Fair scheduling in cellular systems in the presence of noncooperative mobiles

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1 1 Fair cheduling in cellular yte in the preence of noncooperative obile Veeraruna Kavitha +, Eitan Altan R. El-Azouzi + and Rajeh Sundarean Maetro group, INRIA, 2004 Route de Luciole, Sophia Antipoli, France + LIA, Univerity of Avignon, 339, chein de Meinajarie, Avignon, France IISc, Indian Intitute of Science, Bangalore, India Abtract We conider the proble of fair cheduling the reource to one of the any obile tation by a centrally controlled bae tation (BS. The BS i the only entity taking deciion in thi fraework baed on truthful inforation fro the obile on their radio channel. We tudy the well-known faily of paraetric -fair cheduling proble fro a gaetheoretic perpective in which oe of the obile ay be noncooperative. We firt how that if the BS i unaware of the noncooperative behavior fro the obile, the noncooperative obile becoe ucceful in natching the reource fro the other cooperative obile, reulting in unfair allocation. If the BS i aware of the noncooperative obile, a new gae arie with BS a an additional player. It can then do better by neglecting the ignal fro the noncooperative obile. The BS, however, becoe ucceful in eliciting the truthful ignal fro the obile only when it ue additional inforation (ignal tatitic. Thi new policy along with the truthful ignal fro obile for a Nah Equilibriu (NE which we call a Truth Revealing Equilibriu. Finally, we propoe new iterative algorith to ipleent fair cheduling policie that robutify the otherwie non-robut (in preence of noncooperation fair cheduling algorith. I. INTRODUCTION Short-ter fading arie in a obile wirele radio counication yte in the preence of catterer, reulting in tievarying channel gain. Variou cellular network have downlink hared data channel that ue cheduling echani to exploit the fluctuation of the radio condition (e.g. 3GPP HSDPA 2 and CDMA/HDR 8 or 1xEV-DO 1. A central cheduling proble in wirele counication i that of allocating reource to one of any obile tation that hare a coon radio channel. Much attention ha been given to the deign of efficient and fair cheduling chee that are centrally controlled by a bae tation (BS whoe deciion depend on the channel condition of each obile. Thee network ue variou fairne criteria (6, 4 called generalized -fair criteria to deign a cla of paraetric cheduling algorith (which we henceforth call a -fair cheduling algorith or -FSA. One pecial cae, proportional fair haring (PFS, ha been intenely analyzed a applied to the CDMA/HDR yte. See 12, 8, 7, 20, 3, 11, 17. Thee reult are applicable to the 3GPP HSDPA yte a well. Kuhner & Whiting 15 analyzed the PFS algorith uing tochatic approxiation technique and howed that the ayptotic averaged throughput can be driven to optiize a certain yte utility function (u of logarith of offet-rate. See alo Stolyar 21. The BS i the only entity taking deciion in all the above ethod, and the BS depend crucially on truthful reporting of their channel tate by the obile. For exaple, in the frequency-diviion duplex yte, the BS ha no direct inforation on the channel gain, but tranit downlink pilot, and relie on the obile reported value of gain on thee pilot for cheduling. A cooperative obile will truthfully report thi inforation to the BS. A noncooperative obile will however end a ignal that i likely to induce the cheduler to behave in a anner beneficial to the obile. In 13, 14 we analyzed efficient cheduling (a pecial cae with = 0, wherein the cheduler axiize the u throughput at the BS in preence of noncooperation uing a ignaling gae (22. The ignaling gae can be ued only for that pecial cae and an -fair cheduler with > 0 cannot be odeled by a ignaling gae: for -fair cheduler with > 0, the utilitie of the BS are not expected utilitie but are concave cobination of the uer expected utilitie. Further, -fair cheduler (with > 0 ha an inherent feedback in it tructure (ore detail in ection II and thi feedback ake the tudy difficult and different fro the above paper. Thi paper ha contribution to three ain area: Networking Apect: (1 We identify cae where noncooperation reult in an unfair bia in the channel aignent in favor of noncooperative obile, if the bae tation i unaware of the noncooperative behavior. (2 We characterize the liitation of the bae tation, and obtain condition under which even when it i aware of noncooperation, it i not able to hare fairly the reource. (3 We how that the ability to achieve fair haring, in the preence of noncooperation, depend on the paraeter. (4 We deign robut iterative algorith that, under uitable condition, fairly hare the reource even in the preence of noncooperative ignaling. Gae theoretical odeling: (1 We odel a noncooperative obile a a rational player that wihe to axiize it throughput. Since the -fair aignent i related to the axiization of a related utility function, one can view the BS a yet another player. We thu have a gae odel even if there i a ingle noncooperative obile. (2 We forulate three gae of which one i a concave gae. The forulation of the gae turn out to be urpriingly coplex. Except for the pecial cae of = 0 (where the gae can be hown to be equivalent to a atrix gae, the gae are defined over an infinite et of action. We are however able to prove the exitence and characterize the equilibriu policie

2 2 for two gae. (3 The third gae arie when the BS i unaware of noncooperation. BS only repond to the obile, but in a optial way. We could odel thi a a hierarchical gae where the obile are involved in a gae played at the higher level and the BS optiize oe utility at the lower level, unaware of the rationality of the obile. (4 To analyze iterative algorith, we conider a tochatic gae with ayptotic tie liit of the iterative algorith a cot criteria. Deign of the networking protocol baed on tochatic approxiation technique. (1 We analyze the paraetric -fair cheduling algorith (-FSA of 15 in preence of noncooperation. We identified it robutne propertie a a function of. (2 Uing the knowledge of channel and ignal tatitic, one can control the exce utilitie that the obile would have otherwie obtained by noncooperation. Thi i the baic idea behind robut policie. We then ue tochatic approxiation baed approach to cobine etiation (which replace the knowledge required and control to deign robut fair cheduling algorith. We firt otivate the proble uing a iple exaple. A Motivating exaple We conider two uer haring a coon channel. Uer 1 ha two channel tate with utilitie 7 and 3 occurring with probabilitie 0.33 and 0.67 repectively. Uer 2 ha contant channel with utility 4. The BS ha to aign the channel to one of the two uer for every realization of the channel tate and every uch aignent rule reult in a pair of uer average utilitie. The BS ue an -fair cheduler (decribed in the next ection to fair hare thee average utilitie. Firt we aue that both uer cooperate and report their individual channel tate correctly. In figure 1 and 2 (the cae with δ = 0 we plot the average utilitie obtained by uer under -fair cheduler a a function of the fairne paraeter. We ake the following obervation: (1 For every, the BS alway allocate the channel to uer 1 if he i in good tate. (2 For = 0, the expected hare of uer 1 ( i le than that of the uer 2 (( Thi correpond to efficient cheduling point. (3 For all value of, BS allocate the channel to uer 1 only when he i in good tate. (4 The expected hare of uer 1 increae while that of the uer 2 decreae a increae and eventually becoe equal. To achieve thi, the BS tart allocating the channel to the uer 1 even when that uer i in bad tate with increaing probability. The above cenario depend crucially on the truthful reporting of channel by the uer 1. Now, we conider the cenario when uer 1 i noncooperative and trie to increae hi utility. He declare to be in good tate 7 when actually in bad tate 3 with probability δ. BS now oberve the uer 1 to have good channel with better probability 0.33+δ 0.67 and will chedule a before but baed on reported channel condition. In figure 1, 2 we plot the reulting expected utilitie of both the uer a a function of fairne for δ = 0.1, δ = 0.5 repectively. We oberve that the utility of uer 1 for all value of i iproved in coparion with it cooperative utility. Thi alo reduce the utility of the uer 2 below it cooperative hare, utility of obile 1 δ = 0 utility of obile 2 δ = 0 utility of obile 1 δ = 0.1 utility of obile 2 δ = Fig. 1. Uer utilitie veru. for δ = utility of obile 1 δ = 0 utility of obile 2 δ = 0 utility of obile 1 δ = 0.5 utility of obile 2 δ = Fig. 2. Uer utilitie veru. for δ = 0.5 reulting in unfair allocation. Thi effect i een for all value of le than = 1.75, = 6.85 repectively for δ = 0.5, δ = 0.1. However, for alpha greater than the above value, uer 1 loe; in fact it utility get below it cooperative hare, while that of the uer 2 i uch above the later cooperative hare. The above exaple indicate the -fair cheduler: (1 ight be robut againt noncooperation for large value of (2 fail for aller value of. (3 the larger the δ the larger the aount of gain at = 0. (4 the larger the δ the aller the till which the obile gain. A increae, the two uer utilitie converge toward equal value at a rate that directly depend upon the difference at = 0. Thi i the reaon for the above obervation. An iportant point to note here i that, there i no threhold of beyond which the cheduler will be robut to all type of noncooperation, i.e., for all value of δ. However one can gue that for ax in fairne ( = the cheduler will be robut. The tudy of thi noncooperation and deign of robut policie will be the focu of our paper. II. THE PROBLEM SETTING AND -FAIR SCHEDULER The Downlink We conider the downlink of a wirele network with one bae tation (BS. There are M obile copeting for the downlink data channel. Tie i divided into all interval or lot. In each lot, one of the M obile i allocated the channel. Each obile can be in one of the tate h H, where H i finite valued. We aue fading characteritic to be independent acro the obile. Let h := h 1, h 2,, h M t be the vector of channel gain in a particular lot. The channel gain are ditributed according to: p h (h = M i=1 p h i (h i, where {p h ; M} repreent the tatitic of the obile channel. When the obile channel tate i h, it can achieve a axiu utility given by f(h. An exaple of utility i the rate f(h = r( = log(1 + h SNR where SNR capture the noinal received ignal-to-noie ratio under no channel variation. The deciion rule In every lot, the BS ha to ake cheduling deciion, i.e., allocate the downlink lot to one of the M uer, baed on the current realization of the channel tate vector h. For any et C, let P(C be the et of probability eaure on C. With that definition, a BS

3 3 deciion i a function β that aign to any given h an eleent in P({1, 2,, M}, the probability ditribution over the et of uer. Thu, β( h i the probability that the BS chedule current lot to obile given channel tate vector h. The -fairne criterion and cheduler We introduce the well known generalized -fair criterion (4 where the quantity that we wih to hare fairly i the expectation of the rando (intantaneou utilitie correponding to the aignent by the cheduler to the obile: G (β := M Γ (θ (β (1 =1 where θ (β := E h f(h β( h i the expected hare of obile under policy β and where the -fair function i { log(u, for = 1 Γ (u := u 1 1, for 1. One can view β(.., the cheduling policy, a a vector in R B pace, with B := M H, where H i the cardinality of the product pace H = Π M =1H. More preciely the doain of optiization i 1 : { } M D := β(.. : β( h = 1, β( h 0 for all h,. =1 The objective function G given by (1, i concave and continuou in β for each, while the doain D i copact and convex. Hence there alway exit a cooperative -fair cheduling BS trategy β : β (.. arg ax β D G (β. (2 Reark II-1: We ay view the BS chedule a a tatic optiization proble that correpond to a ingle choice of β. Notice that the optial chedule β axiize oe function of the expected hare of utilitie. Thi expected hare depend on aignent at all channel tate, and i therefore a joint optiization proble. Thi feature arie when > 0. When = 0 the proble i eparable, and the olution β ( h for a given h depend only on that h. Indeed, for > 0, the iplicit equation (3 below highlight a certain feedback that i abent in cae with = 0. Thi ake the preent tudy ignificantly different fro our previou work on efficient cheduling with trategic obile (13. Below we how a key (feedback property of fair cheduler. Define β a the vector (fixed point that atifie (if it exit the following: β ( h = 1 { arg ax j dγ (θ j(β f(h j} arg ax j dγ (θ j (β (3 f(h j where dγ (θ j (β := dγ du u=θj(β i the derivative of Γ with repect to (w.r.t. u, evaluated at θ j (β. We now have Lea 1: If there i a β atifying (3, then β i a global axiizer of the objective function in (2 over doain D and hence i an -fair cheduler. 1 For a given channel realization h the BS will chooe the obile randoly according to the eaure β(. h. Let Θ := θ 1 θ M T, Θ(β := θ 1 (β θ M (β T and Θ(D := {Θ(β : β D}. The ap Θ Γ (θ i trictly concave. Hence, there exit an unique axiizer (of the expected aigned hare over the convex et Θ(D: Θ = ax Θ Θ(D Γ (θ. (4 Hence, if there i a β atifying (3 then Θ = Θ(β. Further, any β which i a global axiu of the objective function (2 atifie the efficiency property: whenever f(h > f(h either β ( h, h > β ( h, h (5 or β ( h, h = β ( h, h {0, 1} for all h Π j H j and for all. Proof : Pleae refer to Appendix B. The above Lea 1 give the exact characterization of an optial olution of -fairne proble (3. It further talk about the efficiency of every poible -fair olution (5: the aignent for particular tate (h for any obile increae with the increae in the utility (f(h of the tate. Thi property i ued in the analyi under noncooperation. A part of Lea 1, regarding the poible olution (3, when retricted to proportional fairne, i already tated in 16. Reark II-2: The olution (3 explicitly how the feedback we entioned in Reark II-1. Thi olution ha already been ued in practical cenario (16 to achieve fair cheduling: The -fair olution for the dynaic etting with ergodic channel tate i the optial β that fair hare the tie average utilitie over a ingle realization of a whole aple path 2. In fact, the olution (3 under ergodicity can be ipleented by the following procedure: 1 At any tie lot k, obtain the cheduling deciion uing the current channel vector h k and uing the tie averaged aigned utilitie obtained till the lat tep {θ,k 1 } in place of {θ (β } of (3; 2 Update (in the obviou way the tie averaged aigned utilitie up to tep k {θ,k } uing the current cheduling deciion. III. PROBLEM FORMULATION UNDER NON COOPERATION In every lot, the BS need the knowledge of h for cheduling purpoe. In practice, obile etiate channel h uing the pilot ignal ent by BS. We aue perfect channel etiation. The obile end ignal { } to BS, a indication of the channel gain. Thu BS doe not have direct acce to channel tate h, but intead ha to rely on the obile for it. If the obile have elfih otive, they can ignal a better channel condition to grab the channel even when their channel i bad. The ain purpoe of thi paper i to tudy the effect of thi noncooperation on the -fair cheduler (2. We aue that 2 For ergodic channel under appropriate condition on the function g, 1 li K K K g(h k = E h g(h k=1 We are intereted in a particular function g(h = f(h β( h whoe average i exactly θ (β.

4 4 ignal are choen fro the channel pace itelf, i.e., H for all obile. We hall conider two type of cenario : Hierarchical gae G1: The BS i unaware of the poible noncooperative behavior fro the obile and applie the - fair cheduler (2 to the ignal = 1,, M t (a if they were the true channel value. The obile are aware of BS cheduler, ignal to optiize their own goal. When the bae tation i unaware, we odel thi gae a hierarchical gae with two level: where leader, the noncooperative obile, involve in a gae proble while BS, the follower doe the optiization. In thi gae, there i no coon knowledge: the bae tation doe not know the rationality of the obile. Thi gae i related to that dicued by Auann in 5 through any exaple. For everal year it ha been thought that the auption of coon knowledge of rationality for the player in the gae wa fundaental. It turn out that, in N- player gae, coon knowledge of rationality i not needed a an epiteic condition for equilibriu trategie (ee 5. A gae approach: The BS i odeled a an additional player in a one-hot gae. When the BS becoe aware of the poible noncooperation, it could ipleent better policie to do better. We firt conider a M player gae G2, where the BS chedule till only uing the ignal fro the obile. Becaue of it awarene, it could do better than the ituation of gae G1, but however will not be ucceful in copelling the obile to reveal the truthful ignal (Section V-A. In Section V-B we contruct ore intelligent (which require ore inforation BS policie which would be robut againt noncooperation: the new robut BS policie and the truthful ignal fro the obile for a Nah Equilibriu. Thi we refer a gae G3. We now introduce the iportant concept and definition that are ued in the paper. Thee are ore pecific to the firt two gae cenario. The correponding definition and concept ay vary lightly for the gae G3 and the difference are explained directly in Section V-B. Coon Knowledge : Channel tatitic {p h ; M} of all obile i a coon knowledge, i.e., known to all the obile and the BS. We alo aue that, the inforation about which obile are noncooperative, i a coon knowledge in cae of the lat two gae G2 and G3. If the rational BS doe not know which obile are cooperative, it will treat every obile a noncooperative. Mobile Policie : Soe obile (with indice 1 M 1 where (0 M 1 M are aued to be noncooperative. A policy of obile i a function {µ (. h } that ap a tate h to an eleent in P(H. BS Policie : A policy of the BS i a function which ap every ignal vector to a cheduler β P({1, 2,, M}. Thee policie are ued in ajor part of the paper, while ore coplicated policie are conidered in ection V-B. Utilitie for a given et of trategie : The intantaneou/aple utility of the obile depend only upon the true channel h and the BS deciion β and i given by : U (, h, β = 1 {β=} in{f(h, f( } 3. Define the following to exclude obile : h := h 1,, h 1, h +1,, h M, p h (h := Π j p hj (h j, µ ( h := Π j ;j M1 µ j ( j h j Π j ;j>m1 δ(h j = j. Alo define, µ = {µ ; M 1 } to repreent trategy profile: µ( h := Π 1 j M1 µ j ( j h j Π j>m1 δ(h j = j. With the above definition, each noncooperative uer chooe it trategy µ in uch a way a to axiize it own utility: U (µ, β = E h U (, h, β( µ( h Under the -fair criterion (1, the natural election of utility for BS will be: U BS(µ, β = (6 Γ (U (µ, β. (7 Throughout when arg ax S ha ore than one eleent, by i = arg ax S we ean i arg ax S. By j := arg ax S we ean that j i a choen eleent of arg ax S. ASA, ATA Utilitie : When obile ignal do not atch the true channel value, the gae under conideration will have two iportant average utilitie for any given trategy profile (µ, β : (1 average ignaled utilitie under aignent β (ASA utility, which a (ore intelligent BS can oberve, and (2 average true and aigned (ATA utility, which i the true average utility gained by the obile and whoe value cannot be etiated (a long a the obile i noncooperative at the BS. Thee are defined by U ASA (µ, β := E h U AT A (µ, β := E h f( µ( hβ( in{f(h, f( }µ( hβ( (8. (9 Indeed, it i eay to oberve that the utility of obile i it ATA utility, i.e., U(µ, β = U AT A (µ, β. Truth Revealing Strategy : In the following, by truth revealing trategy at obile we ean the trategy µ T ( h = 1 {=h } for all h, H, which ignal the true channel tate. Let µ T := (µ T 1,, µ T M. Under truthful trategie µ T, ATA and ASA utilitie coincide. For any BS policy β, if trategy profile (µ T, β for a Nah Equilibriu, then we call the NE a a Truth Revealing Equilibriu (TRE. 3 The obile achieve rate f(h even if the BS allocate it a higher rate f( becaue of the inflated ignal ent by the noncooperative obile. The jutification of thi i provided in detail in Appendix C (we ued iilar auption alo in 13, 14.

5 5 Cooperative Share : Bet repone of BS to truthful ignal µ T i any axiizer β of G (1. By Lea 1, the bet repone reult in unique axiu average ATA utilitie, θ c := θ (β = U (µ T, β, (10 which will be referred a Cooperative Share. Contrat between hierarchical optiization and the gae perpective: Recall that coputing a fair aignent by BS involve axiization of (1. Thu in the firt cenario, when obile chooe profile µ, the unaware BS fair hare ASA utilitie under µ by axiizing (12 (given in the next ection. However, what need fair haring i the ATA utilitie. Thi i achieved via the gae perpective, wherein the rational BS trie to fair hare the ATA utilitie gained by the obile. We tudy the variou cenario via three gae entioned above. IV. SCHEDULING UNDER NONCOOPERATION : HIERARCHICAL GAME PROBLEM G1 We conider the cenario in which the BS i unaware of the preence of noncooperative obile. A in cooperative etting, the BS chedule (uing optial cheduler (2 the channel to one of the obile uing the obile ignal, auing the to reflect the channel tate perfectly. The obile, aware of BS cheduler, axiize their utilitie. Utilitie of G1: For any given obile trategy profile µ, let the induced ignal probabilitie be repreented by p, i.e., p ( = h p h(hµ( h. Since the BS oberve p (intead of p h, it aue the expected hare of obile to be θ (µ, β := E p f( β( and hence axiize, UBS ASA (β, µ = Γ (θ (µ, β. (11 One can identify that θ (µ, β are the ASA utilitie. The equilibriu appropriate to thi cenario i Stackelberg Equilibriu. Stackelberg Equilibriu for G1: i a profile (β µ, µ which atifie the following for all : βµ = arg ax U ASA β µ = arg ax U AT A µ BS (β, µ, (12 ( (µ, µ, β (µ,µ.(13 We now preent oe exaple in which a uer deviate unilaterally fro µ T and increae it utility above it cooperative hare, reulting in unfair allocation. Thee exaple do not have TRE for G1, i.e., truthful trategy profile µ T i not a part of any Stackelberg Equilibriu of G1. In particular for (12, we conider -fair cheduler given by (3. Thi cheduler i a widely ued practical olution (ee Reark II-2, -FSA being one of the. A. Ayetric Exaple 1 Proportional fair cheduler ( = 1 : We continue with the otivating exaple given in Section I. Uer 1 ha a ingle tate with utility a. Uer 2 ha 2 tate with repective utilitie given by rb, b and with r > 1. The repective probabilitie to be in one of thee tate are p, (1 p with p (1/(1+r, 1/2. Uing (3, one can eaily etiate β, {θ (β } to be: β (2 a, rb = 1, β (1 a, b = 1, θ 1 (β = a(1 p and θ 2 (β = rbp. (14 Note that θ 1 (β, θ 2 (β are the obile cooperative hare. It i iportant to note here that β atifying (3 exit only if p (1/(1 + r, 1/2 a in thi cae : dγ (θ 2 (β rb = rb dγ (θ 2 (β b = b rbp > rbp < a a(1 p = dγ (θ 1 (β a a a(1 p = dγ (θ 1 (β a. Suppoe uer 2 ignal rb (when actually in tate b with probability q, i.e., µ 2 (rb b = q. Then uer axiu ASA rate (note that β q = β defined in (14 are: U ASA 1 (q, β q = (1 p qa, U AT A 2 (q, β q = rb(p + q repectively whenever rb (p + qrb > a a(1 p q > b rb(p + q. With thi, the obile 2 obtain an iproved ATA utility U2 AT A (q, βq = rbp+bq > θ 2 (β, i.e., obile 2 i ucceful in iproving it utility (above it cooperative hare by ignaling noncooperatively. The axiu poible value of q i q = (0.5 p. 2 Extenion to general : One can extend the above to general, an -fair cheduler atifying (3 exit if, (rb 1 p < a 1 (1 p < r(rb 1 p. Fro above, a increae, p for which (3 exit reduce and thu given (a, r, b, p, there exit a axiu ax, beyond which there doe not exit -fair cheduler of the type (3. However another type of alpha-fair cheduler exit; for exaple for ax-in fairne (when =, θ1 = θ2 a -fair cheduler {β (1 rb, a, β (1 b, a} given by: β a (1 rb, a = rbp + ap ; β (1 b, a = 0 if a(1 p < rbp β a(1 p rbp (1 b, a = (b + a(1 p ; β (1 rb, a = 0 ele. When -fair cheduler (3 exit the noncooperative obile benefit; the axiu q( atifie: (p + q( (rb 1 = a 1 (1 p q(. For exaple with a = 4, r = 3, b = 3, p = 0.33 the axiu for which -fair cheduler (3 exit i 7.9 and uer 1 can benefit by ignaling with q =.05 for all 4. 3 Generalization to ore tate and general : Conider two ayetric uer under the following auption : N.1 The cooperative -fair olution β (3 exit and without lo of generality let 1 = arg ax θ c. N.2 There exit an i > 1 uch that, η := inf dγ (θ1 c f(h 1,i 1 dγ (θ2 c f(h 2 > 0, h 2 H 2 where H 1 = {h 1,1,, h 1,N1 } are arranged uch that f(h 1,1 > f(h 1,2 > > f(h 1,N1.

6 6 Lea 2: Under auption N.1-N.2, there exit a non truth revealing policy µ δ 1 for obile 1 uch that it ATA utility U1 AT A (µ δ 1, (f, β i larger than it cooperative hare θ µ δ 1 c. 1 Proof : The proof i available in Appendix B. B. Syetric Cae We conider a iple yetric two obile exaple. The obile have two tate with utilitie a 1, a 2 occurring repectively with probabilitie p 1, p 2. Let a 1 = ra 2, p 1 = pp 2 with r > 1, p > 0. Under truthful ignaling, by Lea 1, an -fair optial BS policy (for any i given by: β (1 a 1, a 1 = 1/2 = β (1 a 2, a 2, β (1 a 1, a 2 = 1, β (1 a 2, a 1 = 0 with equal cooperative hare ( p θ 1 (β = θ 2 (β 2 = p 1p 2 a 1 + p 2 a ( p = p 2 2 r + 1 2a 2 + pr. 2 Without lo of generality ay obile 1 deviate unilaterally fro hi truthful trategy with µ 1 (a 1 a 2 = t. If obile 1 wa ucceful, hi reported rate would be greater than θ 1 (β and thi rate he would have obtained only when hi declared tate i a 1 with obile 2 being a 2. Thu, obile 1 will be ucceful with axiu ASA utilitie ( = 1: U1 ASA = (p 1 a 1 + p 2 ta 1 p 2 = (p + tp 2 2a 1 and U2 ASA = 1p 1 a 1 + p 2 (1 tp 2 a 2 = (pr + (1 tp 2 p 2 a 2 and the correponding ATA utility, U AT A 1 = (p 1 a 1 + p 2 ta 2 p 2 = (pr + tp 2 2a 2 if the following condition are et: a 1 U1 ASA > a 2 U2 ASA and θ 1 (β < U1 AT A, i.e., if t atifie: 1 1 > (p + tp 2 (pr + (1 tp 2 and p2 r + 1 < t. 2 C. Robutne at large For all value of, fair cheduler fail. However we ee a different phenoenon at higher. A increae to infinity, the fairne increae and the expected hare, i.e., ATA utilitie, of all the obile tend to becoing equal (18, provided all the obile ignal truthfully. However, in preence of noncooperation, it will be the ASA utilitie that tart becoing equal for higher value of. Thi reult in all the cooperative (ATA equal ASA utilitie obile getting equal ATA hare which will be bigger than that for the noncooperative (ATA are trictly le than ASA utilitie obile. Thu the -fair cheduler (2 itelf becoe ore and ore robut toward noncooperation a fairne factor increae, in pite of the BS unawarene of the noncooperation. 4 Thi effect i een 4 However a noticed in otivating exaple, we can only ay the ax in fairne will be robut againt all type of noncooperation fro obile and cannot identify an beyond which the cheduler will be robut. in the otivating exaple a well a in Figure 3 given in a later ection. In Figure 3, the noncooperative obile ATA utility diinihe a increae and goe below it cooperative hare beyond = 1.2 and further, the cooperative obile get ore hare than it cooperative hare for thee large value of. V. SCHEDULING UNDER NONCOOPERATION : GAME THEORETIC STUDY In thi ection the BS know about noncooperative behavior of obile and i conidered a an additional player reulting in the M player gae. A. BS Scheduling policie of ection IV : Gae G2 In contrat to ection IV, the BS know the obile that are noncooperative. The reulting gae i a one-hot concave gae: the utility of obile (6 i linear in it policy µ while that of the BS (7 i continuou and concave in it policy β. By 19, thi gae alway ha a NE 5 (µ, β which atifie, for all, µ = arg ax U µ ((µ, µ, β and β = arg ax U BS(µ, β. β For gae G2 we obtain a babbling equilibriu. We further how that G2 doe not have a TRE. 1 G2 ha Babbling NE : We will now how that thi gae ha a Nah equilibriu where the BS neglect the ignal fro the noncooperative uer. Let h >M1 := h M1+1, h M t repreent the channel tate of the cooperative obile. With θ >M1 (β := E h f(h β ( h >M1, the BS axiize: Γ ( θ >M1 (β. (15 We note here that for any non cooperative obile, θ >M1 (β = Ef(h E h >M 1 β ( h >M 1 for M1. A in Lea 1, there alway exit a β axiizing (15. Call one uch β by β >M1. Chooe any obile profile µ which atifie for all M 1, µ ( h = 0 for all h, with f( < f(h. It i eay to ee that (µ, β >M1 for a Nah Equilibriu. Note here that a noncooperative obile can obtain the utility θ >M1 (β >M1 only if it ignal better than it channel true value (a only in thi cae in{f(h, f( } = f(h and hence the requireent of above condition on the et of obile trategie. Thi i a NE at which the BS ignore the ignal fro the noncooperative obile and i iilar in ene to the Babbling equilibriu defined in the context of ignaling gae (22. Hence we choe to call thi alo a Babbling equilibriu. We end thi ubection with a ueful, practically ipleentable -fair cheduler involving only cooperative ignal 5 Note that when adding further concave contraint the gae reain concave even if the contraint are coupled 19. We thu obtain equilibriu alo for contrained verion of the gae. Exaple of uch contraint are: the (poible weighted u of throughput i bounded by a contant.

7 7 (if exit. Define (if exit, β >M1 ( h >M1 = A >M1 ( h >M1, β = arg ax j 1 { A >M 1(h >M 1,β >M 1 } A >M1 (h >M1, β >M1 dγ (θ >M1 j (βu j with u j = f(h j 1 {j>m1} + Ef(h j 1 {j M1}. Uing iilar tep which obtained (3, one can how that β >M1 i a axiizer of (15. 2 G2 ha No TRE : We now exaine the exitence of the deired TRE. The cae of = 0, the efficient cheduling i tudied in 13. In 13, G2 correponding to efficient cheduling wa odeled by a ignaling gae and it i hown that the gae G2 ha only babbling equilibriu a NE and hence doe not have a TRE. We will now conider the cae > 0. If the M player gae were to have a TRE, the correponding (equilibriu trategy of the BS, by definition the NE, hould be the bet repone to obile truthful trategie µ T and hence will be axiizer of UBS (µt, β = G (β. Hence, the bet repone for truth revealing trategy profile µ T indeed equal one of the axiizer of Lea 1, which atifie the efficiency property (5. Let β be any axiizer of Lea 1. The trategy profile (µ T, β doe not for a NE becaue: Let be any obile with non zero cooperative hare and let h be it channel value with larget utility, i.e., let h = arg ax h H f(h. The obile by changing it policy fro truthful ignal µ T to µ ( h := 1 for all h { = h}, increae it ATA utilitie a by (5 for any h and for any h h H, β ( h, h β ( h, h, and hence, U AT A ((µ T µ, β U AT A (µ T, β = ( p h (h β ( h, h f(h β ( hf(h h = ( p h (hf(h β ( h, h β ( h > 0. h Strict greater than zero reult in the lat line for all > 0, a all the obile obtain non zero utility under an alpha fair cheduler. Thu, the obile can iprove it utility by unilaterally oving away fro µ T, contradicting the definition of NE. Thu the BS, even when aware of the noncooperation, i not ucceful in eliciting the truthful ignal. In the following we contruct ore intelligent policie which induce a TRE. Hence, BS ha to ue ore intelligent cheduling algorith to be robut againt noncooperation. B. Robut BS Policie : Gae G3 ha TRE BS can etiate tatitic p after ufficient obervation of the obile ignal. We ue p to build robut policie for BS which give u the deired TRE. The policy of BS now ap every ordered pair of ignal and ignal tatitic (, p to an ordered pair (Φ, β = {(φ (, p, β(., p } with allocation φ ( f( for all. All the utilitie will change appropriately to include Φ, for exaple: U (µ, (Φ, β = E h in{φ (, f(h }µ( hβ( A profile (µ 1,, µ M 1, (Φ, β i a NE for G3 if, µ = arg ax U µ ((µ, µ, (Φ, β for all (Φ, β = arg ax U BS(µ, (Φ, β. (16 (Φ,β When BS know ignal tatitic, {p }, it can etiate the ASA utilitie for any cheduling policy and for any obile profile µ a: U ASA (µ, (Φ, β = U ASA (p, (Φ, β := E φ(β(. In the above the expectation i w.r.t. p. It can alo etiate their cooperative hare {θ c } of (10 uing it prior knowledge: the channel tatitic. We now propoe a robut policy at the BS which ue both thee average utilitie. The key idea i to deign a policy at BS which doe not allow the (average utility of any obile to be greater than θ c. When a noncooperative obile ue a ignaling trategy to iprove it ATA utility U AT A, even it ASA utility U ASA iprove. The BS can etiate U ASA of each of the obile and hence can ene the increae in the noncooperative obile ASA utility in coparion to it cooperative hare. The BS can enure none of the cooperative obile i allocated ore than it correponding cooperative hare: by allocating only a fraction and not the total ignaled utility at every aple. The fraction to be allocated, i et baed on the preent exce over the cooperative hare a follow: φ (, p, β := ( f( ( U ASA (p, (Φ, β θ c 1 {(f( (U ASA(p,(Φ,β θc >0} (17 for oe large value of. Hence, to enure that none of the obile get ore ASA utility than it cooperative hare, BS need to allocate (chooe Φ = {φ } to atify the following: U ASA (p, (Φ, β = E φ (, p, ββ(. (18 Both the equation (18 and (17 are atified if there exit a fixed point U ASA = U ASA (p, β which atifie: U ASA = E φ β( 1 {φ>0} ; (19 φ := f( ( U ASA θ c. With C f repreenting the upper bound on f, ( f( ( U ASA θ c 1{(f( (U ASA C f + θ c θc >0} for all and U ASA. Thu the ap U ASA φ β( 1 {φ>0} i bounded and continuou alot urely and hence by bounded convergence theore the ap of (19, U ASA E φ β( 1 {φ>0}, i continuou. Thu there exit an U ASA atifying the fixed point equation (19 by Brouwer fixed point theore 6. 6 Brouwer fixed point theore: Every continuou function f fro a cloed ball of a Euclidean pace to itelf ha a fixed point, i.e., an x which atifie x = f(x..

8 8 With the above allocation, ATA utility gained by obile, U AT A (µ, (Φ, β = E h, f gain (h,, p, ββ( (20 f gain (h,, p, β := in{f(h, φ (, p, β}. Equation (19 atifie : U ASA = Ef(β( 1+θc Eβ( 1 1+ E β( 1 (21 with 1 := 1 { U ASA <f(+ θc }. Hence, for any trategy profile (µ, β U ASA (p, (Φ, β θ c = E f( β( 1 { U ASA and hence, U AT A <f(+ θc } θ c 1 + E β( 1{ U ASA <f(+ θc C f o(1/, } (µ, (Φ, β U ASA (µ, (Φ, β θ c + o(1/. The above i true a, f gain (h,, µ, β φ (, µ, β. In other word, with new allocation (19 at BS, no obile can gain o(1/ ore than it cooperative hare for any pair (µ, β. Further, if BS ue any -fair cheduler β of (2, along with allocation policy (19, it i eay to check uing (21 and (20 that under truthful trategie (note p = p h U ASA (µ T, β1 = (µ T, β1 = θ c for all. Alo now, U AT A U ASA (p h, (Φ, β θ c = E h f(h β ( h1 E f(h β ( h 1 + E h β ( h1 = E h f(h β ( h1 { U ASA f(h+ θc } 1 + E h β. ( h1 The above indicate that U ASA (p h, (Φ, β θ c. If it wa trictly le than the cooperative hare, then the indicator in the nuerator of the econd line can never be true and hence there exit only one fixed point, θ c with (µ T, β. We have thu proved: Theore 1: If BS know cooperative hare {θ c } and the ignal tatitic {p }, the M 1 +1 player trategic gae ha an ɛ NE, i.e., TRE: ( µ T, ({φ (, p, β }, β (. In the coing ection, we will turn our attention to iterative algorith which can achieve a deired level of fairne even in the preence of oe noncooperative obile. We begin thi tak by firt tudying -FSA (15. VI. FAIR SCHEDULER ALGORITHM (-FSA Fro thi ection onward the channel tate h a well a the ignaled tate (the tate reported by the obile are continuou rando variable with tationary rate acro the tie, {r,k } k 1 = {f(h,k } k 1, {r,k } k 1 = {f(,k } k 1 for all, atifying the auption of Appendix A 7. Thi ection and the coing ection ue variou type of rate and hence the notation becoe coplicated. Thu a table (in table III of notation pecific to thee two ection i given in Appendix A, where all the rate notation are lited at one place. By auption A.3 of Appendix A, the rate are integrable and hence the ap Θ E h f(h 1 β(1 h, Θ,, E h f(h M β(m h, Θ, β( h, Θ = 1 {=arg ax j dγ (θ jf(h j} {arg ax j dγ (θ j f(h j } ha a fixed point Θ by Brouwer fixed point theore and β (. h := β(. h, Θ exactly atifie (3 and hence i a -fair olution. Thu with continuou rate we alway have fixed point -fair olution (3. We outlined an algorith to ipleent -fair cheduler (3 in Reark II-2 following Lea 1. The -FSA (15, a tochatic approxiation baed fair cheduling algorith, exactly follow thi outline (with Θ k θ := 1,k,, θ M,k, r k := r 1,k,, r M,k : θ,k = θ,k 1 + ɛ k I (r k, Θ k 1r,k θ,k 1 I(r, Θ = 1 {=arg axj dγ (d j+θ jr j} (22 = 1 {=arg axj r j(θ j+d j } where d are all poitive contant (added for tability. While aking deciion {I}, if there are ore than one uer attaining axiu, one of the axiizer i choen by the BS randoly. In 15, Th. 2.2, the author how that {θ,k } of (22, with 1, converge weakly to the unique liit point Θ that atifie E r I(r, Θ = θ for all. A cloe look at thi liit point (when we neglect {d } reveal that I(r, Θ i the -fair cheduler (3 and that Θ are the unique cooperative hare, {θ (β } = {θ c }. Thu, -FSA weakly converge to the unique point (cooperative hare that axiize the -fair criterion (1. A. Convergence of -FSA in preence of noncooperation The -FSA ue ignaled rate, r,k := f(,k and r k = r 1,k,, r M,k t to ake deciion, a in Section IV: θ,k = θ,k 1 + ɛ k I (r k, Θ k 1r,k θ,k 1. Thee ignaled rate reflect the tatitic p (intead of p h, there again i weak convergence, however thi tie to a different attractor correponding to p. It i very eay to ee a in Section IV that, when obile are noncooperative with 7 For undertanding the ayptotic liit of the dynaic algorith of thi ection we will need the reult correponding to the tatic etting of Section II. But, all the reult of Section II correpond to dicrete channel tate and rate. We aue that even for the ore general cae under tudy in thi ection, an -fair olution of the for (3 exit and that the correponding hare {θ c }are unique a in Lea 1. Sufficient condition for thi to occur are under tudy. Thi reult i required for howing that -FSA ayptotically converge to the cooperative hare (i.e., liit axiize the -fair criterion for all. In 15 Theore 2.3 doe thi job approxiately at leat for 1: any other aignent rule reult in a liit Θ with Γ (θ le than that correponding to cheduler {I } of -FSA (22. The iulation of thi ection alo confir the reult we obtained baed on thi auption.

9 9 profile µ, -FSA converge weakly to unique axiu ASA rate, { U ASA (µ, β µ } with β µ defined by (12. B. Failure of -FSA in preence of noncooperation A noted above, the -FSA (22 converge to the axiu ASA utility (under µ which need not be equal to the ATA utility, in the preence of noncooperation. However, to undertand the behavior of (22 in preence of noncooperation, one need to tudy the ayptotic true utilitie gained by the obile under (22. Toward thi, we conider a econd iteration running in parallel with (22, with only the intantaneou ignaled utility r,k replaced by the true intantaneou utility obtained by the obile, r,k := in{r,k, r,k }: θ,k = θ,k 1 + ɛ k I (r k, Θ k 1 r,k θ,k 1 (23 A in 15, one can how that θ,k converge weakly to U AT A (µ, βµ, the ATA utility under (µ, βµ. Thu, the ayptotic liit of -FSA equal axiu ASA utilitie of ection IV while the true utility adaptation (23 converge to the correponding ATA utilitie. Thee tie liit will thu have all the propertie of ection IV: the -FSA will fail for all and will be robut for large a dicued in ection IV. The only difference here i that the channel rate are continuou. C. Nuerical exaple Two ayetric uer are conidered in Figure 3. Let Z(σ 2 be a Rayleigh rando variable with denity f Z (z; σ 2 = ze z2 /2σ 2. Channel tate of uer 1 i conditional Rayleigh ditributed, i.e., h 1 f Z(z; 11 {z 2} dz. P (Z(1 2 Uer 2 ha alot a contant channel, h f Z(z; {z 2} dz. P (Z( The utilitie are the achievable rate f(h = log(1+h. Uer 1 i noncooperative with 1 (h = h(1 δ+2δ with δ = 0.9. We plot the liit of the -FSA, the liit of true utility adaptation (23 a function 8 of. We alo plot the cooperative hare, obtained by the liit of -FSA, i.e., with δ = 0. We oberve that the cooperative hare tend toward equal value a tend to infinity. Uer 1 i ucceful in gaining ore utility in coparion with it cooperative hare for all le than 1.2. Beyond 1.2, uer 1 actually loe and the lo increae a increae. The obervation are iilar to that in otivating exaple and indicate that -FSA i robut only for large. In table I, we conider a yetric exaple. In thi exaple, we conider the dicrete channel of ection IV. We could have contructed exaple with yetric continuou channel tate a in previou exaple and deontrate the failure of -FSA. But we note that the -FSA even with dicrete rate 8 The author in 15 analyze thee algorith only for 1. However we confir uing nuerou exaple that they work in fact for all value of, i.e., when all obile are cooperative the FSA for all converge to the unique hare which axiize the objective function ( FSA θ 1 c Mob 1 θ 2 c Mob 2 ASA Mob 1 (δ=0.9 ASA Mob 2 (δ=0.9 ATA Mob 1 (δ=0.9 ATA Mob 2 (δ= Fig. 3. -FSA : Maxiu ASA and correponding ATA hare veru Robut Fair SA θ 1 c Noncoop Mob θ 2 c Coop Mob ATA Noncoop Mob (δ = 0.9 ATA Coop Mob (δ = 0.9 ATA Noncoop Mob (δ = 0 ATA Coop Mob (δ = Fig. 4. Robut Policy: Maxial ASA and correponding ATA hare. work a explained in thi ection and hence thi exaple i given to deontrate the ae. We conider two uer, both of the having two channel tate with utilitie a1 = 4, a2 = 2 occurring with probabilitie p1 = 0.3, p 2 = 0.7 repectively. In thi exaple we work only with = 1, i.e., the proportional fair cheduler. Both uer have equal cooperative hare, θ 1 (β = θ 2 (β = Hence when both the obile report the channel tate truthfully, under FSA cheduler, the ayptotic throughput of both the obile converge to 1.51, i.e., li k θ,k = 1.51 for = 1, 2. Hence axiu proportionally fair BS (ayptotic utility i UBS = 2log(1.51 = µ 1 (a 1 a 2 True Rate t (U AT A 1, U AT A 2 U AT A log(u AT A 0 (Coop (1.51, (1.62, (1.72, (1.70, TABLE I A SYMMETRIC EXAMPLE IN WHICH FSA FAILS AGAINST NONCOOPERATION The uer 1 becoe noncooperative with µ 1 (a 1 a 2 = t. We ee that the uer 1 i ucceful in grabbing the channel ore often and increaing it utility in coparion with it cooperative hare. The ore he cheat (the ore t i the ore he gain (look at the ayptotic throughput U1 AT A, given in the econd colun in table I. He gain up to 12.5% ore than it cooperative hare. The cooperative uer, uer 2 ha lot due to the non cooperative obile reulting in unfair allocation. VII. ROBUST -FAIR ALGORITHMS : ROBUST FAIR SA We aw that -FSA fail in the preence of noncooperative uer. Hence, we propoe a robutification of -FSA againt noncooperation uing the policie of ubection V-B. In V-B, we propoed BS policie robut againt noncooperation and in thi ection we propoe tochatic approxiation baed algorith to converge toward the ASA utilitie of thoe policie given by (20. The policy of ection V-B require the knowledge of ignal tatitic p, which ha to be etiated. Baically the ethod decribed in thi ection (a i done by FSA cobine the etiation and control uing tochatic

10 10 approxiation baed ethod. We will how robutne of thee policie by uing appropriate gae theoretic tool. Robut Policy 1 : We now propoe a robutification of (22 againt noncooperation in the following : θ,k+1 = θ,k + ɛ k φ,k+1 I ( r k+1, Θ k θ,k φ,k+1 = ax { ( 0, r,k+1 ( θ,k θ c },(24 θ,0 = θ c (25 where the deciion I (r, Θ are ae a that in -FSA (22 and only the allocation Φ k := φ 1,k,, φ M,k T are ade robut. A in the cae of -FSA, to undertand the behavior of thi algorith we need the following iteration which etiate the true utilitie gained by the obile : ˆθ,k+1 = ˆθ,k + ɛ k ˆr,k+1I ( r k+1, Θ k ˆr,k+1 = in { r,k+1, φ },k+1 A. Analyi : ˆθ,k, (26 We analyze the robutne of the propoed algorith uing gae theoretical tool. Fix any. We conider a M 1 +1 player gae with utilitie defined by : U := li ˆθ,k+1 for all and U BS := Γ (U. k We analyze the liit of (26 uing ODE approxiation ethod (for e.g., 15, 9. A a firt tep, we obtain the following ODE approxiation reult. Theore 2: Aue that algorith (24, (26 atify auption A.1, A.2, and A.3 of Appendix A. For any initial condition, (Θ k, ˆΘ k converge weakly to the et of liit point of the olution of the ODE (for all M: θ = h (Θ θ, h (Θ = E φ I (r, Θ, (27 ˆθ = ĥ (Θ ˆθ, ĥ (Θ = E ˆr I (r, Θ. (28 Thee concluion hold whenever ɛ n 0, n ɛ n = and for oe n, li n up 0 l n ɛ n+l /ɛ n 1 = 0. Reark about the proof and the auption : Thi theore can be proved exactly in the ae way a i done for FSA by Theore 2.1 of Kuhner et al 15. The required auption A.1-3 are alo very iilar to that in 15 and thee are atified in alot the ae condition a entioned in 15. Hence, one can upper bound utilitie {U } by upper bounding all the attractor of ODE (28. Any attractor Θ of the ODE (27 atifie θ θ c = E ri (r, Θ I {φ >0} θ c 1 + E. Hence I (r, Θ I {φ >0} θ θ c C f E I (r, Θ I {φ >0} E = C f I (r, Θ I {φ >0} where C f i the upper bound on ignaled rate {r}. Thu, θ θ c + o(1/. Further, any attractor of ODE (28 atifie ˆθ = ĥ (Θ leading to ˆθ θ. Thu for any obile trategy profile µ, U w = ˆθ θ θ c + o(1/. (29 So, none of the uer, no atter what trategy they ue or the other ue, can gain ore than θ c. Under µ T, Θ c = θ1 c,, θm ct i the only zero of RHS of both the ODE (27, (28 a hown uing fixed point analyi in ection V-B (note here that I(r, Θ i the - fair cheduler β ( atifying (3. By virtue of Lea 4, one can eaily how that it will indeed be an attractor (for large enough value of for ODE (27 by howing that the derivative of { h (Θ θ ; M} i negative definite 9 near Θ c. And then it i iediate that Θ c i alo an attractor for the ODE (28. Thu, Θ c, i the only attractor of both the ODE under µ T. Thu U w = θ c for all under µ T. (30 Fro (29, (30, the robut policy (24 at BS together with the truth-revealing policy of uer for an ɛ-ne. Robut Policy 2 : The policie of previou ubection, Robut Policy 1 will not allow the ATA utility of any uer to go above the cooperative hare. Neverthele, when a uer i noncooperative, thee policie ay till reult in a lo for the cooperative uer: the noncooperative uer can till grab the channel fro other uer, without getting a gain becaue of the robut allocation policie (19. To avoid thi proble, we ay robutify the deciion a well: θ,k+1 = θ,k + ɛ k φ,k+1 I ( Φ k+1, Θ k θ,k. (31 The analyi of thi policy would be iilar to the policy 1. We need to change the auption of the Appendix A appropriately (need to replace the deciion I (r, Θ with I (Φ, Θ in all the place for ODE approxiation. However thee policie are ore coplicated and hence further analyi i ore difficult. We will till be able to go through all tep in the analyi exactly in a iilar way, except that we will not be able to how the uniquene of the attractor under truthful trategie µ T. However, we could enure the robutne of thee policie uing the nuerical exaple given below. The exaple alo how that thee policie outperfor Robut 9 Uing iilar tep a thoe ued for deriving (34 of Appendix B, one can eaily ee that ( h(θ θ = 1 E I θ (r, Θ c Θ=Θ c E (r 2 Π k j,i P r(a k (r, Θ c g j (κ(θc j + d j (θ c j + d+1 which i alway negative and whoe agnitude increae a increae for all while for any j ( h(θ θ = θ j Θ=Θ c E (r 2 Π k j,i P r(a k (r, Θc g j (κ(θc j + d j 1 (θ c + d whoe agnitude i bounded independent of. Define the ap H(Θ := h 1 (Θ,, h M (Θ T, the total derivative w.r.t Θ A := D Θ ( H(Θ Θ and the atrix B a a diagonal atrix coniting of only diagonal entire of atrix A. Note that atrix B ha all negative eigenvalue. Now, when 10, Corrollary III 2.6, pp. 63 i applied to atrice A, B (thi corollary copare the eigenvalue of the two atrice we get that atrix A becoe negative definite a the value of increae.

11 11 policy 1 in any way, while Robut policy 1 would be ipler to ipleent. 0.5 Robut Fair SA : Policy Robut Fair SA : Policy Nuerical exaple We continue with the exaple of Figure 3 (in which - FSA failed in Figure 4. We ue Robut Fair SA Policy 1 in place of -FSA. We et = We plot only the ATA utilitie for both value of δ = 0, δ = 0.9. We do not plot the ASA utilitie in thi figure a thee utilitie for all the cae are very cloe to cooperative hare Θ c. We ee that thi policy i indeed robut : 1 the tie liit of {θ,k } (which correpond to ASA utilitie are very cloe to the cooperative hare; 2 the tie liit of the ayptotic true (ATA utilitie are leer than the cooperative hare for the noncooperative obile. It i alo leer for cooperative obile, but the gap between the cooperative hare and the ATA utilitie i uch leer for a cooperative obile (plot correponding to δ = 0.9; 3 when all the obile are cooperative both the ASA a well a ATA utilitie are cloe to the cooperative hare for all the obile (plot correponding to δ = 0. In Figure 5, 6 we copare the two robut policie. Here h 1 f Z(z; 11 {z 2} dz P rob(z(1 2, h 2 f Z(z; 0.51 {z 2} dz P rob(z(0.5 2, f(h = log(1 + h and = Mobile 1, can be noncooperative uing 1 (h = h + (2 hδ with δ = 0.9. We ee fro the figure that both the policie are robut. Even with high value of δ = 0.9 (which indicate large aount of noncooperation both the policie do not allow the ATA utilitie to go beyond the cooperative hare. However the policy 2 i way better than the policy 1 a: 1 the noncooperative obile in fact i punihed by policy 2, it ATA utility i leer than the cooperative hare θ1 c (Figure 6, but the ae i not true for policy 1 (Figure 5. 2 the cooperative obile loe becaue of the noncooperation fro the other obile to uch greater extent in policy 1. Thi goe in line with the extra robutification built into the deciion aking by policy 2. When BS ue policy 1, the noncooperative obile i ucceful in grabbing the channel (alot alway with large value of δ = 0.9, however will not be able to gain uch fro it becaue of the robut allocation (19-(21. When obile i aware that he cant gain fro being noncooperative, he will a well have to tick to truthful ignaling, unle hi intention are that of jaing the other obile (in which cae the BS need to ue policy 2. However the policy 1 i eaier to ipleent than the policy 2 becaue of ipler deciion and ay alo have fater convergence. µ 1 (a 1 a 2 = True Rate U BS = t (U1 AT A, U2 AT A U AT A log(u AT A 0 (Coop (1.51, (1.30, (1.31, (1.32, TABLE II ROBUST POLICY 1 AGAINST NONCOOPERATION EXAMPLE OF TABLE I θ 1 c Mob 1 θ 2 c Mob 2 ASA Mob 1 (δ = 0.9 ASA Mob 2 (δ = 0.9 ATA Mob 1(δ = 0.9 ATA Mob 2(δ = 0.9 ASA Mob 1 (δ = 0 ASA Mob 2 (δ = 0 ATA Mob 1 (δ = 0 ATA Mob 2 (δ = Fig. 5. Robut Policy 1 : Maxial ASA and correponding ATA hare veru θ 1 c Mob 1 θ 2 c Mob 2 ASA Mob 1 (δ = 0.9 ASA Mob 2 (δ = 0.9 ATA Mob 1 (δ = 0.9 ATA Mob 1 (δ = 0.9 ASA Mob 1 (δ = 0 ASA Mob 2 (δ = 0 ATA Mob 1 (δ = 0 ATA Mob 2 (δ = Fig. 6. Robut Policy 2 : Maxial ASA and correponding ATA hare veru. In table II we continue with the yetric exaple of table I wherein the FSA fail. We ee once again that (even with dicrete and yetric condition the Robut Fair SA policy 1 i robut againt noncooperation, it doe not allow the noncooperative uer to iprove it ayptotic throughput. VIII. CONCLUSIONS We tudied centralized downlink traniion in a cellular network in the preence of noncooperative obile. Uing - fair cheduler, the BS ha to aign the lot to one of the any obile baed on truthful inforation fro obile about their tie-varying channel gain. A noncooperative obile ay irepreent it ignal to the BS o a to axiize hi throughput. We odeled a noncooperative obile a a rational player who wihe to axiize hi throughput. For thi gae, we identified everal cenario related to the awarene of BS. When the BS i unaware of thi noncooperative behavior, we odel thi gae a hierarchical gae with two level. We identify that, the preence of noncooperative uer, reult in an -fair bia in the channel aignent for all value of. A increae, an -fair cheduler becoe ore and ore robut to noncooperation irrepective of the awarene of BS and a ax-in fair cheduler i alway robut. When the BS i aware of the noncooperative obile, we characterized a babbling equilibriu which i obtained when both the BS and the noncooperative player ake no ue of the ignaling opportunitie. Thi gae ha no TRE (Truth Revealing Equilibriu. Uing additional knowledge of the tatitic of the ignal oberved at the BS, we built new robut policie to elicit the truthful ignal fro obile and achieve a Truth Revealing Equilibriu. We then tudied the popular, iterative fair cheduling algorith (which we called -FSA analyzed by Kuhner and Whiting in 15. We howed that - FSA fail under noncooperation. Finally, we propoed iterative robut fair haring to robutify the -FSA in the preence of noncooperation. Acknowledgent Thi project wa upported by the Indo-French Center for the Prootion of Advanced Reearch (IFCPAR, project IT-1 and by the INRIA aociation progra DAWN. The french co-author have alo been upported by the Bionet European project and the RNRT-ANR WiNEM project.