1.d: Every user w emits its reduced list ofpreference P ′(w);

7 Step 2 (BDAA);8 Step 2.a, Mobiles proposals: According to P ′(w),

every unengaged user w proposes to its most preferredacceptable AP for which it has not yet proposed. If thisAP was engaged in a coalition, all players of thiscoalition are marked unengaged ;

9 Step 2.b, Lists update: Every AP f updates its listwith the set of its proposers:L(f)←− L(f) ∪ proposers and L∗(f)←− L(f);

10 Step 2.c, Counter-proposals: Every AP f computesthe set of coalitions with users in the dynamic listL∗(f) and counter-proposes to the users of their mostpreferred coalition according to P#(f);

11 Step 2.d, Acceptance/Rejections: Based on thesecounter-proposals and the best achievable payoffsoffered by APs in Step 1.c to which they have not yetproposed, users accept or reject the counter-proposals;

12 Step 2.e: If all users of the most preferredcoalition accept the counter-proposal of an AP f ,all these users and f defect from their previouscoalitions;

13 all players of these coalitions are markedunengaged;

14 users that have accepted the counter-proposal andf are marked engaged in this new coalition;

15 Step 2.f: Every unengaged AP f updates itsdynamic list by removing users both havingrejected the counter-proposal and being engaged toanother AP:

16 L∗(f)←− L∗(f)\engaged rejecters;17 Step 2.g: Go to Step 2.c while the dynamic list L∗ of

at least one AP has been strictly decreased (in thesense of inclusion) in Step 2.f;

18 Step 2.h: Go to Step 2.a while there are unengagedusers that can propose;

19 Step 2.i: All players engaged in some coalition arematched.

APs preferences over groups of users and users prefer-ences over individual APs is a stable matching mech-anism for the many-to-one matching games with com-plementarities, peer effects considered in this paper (seeAlgorithm 1: Backward Deferred Acceptance).

BDAA is similar to the DAA in many aspects. Itinvolves two sets of players that have to be matched.Every player from one side has a set of unacceptableplayers from the other side. In our case, an AP and amobile user are acceptable to each others if the useris under the AP coverage. As in DAA, the algorithmproceeds by proposals and corresponding acceptances orrejections. The main difference resides in the notion ofcounter-proposals, introduced to tackle the problem ofcomplementarities.

The block diagram representation of the algorithmis shown in Figure (2). In block P# the APs emittheir preferences over the coalitions. In block P

′

the mobiles emit their preferences over the APs. Inblock Proposals the mobiles propose to the APs. Inblock counter-proposals the APs counter-propose. Thecounter-proposing round continues up to convergence.The next proposing round starts.We enter the details of the algorithm. Having the infor-mation of the data rates with users under their coverage,APs are able to compute all the possible coalitionsthey can form and the corresponding allocation vectors(throughputs). They can thus build their preference lists(Steps 1b). Then, every AP f transmits to each of itsacceptable users the maximum achievable throughput(based on MAC layer and virtual mechanism) it canachieve in the coalitions it can form with f (Step 1.c).Every user w can thus build its reduced list of prefer-ences over individual APs: w prefers fi to fj if the max-imum achievable throughput with fi is strictly greaterthan its maximum achievable throughput with fj (Step1.d). BDAA then proceeds by rounds during which usersmake proposals, AP make counter-proposals and usersaccept or reject (from Step 2.a to Step 2.h). Every APthat receives a new proposal shall reconsider the set of itsopportunities and is thus marked unengaged (Step 2.a).L(f) is the list of all users that have proposed at leastonce to AP f . L∗(f) is a dynamic list that is reinitializedto L(f) before every AP counter-proposal (Step 2.b).In each round of the algorithm, every unengaged userproposes to its most preferred AP for which it has not yetproposed (Step 2.a). Every AP receiving proposals addsthe proposing players to its cumulated list of proposersand reinitializes its dynamic list (Step 2.b). Using P#(f)it then searches for its most preferred coalition involvingonly users from the dynamic list and emits a counter-proposal to these users. This counter-proposal containsthe throughput every user can achieve in this coalition(Step 2.c). Each user compares the counter-proposals itjust received with the best achievable payoffs obtainedwith the APs it has not proposed to yet (Step 2.d). If oneof these best achievable payoffs is strictly greater thanthe best counter-proposal, the users rejects the counter-proposals and continues proposing (Step 2.d, Step 2.h).Otherwise, the user accepts its most preferred counter-proposal (Step 2.d). Given a counter-proposal, if all usersaccept it, then they are engaged to the AP. All coalitionsin which these users and the AP were engaged are brokenand their players are marked unengaged (Step 2.e). If atleast one user does not agree, then the AP is unengaged(Step 2.e), it updates its dynamic list by removing themobiles having rejected its counter-proposal and beingengaged to another AP (Step 2.f). The counter-proposalscontinues up to the point when no AP can emit anynew counter-proposal (Step 2.g). The current round endsand the algorithm enters a new round (Step 2.h). Thealgorithm stops when no more users are rejected (Step2.h). A stable matching is obtained (Step 2.i).

P# P′ Proposals Counter-proposals

µ

counter-proposing loop

proposing loop

Fig. 2. Block diagram of the BDAA.

Remark: In Step 1.d, users’ reduced preferences areoptimistic preferences providing the users the incentiveto propose to new APs as long as their correspondingbest case is better than the received proposals.

Remark: We see from the above description thatthere are mainly two mechanisms that allow to takeinto account complementarities and peer effects. First,information is transferred from APs to users via through-put values and every throughput computed by an APtakes into account the composition of the cell and so thecomplementarities and peer effects between the nodes ofthis coalition. Second, an AP is always able to reconsiderall possible coalitions with users having proposed to itso far, even those that have been rejected before. Thisis the main difference with DAA, where a rejected useris never reconsidered because of the absence of groupeffects.

Proposition 4. Given a many-to-one matching game,BDAA converges, i.e., outputs a matching in a finitenumber of steps.

Proof. See Appendix D in [36].

Proposition 5. Suppose the family of coalitions C asdefined in (1), and a sharing rule as defined in propo-sition 2. Furthermore assume a tie-breaking rule suchthat there is no indifference (strict preferences). BDAAconverges to the unique core stable matching.

Proof. See Appendix D in [36].

Proposition 6. The complexity of BDAA is O(n5) inthe number of proposals of the players, where n =max(F,W ).

Proof. First we give an upper bound on the number ofproposals emitted by the mobile users, then we givean upper bound on the number of proposals emittedby the APs. In at most F proposals, every mobileuser has proposed to all the APs. Thus, in at mostF × W proposals, the mobile users have proposed toall the APs. At a given round of proposals, the counter-proposals ends at Step 2.g when no dynamic list hasbeen strictly decreased. These lists contains at mostW users. In the worst case, a single dynamic list isdecreased by one user, and thus after W × F rounds ofcounter-proposals, all dynamic lists are empty. The roundof counter-proposals stops. At each round of counter-proposals, F APs counter-proposes. Thus, in at mostF × W × F the APs have emitted all their counter-proposals. We obtain that the total number of proposals

(both mobile users proposals and APs counter-proposals)cannot exceed F 3 ×W 2. The complexity of BDAA isO(n5) where n = max(F,W ).

In Appendix E in [36], we provide an interpretationof BDAA in the economic framework. In Appendix F[36], we give an example of application of the BDAA.

Remark: Note APs perform at initialization and atevery counter-proposal search and sort operations on theset of coalitions. As the number of possible coalitionsis O(2W ) in the worst case, exponential complexityin number of elementary operations arises in the gen-eral case (a well known problem in coalition games).However, with dynamic lists, an AP considers onlycoalitions with users that have proposed to it so far. Thisreduces the dimension of the search space in practicalimplementations. Also, in the special case of WiFi, bestcoalitions are obtained with maximum data rates users sothat there is no need to sort all coalitions to find the best.In the special case of WiFi with the proposed control inCorollary 1, coalitions of size closer to the target loadare always preferred and within a class of equal sizecoalitions, those with best data rate users are preferred.Again, the sorting of all coalitions is not required. Inthe two WiFi cases, AP operations on coalitions are thuspolynomial.

Remark: User association is decentralized in thesense that both APs and mobile users take decisionsbased on their preferences. As in DAA, the associationcan be performed by a central entity but it is notnecessary. The resource allocation phase is the resultsof the WiFi MAC protocol, which is decentralized. Interms of information, every AP only needs to know thecoalitions to which it can participate at initialization. InWiFi, only users under coverage are involved so that theneeded information remains local.

VI. NUMERICAL RESULTS

A. Simulations Parameters and Scenarios

The numerical computations are performed under theassumption of equal packet sizes and saturated queues.Under this assumption the sharing rule is equal sharing.Analytical expressions of the throughputs (individual andtotal throughputs) are taken from [18] with the parame-ters of Table II. We further assume that a node compliantwith a IEEE 802.11 standard (in chronological order: b,g, n) is compliant with earliest ones. By convention, if allnodes of a cell have the same data rate, we use the MACparameters of the standard whose maximum physical

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Fig. 3. (a) Scenario 1: A spatial distribution of APs (smallest red circles) F = f1, . . . , f5 and devices (black points) W = w1, . . . , w20.(b) Scenario 2: A spatial distribution of APs (smallest red circles) F = f1, . . . , f5 and devices (black points) W = w1, . . . , w10. Circlesshow the coverage areas corresponding to different data rates. (c) A stable matching in the uncontrolled case for Scenario 1.

802.11n 802.11g 802.11bParameter value unit

Θ 300, 54, 11 54, 11 11 Mbits/sslot duration 9 9 20 µs

T0 3 5 50 slotsTC 2 10 20 slotsL 8192 8192 8192 bitsK 2 2 2b0 16 16 16p 2 2 2

TABLE IISIMULATION PARAMETERS.

data rate is the common data rate. Otherwise, we usethe MAC parameters of the standard whose maximumphysical data rate is the lowest data rate in the cell.

Assume the spatial distributions of nodes of Fig-ures 3(a) and (b). The first scenario (a) shows thecase of 5 APs with a uniform spatial distribution of20 mobile users. The second scenario (b) has non-uniform distribution of 10 mobile users in the plane.The green (inner), red (intermediate) and black (outer)circles show the spatial region where the mobiles achievea data rate of 300 Mbits/s, 54 Mbits/s and 11 Mbits/srespectively. Scenario 2 exhibits a high overlap betweenAP coverages.

B. Numerical Work

1) No mechanism: We show in Figure 3(c) a stablematching. No associated player has an incentive todeviate and form a coalition of size superior to two. Thefigure shows the natural incentives of the system in form-ing low cardinalities coalitions with good compositions.This can also be observed on Figure 4 which shows theindividual throughputs obtained in the coalitions. Thecoalitions are sorted by cardinalities from low to high.In plot (a) no mechanism is used. In plot (b) a gaussiantax rate is applied. See Section VI-B2.

Figure 3(c) and Figure 4(a) show the natural incentivesof the system in forming low cardinalities coalitions withgood compositions. As a result, a one-to-one matchingis obtained. Using our mechanism, this structure ofthroughputs will be changed (as in Figure 4(b)) tomove the incentives according to q and thus provide the

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Fig. 4. (a) Scenario 1. Structure of the payoffs in the uncontrolledmatching game. (b) Scenario 1. Structure of the payoffs in thecontrolled matching game with a multiplicative tax rate of varianceσf = 0.3, ∀f ∈ F .

players the incentives to associate according to a many-to-one matching rather than a one-to-one.

2) Gaussian Tax Rate in Cardinalities: As an exam-ple of family of cost functions, we can use multiplicativesymmetric unimodal cost functions. The multiplicativecost functions are commonly called tax rates and aredefined such that for any AP f ∈ F and any coalitionC containing f , we must have:

v(C) = Ω(v(C)) , cf (|C|)v(C) (8)We particularly consider Gaussian tax rates such that:

v(C) = e−

(|C|−qf )2

2σ2f v(C) (9)

where σf is the variance of the function cf . The Gaus-sian cost function is convenient in the sense that it doesnot penalize the mean-sized coalitions and it provides a

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Fig. 5. Controlled matching game in scenario 1. Comparison of the association obtained from (a) BDAA, (b) a global optimum for Gaussiancosts with variance σ = 0.2, (c) a global optimum without costs.

great amount of flexibility by the way of its variance.Decreasing or increasing the variance σf indeed allowsfor a strict or relaxed control of the incentives for thetarget loads. Focusing on the first scenario (Figure 3 (a)),we consider the three matchings shown in Figure 5. Thefirst one (a) is the stable matching resulting from themechanism (including BDAA and Gaussian costs); Thesecond matching (b) maximizes the sum of modifiedthroughputs (i.e. including Gaussian costs); The thirdmatching (c) maximizes the sum of unmodified through-puts (i.e. without costs).

We first observe that the proposed mechanism in-duces a stable matching with a drastic reduction of theunemployment problem w.r.t. the result of Figure 3(c).The natural incentives of the system resulting in a one-to-one matching have been countered and a many-to-one matching is obtained. The unemployment has beenreduced from 73% to 5% in this particular scenario. Thesecond point to be raised is that the proposed mechanismallows to obtain (with a polynomial complexity) a stablematching with a high modified total throughput, close tothe optimal modified total throughput that is however notstable. For this scenario, we achieve through our mech-anism 99% of the total modified maximum throughput(see Figure 5(b)). This means that the cost for stabilityis very small in this particular scenario. Furthermore,the total throughput performance of the system at theMAC layer (i.e. unmodified throughputs obtained in

block MAC of the block diagram representation of themechanism, see Figure 1) is 97% the total unmodifiedmaximum throughput (see Figure 5(b)) and 47% of thetotal maximum throughput of the uncontrolled system(see Figure 5(c)). This quantifies the cost for control,stability and low unemployment in this scenario. Thethird point is that the target loads have been enforced bythe mechanism (via the cost function) since the targetload vector is q = (8.0, 4.5, 3.33, 3.83, 4.33) (obtainedby Nash bargaining4 over the share [0, 1] of the playersat the intersection of the coverages of the APs) and theformed coalitions are of sizes 8, 4, 3, 4 and 4.

We go into more details on the difference betweenthe target load vector and the integer-sized coalitions inthe stable matching. Focusing on AP3 with the targetload q3 = 3.33, one may observe that in case of aGaussian cost function with unit variance, the conditionfor an integer target load 3 is only satisfied for sizesof coalitions superior or equal to 4. This meaning thatthe use of a gaussian cost function centered on 3.33 andunit variance even though increasing the penalty with thedistance in sizes to qf cannot guarantee the systematicincentive to form coalitions of size 3 with AP3. Thereexists some coalitions of size 2 giving the players moreindividual throughputs than the worst coalition of size 3.In such case, the players will have the incentive to form

4Achieves a proportional fair allocation in the utility space of theAPs. Induces the number of players to be connected to each AP.

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Fig. 6. (a)Unemployment rates, (b)social welfares (the social welfare of a matching is measured as the total throughput of the system atequilibrium) and (c)computation times of BDAA over a sample of 50 scenarios obtained by spatial random uniform distribution of the mobiledevices. APs are spatially distributed as in Scenario 1. For each plot, the red line gives the empirical mean m of the sample and the greendotted lines the interval [m− σ, m+ σ] where m is the empirical mean of sample and σ is the standard deviation.

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Fig. 7. (a)Unemployment rates, (b)social welfares and (c)computation times of BDAA over a sample of 50 random networks obtained by spatialrandom uniform distribution of the mobile devices and APs. For each plot, the red line gives the empirical mean m of the sample and the greendotted lines the interval [m− σ, m+ σ] where m is the empirical mean of sample and σ is the standard deviation.

the coalition with the highest individual value amongthose of cardinalities 2 and 3. In Figure 6, we show theperformance of the mechanism over a set of 50 scenariosgenerated by spatial random uniform distribution of themobile devices. The APs are spatially distributed as inScenario 1 (see Figure 7 for a random distribution ofboth the mobile devices and APs). The red line showsthe empirical mean of the sample and the green dottedlines show the interval [m − σ, m + σ] where m isthe empirical mean of sample and σ is the standarddeviation. The empirical mean of the unemployment rateis 6%, the mean modified social welfare is 61Mbits/s andthe mean computation time of BDAA is 0.45s. Observethat in 22% of the realizations the unemployment isnull and that in 70% of the realizations it is belowthe mean. In terms of computation times of BDAA, themean performance is reasonably low (0.45s to match 20mobiles to 5 APs) and the algorithm performs even betterin 62% of the scenarios.

In Figure 7, we show the performance of the mech-anism over a set of 50 scenarios generated by spatialrandom uniform distribution of the mobile devices andAPs. The red lines show the empirical mean of thesample and the green dotted lines show the interval

[m−σ, m+σ] where m is the empirical mean of sampleand σ is the standard deviation. The empirical meanof the unemployment rate is 8%, the mean modifiedsocial welfare is 60Mbits/s and the mean computationtime of BDAA is 0.51s. Observe that in 22% of therealizations the unemployment is null and that in 56%of the realizations it is below the mean. In terms ofcomputation times of BDAA, the mean is higher thanin the previous case but the algorithm performs betterthan the mean in 68% of the scenarios.

In Figure 8, plot (a), we show the ratios of the modi-fied (mechanism level) social welfare (by definition, thetotal throughput resulting from BDAA) to the maximumtotal modified throughput. The mean performance ofBDAA achieves 96% of this maximum. Furthermore,observe that the global maximum is achieved by BDAAin 46% of the random networks. The ratio is below m−σin only 10% of the cases. In Figure 8, plot (b), weshow the ratios of the unmodified (MAC level) socialwelfare to the unmodified total throughput induced atthe matching maximizing the total modified throughput.The mean performance of BDAA achieves 97% of thisunmodified total throughput. Finally, observe that insome cases, the ratio is even higher than one. This

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Fig. 8. (a)Ratios of the modified social welfares to the maximum modified (mechanism level) total throughput, (b)Ratios of the unmodified(MAC level) social welfares to the unmodified total throughputs corresponding to the matching with maximum modified total throughput. Sampleof 50 random networks obtained by spatial random uniform distribution of the mobile devices and APs. For each plot, the red line gives theempirical mean m of the sample and the green dotted lines the interval [m − σ, m + σ] where m is the empirical mean of sample and σ isthe standard deviation.

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(b) Matching resulting from the best-RSSIscheme.

Fig. 9. Comparison of the association obtained from (a) BDAA and (b) the best-RSSI scheme in scenario 2. These two figures show the APat center (zoom).

means that BDAA gives a total throughput at the MAClevel that is superior to the (unmodified) total throughputresulting from the maximization at the mechanism level(modified values) while the ratio was inferior to one inthe modified case. This may come from the fact thatin some cases, the equilibrium point (stable matchingresulting from BDAA) may contain coalitions with lowermodified worths (because of the penalization) w.r.t. thosein the global maximum but higher worths at MAC level(real unpenalized setting).

To conclude, we compare our approach to the best-RSSI scheme in Scenario 2. The two matchings are com-pared Figure 9. We observe that the load is effectivelyshared among the APs and that the individual through-puts are greatly increased from 527 kbits/s when usingbest-RSSI to 1.64 Mbits/s for the coalition with AP1,1.93 Mbits/s for the coalition with AP2, 2.59 Mbits/sfor the coalition with AP3, 1.64 Mbits/s for the coalitionwith AP4 and 2.59 Mbits/s for the coalition with AP5.The individual performances are multiplied by a factor3 to 5.

VII. CONCLUSION

In this paper, we have presented a novel AP asso-ciation mechanism in multi-rate IEEE 802.11 WLANs.We have formulated the problem as a coalition matchinggame with complementarities and peer effects and wehave provided a new practical control mechanism thatprovides nodes the incentive to form coalitions bothsolving the unemployment problem and reducing theimpact of the anomaly in IEEE 802.11. Simulationresults have shown that the proposed mechanism canprovide significant gains in terms of increased through-put by minimizing the impact of the anomaly throughthe overlapping between APs. We have also proposed apolynomial complexity algorithm for computing a stablestructure in many-to-one matching games with comple-mentarities and peer effects. This work is a first stepin the field of controlled coalition games for achievingcore stable associations in distributed wireless networks.Further works includes for example the study of adynamic number of users or the impact of interference.

REFERENCES

[1] J.F. Nash, The Bargaining Problem, Econometrica, Vol. 18, No.2, pp. 155-162, April 1950.

[2] J.F. Nash, Two-Person Cooperative Games, The RAND Corpo-ration, P-172, August 1950.

[3] D. Gale and L.S. Shapley, College Admissions and the Stabilityof Marriage, The American Mathematical Monthly, Vol. 6, No.1, pp. 9-15, January 1962.

[4] J.C. Harsanyi, A Simplified Bargaining Model for the n-PersonCooperative Game, International Economic Review, Vol. 4, No.2, pp. 194-220, May 1963.

[5] K.W. Pratt, Risk Aversion in the Small and in the Large,Econometrica, Vol. 32, No.1/2, pp. 122-136, January-April 1964.

[6] R.J. Aumann and M. Kurz, Power and Taxes, Econometrica, Vol.45, No. 5, pp. 1137-1161, July 1977.

[7] A.E. Roth and M.A.O. Sotomayor, Two-Sided Matching A StudyIn Game-Theoritic Modeling and Analysis, Econometric SocietyMonographs, No. 18, Cambridge University Press, 1990.

[8] S. Lasaulce and H. Tembin, Game Theory and Learning forWireless Networks: Fundamentals and Applications, AcademicPress, 2011.

[9] S. Tijs, Introduction to Game Theory, Texts and Readings inMathematics, Hindustan Book Agency, 2011.

[10] G. Tan and J.V. Guttag, Time-based Fairness Improves Per-formance in Multi-Rate WLANs, USENIX Annual TechnicalConference, June 2004.

[11] A.V. Babu and L. Jacob, Performance Analysis of IEEE 802.11Multirate WLANs: Time Based Fairness vs Throughput BasedFairness, IEEE Int. Conf. on Wireless Networks, Communica-tions and Mobile Computing, June 2005.

[12] J. Foncel and N. Treich, Fear of Ruin, Journal of Risk andUncertainty, Vol. 31, No. 3, pp. 289-300, December 2005.

[13] T. Korakis, O. Ercetin, S. Krishnamurthy, L. Tassiulas, and S.Tripathi, Link Quality Based Association Mechanism in IEEE802.11h Compliant Wireless LANs, WiOpt RAWNET Workshop,April 2005.

[14] A. Kumar and V. Kumar, Optimal Association of Stations andAPs in an IEEE 802.11 WLAN, National Conference on Com-munications, January 2005.

[15] C. Touati, E. Altman, J. Galtier, Generalized Nash BargainingSolution for Bandwidth Allocation, Computer Networks, Vol. 50,No. 17, pp. 3242-3263, December 2006.

[16] A. Banchs, P. Serrano, H. Oliver, Proportional Fair ThroughputAllocation in Multirate IEEE 802.11e Wireless LANs, WirelessNetworks, Vol. 13, No. 5, pp 649-662, October 2007.

[17] F. Echenique and M.B. Yenmez, A Solution to Matching withPreferences over Colleagues, Vol. 59, Issue 1, pp. 46 - 71, April2007.

[18] A. Kumar and E. Altman and D. Miorandi and M. Goyal, NewInsights From a Fixed-Point Analysis of Single Cell IEEE 802.11WLANs, IEEE INFOCOM, March 2005.

[19] Y. Bejerano, and H. Seung-jae and L. Li, Fairness and Load Bal-ancing in Wireless LANs Using Association Control, IEEE/ACMTrans. on Networking, Vol. 15, No. 3, pp. 560-573, June 2007.

[20] E. Altman and K. Avrachenkov and A. Garnaev, Generalized α-Fair Resource Allocation in Wireless Networks, IEEE Conferenceon Decision and Control, December 2008.

[21] Lin Chen and J. Leneutre, A Game Theoretic Framework ofDistributed Power and Rate Control in IEEE 802.11 WLANs,IEEE J. on Selected Areas in Communications, Vol. 26, No. 7,pp. 1128-1137, September 2008.

[22] H. Gong and K. Nahm and Jong Won Kim, Distributed FairAccess Point Selection for Multi-Rate IEEE 802.11 WLANs,IEEE CCNC, January 2008.

[23] B. Escoffier and J. Lang and M. Ozturk, Single-peaked consis-tency and its complexity, 18th European Conference on ArtificialIntelligence, July 2008.

[24] T. Bonald, A. Ibrahim and J. Roberts, The Impact of Associationon the Capacity of WLANs, WiOpt, June 2009.

[25] F. Xu, C.C. Tan, Q. Li, G. Yan, and J. Wu, Designing a PracticalAccess Point Association Protocol, IEEE INFOCOM, Mar. 2010.

[26] H. Keinanen, Algorithms for coalitional games, PhD Thesis,Turku School of Economics, 2011.

[27] M. Pycia, Stability and Preference Alignment in Matching andCoalition Formation, Econometrica, Vol. 80, No. 1, pp. 323-362,January 2012.

[28] F. Pantisano, M. Bennis, W. Saad, S. Valentin, and M. Debbah,Matching with Externalities for Context Aware User Cell As-sociation in Small Cell Networks, IEEE Globecom, December2013.

[29] K. Hamidouche, W. Saad, and M. Debbah, Many-to-many Match-ing Games for Proactive Social-Caching in Wireless Small CellNetworks, WiOpt WNC3 workshop, May 2014.

[30] W. Saad, Z. Han, R. Zeng, M. Debbah, and H. Vincent Poor,A College Admissions Game for Uplink User Association inWireless Small Cell Networks, IEEE INFOCOM, April 2014.

[31] M. Touati, R. El-Azouzi, M. Coupechoux, E. Altman, J.-M. Kelif,Core stable algorithms for coalition games with complementar-ities and peer effects, Workshop NetEcon, ACM Sigmetrics &EC, June 2015.

[32] M. Touati, R. El-Azouzi, M. Coupechoux, E. Altman, J.-M. Kelif,Controlled Matching Game for User Association and ResourceAllocation in Multi-Rate WLANs, 53rd Annual Allerton Con-ference on Communication, Control, and Computing, September2015.

[33] M. Touati, R. El-Azouzi, M. Coupechoux, E. Altman, J.-M. Kelif,About Joint Stable User Association and Resource Allocationin Multi-Rate IEEE 802.11 WLANs, Performance EvaluationReview, Vol. 43, Issue 2, pp. 30-31, September 2015.

[34] N. Namvar and F. Afghah, Spectrum sharing in cooperativecognitive radio networks: A matching game framework, 49thIEEE Annual Conf. on Information Sciences and Systems (CISS),March 2015

[35] R. Mochaourab, B. Holfeld and T. Wirth, Distributed ChannelAssignment in Cognitive Radio Networks: Stable Matching andWalrasian Equilibrium, in IEEE Transactions on Wireless Com-munications, Vol. 14, No. 7, pp. 3924-3936, July 2015.

[36] M. Touati, R. El-Azouzi, M. Coupechoux, E. Altman, J.-M. Kelif,Controlled Matching Game for Resource Allocation and UserAssociation in WLANs, Technical Report available at http://arxiv.org/abs/1510.00931, 2016.

Mikael Touati Mikael Touati is a researchengineer at Orange Labs (Chatillon, France).He received the PhD degree in Computerand Network Science from Telecom Paris-Tech (Paris) and Orange Labs in 2016. Heis working on game theory and its appli-cations to networks. He both received theEngineering Degree from SUPELEC and aMSc in wireless communications from in2013. In 2013, he was the recipient for theSEE Andre Blanc-Lapierre Regional prize

(Ile-de-France) and National prize (France).

Rachid El-Azouzi Rachid El-Azouzi re-ceived the Ph.D. degree in Applied Math-ematics from the Mohammed V University,Rabat, Morocco (2000). Since 2003, he hasbeen a researcher at the University of Avi-gnon, France. His research interests are, Net-working Games, Biologically Inspired Net-works, Wireless MAC protocols design andevaluation, Intelligent wireless networks andlearning algorithms.

Marceau Coupechoux has been workingas an Associate Professor at Telecom Paris-Tech since 2005. Currently, at the Computerand Network Science department of TelecomParisTech, he is working on cellular net-works, wireless networks, ad hoc networks,cognitive networks, focusing mainly on radioresource management and performance eval-uation.

Eitan Altman is a researcher at INRIA inSophia-Antipolis, France. He has been in theeditorial boards of several scientific journals:Wireless Networks (WINET), Computer Net-works (COMNET), Computer Communica-tions (Comcom), J. Discrete Event DynamicSystems (JDEDS), SIAM J. of Control andOptimisation (SICON), Stochastic Models,and Journal of Economy Dynamic and Con-trol (JEDC). He received the best paper awardin the Networking 2006, in Globecom 2007,

in IFIP Wireless Days 2009 and in CNSM 2011 (Paris) conferences.His areas of interest include network engineering games, social net-works and their control. He received in 2012 the Grand Prix de FranceTelecom from the French Academy of Sciences.

Jean-Marc Kelif is a research engineer inwireless communications at Orange Labs(Chatillon, France). He received the Ph.Ddegree from Telecom ParisTech. His currentresearch interests include the performanceevaluation, modeling, dimensioning and opti-mization of wireless telecommunication net-works.