Optimal Cost Sharing Protocols for Scheduling Games

Transcription

1 Optiml Cost Shring Protocols for Scheduling Gmes Philipp von Flkenhusen Berlin Institute of Technology Institute of Mthemtics Tobis Hrks Berlin Institute of Technology Institute of Mthemtics ABSTRACT We consider the problem of designing cost shring protocols to minimize the price of nrchy nd stbility for clss of scheduling gmes. Here, we re given set of plyers, ech ssocited with job of certin non-negtive weight. Any job fits on ny mchine, nd the cost of mchine is non-decresing function of the totl lod on the mchine. We ssume tht the privte cost of plyer is determined by cost shring protocol. We consider four nturl design restrictions for fesible protocols: stbility, budget blnce, seprbility, nd uniformity. While budget blnce is selfexplntory, the stbility requirement sks for the existence of pure-strtegy Nsh equilibri. Seprbility requires tht the resulting cost shres only depend on the set of plyers on mchine. Uniformity dditionlly requires tht the cost shres on mchine re instnce-independent, tht is, they reminthesmeevenifnewmchinesreddedtoorremoved from the instnce. We cll cost shring protocol bsic, if it stisfies only stbility nd budget blnce. Seprble nd uniform cost shring protocols dditionlly stisfy seprbility nd uniformity, respectively. For n-plyer gmes we show tht mong ll bsic nd seprble cost shring protocols, there is n optiml protocol with price of nrchy nd stbility of precisely H n = n i= /i. For uniform protocols we present strong lower bound showing tht the price of nrchy is unbounded. Moreover, we obtin severl results for specil cses in which either the cost functions re restricted, or the job sizes re restricted. As byproduct of our nlysis, we obtin complete chrcteriztion of outcomes tht cn be enforced s pure-strtegy Nsh equilibrium by bsic nd seprble cost shring protocols. Ctegories nd Subject Descriptors J.4 [Computer Applictions]: Socil nd Behviorl Sciences Economics; I.2.8 [Artificil Intelligence]: Problem Solving, Control Methods, nd Serch Scheduling Permission to mke digitl or hrd copies of ll or prt of this work for personl or clssroom use is grnted without fee provided tht copies re not mde or distributed for profit or commercil dvntge nd tht copies ber this notice nd the full cittion on the first pge. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission nd/or fee. EC, June 5 9, 20, Sn Jose, Cliforni, USA. Copyright 20 ACM //06...$0.00. Generl Terms Theory Keywords Cost shring, scheduling, congestion gmes. INTRODUCTION Congestion gmes ply fundmentl role for mny pplictions, including trffic networks, telecommuniction networks nd economics. In congestion gme, there is set of resources nd pure strtegy of plyer consists of subset of resources. The cost of resource depends only on the number of plyers choosing the resource, nd the privte cost of plyer is the sum of the costs of the chosen resources. Under these ssumptions, Rosenthl proved the existence of pure Nsh equilibrium (PNE for short) [20]. An importnt question in congestion gmes is the degree of suboptimlity cused by selfish resource lloction. Koutsoupis nd Ppdimitriou [7] introduced mesure to quntify the inefficiency of Nsh equilibri which they termed the price of nrchy. The price of nrchy is defined s the worst-cse rtio of the cost of Nsh equilibrium over the cost of system optimum. In the pst decde, considerble progress hs been mde in exctly quntifying the price of nrchy for mny interesting clsses of gmes. In the context of nontomic network routing gmes, the price of nrchy for specific clsses of cost functions is well understood, see Roughgrden nd Trdos [23], Roughgrden [2] nd Corre, Schulz, nd Stier-Moses []. (For n overview of these results, we refer to the book by Roughgrden [22].) Awerbuch et l. [4], Christodoulou nd Koutsoupis [9], Alnd et l. [2] nd Bhwlkr et l. [5] derived severl tight bounds on the price of nrchy for weighted nd unweighted congestion gmes with specific clsses of ltency functions. Despite these bounds, it is known tht the price of nrchy for generl ltency functions is unbounded even on simple prllel-rc networks [5, 23]. Motivted by the fct tht pure Nsh equilibri my be very inefficient even in prllel-rc networks, we focus in this pper on the design of cost shring methods s mens to leverge the resulting price of nrchy. The concrete scenrio tht we consider is the problem of scheduling jobs on prllel mchines. We re given set of plyers, ech ssocited with job of certin non-negtive weight. Any job fits on ny mchine, nd the cost of mchine is nondecresing function of the totl lod on the mchine. We ssume tht the privte cost of plyer is determined by 285

2 cost shring method. For instnce, simple cost shring method tht hs been nlyzed in the forementioned literture is verge cost shring, see [5,, 23]. In lmost ll settings in the theory nd prctice of mechnism or protocol design, designer my only choose protocols out of set of fesible protocols. Therefore, we hve to precisely define the design spce of fesible protocols. To this end, we define the following four properties listed below which re defined more formlly in Section 2. These properties hve been introduced first by Chen et l. [8] in the context of the design of cost shring protocols for network design gmes ( forml definition of these gmes will be given in Section.2).. Stbility: There is t lest one pure strtegy Nsh equilibrium in ech scheduling gme induced by the cost shring protocol. 2. Budget-blnce: For every outcome of scheduling gme induced by the cost shring protocol, the cost of ech resource is exctly covered by the collected cost shres of the plyers using the resource. 3. Seprbility: In ech scheduling gme induced by the cost shring protocol, the cost shres of ech resource re completely determined by the set of plyers tht use it. 4. Uniformity: Across ll scheduling gmes induced by the cost shring protocol, the cost shres of resource (for ech potentil set of users) depend only on the resource cost, nd not on the set of vilble resources. A cost shring method is clled bsic if it stisfies ()-(2), seprble if it stisfies ()-(3), nd uniform if it stisfies ()- (4). We briefly discuss the bove four properties nd refer to [8] for more detiled tretment. The condition (2) is the lest controversil in the context of cost shring protocols. The stbility condition () requires the existence of t lest one Nsh equilibrium in pure strtegies. While this requirement restricts the serch spce for cost shring protocols, it is certinly the solution concept of choice when mixed or correlted strtegies hve no meningful physicl interprettion in the gme plyed; see lso the discussion in Osborne nd Rubinstein [9, 3.2] bout critics of mixed Nsh equilibri. While condition (3) seems restrictive, it is crucil for prcticl pplictions in which cost shring methods hve only locl informtion bout their own resource usge (see for instnce the TCP/IP protocol design, where routers drop pckets bsed on some function of the number of pckets in the queue, see [24]). Uniformity (4) is the strongest nd perhps the most problemtic design restriction. A uniform protocol is not only seprble but lso strongly locl in the sense tht the cost shres of resource re independent of the set of resources vilble to the gme designer. This property my be crucil for systems in which the resources cn be dded or removed over time nd reconfigurtion of the system (chnging the cost shring protocol) is too costly. The gol of this pper is twofold. On the one hnd side, we wnt to systemticlly nlyze the chievble worst-cse efficiency of Nsh equilibri by bsic, seprble nd uniform cost shring protocols in the context of scheduling gmes. Besides this worst-cse perspective, we lso sk lrger question: Which outcomes cn ctully be enforced s pure Nsh equilibri? More precisely, we cll n outcome of scheduling gme wekly-enforceble if there is bsic protocol tht induces the outcome s pure Nsh equilibrium. We cll n outcome of scheduling gme strongly-enforceble if there is bsic protocol tht induces the outcome s the most expensive pure Nsh equilibrium.. Our Results We study protocol design problems in the context of scheduling gmes, where the gol is to minimize the induced price of nrchy nd the price of stbility. Our results for these problems cn be summrized s follows. Among ll bsic nd seprble protocols, we provide n optiml protocol minimizing the resulting price of nrchy nd price of stbility simultneously. For n-plyer scheduling gmes, the optiml vlue of the price of nrchy nd stbility is precisely the n-th hrmonic number H n = n i= /i. Moreover, we obtin complete chrcteriztion of weklyenforceble outcomes. This chrcteriztion cn be used for designing cost shring protocols minimizing the price of stbility with respect to n rbitrry objective function. We lso derive sufficient conditions for n outcome to be strongly-enforceble. Our proof of this result is constructive by providing cost shring protocol tht strongly enforces n outcome stisfying the sufficient conditions. We then show tht this protocol gives rise to n optiml cost shring protocol minimizing the price of nrchy nd stbility s mentioned bove. For scheduling gmes with cost functions tht hve non-decresing per-unit costs, we derive n optiml cost shring protocol with price of nrchy equl to. We remrk tht this ssumption is quite wek insofr s nondecresing nd convex cost functions stisfy non-decresing per-unit costs. We lso study the chievble price of nrchy of uniform cost shring protocols. We show tht there is no uniform cost shring protocol with bounded price of nrchy. This bound even holds for fmily of instnces with only 3 plyers, t most 3 mchines nd cost functions with nondecresing costs per unit. Only for instnces in which the demnds re integer multiples of ech other, we present cost shring protocol with bounded price of nrchy of n..2 Relted Work There is lrge body of work on scheduling gmes (or singleton congestion gmes) with unweighted nd weighted plyers [, 2, 3, 4, 5, 8]. Most of these ppers study the existence nd price of nrchy of pure Nsh equilibri for the uniform cost shring protocol in which the privte cost of every plyer is equl to the cost on the resource. These works, however, do not consider the design perspective of cost shring protocols. Christodoulou et l. [0] nd followup ppers such s [6, 6] study coordintion mechnisms nd their price of nrchy in scheduling gmes in which n plyers ssign tsk to one of m mchines. Rther thn pying shre of the resulting cost of mchine s in our scenrio, the plyers in these gmes consider the completion time of their respective job s privte cost. This completion time depends on the sequence in which the jobs on mchine re processed which in turn is given by the coordintion mechnism. The notion of privte cost in these ppers estblishes n entirely different set of Nsh equilibri compred to our work nd hence their results concerning the price of nrchy re unrelted to ours. Our work is motivted by the pper by Chen et l. [8]. 286

3 2. MODEL AND PROBLEM STATEMENT A scheduling model is represented by tuple (N,M,d,c). In this pper, the uthors study the design of cost shring protocols for network design gmes, see lso Anshelevich et l. [3] nd Chen nd Roughgrden [7] for erlier work on network design gmes. In network design gme, ech plyer i hs unit demnd tht she wishes to send long pth in (directed or undirected) network connecting her source node s i to her terminl node t i. Every edge hs constnt non-negtive cost nd the problem is to design seprble or uniform cost shring protocol so s to minimize the price of nrchy nd stbility in this setting. Our pproch follows their led in terms of the fesible protocol spce, but we pply cost shring protocols to the structurl different clss of scheduling models. On the one hnd, such scheduling models re more generl in the sense tht we llow rbitrry non-decresing cost functions insted of constnt costs on the resources. Moreover, in contrst to [8], we llow plyers to hve different non-negtive weights. On the other hnd, scheduling models re more restricted in the sense tht we consider reltively simple strtegy spce for the plyers, tht is, pure strtegy for plyer is simply single resource. Moreover, in contrst to [8], our gmes re symmetric, tht is, every plyer hs ccess to every resource. These structurl differences result in different pproches nd lso the results of [8] re different to ours. For exmple, while Chen et l. [8] proved bounds on the price of nrchy for uniform protocols of order Θ(log(n)), Θ(polylog(n)), nd n for undirected single-sink instnces, undirected multicommodity instnces, nd directed single-sink instnces, respectively, we show tht for scheduling gmes such results re impossible. The price of nrchy for uniform protocols inducing scheduling gmes is unbounded. Finlly, it is worth noting tht, while [8] nlyzed seprble nd uniform protocols, we dditionlly nlyze the lrger clss of bsic protocols. Here, N = {,...,n} is nonempty set of plyers nd M = {,..., m} is nonempty set of mchines. Every plyer is ssocited with tsk of weight d i nd d = (d,...,d n) is the combined weight vector. Every mchine M hs n ssocited non-negtive nd non-decresing cost function c : R + R +. We ssume c (0) = 0 for ll M. The vector of cost functions is denoted by c =(c,...,c m). Given scheduling model (N,M,d,c), we ssocite strtegic gme represented by the tuple (N,X,ξ). Here, it is ssumed tht every tsk fits on every mchine, thus, the set of pure strtegies for plyer i N is X i = M nd the overll strtegy spce is X = M n. The outcomes x =(x,...,x n) M n re vectors of mchines where the strtegy plyed by plyer i is mchine x i. The privte cost of plyer i N in such n outcome x is determined by the cost shring method ξ i : X R +. A cost shring protocol Ξ : (N,M,d,c) ξ provides every scheduling model with vector ξ =(ξ i) i N of such cost shring methods. For x M n, the set of plyers using some mchine M is denoted by S (x) :={i N : x i = } nd for M, the lod on mchine is defined s l (x) := di. The cost of n outcome is defined s C(x) := M c(l(x)). Abusing nottion, we will often write c (x) insted of c (l (x)). We consider cost minimiztion gmes, thus, when choosing her strtegy, ech plyer strives to minimize her resulting privte cost ξ i(x). We sy tht the gme (N,X,ξ) on scheduling model (N,M,d,c) is induced by the protocol Ξ. An importnt solution concept in non-coopertive gme theory for the nlysis of strtegic gmes re pure Nsh equilibri. Using stndrd nottion in gme theory, for n outcome x M n we denote by (, x i) :=(x,...,x i,,x i+,...,x n) M n the outcome tht rises if only plyer i devites to strtegy. Definition 2.. (Pure Nsh Equilibrium) Let (N,X,ξ) be scheduling gme. The outcome x is pure Nsh equilibrium if no plyer i cn strictly reduce her privte cost by unilterlly moving to different mchine, tht is, for ll i N ξ i(x) ξ i(, x i) for ll M. (2.) Two well estblished concepts tht quntify the efficiency of Nsh equilibri re the price of nrchy nd the price of stbility. The price of nrchy mesures the lrgest possible rtio of the cost of Nsh equilibrium nd the cost of n optiml outcome. The price of stbility mesures the smllest rtio of the cost of Nsh equilibrium nd the cost of n optiml outcome. For cost shring protocol Ξ, we define by PoA(Ξ) nd PoS(Ξ) the corresponding worst cse price of nrchy nd price of stbility cross gmes induced by protocol Ξ. The min gol of this pper is to design cost shring protocols tht minimize the price of nrchy nd price of stbility, respectively. Of course, the ttinble objective vlues crucilly depend on the design spce tht we permit. The following properties hve been first proposed by Chen et l. [8] in the context of designing cost shring methods for network design gmes. Definition 2.2. (Properties of cost shring protocols) A cost shring protocol Ξ is. stble if it induces only gmes tht dmit t lest one pure Nsh equilibrium. 2. bsic if it is stble nd dditionlly budget blnced, i.e. if it ssigns ll scheduling models (N,M,d,c) with cost shring methods (ξ i) i N such tht c (x) = ξ i(x) for ll M,x M n. (2.2) This property requires c (0) = 0 for unused mchines, which we will ssume in the pper. 3. seprble if it is bsic nd if it induces only gmes (N,M n,ξ)forwhichinnytwooutcomesx, x M n S (x) =S (x ) ξ i(x) =ξ i(x ) i S (x), M. 4. uniform if it is seprble nd if it ssigns ny two models (N,M,d,c), (N,M,d,c ) with cost shring methods (ξ i) i N nd (ξ i) i N such tht the following condition holds. For ll M M with c = c nd ll outcomes x M n, x M n S (x) =S (x ) ξ i(x) =ξ i(x ) for ll i S (x). Informlly, seprbility mens tht in n outcome x the vlue ξ i(x) depends only on the set S xi (x) of plyers shring mchine x i nd disregrds ll other informtion contined in 287

4 x. Still, seprble protocols cn ssign cost shring methods tht re specificlly tilored to the given scheduling model, for exmple bsed on n optiml outcome. Uniform protocols re not llowed to do this, they even disregrd the lyout of the model nd ssign the sme cost shring methods when mchines re dded to or removed from the model. We denote by B n, S n nd U n the set of bsic, seprble nd uniform cost shring protocols for scheduling gmes with n plyers, respectively. We obtin the following optimiztion problems tht we ddress in this pper. min Ξ B n PoA(Ξ), min Ξ B n PoS(Ξ), min Ξ S n PoA(Ξ), min Ξ S n PoS(Ξ), min PoA(Ξ), min PoS(Ξ). Ξ U n Ξ U n 3. BASIC AND SEPARABLE PROTOCOLS We strt with studying bsic nd seprble cost shring protocols. While our gol is to find cost shring protocol minimizing the induced PoA nd PoS, we first study the issue of enforcebility of pure Nsh equilibri by bsic nd seprble cost shring protocols. To be more precise, given scheduling model (N,M,d,c), we first sk which outcomes x M n cn be enforced s pure Nsh equilibri by some bsic or seprble cost shring protocol. We will differentite between wekly enforceble outcomes nd strongly enforceble outcomes, see the definition below. Definition 3.. (Enforceble outcomes) Consider scheduling model (N,M,d,c) nd n outcome x M n. i) x is wekly-enforceble if there exists bsic cost shring protocol Ξ such tht x is Nsh equilibrium in the gme (N,M n,ξ) induced by Ξ. ii) x is seprble wekly-enforceble if there exists seprble cost shring protocol Ξ such tht x is Nsh equilibrium in the gme (N,M n,ξ) induced by Ξ. iii) x is strongly-enforceble if there exists seprble cost shring protocol Ξ such tht x is the most expensive Nsh equilibrium in the gme (N,M n,ξ) induced by Ξ, i.e. C(x ) C(x) for ll Nsh equilibri x M n. In the following section, we will give n exct chrcteriztion of wekly-enforceble nd seprble wekly-enforceble outcomes. This chrcteriztion provides structurl property tht cn be used to design cost shring protocols for minimizing the price of stbility for rbitrry objective functions. Throughout this section, the plyers re ssumed to be ordered by non-decresing weights: d d 2 d n. 3. Wekly-Enforceble Outcomes This section provides n exct chrcteriztion of weklyenforceble outcomes. Our chrcteriztion relies on the notion of dechrged outcomes defined below. Definition 3.2. (Wekly dechrged outcome) Consider scheduling model (N,M,d,c). A mchine M is wekly dechrged in n outcome x M n if c (x) min c b(b, x i). (3.) The outcome x itself is clled wekly dechrged if ll mchines re wekly dechrged. We further introduce the wek x-enforcing protocol. Definition 3.3. (Wek x-enforcing protocol) The wek x- enforcing protocol tkes s input wekly dechrged outcome x. We use x to define for ny outcome z nd mchine the sets S(z) 0 :={i S (z) S (x)} (home plyers on ) nds(z) :={i S (z)\s (x)} (foreign plyers on ). Then, the wek x-enforcing protocol ssigns for ll i N,z M n the following cost shring methods min c b(b, x i) min c b(b, x j) cx (x), i j S xi (x) if S zi (z) =S zi (x) ndc xi (x) > 0, ξ i(z) := c zi (z), if Sz i (z) nd i =minsz i (z), c zi (z), if S z i (z) =, S zi (z) S zi (x) nd i =mins zi (z), 0, else. Informlly, if S (z) =S (x), the plyers on mchine shre the cost proportionl to their opportunity cost (cost of chnge) in outcome x. Otherwise, the smllest foreign plyer (deviting from outcome x) or, if there re none, the smllest home plyer (not deviting) pys the entire cost of the mchine. Observe tht in wekly dechrged outcomes x we hve j S min (x) c b (b, x j) > 0 for ll M with c (x) > 0 nd thus the protocol is well defined. We re now redy to stte our first min result. Theorem 3.4. For ny scheduling model (N,M,d,c) nd outcome x, the following sttements re equivlent. (i) the outcome x is wekly dechrged, (ii) the outcome x is wekly-enforceble, (iii) the outcome x is seprble wekly-enforceble. Observe tht (iii) (ii) holds becuse by definition seprble protocols re subclss of bsic protocols. We prove (i) (iii) nd (ii) (i) by two lemms. Lemm 3.5. For every wekly dechrged outcome x, the wek x-enforcing protocol is seprble cost shring protocol nd wekly enforces x. Proof. Budget blnce nd seprbility of the cost shring methods re cler from the definition of the protocol, thus, we prove only tht x is Nsh equilibrium. For ll mchines M with c (x) > 0 we re in the first cse of the definition of the protocol, thus, we obtin ξ i(x) = min c b(b, x i) min c b(b, x c(x) j) j S (x) min c b(b, x i) min ξi(b, x i) for ll i S(x), \{} where the first inequlity holds becuse outcome x is wekly dechrged. For ll other mchines M, wehveξ i(x) = c (x) = 0 for ll i S (x) nd thus x is pure Nsh equilibrium. Lemm 3.6. Consider the scheduling model (N,M,d,c). Then, ny wekly-enforceble outcome x is wekly dechrged. 288

5 Proof. Sy x is Nsh equilibrium under the bsic protocol Ξ tht ssigns cost shring methods ξ. Then ξ i(x) min ξ i(b, x i) for ll i N nd hence due to budget blnce of Ξ, c (x) = ξ i(x) min ξi(b, x i) min c b(b, x i) for ll mchines M. Thus,x is wekly dechrged. The bove chrcteriztion hs direct consequence for the design of cost shring protocols so s to minimize the price of stbility with respect to n rbitrry objective function over the strtegy spce. As formlized below, by Theorem 3.4 this problem reduces to solving well-structured finite-dimensionl optimiztion problem. Corollry 3.7. Let (N,M,d,c) be scheduling model nd let F : M n : R be socil welfre function. Then, min ξ Bn PoS(ξ; F ) nd min ξ Sn PoS(ξ; F ) cn be reduced to solving the optimiztion problem min F (x) s.t. c(x) x M n min c b(b, x i) M. 3.2 Strongly-Enforceble Outcomes In this section, we turn to strongly-enforceble outcomes. We present slightly extended protocol tht we term the the strong x-enforcing protocol. We will show tht outcomes tht re strongly dechrged ( definition will follow shortly) re strongly-enforceble by this protocol. Definition 3.8. (Strongly Dechrged Outcome) Consider scheduling model (N,M,d,c). A mchine M is strongly dechrged if it is wekly dechrged nd dditionlly c (x) < min c b(b, x i), if S (x) > ndc (x) > 0. (3.2) Mchines tht re not strongly dechrged re clled chrged. The outcome x is clled strongly dechrged if ll mchines re strongly dechrged. We now introduce the strong x-enforcing protocol. Definition 3.9. (Strong x-enforcing protocol) The strong x-enforcing protocol tkes s input strongly dechrged outcome x. As before, we use x to define for ny outcome z nd mchine the sets S(z) 0 nds(z). Additionlly, we define the set S(z) 2 :={i S (z)\s (x) :c xi (x) =0} tht we term strong foreign plyers on. Then, the strong x- enforcing protocol ssigns for ll i N,z M n, the following cost shring methods: min c b(b, x i) min c b(b, x j) cx (x), i j S xi (x) if S zi (z) =S zi (x) ndc xi (x) > 0, ξ i(z) := c zi (z), if Sz 2 i (z) nd i =minsz 2 i (z), c zi (z), if S 2 z i (z) =, S z i (z), i =mins z i (z), c zi (z), if Sz i (z) =, S zi (z) S zi (x),i=mins zi (z) 0, else. The protocol works lmost the sme s the wek x-enforcing protocol, it only ccounts differently for strong foreign plyers. Theorem 3.0. If n outcome x is strongly dechrged, then the strong x-enforcing protocol is seprble nd strongly enforces x. Proof. Seprbility follows from the definition of the strong x-enforcing protocol. We only show tht for ny Nsh equilibrium z x we hve C(z) C(x). To this end, fix such z nd let i := min {j N : z j x j} (3.3) be the smllest plyer who devites from x. First, note tht for ll j>i, { ξ j(z i,z j) =0,ifc xj (x) > 0 ξ j(z) (3.4) ξ j(x j,z j) =0,ifc xj (x) =0, becuse z is Nsh equilibrium. Hence, c (z) = 0 for ll z i with foreign plyers S (z). (3.5) Also, c (z) c (x) for ll mchines z i tht only hve home plyers S 0 (z) = S (z), becuse for these mchines l (z) l (x). Thus, we lredy hve c (z) c (x) for ll mchines z i. (3.6) If there is strong foreign plyer on z i,thenevenc zi (z) = 0 nd we re done. Thus, from now on we ssume tht there re no strong foreign plyers on z i. We cn bound c zi (z) from bove using the Nsh inequlity c zi (z) =ξ i(z) ξ i(x i,z i). The remining proof focuses on bounding the vlue ξ i(x i,z i) frombove. The vlue of ξ i(x i,z i) ssigned by the x-enforcing protocol depends on S xi (x i,z i) ndc xi (x), for which there re three possibilities, ccording to the definition of the strong x-enforcing protocol.these cses re. S xi (x i,z i) =S xi (x) ndc xi (x) > 0, where the protocol returns ξ i(x i,z i) =ξ i(x). 2. S xi (x i,z i) S xi (x) ndi =mins xi (x i,z i), where the protocol returns ξ i(x i,z i) =c xi (x i,z i). 3. All cses in which the protocol returns ξ i(x i,z i) =0. In ech cse we will find c xi (z) +c zi (z) c xi (x) +c zi (x) nd thus with (3.6) we hve C(z) C(x), which proves the Theorem. Note tht (3.6) lredy implies c xi (z) c xi (x). We begin with Cse. The condition c xi (x) > 0 implies tht if there is some strong foreign plyer j>i(with z j x j nd c xj (x) =0),thenc zi (z) =ξ i(z) ξ i(z j,z i) =0nd we re done. Thus, we will in the following ssume tht there re no strong foreign plyers t ll. If ξ i(x) =0,we obtin 0 = ξ i(x) =ξ i(x i,z i) ξ i(z) =c zi (z), becuse we re in Cse. Thus, we will lso ssume ξ i(x) > 0. (3.7) We now compre the lloction of lod in the outcomes z nd x, respectively. First, we consider mchines z i, 289

6 which host foreign plyers j S (z)\s (x). For these foreign plyers we obtin min c b(b, x j) min c b(b, x i) ξ i(x) (3.8) (3.8b) > 0. (3.8c) Observe tht (3.3) implies j>ind hence (weights re ordered non-decresingly) d j d i. As the cost functions re non-decresing, the first inequlity (3.8) follows. Inequlity (3.8b) holds since x is dechrged. The lst inequlity (3.8c) follows from (3.7). We conclude for mchine c (, x j) ξ j(, x j) ξ j(x) (3.9) c xj (x) = min c b(b, x k ) min c b(b, x j) (3.0) k S xj (x) > 0=c (z), (3.) where (3.9) holds becuse x is Nsh equilibrium nd (3.0) stems from the definition of the protocol becuse there re no strong foreign plyers nd hence c xj (x) > 0. The inequlity (3.) holds becuse of (3.8) nd the equlity holds becuse of (3.5). Hence, there must be non-empty set of plyers S (x)\s (z). These plyers cnnot be strong foreign plyers, thus c (x) > 0. With c (z) =0ndc (x) > 0 we hve l (x) >l (z) for ll mchines z i with foreign plyers. For ll mchines without foreign plyers we know l (x) l (z) nd for mchine x i even l xi (x) =l xi (z)+d i becuse we re in Cse. Since the totl lod is the sme in x nd z, wehveformchinez i Consequently, l zi (z i,x i) =l zi (x)+d i l zi (z). (3.2) ξ i(z) =c zi (z) c zi (z i,x i) (3.3) c xi (x) min c b(b, x min c b(b, x i) (3.4) j) j S xi (x) = ξ i(x) =ξ i(x i,z i), (3.5) where the first inequlity (3.3) holds becuse of (3.2) nd the second inequlity (3.4) becuse x is dechrged nd c zi (z i,x i) min c b (b, x i). Equlity (3.5) holds by the definition of the strong x-enforcing protocol for Cse nd the lst eqution holds becuse we ssume Cse. If S xi (x) >, then inequlity (3.4) is strict, becuse x is strongly dechrged (i.e., (3.2) holds) which implies ξ i(z) > ξ i(x i,z i). This contrdicts the fct tht z is Nsh equilibrium. Thus, S xi (x) ={i} nd c zi (z) =ξ i(z) ξ i(x i,z i) = c xi (x i,z i) =c xi (x). Moreover, using c xi (z) =0,becuse l xi (z) = 0, we obtin c xi (z) +c zi (z) c xi (x) +c zi (x) s desired. Cse 2 is S xi (x i,z i) S xi (x) ndi =mins xi (x i,z i). Here, we obtin c zi (z) =ξ i(z) ξ i(x i,z i) =c xi (x i,z i) c xi (x), where the first inequlity holds becuse z is Nsh equilibrium. The second inequlity holds becuse Cse 2 implies l xi (x i,z j) l xi (x). We lso get c xi (z) = ξ j(z) ξ j(z i,z j) =0. j S xi (z) j S xi (z) This inequlity is result of (3.4), becuse in this cse ll plyers j S xi (z) hvehigherindexj>i. Consequently, we hve gin c xi (z)+c zi (z) c xi (x)+c zi (x). Finlly, we exmine Cse 3 where the protocol returns ξ i(x i,z i) = 0 nd thus for the Nsh equilibrium z we hve c zi (z) =ξ i(z) ξ i(x i,z i) = 0. Agin, c xi (z) +c zi (z) c xi (x)+c zi (x). 3.3 An Optiml Protocol Using the insights gined in the previous sections, we show tht mong ll bsic nd seprble protocols, the strong x-enforcing protocol gives rise to n optiml protocol simultneously minimizing the price of nrchy nd stbility. Our min result involves the n-th hrmonic number H n = n i= i. Theorem 3.. min PoA(Ξ) = min PoS(Ξ) = min PoA(Ξ) Ξ B n Ξ B n Ξ S n = min PoS(Ξ) = H n. Ξ S n We will prove the theorem by two subsequent lemms. In the first lemm, we prove tht H n is lower bound on the price of stbility for every bsic cost shring protocol. We then continue by presenting n lgorithm tht returns for ny scheduling model strongly dechrged outcome of cost t most H n times the cost of n optiml outcome. Together with the strong x-enforcing protocol we conclude tht the price of nrchy of the thus defined protocol is precisely H n. Lemm 3.2. For bsic cost shring protocols on scheduling models with n plyers nd non-decresing cost functions, the price of stbility is t lest H n. This lower bounds holds even for models with unit demnds. Proof. Consider the scheduling model (N,M,d,c) with n plyers tht hve unit demnd d i =forlli N nd n mchines with cost functions s in Tble. Tble : Cost functions for mchines used in the proof of Lemm 3.2 l c (l) c 2 (l)... c i (l)... c n (l) ɛ i > +ɛ n... n... n for some smll ɛ>0. The only optiml outcome is clerly y =(,..., )with C(y) =+ɛ. Anoutcomez cn only be Nsh equilibrium if it is wekly dechrged (Lemm 3.6). We show tht the chepest wekly dechrged outcomes re those in which ech mchine is used by exctly one plyer, which ll hve the sme cost s x =(,..., n). It is esy to see tht outcome x is dechrged nd with C(x) = n i= = Hn this proves i the lemm. If in n outcome z some mchine other thn is used by multiple plyers, then C(z) n, thus such outcomes re more expensive thn x. If in outcome z multiple plyers use mchine,syk plyers, then there re t lest k unused mchines nd for the chepest of these, sy mchine â, wehvecâ().thus,zis not wekly dechrged s k c (z) =+ɛ>= k min c b(b, z i). i S (z) i S (z) 290

7 Algorithm Find strongly dechrged outcome x : k {stepnumber} 2: x y {strts with optiml outcome y} 3: t i 0 for ll i N {stores when plyer ws lst moved} 4: while there re chrged mchines do 5: k rgmx {c (x k ): M is chrged} {select the most expensive chrged mchine} 6: if min {c b (b, x k i) :b M} =0 fori =mins k(x k ) then 7: {i cn move to cost-free mchine, clled Zero-move} 8: i k min S k(x k ) {select smllest plyer} 9: else if mx {t i : i S k(x k )} > 0 then 0: { plyeron k ws moved before, clled Shuffle} : i k rgmx {t i : i S k(x k )}{select lst moved plyer} 2: else 3: {no foreign plyers on k, clled Kick-off } 4: i k min S k(x k ) {select smllest plyer} 5: end if 6: b k rgmin {c b (b, x k i ):b M} {select chepest k vilble mchine} 7: 8: x k+ (b k,x k i ) {move plyer} k t i k k {store stepnumber} 9: k k +{iterte} 20: end while 2: return x x k Altogether, only such outcomes in which ll mchines re used by exctly one plyer re chep wekly-enforceble outcomes. While the previous Lemm showed tht there sometimes re no wekly-enforceble outcomes cheper thn H n times the cost of n optiml outcome, the following lemm shows tht we lwys find strongly dechrged outcomes of t most H n times the cost of n optiml outcome. Lemm 3.3. Any scheduling model (N,M,d,c) with n optiml outcome y hs strongly dechrged outcome x with C(x) H n C(y) = n k= C(y). k Proof. The desired outcome x is found by Algorithm. The lgorithm tkes s input n optiml outcome y. Inech cycle k of the lgorithm s min loop (lines 4-20), plyer i k on the most expensive chrged mchine k is selected (line 5) nd moved to the chepest vilble mchine b k (lines 6, 7). If possible, the lgorithm selects plyer who cn be moved to cost-free mchine, this is clled Zero-move (line 6). Otherwise, it selects plyer tht hs been moved before in lst-in/first-out scheme which is mintined through the vribles t i tht store the cycle in which ech plyer ws lst moved. Such moves re clled Shuffles (line 9). If neither Zero-move nor Shuffle is possible, the smllest plyer on the mchine is selected, which is clled Kick-off (line 2). The lgorithm termintes when no chrged mchines re left. First, we show tht the lgorithm termintes. To this end, observe tht Shuffles re only performed when Zero-moves re not possible. Hence, if in cycle k Shuffle is performed, the following inequlities hold. min c b(b, x k j) > 0 for ll j S k(x k ). (3.6) We now consider two cses. For S k(x k ) =,weobtin c k(x k ) > min b(b, x k ik) (3.7) = c b k(b k,x k i k)=c b k(xk+ ), (3.8) where (3.7) follows becuse k is chrged in x k. Equlity (3.8) follows since Algorithm moves i k to the chepest vilble mchine. If S k(x k ) >, then we obtin c k(x k ) j S k (x k ) min c b(b, x k j) (3.9) > min c b(b, x k ik) (3.20) = c b k(b k,x k i k)=c b k(xk+ ), where (3.9) is vlid becuse k is chrged in x k. The second inequlity (3.20) holds becuse of (3.6) nd the equlities follow s bove. In both cses, Shuffle moves the plyer to strictly cheper mchine. To see tht the lgorithm termintes, we will now follow some plyer i over the course of the lgorithm. Ech Zero-move nd ech Shuffle tke her to strictly cheper mchine. If the plyer is moved in cycle k ndisnextmovedbyshuffleincycle l, the cost of her mchine x k+ i = x l i my increse in the mentime s other plyers rrive on tht mchine. The lgorithm ssures by its lst-in/first-out mechnism tht these other plyers hve been moved gin before the Shuffle in cycle l nd consequently the cost hs decresed to the originl level c x k+(x k+ ) c x l (x l ). Since only mchines with i i positive costs cn be chrged, this implies tht fter Zeromove, the plyer will never gin be considered for Shuffles. Hence, plyer cn be moved by t most one Kick-off, fterwrds sequence of Shuffles nd therefter only Zero-moves. The sequence of Shuffles is finite becuse ech Shuffle tkes the plyer to strictly cheper mchine. Once the plyer hs hs been moved by Zero-move, further Zero-moves re only possible if in between some other plyer rrives on the plyer s mchine vi Kick-off or Shuffle, but gin this is only finitely often possible. Altogether, ech plyer cn only be moved finitely often nd thus the lgorithm termintes fter finite number of cycles. To complete the proof, we show tht the finl outcome x hs cost C(x) H n C(y). The concept of this finl prt of the proof is tht in outcome x the cost of every used mchine is determined by the plyer who hs lst moved there or, if there re no such plyers, the home plyers. For this, some new nottion is needed. Let p i,i N, correspond to the position (by index) of plyer i on her optiml mchine y i, i.e., on ny mchine we hve p j =forplyerj =mxs (y), p j =2forj =mx(s (y)\{j}) nd so on. Consequently, when some plyer i performs her Kick-off in cycle k, there re p i plyers shring her mchine k = y i t tht moment nd she is the smllest of them. We obtin for mchine b k 29

8 tht she is moved to c b k(x k+ )=c b k(b k,x k i) =min c b(b, x k i) min p c b(b, x k j) (3.2) i j S k (x k ) p i c k(x k ) (3.22) p i c yi (y), (3.23) where the first inequlity (3.2) is vlid becuse i is the smllest of the p i plyers on mchine k in step k, thesecond inequlity (3.22) holds becuse k is chrged in x k nd the lst inequlity (3.23) holds becuse there re no foreign plyers on k = y i nd hence l k(x k ) l k(y) =l yi (y). Since Shuffles nd Zero-moves ssign plyer i to cheper mchines, fter her lst move in cycle k, she is on mchine b k t cost c b k (xk ) p i c yi (y). Altogether, in the finl outcome x, the cost of mchine M to which plyers hve been moved is determined by the lst plyer who ws moved there, tht is, i := rgmx {t i : i S (x)}. Wethus obtin c (x) p i c yi (y). For mchines M tht re used in x but where no plyer hs been moved, the plyer i := mx S (y) withp i = is still on mchine. In this cse, the cost is bounded from bove by c (x) c (y) = p i c yi (y). Unused mchines M hve cost c(x) =0. Altogether, we obtin c (x) = p i c yi (y) for ll M with l (x) > 0, nd c (x) = 0 for ll M with l (x) =0. This yields the desired bound for the cost of outcome x, becuse now every used mchine M hs unique plyer i tht determines the mchine s cost. We obtin C(x) = c (x) M M l (x)>0 p i c yi (y) i N p i c yi (y) M H pmx c (y) =H pmx C(y) H n C(y), where p mx := mx { S (y) : M}. Observe tht the bound for the price of nrchy obtined here cn be much lower thn H n for scheduling models tht hve optiml outcomes, where the plyers re scttered over the mchines nd where therefore p mx is smller thn n. Remrk 3.4. While Lemm 3.3 shows tht n optiml outcome cn be turned into strongly dechrged outcome of cost t most H n times the cost of n optiml outcome, this holds true more generlly: Algorithm turns every outcome into strongly dechrged outcome with cost increse of fctor t most H n. This my be useful if the computtion of n optiml outcome is not possible in polynomil time. Still, Algorithm does not run in polynomil time nd this issue deserves further ttention. 3.4 Non-Decresing Cost per Unit In this section we require tht the cost functions re nonnegtive, non-decresing nd the per-unit costs c(x) re nondecresing with respect to the lod l(x). Such functions re l(x) still quite rich s they contin non-negtive, non-decresing nd convex functions. We introduce the opt-enforcing protocol for which we prove price of nrchy of. The intuition behind this protocol is similr to the x-enforcing protocols: mke ll undesired outcomes unstble by chrging some plyer very high price. Definition 3.5. (opt-enforcing protocol) Given scheduling model (N,M,d,c) theopt-enforcing protocol tkes s input n optiml outcome y. We gin denote for ny outcome z nd mchine the set of foreign plyers on by S(z) ={i S (z)\s (y)}. Then, the opt-enforcing protocol ssigns the cost shring methods d i cz (z) i l zi (z), if S z i = (3.24) ξ i(z) := c zi (z), if Sz i nd i =minsz i (z) (3.24b) 0, else. (3.24c) Under the opt-enforcing protocol, the plyers shre the cost proportionl to their job weights on ll mchines without foreign plyers. On mchines with foreign plyers, the foreign plyer with the smllest index pys the entire cost of the mchine. Theorem 3.6. The opt-enforcing protocol is seprble nd hs price of nrchy of. Proof. Budget blnce nd seprbility re cler from the definition of the privte cost functions. For stbility it cn esily be verified tht for n instnce (N,M,d,c) the optiml outcome y is Nsh equilibrium. We only proof the bound on the price of nrchy, showing tht ll Nsh equilibri x re optiml outcomes using the Nsh inequlities ξ i(x) ξ i(y i,x i) for ny i N. Two cses re to be considered for such mchine y i: either it hosts foreign plyers Sy i (x) orsy i (x) =. If there re foreign plyers on y i, then one of them will py for the entire cost there nd hence (3.24c) gives ξ i(x) ξ i(y i,x i) = 0. If there re no foreign plyers on y i,thend i l yi (y i,x i) l yi (y) yields c yi (y i,x i) l yi (y cy (y) i i,x i) l yi (y), becuse the cost per unit is non-decresing. Plugging this into (3.24) we hve ξ i(x) ξ i(y i,x i) ξ i(y). In both cses ξ i(x) ξ i(y) for ll i N nd thus C(x) = ξ i(x) ξ i(y) =C(y) i N i N which implies tht every Nsh equilibrium x is lso n optiml outcome. 4. UNIFORM PROTOCOLS The seprble protocols tht we introduced so fr were lwys tilored to some desirble outcome, either n enforceble outcome or even n optiml outcome. Since uniform protocols need to ssign cost shring methods independent of the set M, they cnnot be bsed on specific outcomes. We show in this section tht uniformity leds in generl to n unbounded price of nrchy. Only for gmes in which the demnds re integer multiples of ech other we introduce the semi-ordered protocol tht gives price of nrchy of n. The question of min ξ Un PoS(ξ) remins open. 4. Lower Bound Theorem 4.. There is no uniform protocol for which the price of nrchy hs n upper bound. This holds even for models with t most 3 plyers, 3 mchines nd nondecresing costs per unit. 292

9 Proof. The essence of uniform protocols is tht dding mchines to or removing them from the model does not chnge the cost shres of plyers using certin mchine, s long s the plyer set nd the weight vector remin the sme. This motivtes the definition of cost shre functions ˆξ i tht return the privte cost ξ i of plyer i s function of the mchine tht she uses nd the set of plyers S N shring the mchine. ˆξ i(, S) :=ξ i(x) M,S N,i S, x M n : S (x) =S. (4.) Nsh equilibri cn be expressed vi cost shre functions s follows. For ll i N, M it holds tht ˆξ i(x i,s xi (x)) ˆξ i(, S (x) {i}). (4.2) For the proof of the theorem, we propose number of scheduling models nd show tht for ny uniform cost shring protocol t lest one of these models hs Nsh equilibrium of more thn q times the cost of n optiml outcome for rbitrry q 2. Throughout the entire proof, the plyer set will lwys be N = {, 2, 3} with weights d =(4, 3, 2). The mchines will be subset of M = {,..., 7} with cost functions s outlined in Tble 2. Tble 2: Cost functions used in proof of Theorem 4. l c (l) c 2 (l) c 3 (l) c 4 (l) c 5 (l) c 6 (l) c 7 (l) q q 3 0 q q 2q 3 q 3 2q 4 q q 3 q 5 2q 3 6 q 3 q 5 7 q 4 First, consider the model with mchines M = {, 2} nd their respective cost functions. The optiml outcome y =( 2,, )hscostc(y )=q + 2, while the outcome x =(, 2, 2)hscostC(x )=q 3 +. Either x is Nsh equilibrium nd hence the protocol hs price of nrchy greter thn q or or one of the three plyers cn reduce her privte cost by choosing different mchine, which results by (4.2) in the following three cses. ) ˆξ ( 2, {, 2, 3}) < ˆξ (, {}) =. In this cse, consider the model with mchines M 2 = { 2, 3}. The optiml outcome y 2 =( 3, 2, 2)withC(y 2)=q 3 + is not Nsh equilibrium nd due to stbility some other outcome x with cost C(x) q 5 hs to be Nsh equilibrium. b) ˆξ 2(, {, 2}) < ˆξ 2( 2, {2, 3}) q 3. In this cse, consider M 3 = {, 2, 4}. The optiml outcome y 3 = (, 2, 4) hs cost C(y 3) =, while the outcome x 3 =( 2,, 4)hscostC(x 3)=q. Either x 3 is Nsh equilibrium or, gin, one of the plyers cn reduce her privte cost by choosing different mchine, which leds to the following cses. b.) ˆξ (, {, 2}) < ˆξ ( 2, {}) = q. This contrdicts b), tht is ˆξ (, {, 2}) = c (d + d 2) ˆξ 2(, {, 2}) >q 4 q 3 >q. b.2) ˆξ ( 4, {, 3}) < ˆξ (, {}) =q. In this cse, consider M 4 = { 2, 4, 5}. The optiml outcome y 4 =( 2, 5, 4)withC(y 4)=q is not Nsh equilibrium nd due to stbility some other outcome x with cost C(x) q 3 hs to be Nsh equilibrium. b.3) Plyers 2 nd 3 cnnot reduce their privte cost s ξ 2(x 3)=c (x 3)=0ndξ 3(x 3)=c 4 (x 3)=0. c) ˆξ 3(, {, 3}) < ˆξ 3( 2, {2, 3}) q 3. In this cse, consider M 5 = {, 2, 6}. The optiml outcome y 5 = ( 2,, )hscostc(y 5)=q +, while the outcome x 5 =(, 2, 6)hscostC(x 5)=q 2 +. Eitherx 5 is Nsh equilibrium or, gin, one of the plyers cn reduce her privte cost by choosing different mchine. c.) ˆξ ( 2, {, 2}) < ˆξ (, {}) =. In this cse, consider gin M 3 = {, 2, 4}. The optiml outcome y 3 =(, 2, 4)withC(y 3)=isnot Nsh equilibrium nd due to stbility some other outcome x with cost C(x) q hs to be Nsh equilibrium. c.2) ˆξ ( 6, {, 3}) < ˆξ (, {}) =. In this cse, consider M 6 = { 3, 5, 6}. The optiml outcome y 6 = ( 3, 5, 6)withC(y 6) = q 2 + is not Nsh equilibrium nd due to stbility some other outcome x with cost C(x) q 3 hs to be Nsh equilibrium. c.3) Plyer 2 cnnot reduce her privte cost becuse ξ 2(x 5)=c 2 (x 5)=0. c.4) ˆξ 3(, {, 3}) < ˆξ 3( 6, {3}) = q 2. In this cse, consider M 7 = {, 6, 7}. The optiml outcome y 7 = (, 7, 6)withC(y 7) = q 2 + is not Nsh equilibrium nd due to stbility some other outcome x with cost C(x) q 3 hs to be Nsh equilibrium. c.5) ˆξ 3( 2, {2, 3}) < ˆξ 3( 6, {3}) = q 2. This extends the originl ssumption from c) ˆξ 3(, {, 3}) < ˆξ 3( 2, {2, 3}) <q 2 nd therefore implies c.4). Altogether, every uniform cost shring protocol llows in t lest one of the nlyzed cses Nsh equilibrium of t lest q times the cost of n optiml outcome for n rbitrry q 2. Consequently, the price of nrchy is not bounded. 4.2 Models with Restricted Weights Although uniform protocols in generl do not llow bound on the price of nrchy, the following clss of gmes permits uniform cost shring protocols with bounded price of nrchy. We ssume tht the plyer s weights re either uniform, i.e. d =... = d n, or they re multiples of ech other. In the following, we propose semi-ordered protocol tht hs price of nrchy of t most n for such gmes. In this section, we ssume tht the plyers re indexed with their weights in non-incresing order: d d 2... d n. The semi-ordered protocol lets the plyers one fter the other choose mchine nd lets them py only for the dditionl cost they cuse on tht mchine, thus mking the choice of plyer i independent of the choices of ll plyers j>i. Definition 4.2. (Semi-ordered Protocol) The semi-ordered protocol ssigns for ll i N ( ) ( ξ i(x) :=c xi d j c xi d j ). (4.3) j S xi (x):j i j S xi (x):j<i 293

10 Theorem 4.3. The semi-ordered protocol is uniform nd its price of nrchy is t most n for instnces, where the plyers weights re multiples of ech other, i.e. d i = q i d i+i for ll i<nnd some q i N. Proof. Budget blnce, seprbility nd uniformity of the cost shring methods re cler. A Nsh equilibrium cn be found by sking the plyers in the order of their index to choose mchine tht minimizes their privte cost considering the choice of ll previous plyers. For proving the bound on the price of nrchy, consider model (N,M,d,c) which fulfills the restriction on the plyers weights. Suppose y is n optiml outcome nd x Nsh equilibrium. First, we show tht ξ i(x) mx j i ξ j(y) holdsforlli N, which is motivted by the ide tht plyer cn lwys choose mchine tht she or one of the lrger plyers hd chosen in the optiml outcome. To this end, fix plyer i N. On some mchine {y,...,y i } there is in outcome x less lod from the first i plyers thn in the optiml outcome y from the first i plyers. Due to the restriction on the plyers weights this difference in lod on mchine hs to be t lest d i yielding j S (x):j<i d j + d i j S (y):j i Also, there is plyer k i (hence d k d i), k S (y) who in outcome y uses mchine nd for whom due to the weight restrictions d j d j < d j + d i d j. j S (y):j<k j S (x):j<i j S (x):j<i d j. Combining the bove inequlities with (4.3) yields ξ i(x) ξ i(, x i) ( = c c ( This implies: C(x) = i N proving the clim. j S (x):j<i j S (y):j k = ξ k (y) mx j i ξj(y). ξ i(x) i N mx j i d j + d i ) c ( d j ) c ( j S (x):j<i j S (y):j<k j S (y):j k d j ) d j ) ξj(y) n mx ξj(y) n C(y) j N 5. REFERENCES [] H. Ackermnn, H. Röglin, nd B. Vöcking. Pure Nsh equilibri in plyer-specific nd weighted congestion gmes. Theor. Comput. Sci., 40(7): , [2] S. Alnd, D. Dumruf, M. Giring, B. Monien, nd F. Schoppmnn. Exct price of nrchy for polynomil congestion gmes. In Proc. of STACS, pges , [3] E. Anshelevich, A. Dsgupt, J. Kleinberg, É. Trdos, T. Wexler, nd T. Roughgrden. The price of stbility for network design with fir cost lloction. SIAM J. Comput., 38(4): , [4] B. Awerbuch, Y. Azr, nd A. Epstein. The price of routing unsplittble flow. In Proc.ofSTOC, pges ACM, [5] K. Bhwlkr, M. Giring, nd T. Roughgrden. Weighted congestion gmes: Price of nrchy, universl worst-cse exmples, nd tightness. In Proc. of ESA, pges Springer, 200. [6] I. Crginnis. Efficient coordintion mechnisms for unrelted mchine scheduling. In Proc. of SODA, pges , [7] H.-L. Chen nd T. Roughgrden. Network design with weighted plyers. Theor. Comput. Syst., 45(2): , [8] H.-L. Chen, T. Roughgrden, nd G. Vlint. Designing network protocols for good equilibri. SIAM J. Comput., 39(5): , 200. [9] G. Christodoulou nd E. Koutsoupis. The price of nrchy of finite congestion gmes. In Proc. of STOC, pges 67 73, [0] G. Christodoulou, E. Koutsoupis, nd A. Nnvti. Coordintion mechnisms. In Proc.ofICALP, pges Springer, [] J. R. Corre, A. S. Schulz, nd N. E. Stier-Moses. Selfish routing in cpcitted networks. Mth. of Oper. Res., 29:96 976, [2] E. Even-Dr, A. Kesselmn, nd Y. Mnsour. Convergence time to Nsh equilibrium in lod blncing. ACM Trns. Algorithms, 3(3):32, [3] D. Fotkis, S. Kontoginnis, E. Koutsoupis, M. Mvronicols, nd P. Spirkis. The structure nd complexity of Nsh equilibri for selfish routing gme. In Proc. of ICALP, pges 23 34, [4] M. Giring, B. Monien, nd K. Tiemnn. Routing (un-) splittble flow in gmes with plyer-specific liner ltency functions. In Proc of ICALP, pges 50 52, [5] S. Ieong, R. McGrew, E. Nudelmn, Y. Shohm, nd Q. Sun. Fst nd compct: A simple clss of congestion gmes. In Proc. AAAI, pges , [6] N. Immorlic, L. Li, V. Mirrokni, nd A. Schulz. Coordintion mechnisms for selfish scheduling. Theor. Comput. Sci., 40(7): , [7] E. Koutsoupis nd C. H. Ppdimitriou. Worst-cse equilibri. In STACS, pges , 999. [8] I. Milchtich. Congestion gmes with plyer-specific pyoff functions. Gmes Econom. Behv., 3(): 24, 996. [9] M. Osborne nd A. Rubinstein. A Course in Gme Theory. MIT Press, 994. [20] R. Rosenthl. A clss of gmes possessing pure-strtegy Nsh equilibri. Inter. J. Gme Theory, 2():65 67, 973. [2] T. Roughgrden. The price of nrchy is independent of the network topology. Journl of Computer nd System Sciences, 67:34 364, [22] T. Roughgrden. Selfish Routing nd the Price of Anrchy. The MIT Press, [23] T. Roughgrden nd É. Trdos. How bd is selfish routing? J. ACM, 49(2): , [24] R. Sriknt. The Mthemtics of Internet Congestion Control. Birkhäuser Boston,