Improving the Efficiency of Load Balancing Games through Taxes

Transcription

1 Improvng the Effcency of Load Balancng Games through Taxes Ioanns Caraganns, Chrstos Kaklamans, and Panagots Kanellopoulos Research Academc Computer Technology Insttute and Dept. of Computer Engneerng and Informatcs Unversty of Patras, 6500 Ro, Greece Abstract. In load balancng games, there s a set of avalable servers and a set of clents; each clent wshes to run her ob on some server. Clents are selfsh and each of them selects a server that, gven an assgnment of the other clents to servers, mnmzes the latency she experences wth no regard to the global optmum. In order to mtgate the effect of selfshness on the effcency, we assgn taxes to the servers. In ths way, we obtan a new game where each clent ams to mnmze the sum of the latency she experences and the tax she pays. Our obectve s to fnd taxes so that the worst equlbrum of the new game s as effcent as possble. We present new results concernng the mpact of taxes on the effcency of equlbra, wth respect to the total latency of all clents and the maxmum latency makespan). Introducton Load balancng games are specal cases of the well-known congeston games ntroduced by Rosenthal [8]. A congeston game Π conssts of a set E of resources, each resource e havng a non-negatve and non-decreasng latency functon f e defned over non-negatve numbers, and a set of n players. Each player has a weght or demand) w and can select among a set of permssble strateges S E where each strategy of player s a set of resources). In general, players may follow mxed strateges,.e., use a probablty dstrbuton over ther permssble strateges. An assgnment A = A,..., A n ) s a vector of strateges, one possbly mxed) strategy for each player. We mostly refer to pure assgnments,.e., assgnments where each player selects a sngle strategy wth probablty. The cost of a player at an assgnment A s defned as cost A) = e A f e n e A)), where n e A) s the total weght of players usng resource e n A. The socal cost of an assgnment can be ether the weghted total cost over all players or the maxmum latency makespan) over all resources. A pure resp., mxed) assgnment s a pure resp., mxed) Nash equlbrum f no player has any ncentve to unlaterally devate to another strategy, Ths work was partally supported by the European Unon under IST FET Integrated Proect 0596 AEOLUS and by a Caratheodory grant from the Unversty of Patras.

2 .e., cost A) cost A, s) for any player and for any pure resp., mxed) strategy s, where A, s) s the assgnment obtaned f ust player devates from A to s. In lnear congeston games, the latency functon of resource e s of the form f e x) = α e x + β e wth non-negatve constants α e and β e. We use the terms weghted and unweghted to dstngush between the cases where the clents have dfferent or dentcal weghts. Load balancng games are congeston games where the strateges of players are sngleton sets. In load balancng termnology, we use the terms server and clent nstead of the terms resource and player. The set of strateges of a clent contans the servers that are permssble for the clent. A load balancng game s called symmetrc when all servers are permssble for any clent. Usually, the servers of load balancng games have lnear latency functons; an mportant specal case s that of related servers where the latency functon of server s of the form f x) = α x, wth α > 0. Motvated by [8], we use the term graph balancng games to denote asymmetrc load balancng games where each clent s unweghted and has at most two permssble servers, and all servers have dentcal lnear latency functons. Snce players act selfshly, load balancng games may reach assgnments that do not mnmze the socal cost. We use the noton of the prce of anarchy ntroduced n [5, 7] to quantfy the degradaton of the overall system performance. In partcular, the prce of anarchy of a game Π s the maxmum over all pure or mxed) Nash equlbra of the rato of the socal cost of a pure or mxed) Nash equlbrum over the socal cost of the optmal assgnment. A vast amount of the lterature see [0, 5] and the references theren) studes the complexty of computng equlbra of best and worst socal cost and provdes bounds on the prce of anarchy for varous games that can be thought of as specal cases of congeston games such as load balancng games, when the socal cost s defned as the makespan or the weghed total latency. Awerbuch et al. [] and, ndependently, Chrstodoulou and Koutsoupas [] prove tght bounds on the prce of anarchy of congeston games wth respect to the weghted total latency. Among other results concernng polynomal latency functons, they show that the prce of anarchy of pure Nash equlbra n unweghted lnear congeston games s 5/ whle for mxed Nash equlbra or pure Nash equlbra of weghted players t s.68. These bounds carry over to load balancng games [] and can be mproved for nterestng specal cases [, 6, ]. The prce of anarchy of log m weghted load balancng games on m related servers s Θ log log log m ) [7] over mxed Nash equlbra wth respect to the makespan. A better tght bound of Θ log m log log m ) s known for pure Nash equlbra as well as for mxed Nash equlbra at dentcal servers [7, ]. In order to downscale the effect of selfshness to performance, we assgn taxes to the servers. Formally, a tax functon δ : E Q + Q + assgns a tax δ w) to each clent of weght w that wshes to use server E. Furthermore, we assume that clents are not equally senstve to taxes. In partcular, clent has a tax senstvty γ > 0. Assumng selfsh behavor of the clents, we obtan a new extended game Π, δ) where each clent now ams to mnmze the expected

3 latency she experences plus her dsutlty due to the taxes she pays at the server she uses. Ths dsutlty equals γ δ w ) when clent selects server. Agan, an assgnment y s a pure Nash equlbrum for the extended game f no player has an ncentve to unlaterally change her strategy,.e., cost y) + γ δ s w ) cost y, s )+γ δ s w ) for any clent that s on server s under the assgnment y, where y, s s the assgnment produced when clent moves from s to s. Lke n our prevous work [3] on the topc and motvated by [6], we consder both refundable and non-refundable taxes. In the former case, we assume that the collected taxes can be feasbly returned drectly or ndrectly) to the players e.g., as a lump-sum refund ) and therefore the overall system dsutlty depends only on the socal cost. However, refundng the collected taxes could be logstcally or economcally nfeasble; the latter case models ths scenaro. We wll say that a functon δ : E Q + Q + s a ρ-pure-effcent refundable tax for the load balancng game Π f the socal cost for any pure Nash equlbrum of the extended game s at most ρ tmes the socal cost of the optmal assgnment. Smlarly, a functon δ : E Q + Q + s a ρ-pure-effcent non-refundable tax for the load balancng game Π f the socal cost plus the total dsutlty due to taxes at any pure Nash equlbrum s at most ρ tmes the socal cost of the optmal assgnment. Smlar defntons apply to the case of mxed Nash equlbra. The problem of computng optmal taxes has receved sgnfcant attenton n the economcs and transportaton scence lterature; the man underlyng model n these studes s that of non-atomc congeston games []. These games dffer from the atomc games that we consder n that each player controls an nfntesmal amount of demand, and, therefore, the actons of a sngle player cannot affect the overall system performance. The results about taxes n nonatomc congeston games see for example [5, 6, 0, 3]) do not carry over to the atomc model. In our prevous work [3], we presented among several negatve and postve results on the nfluence of taxes, under the assumpton that all clents are equally senstve to taxes) -mxed-effcent refundable taxes wth respect to the weghted total latency for lnear atomc congeston games and a pure-optmal tax functon for symmetrc load balancng games; ths latter result was extended by Fotaks and Spraks [] to also hold for network congeston games on seres-parallel graphs. Swamy [3] studed more general e.g., polynomal) latency functons for the case of atomc congeston games wth splttable demands and presented taxes that ensure that the optmal assgnment s a pure Nash equlbrum. In ths paper we show the followng results concernng non-refundable taxes. For the case of graph balancng games and unweghted clents wth dfferent senstvtes, we present an.68-pure-effcent tax functon. Ths s the frst class of asymmetrc load balancng games for whch an upper bound better than s acheved, whle we note that the lower bound of /0 presented n [3] also holds for graph balancng games. Recall that the prce of anarchy of these games can be at least.0 []. Our tax functon explots the structure of graph balancng games and also uses the optmal assgnment whch can be computed n polynomal tme. Then, we consder symmetrc load balancng games wth

4 unweghted clents and servers wth polynomal latency functons of degree p. We prove a negatve result that no non-refundable ) tax functon can be better than p+)+/p p+) +/p p -pure-effcent,.e., O p ln p -pure-effcent. Note that ths lower bound matches the known upper bound on the prce of anarchy of these games whch s a corollary of the relaton to symmetrc non-atomc congeston games n [] and the upper bounds of [9]. Next, we focus on the makespan as the socal cost. For the case of pure Nash equlbra and weghted clents on m related servers, we present a -pure-effcent tax functon, greatly mprovng upon the ) bound on the prce of anarchy presented n [7]. The tax functon s defned usng a partcular fractonal schedule of clents to servers. We also present a lower bound that shows that ths tax functon s best possble. Fnally, for mxed Nash equlbra we observe that the ntroducton of taxes does not Θ log m log log m mtgate sgnfcantly the mpact of selfshness, snce no better than O log m log log m )- mxed-effcent taxes exst, even for games wth unweghted clents on dentcal servers. The rest of the paper s structured as follows. We begn by presentng, n Secton, our result concernng graph balancng games. We contnue n Secton 3 wth the negatve result about non-refundable taxes n symmetrc load balancng games wth polynomal latency functons. The results concernng the obectve of mnmzng the makespan are presented n Secton. Effcent Taxes for Graph Balancng Games In ths secton, we present.68-pure-effcent tax functons for graph balancng games. Ths s the frst subclass of asymmetrc load balancng games whch s proved to have better than -pure effcent taxes. The tax functon s smple and explots the structure of the game. We wll assgn very small taxes to the servers so that each server s assgned a dfferent tax. So, although we prove the result gnorng the total taxes pad by the clents, ths quantty can become arbtrarly small and our result carres over to non-refundable taxes by addng an ɛ factor to the effcency. Consder a graph balancng game wth a set of clents U wth U = n) and let ˆδ be such that 0 < ˆδ / max γ, where γ s the tax senstvty of clent. Frst, we compute an optmal assgnment and denote by o the number of clents usng server n ths assgnment. Ths computaton can be done n polynomal tme by a natural reducton to a mnmum cost flow problem on a sngle-source network, smlar to the reducton presented n [9] for computng an equlbrum of symmetrc congeston games that mnmzes Rosenthal s potental. Then, we consder the graph havng a node for each server and an edge between two dfferent nodes and for each clent that has servers and as permssble servers. For each such edge correspondng to a clent, we defne the edge s optmal node to be the endpont correspondng to the server that clent uses n the optmal assgnment. We compute an orentaton of the edges so that the correspondng drected graph s acyclc. Then, ether ths drected graph or the one n whch all edges have opposte drectons have the followng property: at

5 most half of the edges pont to ther non-optmal node. We { select the orentaton that has ths property and assgn dfferent taxes from ˆδ, ˆδ, } n n..., n n ˆδ, ˆδ to the nodes/servers so that for any edge drected from to, t s δ > δ. Now, consder any pure Nash equlbrum of the extended game and let n denote the number of clents usng server. For a clent, denote by the server that clent uses n the pure Nash equlbrum and let be the other permssble server of clent = f the clent has only one permssble server). Snce no clent has an ncentve to change her strategy, t s n + γ δ n + + γ δ. Ths means that n n + + γ δ δ ) and n n + snce n and n are ntegers and γ δ δ ) < by the defnton of the tax functon. Ths nequalty holds for any clent and we conclude that any pure Nash equlbrum of the extended game s also a pure Nash equlbrum for the orgnal game. We now show that any pure Nash equlbrum for the extended game s a -PNE for the orgnal game,.e., a pure Nash equlbrum that satsfes the property n n o + o ). ) For each clent, we denote by and the servers she uses n the pure Nash equlbrum and n the optmal assgnment, respectvely. Denote by S the set of clents such that =. Denote by F the set of clents such that and δ > δ. Then, the condton n + γ δ n + + γ δ, mplyng that clent has no ncentve to use the server she uses n the optmal assgnment, mples that n n, snce n and n are ntegers and δ > δ. The condton n n + holds for any clent not belongng n S and F snce the pure Nash equlbrum for the extended game s also a pure Nash equlbrum for the orgnal game. By the defnton of the tax functon, we have that U \ F S) U / = o. By consderng the equlbrum condtons for all clents, we have n n = = : = n = n + n + S : = F : = n + n + S : = F : = n = n + U \ F S) = : = n o + o = n o + o ). U\F S) : = U\F S) n : = n + )

6 Ths completes the proof of nequalty ). In our analyss, we wll consder all -PNE for the orgnal graph balancng game, and we wll show that ther prce of anarchy s at most + 5. We wll need the followng techncal clam. Lemma. For any non-negatve ntegers x and y, 3 5 x y xy ) y x. Theorem. For any graph balancng game, the tax functon descrbed above s a pure-effcent tax. Proof. We wll show that the prce of anarchy of any -PNE of a graph balancng game s at most + 5. Agan, we denote by n and o the number of clents n server n the -PNE and n the optmal assgnment, respectvely. By nequalty ) and snce n = o, we have that the socal cost s n = = = 3 5 n o + o ) n o + o ) o ) n o ) o n 3 5 n ) 5 o n where the frst equalty follows snce n = o, the second and thrd equaltes are obvous, and the second nequalty follows by Lemma. We obtan that the prce of anarchy s n o o = + 5. Broadenng the class of load balancng games that admt better than -pureeffcent taxes or even pure-optmal taxes) s an nterestng open problem. 3 Non-refundable Taxes n Symmetrc Load Balancng We now proceed to answer n a negatve way a queston posed n [3] concernng non-refundable taxes n symmetrc load balancng games,.e., whether taxes n

7 can dmnsh the effect of selfshness. Our followng theorem suggests that taxes do not help n the case of symmetrc load balancng wth polynomal latency functons of degree p, snce for any tax functon, the prce of anarchy of the extended game n these games s p+) +/p p+) +/p p O p ln p ). Clearly, our result also demonstrates that the known upper bound on the prce of anarchy of such games wthout taxes) s tght. Theorem. For any p and any ɛ > 0, there exsts a symmetrc load balancng game wth polynomal latency functons of degree p that does not admt better than ρ ɛ)-pure-effcent non-refundable taxes where ρ = ) O. p ln p p+)+/p p+) +/p p Proof. Let k be an nteger and defne λ = kp+ k p k ) p+ k. We have λ = ) ) ) k p + k p k p+ k and, snce k and p, t s k p < λ < k p. ) /p Defne y λ = k p+. Snce p and λ < k p, t s y k. Consder a game wth k clents where each clent has γ =, and k + servers 0,,..., k. Server 0 has latency functon x p whle each of the other k servers has latency functon λx p. The assgnment n whch server 0 has k y clents, y among the other servers have exactly one clent and any other server s empty has cost opt = k y ) p+ + y λ. In the absence of taxes, the assgnment where all clents select server 0 s a pure Nash equlbrum snce each of them has a latency of k p and, n case a clent decdes to choose another server, she would face latency λ > k p. The cost of ths equlbrum s cost = k p+ and the prce of anarchy s P oa cost opt = k p+ k y ) p+ + y λ. Therefore, n order to avod ths assgnment as an equlbrum of the extended game, we have to assgn taxes n such a way that at least one clent has an ncentve to change her choce. So, wthout loss of generalty, we assume that there s a tax functon δ, for whch t holds that δ 0 w) = α and δ w) = 0, for any k. Note that, for any α λ k p, the aforementoned assgnment remans a pure Nash equlbrum of the extended game, snce any clent at server 0 would have a cost of k p +α λ. Now, assume that α = λ k p +ɛ for any ɛ > 0. Then, any clent would have a ncentve to leave server 0 and move to another server. Then, assumng that one clent moves, the total cost cost latency plus

8 taxes) of the resultng assgnment would be cost = k ) p+ + α k ) + λ > k ) p+ + λ k p ) k ) + λ = k ) p+ + λ k ) k p k ) + λ = λk + k ) p+ k p+ + k p = k p+ = cost. Applyng smlar reasonng, t s not hard to see that by ncreasng δ 0 w) = α even more so that more clents have an ncentve to leave server 0, the total cost smlarly ncreases. Therefore, the total cost s mnmzed by settng α = 0. λ y /p. Observe that lm k k = and lm p k k = p+) Hence, lm k k p+ k y ) p+ + y λ = lm k = p+ y k )p+ + y λ k p+ ) +/p + p + )+/p = p + ) +/p p = ρ. p+ ) /p Hence, for any ɛ > 0, by settng k to a suffcently large value, we obtan that the prce of anarchy becomes at least ρ ɛ. Mnmzng the Makespan In ths secton we focus on the makespan as the socal cost. We consder the wellknown case where servers are related,.e., server has latency functon α x. Our upper bound uses tax functons that assgn to each server a tax of ether 0 or. In ths settng, there s no dfference between refundable and non-refundable taxes snce no clent s assgned to a server where t has to pay an nfnte tax. Furthermore, the tax senstvty of each clent does not affect her behavor. Denote by n the number of clents and by m the number of servers. We assume that the servers are sorted n non-decreasng order of α.e., α α + ) and clents are sorted n non-ncreasng order of ther weght.e., w w + ). We defne the followng procedure that produces fractonal schedules of makespan w T. Observe that the quantty w s a lower bound on the makespan /α /α of any fractonal schedule; the numerator s the total weght of the clents and the denomnator s the capacty of all servers.

9 . set =, =, and t = 0;. whle n do 3. f T t α w then. put the remanng weght of clent at server ; 5. set t = t + α w and = + ; 6. else 7. put weght T t α of clent at server ; 8. set w = w T t α, = +, and t = 0; What the above procedure s dong s to consder each clent accordng to ther orderng) and put as much of her weght as possble to the server of smallest ndex so that the latency does not exceed T. Ths wll end up wth a fractonal schedule n whch there exsts a server such that the latency of all servers s exactly T, f < m) the latency of server + s at most T, and the latency of all servers > + f any) s 0. Each clent occupes consecutve servers and, furthermore, at most one clent may have non-zero weghts n two specfc consecutve servers. Gven a value of T, the schedule produced by the procedure above s called -feasble f for any clent and any two consecutve servers and +, t holds that α w + α +w + T, where w denotes the weght of clent assgned w to server. We start wth the value T = /α and run the procedure. If the schedule produced s -feasble, we stop. Otherwse, we ncrease T untl the schedule produced by the procedure s -feasble. Let T be the correspondng value of T.e., the mnmum value for whch the schedule produced s -feasble). w Clearly, f T >, there wll be at least one clent and two consecutve /α servers and + such that α w + α +w + = T. We now descrbe the tax functon. We partton the clents nto groups accordng to ther weght, so that two clents and belong to same group when w = w. We denote by wg the weght correspondng to group g. Let S g denote the set of servers that contan non-zero weghts of clents belongng to group g n the fractonal schedule of makespan T. If S g =, then we set δ wg) = 0 for the server S g and δ wg) = for any other server / S g. Otherwse, when S g >, we dstngush between two cases dependng on whether the last server of S g.e., the one wth the larger ndex) contans only clents of group g or also clents of dfferent groups. In the frst case, we set δ wg) = 0 for any server S g and δ wg) = for any other server / S g, whle n the second case, we set δ wg) = 0 for the S g servers of S g wth smallest ndex and δ wg) = for any other server. In any case, we denote wth g the set of servers for whch δ wg) = 0. We show the followng result. Theorem 3. For any symmetrc load balancng game on related servers, the above tax functon s -pure-effcent wth respect to the makespan.

10 Proof. Consder the -feasble fractonal schedule of makespan T produced as above. We frst show that the optmal assgnment has makespan at least T. Ths clearly holds f T = w. Otherwse, there wll be a clent and two /α consecutve servers and + such that α w +α +w + = T. Then, all clents wth smaller ndex than are fractonally scheduled at servers,..., whch have latency exactly T. In any ntegral schedule, ether all of the clents,..., wll be scheduled to servers,..., or some of them wll be scheduled at some server wth larger ndex than. In the frst case, the makespan wll be at least T snce the total weght of clents assgned to servers,..., does not decrease compared to the fractonal schedule. In the second case, a clent of weght at least w wll be assgned to a server wth α α + α. Ths server wll have latency at least α w α w + α +w + = T. Now, we show that there exsts an ntegral schedule wth makespan at most T n whch each clent n group g selects a server from the set g. The clents that have non-zero weght n the server of S g wth the smallest ndex are scheduled n ths server. Each other clent of group g for whch the server wth largest ndex contanng a non-zero amount of her weght n the fractonal schedule s s scheduled at server. In ths way, the total weght of any server may ncrease by at most the weght of clents n server + n the fractonal schedule. Snce α α +, the latency at server wll not exceed T. Observe that the tax functon essentally dvdes the orgnal game nto subgames n the followng sense. In any pure Nash equlbrum, the clents of group g wth g = are forced to use the server of g. The clents of group g wth g > play a symmetrc game wth lnear latency functons at server of the form α x + β. Here, β denotes the latency at server due to clents not belongng to group g whch are forced to use server. Furthermore, by the defnton of the tax functon, the sets g wth sze more than are dsont and, hence, the correspondng sets of clents do not nterfere. It s not hard to see that any equlbrum n each subgame of clents of group g has the mnmum possble ntegral makespan,.e., at most T. Ths completes the proof of the theorem. The next theorem states that ths tax functon s best possble. The proof s omtted; t wll appear n the fnal verson. Theorem. For any ɛ > 0, there exsts a load balancng game on m dentcal servers that does not admt better than ɛ)-pure-effcent taxes wth respect to the makespan. Unfortunately, taxes cannot sgnfcantly mprove the prce of anarchy wth respect to the makespan over mxed Nash equlbra. To show ths, we use a constructon that we have also used n [3] to lower-bound the effcency of taxes at mxed Nash equlbra wth respect to the total latency. The constructon apples to symmetrc load balancng games wth dentcal clents and dentcal servers and the proof follows by a standard balls-to-bns argument.

11 Consder a tax functon δ. Wthout loss of generalty, we assume that δ δ for <. Let k be equal to m f + m = δ m > δ m, otherwse k s equal to the largest nteger such that m + k = δ k δ k+. Let D = k = δ. Consder the followng assgnment y for all clents. Clent uses server wth probablty y = k + D km ) δ m f k and y = 0 otherwse. Notce that all clents have the same probablty dstrbuton. It can be verfed that y s a mxed Nash equlbrum of the extended game. In order to compute the expected makespan, t suffces to observe that t s the expectaton of the maxmum number of balls at any bn when m balls are thrown ndependently at m bns accordng to the probablty dstrbuton y. It log m log log m ) s well-known e.g., see []) that ths expectaton s mnmzed to Θ when y s the unform dstrbuton. Thus, we obtan the followng statement. Theorem 5. There exsts a symmetrc load balancng game on m servers that does not admt better than Ω ln m ln ln m) -mxed-effcent taxes wth respect to the makespan. Note that ths bound matches the prce of anarchy of symmetrc load balancng wth dentcal servers [7, ]. The prce of anarchy for related servers s slghtly hgher [7]. We leave as an open problem whether taxes can mprove the prce of anarchy wth respect to the makespan n ths partcular case and, more mportantly, n the more general case of congeston games. References. B. Awerbuch, Y. Azar, and A. Epsten. The prce of routng unsplttable flow. In Proceedngs of the 37th Annual ACM Symposum on Theory of Computng STOC 05), pp , I. Caraganns, M. Flammn, C. Kaklamans, P. Kanellopoulos, and L. Moscardell. Tght bounds for selfsh and greedy load balancng. In Proceedngs of the 33rd Internatonal Colloquum on Automata, Languages and Programmng ICALP 06), LNCS 05, Sprnger, Part I, pp. 3-3, I. Caraganns, C. Kaklamans, and P. Kanellopoulos. Taxes for lnear atomc congeston games. In Proceedngs of the th Annual European Symposum on Algorthms ESA 06), LNCS 68, Sprnger, pp. 8-95, G. Chrstodoulou and E. Koutsoupas. The prce of anarchy of fnte congeston games. In Proceedngs of the 37th Annual ACM Symposum on Theory of Computng STOC 05), pp , R. Cole, Y. Dods, and T. Roughgarden. Prcng network edges for heterogeneous selfsh users. In Proceedngs of the 35th Annual ACM Symposum on Theory of Computng STOC 03), pp , R. Cole, Y. Dods, and T. Roughgarden. How much can taxes help selfsh routng? Journal of Computer and System Scences, 73), pp. -67, A. Czuma and B. Vöckng. Tght bounds for worst-case equlbra. ACM Transactons on Algorthms, 3), 007.

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