Tight bounds for selfish and greedy load balancing

Transcription

1 Tight bounds for selfish and greedy load balaning Ioannis Caragiannis Mihele Flammini Christos Kaklamanis Panagiotis Kanellopoulos Lua Mosardelli Deember, 009 Abstrat We study the load balaning problem in the ontext of a set of lients eah wishing to run a ob on a server seleted among a subset of permissible servers for the partiular lient. We onsider two different senarios. In selfish load balaning, eah lient is selfish in the sense that it hooses, among its permissible servers, to run its ob on the server having the smallest lateny given the assignments of the obs of other lients to servers. In online load balaning, lients appear online and, when a lient appears, it has to make an irrevoable deision and assign its ob to one of its permissible servers. Here, we assume that the lients aim to optimize some global riterion but in an online fashion. A natural loal optimization riterion that an be used by eah lient when making its deision is to assign its ob to that server that gives the minimum inrease of the global obetive. This gives rise to greedy online solutions. The aim of this paper is to determine how muh the quality of load balaning is affeted by selfishness and greediness. We haraterize almost ompletely the impat of selfishness and greediness in load balaning by presenting new and improved, tight or almost tight bounds on the prie of anarhy of selfish load balaning as well as on the ompetitiveness of the greedy algorithm for online load balaning when the obetive is to minimize the total lateny of all lients on servers with linear lateny funtions. In addition, we prove a tight upper bound on the prie of stability of linear ongestion games. A preliminary version of the results in this paper appeared in Proeedings of the rd International Colloquium on Automata, Languages and Programming ICALP 06), LNCS 468, Springer, pp , 006. This work was partially supported by the European Union under IST FET Integrated Proet AEOLUS and COST Ation 9 GRAAL, and by a Caratheodory basi researh grant from the University of Patras. Researh Aademi Computer Tehnology Institute & Department of Computer Engineering and Informatis, University of Patras, 6500 Rio, Greee. aragian@eid.upatras.gr Dipartimento di Informatia, Università di L Aquila, Via Vetoio, Coppito 6700, L Aquila, Italy. flammini@di.univaq.it Researh Aademi Computer Tehnology Institute & Department of Computer Engineering and Informatis, University of Patras, 6500 Rio, Greee. kakl@eid.upatras.gr Researh Aademi Computer Tehnology Institute & Department of Computer Engineering and Informatis, University of Patras, 6500 Rio, Greee. kanellop@eid.upatras.gr Department of Sienes, University of Chieti-Pesara, Viale Pindaro 4, 657, Pesara, Italy. mosardelli@si.unih.it

2 Introdution We study the load balaning problem in the ontext of a set of lients eah wishing to run a ob on a server seleted among a subset of permissible servers for the partiular lient. We onsider two different senarios. In the first, alled selfish load balaning or load balaning games), eah lient is selfish in the sense that it hooses, among its permissible servers, to run its ob on the server having the smallest lateny given the assignments of the obs of other lients to servers. In the seond senario, alled online load balaning, lients appear online and, when a lient appears, it has to make an irrevoable deision and assign its ob to one of its permissible servers. Here, we assume that the lients are not selfish and aim to optimize some global obetive but in an online fashion i.e., without any knowledge of lients that may arrive in the future). A natural loal optimization riterion that an be used by eah lient when making its deision is to assign its ob to the server that gives the minimum inrease of the global obetive. This gives rise to greedy online solutions. The aim of this paper is to answer the question of how muh the quality of load balaning is affeted by selfishness and greediness. Load balaning games. Load balaning games are speial ases of the well-known ongestion games introdued by Rosenthal [9] and studied in a sequene of papers [6, 0,, 4, 6, 8, 6, 0, ]. In ongestion games there is a set E of resoures, eah resoure e having a non-negative and non-dereasing lateny funtion f e defined over non-negative numbers, and a set of n players. Eah player i has a set of strategies S i E eah strategy of player i is a set of resoures). An assignment A = A,..., A n ) is a vetor of strategies, one strategy for eah player. The ost of a player for an assignment A is defined as osti) = e A i f e n e A)), where n e A) is the number of players using resoure e in A, while the soial ost of an assignment is the total ost of all players. An assignment is a pure Nash equilibrium if no player has any inentive to unilaterally deviate to another strategy, i.e., ost i A) ost i A i, s) for any player i and for any s S i, where A i, s) is the assignment produed if ust player i deviates from A i to s. This inequality is also known as the Nash ondition. In weighted ongestion games, eah player has a weight w i and the lateny of a resoure e depends on the total weight of the players that use e. For this ase, a natural soial ost funtion is the weighted sum of the osts of all players or the weighted average of their osts). In linear ongestion games, the lateny funtion of resoure e is of the form f e x) = α e x + β e with non-negative onstants α e and β e. Load balaning games are ongestion games where the strategies of players are singleton sets. In load balaning terminology, we use the terms server and lient instead of the terms resoure and player. The set of strategies of a lient ontains the servers that are permissible for the lient. A load balaning game is alled symmetri when all servers are permissible for any lient. We evaluate the quality of solutions of a load balaning game by omparing the soial ost of Nash equilibria to the ost of the optimal assignment i.e., the minimum ost). We use the notions of prie of anarhy introdued in a seminal work of Koutsoupias and Papadimitriou [] see also [7]) and prie of stability [] or optimisti prie of anarhy) defined as follows. The prie of anarhy/stability of a load balaning game is defined as the ratio of the maximum/minimum soial ost over all Nash equilibria over the optimal ost. The prie of anarhy/stability for a lass of load balaning games is simply the highest prie of anarhy/stability among all games belonging to that lass. The papers [5, 7, 8, 9,, 5] study various games whih an be thought of as

3 speial ases of ongestion games with respet to the omplexity of omputing equilibria of best/worst soial ost and the prie of anarhy when the soial ost is defined as the maximum lateny experiened by any player. The soial ost of the total lateny has been studied in [6, 0, 4, ]. The authors in [4] study symmetri load balaning games with linear lateny funtions and show tight bounds on the prie of anarhy of 4/ for different servers and 9/8 for idential servers with weighted lients. The prie of anarhy of symmetri load balaning games with polynomial or onvex lateny funtions is studied in [0, ]. In two papers, Awerbuh et al. [6] and Christodoulou and Koutsoupias [0] prove tight bounds on the prie of anarhy of ongestion games with linear lateny funtions. Among other results, they show that the prie of anarhy of pure Nash equilibria is 5/ while for mixed Nash equilibria or pure Nash equilibria of weighted lients it is Tight bounds on the prie of anarhy of ongestion games with polynomial lateny funtions are presented in []; these improve previous results in [6, 0]. Does the fat that load balaning games are signifiantly simpler than ongestion games in general have any impliations for their prie of anarhy? We give a negative answer to this question for linear lateny funtions by showing that the 5/ upper bound as well as the + 5 upper bound for weighted lients) is tight. This is interesting sine the upper bounds for ongestion games as well as an earlier upper bound of 5/ proved speifially for load balaning []) are obtained using only the Nash inequality i.e., the inequality obtained by summing up the Nash ondition inequalities over all players strategies) and the definition of the soial ost. So, it is somewhat surprising that load balaning games are as general as ongestion games in terms of their prie of anarhy and that the Nash inequality provides suffiient information to haraterize their prie of anarhy. An important speial ase of load balaning is when servers have idential linear lateny funtions. Here, better upper bounds on the prie of anarhy an be obtained. Note that this is not the ase for ongestion games sine, as it was observed in [0], any ongestion game an be transformed to a ongestion game on idential resoures and, hene, the lower bounds of [6, 0] hold for ongestion games with idential resoures as well). Suri et al. [] prove that the prie of anarhy of selfish load balaning on idential servers is between.0067 and + /.547. Again, the upper bound is obtained by using the Nash inequality and the definition of the soial ost. We improve this result by showing that the lower bound is essentially tight. Besides the Nash inequality, our proof also exploits strutural properties of games with high prie of anarhy. We argue that suh games an be represented as a direted graph alled the game graph) and, then, strutural properties of suh a game follow as strutural properties of this graph. Furthermore, for weighted lients and idential servers, we prove that the prie of anarhy is at least 5/. Tight bounds on the prie of stability of load balaning games have been proved in []. The prie of stability of linear ongestion games has been studied in [] where it was shown that it lies between + /.577 and.6. The tehnique used to obtain the upper bound is to onsider pure Nash equilibria with potential not larger than the potential of the optimal assignment and bound their soial ost in terms of the optimal ost using the Nash inequality. Using the same tehnique but with a more refined analysis, we show that the lower bound is tight. Greedy load balaning. From the algorithmi point of view, load balaning has been studied extensively, inluding papers studying online versions of the problem e.g., [, 4, 5,

4 7, 9,, 8,, ]). In online load balaning, lients appear in online fashion; when a lient appears, it has to make an irrevoable deision and assign its ob to a server. In our model, servers have linear lateny funtions and the obetive is to minimize the total lateny, i.e., the sum of the latenies experiened by all lients. Clients may also own obs with nonnegative weights; in this ase, the obetive is to minimize the weighted sum of the latenies experiened by all lients. A natural greedy algorithm proposed in [5] for this problem is to assign eah lient to the server that yields the minimum inrease to the total lateny ties are broken arbitrarily). This results to greedy assignments. Given an instane of online load balaning, an assignment of lients to servers is alled a greedy assignment if the assignment of a lient to a server minimizes the inrease in the ost of the instane revealed up to the time of its appearane. Following the standard performane measure in ompetitive analysis, we evaluate the performane of this algorithm in terms of its ompetitiveness or ompetitive ratio). The ompetitiveness of the greedy algorithm on an instane is the maximum ratio of the ost of any greedy assignment over the optimal ost and its ompetitiveness on a lass of load balaning instanes is simply the maximum ompetitiveness over all instanes in the partiular lass. The performane of greedy load balaning with respet to the total lateny has been studied in [5, ]. Awerbuh et al. [5] onsider a more general model where eah lient owns a ob with a load vetor denoting the impat of the ob to eah server i.e., how muh the assignment of the ob to a server will inrease its load) and the obetive is to minimize the L p norm of the load of the servers. In the ontext similar to the one studied in the urrent paper, their results imply a upper bound. This result applies also in the ase of weighted lients where the obetive is to minimize the weighted average lateny. Suri et al. [] onsider the same model as ours and show upper bounds of 7/ and for different servers and idential servers, respetively. In a way similar to the study of the prie of anarhy of ongestion games, [] develops a greedy inequality whih is used to obtain the upper bounds on ompetitiveness. They also present a lower bound of.08 for the ompetitiveness of greedy assignments in the ase of idential servers. Christodoulou et al. [] have analyzed a different than greedy online algorithm for load balaning and proved that it has ompetitiveness at most The main question left open by the work of [] is whether the existene of different servers does hurt the ompetitiveness of greedy load balaning. We give a positive answer to this question as well. By a rather ounterintuitive onstrution, we show that the 7/ upper bound of [] is tight. This is interesting sine it indiates that the greedy inequality is powerful enough to haraterize the ompetitiveness of greedy load balaning. We also onsider the ase of idential servers where we almost lose the gap between the upper and lower bounds of [] by showing that the ompetitiveness of greedy load balaning is between 4 and In the proof of the upper bound, we use the greedy inequality but, more importantly, we also use arguments for the struture of greedy and optimal assignments of instanes that yield a high ompetitiveness. In a similar way to the ase of selfish load balaning, we argue that suh instanes an be represented as direted graphs alled greedy graphs) that enoy partiular strutural properties. In the ase of weighted lients, we present a tight lower bound of + on idential servers mathing the upper bound of [5]. The results presented in this paper are summarized in Table. 4

5 Problem Measure Result Comments unweighted prie of stability, + / Setion, Theorem. ongestion game upper bound Mathes a lower bound from [] unweighted prie of anarhy,.5 Setion., Theorem 4. load balaning lower bound Mathes an upper bound from [6, 0] for unweighted ongestion games unweighted prie of anarhy,.0 Setion.. load balaning, upper bound Mathes a lower bound from [] idential servers unweighted ompetitiveness 7/ Setion 4., Theorem 0. load balaning of greedy, Mathes an upper bound from [] lower bound unweighted ompetitiveness Setion 4., Theorem. load balaning, of greedy, Improves an upper bound from [] idential servers upper bound unweighted ompetitiveness 4 Setion 4., Theorem 4. load balaning, of greedy, Improves a lower bound from [] idential servers lower bound weighted prie of anarhy, + 5 Setion 5, Theorem 5. load balaning lower bound Mathes an upper bound from [6] for ongestion games weighted prie of anarhy,.5 Setion 5, Theorem 6 load balaning, lower bound idential servers weighted ompetitiveness + Setion 5, Theorem 7. load balaning, of greedy, Mathes an upper bound from [5] idential servers lower bound for unrelated servers Table : Summary of our results. Roadmap. The rest of the paper is strutured as follows. We present the bounds on the prie of stability of linear ongestion games in Setion. The bounds on the prie of anarhy of selfish load balaning are presented in Setion, while the bounds on the ompetitiveness of greedy load balaning are presented in Setion 4. In Setion 5 we present extensions of the results to selfish and greedy load balaning when lients are weighted and onlude with open problems in Setion 6. The prie of stability of linear ongestion games We present a tight upper bound on the prie of stability of linear ongestion games. Our proof uses the main idea in the proof of [] and bounds the soial ost of any Nash equilibrium having a potential smaller than the potential of the optimal assignment. In the proof we also 5

6 make use of the Nash inequality whih together with the inequality on the potentials yields the upper bound. However, the two inequalities may not be equally important in order to ahieve the best possible bound and this is taken into aount in our analysis. We now state the Nash inequality see for example [0,, ]) as applied to our setting. It follows by summing the Nash ondition inequalities of all lients. Lemma Nash inequality) For any ongestion game, where eah resoure e has lateny funtion f e x) = α e x + β e, with a pure Nash equilibrium and an optimal assignment of n e and o e players at eah resoure e, respetively, it holds that αe n ) e + β e n e o e α e n e + α e + β e ). e We use Rosenthal s potential funtion [9]. We remind that, assuming a strategy profile A for a ongestion game with linear lateny funtion f x) = α x+β, we define the potential of the strategy to be P ota) = m n A) = i= f i), where m is the number of resoures, and n A) denotes the number of lients using resoure in A. By its definition, the potential funtion has the property that for any two assignments differing only in the strategy of a single lient, the differene of the potentials and the differene of the ost experiened by that lient in the two assignments have the same sign. Furthermore, the potential funtion has loal minima at pure Nash equilibria and, in order to establish an upper bound on the prie of stability, it suffies to bound the soial ost of pure Nash equilibria whose potential is less than or equal to the potential of the optimal assignment. In our proof, we will need the following tehnial lemma. Lemma For any non-negative integers x, y and γ =, it holds that ) γ γ)xy + y γx + γy x + + γ)y. Proof: Define the funtion gx, y) as the subtration of the left part from the right part in the above inequality. Substituting γ we have gx, y) = 7 4 ) x + y 4 ) xy y + ) x = ) ) ) x y + + y 4. In order to prove the lemma, it suffies to show that gx, y) 0 for any non-negative integer values of x and y. First, we observe that if y it is ) y 4 0. Hene, ) ) gx, y) 0 for any integer y. Also, gx, 0) = x for any integer x 0. For y =, by trivial alulations we obtain that the paraboli funtion ) ) gx, ) = x is equal to zero for x = 0 and x =. So, it is non-negative for any non-negative integer value of x. We are now ready to prove the following result. A mathing lower bound is presented in []. e 6

7 Theorem The prie of stability of ongestion games with linear lateny funtions is at most + /. Proof: Consider a linear ongestion game, an optimal assignment and a pure Nash equilibrium of not larger potential. We will show that the soial ost of this Nash equilibrium and, as a onsequene, the soial ost of the best Nash equilibrium) is no more than + times the ost of the optimal assignment. Denote by n and o the number of lients using resoure in the Nash and optimal assignment, respetively. By the inequality of the potentials, we obtain that n m f i) = i= m α n + ) n + β n ) = o m f i) = i= m α o + ) o + β o ) = m α n ) m ) m + β n α o + o n + β o n ) ) = By the Nash inequality, we obtain that = m α n ) m + β n α n + ) o + β o ) ) = = Let γ =. By multiplying ) by γ and ) by γ and adding them and using Lemma, we obtain that m α n ) m + β n α γ) n o + o γn + γo ) = = = + m = m β + γ) o γn ) = α γ ) n + + γ) o ) = m ) ) γ + β n + + γ) o = ) γ m α n ) m + β n + + γ) α o ) + β o. = = Therefore, the prie of stability is at most m = α n + β n ) m = α o + β o ) + γ γ ) = +. 7

8 Bounds on the prie of anarhy In this setion, we present tight bounds on the prie of anarhy. We first show that the known upper bound of Suri et al. [] on the prie of anarhy of load balaning games on different servers with linear lateny funtions is tight Setion.). Then, we present better upper bounds in the ase of idential servers starting from simple bounds that already improve the previous results from [] Setion.) and onluding with a omputer-assisted proof in Setion.) whih essentially yields an upper bound mathing the orresponding lower bound of []. For the study of the prie of anarhy of non-symmetri load balaning games, we an onsider games in whih eah lient has at most two strategies. This is learly suffiient when proving lower bounds. In order to prove upper bounds, we observe that for any game, there exists another game with at most two strategies per lient whih has the same prie of anarhy. Given any load balaning game, let O and N be the optimal assignment and the Nash equilibrium that yields the worst soial ost for this game, respetively. The game with the same lients and servers in whih eah lient has its strategies in O and N as strategies also has the same optimal assignment and the same Nash equilibrium and, onsequently, the same prie of anarhy). We represent suh games as direted graphs alled game graphs) having a node for eah server and a direted edge for eah lient; the diretion of eah edge is from the strategy of the lient in the optimal assignment to the strategy of the lient in the Nash equilibrium. A self-loop indiates that the lient has the same strategy in the optimal assignment and the Nash equilibrium.. Servers with different lateny funtions The next theorem states that the upper bound of 5/ presented in [] and also implied by the results in [6, 0] for linear ongestion games) is tight. This bound was known to be tight for linear ongestion games in general but the onstrutions in the lower bounds in [6, 0] are not load balaning games. Theorem 4 For any ϵ > 0, there is a load balaning game with linear lateny funtions whose prie of anarhy is at least 5/ ϵ. Proof: We onstrut a game graph G onsisting of a omplete binary tree with k + levels and k+ nodes with a line of k + edges and k + additional nodes hung at eah leaf. So, graph G has k + levels 0,..., k +, with i nodes at level i for i = 0,..., k and k nodes at levels k +,..., k +. The servers orresponding to nodes of level i = 0,..., k have lateny funtions f i x) = /) i x, the servers orresponding to nodes of level i = k,..., k have lateny funtions f i x) = /) k /) i k x, and the servers orresponding to nodes of level k + have lateny funtions f k+ x) = /) k /) k x. Consider the assignment where all lients selet servers orresponding to the endpoint of their orresponding edge whih is loser to the root of the game graph. This assignment is a Nash equilibrium, sine servers orresponding to nodes of level i = 0,..., k have two lients and lateny /) i, servers orresponding to nodes of level i = k,..., k have one lient and lateny /) k /) i k, and servers orresponding to nodes of level k + have no lient. Therefore, due to the definition of the lateny funtions, a lient assigned to a server orresponding to a node of level i = 0,..., k would experiene exatly the same lateny if it hanged its deision and hose the server orresponding to the node of level i +. 8

9 The ost of this assignment is ost = k 4 i /) i + i=0 k i=k = 5 4/) k /) k. k /) k /) i k To ompute an upper bound for the ost of the optimal assignment, it suffies to onsider the assignment where all lients selet the servers orresponding to nodes whih are further from the root. We obtain that the ost opt of the optimal assignment is opt k i /) i + i= = 6 4/) k 4. k i=k k /) k /) i k + k /) k /) k Hene, for any ϵ > 0 and for suffiiently large k, the prie of anarhy of the game is larger than 5/ ϵ.. Idential servers In the ase of idential servers with linear lateny funtions we an show a tight bound on the prie of anarhy of approximately.0067; a mathing lower bound has been presented in []. First, we present the main idea in our analysis to obtain a slightly weaker result whih already improves the previously known upper bound of []. Then, we further improve our analysis. We will upper-bound the ratio of the soial ost of the worst Nash equilibrium to the optimal soial ost of games with at most two strategies per lient whih satisfy a partiular property. We say that server is of type n /o meaning that it has n lients in the Nash equilibrium and o lients in the optimal assignment equivalently, server has in-degree n and out-degree o in the game graph). We first show that for any game we an onstrut another game that has at least the same prie of anarhy and, furthermore, satisfies the following -neighborhood property: in the game graph, the inoming edge of any server of type / originates from a server of type 0/. Then, the idea behind the proof is to aount for the ontribution of servers of type / and 0/ in the soial ost together. In general, lateny funtions would be of the form fx) = αx + β where α > 0 and β 0. Then, the prie of anarhy is given by the ratio αn +βn ) αo +βo ) whih is at most n. Hene, o without loss of generality, we may assume that the lateny funtion is of the form fx) = x. In the proof of our weakest bound Theorem 6), we make use of the following tehnial lemma. Lemma 5 Let ψ = 6+ 6, ξ = 7 0 and define the funtions gx, y) = xy + + ξ) y ξx and hx, y) = 4ψ x + ψy. For any non-negative integers x, y suh that either x or y, it holds that gx, y) hx, y). Furthermore, g0, ) + g, ) = h0, ) + h, ). Proof: We start by noting that g0, ) + g, ) = ξ + = and h0, ) + h, ) = ψ + 4ψ = Define the funtion fx, y) = hx, y) gx, y) = ψ x ψy + ξ ) ψ + ξψ ξ ) y ξ ψ. 9

10 In order to prove the lemma, it suffies to show that fx, y) 0 for any non-negative integer values of x and y when either x or y. First, observe that if y, then y ξ ψ ξψ ξ = 6.4, whih implies that ξψ ξ ) y ξ ψ 0. Hene, fx, y) 0 for ) any integer y. Also, fx, 0) = ψ x + ξ ψ ξ ψ 0 for any integer x 0. For y =, by straightforward alulations we obtain that the paraboli funtion fx, ) = ψ x + ξ ) ) ψ + ψξ ξ ξ ψ is positive for x = 0, negative for x = and equal to zero for x =. So, it is non-negative for any non-negative integer value of x besides. Theorem 6 The prie of anarhy of selfish load balaning on idential servers is at most Proof: Consider a load balaning game on servers with lateny funtion fx) = x and lients having at most two strategies. Without loss of generality, we may assume that the game satisfies the -neighborhood property, i.e., the inoming edge of any server of type / originates from a server of type 0/ in the game graph. If this is not the ase, we show how to onstrut another game with not smaller prie of anarhy. If a lient had server as its only strategy this orresponds to a self-loop in the orresponding game graph), then we may onstrut a new game by exluding server and lient from the original one; the new game has worse prie of anarhy sine both the soial ost of the optimal assignment and the soial ost of the Nash equilibrium are dereased by. So, let and be the servers to whih server is onneted orresponding to lients and seleting servers and in the optimal assignment and servers and in the Nash assignment, respetively. Clearly, n, sine, otherwise, lient would have an inentive to use server in the Nash equilibrium. Assume that server is of type n /o for n > 0 or o >. If n > 0, we an onstrut a new game by exluding server and substituting lients and by a lient seleting server in the optimal assignment and server in the Nash assignment. The new game has worse prie of anarhy, sine both the ost of the optimal assignment and the ost of the Nash equilibrium are dereased by. If o >, then we an add a new server and hange the strategy of lient to {, }. The new game has worse prie of anarhy sine the ost of the Nash equilibrium remains the same, while the ost of the optimal assignment dereases. Now, denote by F the set of servers of type / and by S the set of servers of type 0/ whih are onneted through an edge to a server in F in the game graph. Also, for eah server in F we denote by S) the server of S from whih the lient destined for originates. By the Nash inequality, we obtain that n o n + o ) and, sine n = o, we 0

11 have that n = = n o + o ) ) n o o 7 n 0 0 gns), o S) ) + gn, o ) ) gn, o ) + F S F hns), o S) ) + hn, o ) ) hn, o ) + F S F = 6 0 n o where the first equality follows sine n = o, the seond equality follows by the definition of funtion g, the seond inequality follows by Lemma 5, and the last equality follows by the definition of funtion h. We obtain that the prie of anarhy is n o.. Tightening the analysis The main idea in order to improve the analysis in the proof of Theorem 6 is to strengthen the properties of the games that have to be onsidered and aount for the ontributions of servers of type / together with the servers in their neighborhood in the game graph in a way that is well defined below). By extending the neighborhood that we onsider, we obtain better and better upper bounds whih onverge to the lower bound of.0067 presented in []. We present the formal proof for an upper bound of and a series of better bounds obtained using more ompliated omputer-assisted proofs. Consider the game graph of the game satisfying the -neighborhood property. We all a 4-path any direted path of at most four nodes in the game graph starting with a server of type 0/ and having a server of type / as its seond node. Let p be suh a path starting with the server 0 and ontaining server as its seond node. Denote by the third server in the path p and assume that is of type n /o. The path may terminate at server if it has no outgoing edges in the game graph. Otherwise, path p has a fourth server of type n /o. An example of a 4-path is presented in Figure. We say that a game satisfies the 4-neighborhood property if in any 4-path, the third server has n = and the fourth server, if it exists, has n = and neither of them has any self-loop. We show that given any game that satisfies the -neighborhood property and has prie of anarhy at least 5/, there exists a game satisfying the 4-neighborhood property whih has at least the same prie of anarhy. Consider a 4-path onsisting of servers 0,, and possibly. Sine is onneted to at least server, it has n > 0. If n, then lient between and would have an inentive to use server in the Nash equilibrium. If n =, then we

12 0/ / 0 n / o n / o Figure : An example of a 4-path. an replae the lients and by a new lient seleting server 0 in the optimal assignment and server in the Nash assignment and obtain a game with higher prie of anarhy, sine both the ost of the optimal assignment and of the Nash equilibrium are dereased by. If server exists, then sine is onneted to at least server, it has n. If n =, then we an introdue a new server and hange the strategy set of lient to {, } to obtain another game in whih server is onneted to a server of type 0/. The soial ost of the Nash assignment is the same while the optimal soial ost does not inrease. If n =, we distinguish between two ases for o. If o =, then we may remove servers 0,, and and lients, and and hange the strategy of the lient 4 whih is the seond lient onneted to server in the Nash equilibrium so that it onnets to instead of. In this way, the soial ost of the Nash assignment is dereased by 5 while the optimal soial ost is dereased by ; overall the prie of anarhy inreases sine the original game had prie of anarhy larger than 5/. If o >, we remove the lient, hange the strategy of lient 4 so that it onnets to server in the Nash equilibrium and introdue two new servers 0 and of types 0/ and /, respetively, and lients 5 and 6 onneting 0 to and to, respetively. The soial ost of the Nash assignment is inreased by while the optimal ost dereases by at least. Overall, the prie of anarhy inreases. Clearly, if n 4, then lient would have an inentive to use server. It remains to show that neither nor have self-loops. We will atually show that any game whose game graph has self-loops at nodes of in-degree or an be onverted to a game without suh self-loops with higher prie of anarhy. Lemma 7 For any game, there exists a game having at least the same prie of anarhy and whose game graph has no self-loops at nodes with in-degree and. Proof: Starting from a game whose game graph has self-loops at some nodes of in-degree respetively, ), we will onstrut another game whose game graph has no self-loops at nodes of in-degree resp., ) and has higher prie of anarhy. Consider a game with game graph G that has t self-loops at nodes of in-degree resp., ). Construt the graph G by first putting twelve resp., twenty) opies of G. Denote by L the set of self-loops at nodes of G with in-degree resp., ). For eah self-loop in L we apply the following proedure in order to augment G. Let v be the node of G with in-degree resp., ) having the self-loop. Connet the outgoing edges of twelve resp., twenty) onstrutions like the one in Figure a resp., Figure ) to the twelve resp., twenty) opies of node v, with one outgoing edge onneted to eah opy. Furthermore, onnet the twelve resp., twenty) opies of v to the input edges of the onstrution of Figure b resp., Figure d) and remove the twelve resp., twenty) opies of the self-loop from G. An example is depited in Figure e. We first show that graph G is a game graph, i.e., the assignment where eah lient selets the server to whih the orresponding edge points to is a Nash equilibrium. Notie that eah

13 a) b) e) d) ) Figure : a-d) Construtions used in the proof of Lemma 7. e) Conneting the opies of a node with a self-loop and in-degree in the original game graph with the onstrutions in a) and b). opy of a node of G has in- and out-degree in G equal to those of the orresponding node in G. Hene, a lient orresponding to any edge of G whih is a opy of an edge in G has no inentive to deviate sine the lient orresponding to the edge in G had no inentive to deviate either). Also, edges with at least one endpoint in the onstrutions of Figures a, b,, and d point from a node of in-degree i to a node of in-degree i + for i = 0,,..., 4). Hene, the orresponding lient would experiene the same lateny if it hanged its strategy and, hene, no suh lient has an inentive to deviate either. We will now show that the new game has higher prie of anarhy than the original one. In order to show this, we will also use the fat that the prie of anarhy of the original game is at most whih follows by Theorem 6. Denote by ost the soial ost of the Nash equilibrium of the original game where eah lient selets the server to whih the orresponding edge points in the game graph G) and by opt the ost of the assignment where eah lient selets the server from whih the orresponding edge originates in the game graph G. For the ase of nodes of in-degree, the ost of the Nash equilibrium for the new game is ost + t + 4t + t 4 = ost + 6t ). The first term omes from the ontribution of the nodes in the twelve opies of G, the seond term omes from the ontribution of the nodes in the onstrutions of Figure a, while the last two terms ome from the ontribution of nodes in the onstrution of Figure b. Similarly, the assignment where all lients selet the server at the origin of the orresponding edge has ost opt + 4t + 4t = opt + 7t ). The first term omes from the ontribution of the nodes in the twelve opies of G, the seond term omes from the ontribution of nodes in the onstrutions of Figure a, while the last term omes from the ontribution of nodes in the onstrutions of Figure b.

14 So, the prie of anarhy of the new game is at least ost + 6t/ opt + 7t/ > ost opt, sine, by Theorem 6, it is ost/opt < 6/7. Respetively, for the ase of nodes with in-degree, the ost of the Nash equilibrium of the new game is 0 ost + 40t + 0t + 5t 4 + t 5 = 0 ost + 45t ) 4 while the assignment where all lients selet the server at the origin of the orresponding edge has ost 0 opt + 00t + 5t = 0 opt + t ). 4 So, the prie of anarhy of the new game is at least ost + 45t/4 opt + t/4 > ost opt, sine, by Theorem 6, it is ost/opt < 45/. Given a server in a 4-path p of a game satisfying the 4-neighborhood property, we define part, p) as follows. For servers 0,, and if it exists), it is part 0, p) = part, p) = part, p) = {, if o = 0 o, if o > 0 {, if o = 0 o, if o > 0 part, p) = 6. Denote by P 4 the set of all 4-paths in the game graph. The above definition satisfies that p P 4 part, p), for eah server and, in partiular, p P 4 part, p) =, for eah server of type /. Intuitively, we amortize the ontribution of a server to the soial ost over the 4-paths ontaining that server, and part, p) is the fration of the ontribution of that we assign to path p. In our proof of the stronger upper bound, we use the following two tehnial lemmas. Let ψ = and ξ = We define the funtions gx, y) = xy + + ξ)y ξx and hx, y) = 4ψ x + ψy. Lemma 8 For any non-negative integers x, y suh that either x or y, it holds that gx, y) hx, y). Proof: Define the funtion fx, y) = hx, y) gx, y) = ψ x ψy + ξ ) ψ + ξψ ξ ) y ξ ψ. 4

15 In order to prove the lemma, it suffies to show that fx, y) 0 for any non-negative integer values of x and y when either x or y. First, observe that if y, then y ξ ψ ξψ ξ.7, whih implies that ξψ ξ ) y ξ ψ 0. Hene, fx, y) 0 for ) any integer y. Also, fx, 0) = ψ x + ξ ψ ξ ψ 0 for any integer x 0. For y =, by straightforward alulations we obtain that the paraboli funtion fx, ) = ψ x + ξ ) ) ψ + ψξ ξ ξ ψ is positive for x = 0, negative for x = and equal to zero for x =. So, it is non-negative for any non-negative integer value of x besides. Lemma 9 For any 4-path p, it holds that part, p)gn, o ) part, p)hn, o ). p p Proof: We onsider a 4-path p with servers 0,, of type 0/, /, /o and a fourth server of type /o if o > 0. We distinguish between the ases o = 0 and o > 0. In the first ase, by simple alulations, we show that g0, )+g, )+ g, 0) h0, )+ h, ) + h, 0). In the seond ase, we have to show that g0, ) + g, ) + g, o ) + o 6 g, o ) h0, ) + h, ) + h, o ) + o 6 h, o ). By straightforward alulations, we obtain that g, o ) h, o ) g, ) h, ) = 0 for any o, and that g, o ) h, o ) g, ) h, ) for any o 0. So, the proof ompletes by showing that g0, ) + g, ) + 6 g, ) h0, ) + h, ) + 6h, ). By using the fat that n = o, the definition of funtions g and h, and Lemmas 8 and 9, we obtain that n n o + o ) = n o + + ξ)o ξn ) = gn, o ) = gn, o )part, p) + part, p) gn, o ) p P 4 p p P4 part, p) hn, o ) p P 4 p P4 = hn, o )part, p) + p hn, o ) = n + ψ 4ψ o whih yields that the prie of anarhy is n 4ψ o 4ψ = The analysis an be extended by onsidering games satisfying the κ-neighborhood property for κ > 4. We all a κ-path any direted path of at most κ nodes in the game graph starting with a server of type 0/ and having a server of type / as its seond node. A game satisfies the κ-neighborhood property if it satisfies the κ )-neighborhood property and for any 5

16 κ-path in the game graph, the κ-th node, if it exists, has in-degree κ and no self-loops. Again, it an be shown that for any game there exists a game satisfying the κ-neighborhood property having at least the same prie of anarhy. In order to upper-bound the prie of anarhy, we define part, p) whih denotes how muh server partiipates in the κ-path p and the funtions g κ x, y) = xy + + ξ κ )y ξ κ x and h κ x, y) = 4ψ κ x + ψ κ y. We seek for values of ψ κ and ξ κ so that the funtions g κ and h κ satisfy lemmas similar to Lemmas 8 and 4ψκ 9 that minimize 4ψ κ. Then, the analysis ontinues in the same way as in the proof above. We have implemented the proof in a C program for values κ = 5,..., 5. The bounds obtained are depited in Figure. κ Upper Bound Lower Bound Figure : Upper bounds obtained for κ = 4,..., 5. The third olumn has the lower bounds obtained by the onstrutions of [] with κ + levels. For κ = 4,..., 5, by appropriately defining part, p), the κ-paths that make the inequality of Lemma 9 tight onsist of the first κ servers with types in the following sequene 0/, /, /, /, 4/, 5/, 6/, 7/, 8/, 9/, 0/, /4, /4, /4, 4/5. These are essentially the lower bound onstrutions of []. For κ 4, suh a onstrution with κ + levels has the servers in the first κ levels to be of type in the above sequene while servers of the last level κ are of type κ/0. 4 Greedy load balaning In this setion we study greedy load balaning by fousing on servers with linear lateny funtions. Similarly to the ase of selfish load balaning, in the study of the ompetitiveness of greedy load balaning we onsider load balaning instanes in whih eah lient has at most two strategies. This is learly suffiient when proving lower bounds. In order to prove upper bounds, we observe that for any instane, there exists another instane with at most two strategies per lient for whih greedy has the same ompetitiveness. Given any load balaning instane, let O and N be the optimal assignment and the greedy assignment of the highest ost for this instane, respetively. The instane with the same lients and servers in whih eah lient has its strategies in O and N as strategies also has the same optimal assignment and the same greedy assignment and, onsequently the same ompetitiveness). We represent suh instanes as direted graphs alled greedy graphs) having a node for eah 6

17 server and a direted edge with timing information for eah lient; the diretion of eah edge is from the strategy of the lient in the optimal assignment to the strategy of the lient in the greedy assignment and the timing information denotes the time the lient appears. The ost of an assignment is again the total lateny. When eah server has been assigned n lients and its lateny funtion is f x) = α x + β, then the total ost of the greedy assignment equals ) α n + β n. As disussed in [], the greedy algorithm does not neessarily lead to equilibrium assignments. In the greedy algorithm, eah lient is essentially hoosing the best possible server at the time it makes its deision. When all servers have the same lateny funtion fx) = αx + β, eah lient is simply hoosing the server with the minimum number of lients. 4. Different servers First, we show that the upper bound of [] for different servers is tight. Theorem 0 For any ϵ > 0, greedy load balaning has ompetitiveness at least 7/ ϵ. Proof: We first present an instane that yields a lower bound arbitrarily lose to 5. We onstrut an instane I k α, t) represented by a greedy graph whih is a omplete binary tree with k levels 0,,..., k and with its edges direted towards the root. Denote by S i the set of servers at level i. The root R has lateny funtion f R x) = α R x = αx and the lateny funtions f s x) = α s x for the other nodes are defined as follows: For i = 0,..., k, given a server s at level i, its left hild ls) has α ls) = α s /5 and the right hild rs) has α rs) = α s /5. Given a server s at level k, its left hild ls) has α ls) = α s and its right hild rs) has α rs) = α s. These definitions yield s S i α s = α4/5) i for i = 0,..., k, and s S k α s = 4α4/5) k. Given any non-leaf server s, the lient onneting s with its left hild appears prior to the lient onneting s with its right hild and, if s is not the root, the lient onneting s to its parent appears after the lients onneting s with its hildren. The lient onneting the root with its right hild has timing t and is the lient that arrives last. Consider the assignment where eah lient selets the server orresponding to the node loser to the root. We will show that this is a greedy assignment. Indeed, onsider the two lients and onneting a server s to ls) and rs), respetively, and reall that appears before and, furthermore, the lient onneting s to its parent if s R) appears after. We first onsider the ase when ls) and rs) are leaves. When appears, both ls) and s have zero lients and the same lateny funtions, and, therefore, we an assign to s, sine the inrease of the ost is α s for both hoies. Moreover, when appears, we an also assign it to s sine rs) has zero lients and lateny funtion f rs) x) = α s x and s has one lient and lateny funtion f s x) = α s x. Thus, the inrease in the ost is α s for both hoies. Now, assume that ls) and rs) are not leaves. When appears, server ls) has already two lients and lateny funtion f ls) x) = α s x/5, while server s has zero lients and lateny funtion f s x) = α s x. Therefore, we an assign to s, sine the inrease in the ost is α s for both hoies. Moreover, when appears, we an also assign it to s sine rs) has two lients and lateny funtion f rs) x) = α s x/5, while server s has one lient and lateny funtion f s x) = α s x. Thus, the inrease in the ost is α s for both hoies. In order to bound the optimal ost it suffies to onsider the assignment of eah lient to the server whih is loser to the leaves. Denote by opti k α, t)) and gri k α, t)) the optimal ost and the ost of the greedy assignment of an instane I k α, t), respetively. We have opti k α, t)) k i= s S i α s = 4α and gri k α, t)) = 4 k i=0 s S i α s = 0α 4/5) k ). 7

18 We will now use the instane desribed above to obtain the 7/ lower bound. We onstrut a greedy graph onsisting of a omplete binary tree with k levels and with its edges direted towards the root. Denote by S i the set of servers of level i. The root R has lateny funtion f R x) = x. The lateny funtions for the other servers are defined as follows. For i = 0,..., k, given a server s at level i, its left hild ls) has α ls) = α s /7 and the right hild rs) has α rs) = 5α s /7. Given a server s at level k, its left hild ls) has α ls) = α s /5 and its right hild rs) has α rs) = α s. These definitions yield s S i α s = 8/7) i for i = 0,..., k, and s S k α s = 8 5 8/7)k. Given any server s whih is not a leaf, the lient onneting s to its left hild ls) has timing t ls) and the lient onneting s to its right hild rs) has timing t rs) with t ls) < t rs). If s is not the root, the lient onneting s to its parent appears after the lients onneting s to its hildren. We augment the instane as follows: For eah leaf s of the binary tree onneted with its parent node through an edge of timing t, we inlude a opy I s, k of instane I k α s /5, t ) and a opy I s, k of instane I k α s /5, t ) whose roots R s, and R s, are onneted through edges of timing t and t with s. For eah non-leaf node s, we inlude a opy Ik s of instane I k α s /5, t ls) ) whose root R s is onneted through an edge of timing t ls) with s, where t ls) is the timing of the edge onneting s with its left hild in the binary tree. The onstrution is depited in Figure 4. α t t α/5 α/5 α/5 9α/ α t /7 /5 t t 5/ α /49 5/49 /7 t α/5 α α α α/5 t t α/5 9α/5 α/5 α/5 Figure 4: The onstrution in the proof of Theorem 0. The instane Iα, t) is depited at the left and is used as a triangle in the onstrution at the right part. Consider the assignment where eah lient selets the server orresponding to the node loser to the root. We will show that this is a greedy assignment by arguing about the hoies of all different sets of lients. We begin by onsidering lients orresponding to edges inside the opies of Ik s when s in not a leaf and Is, k and I s, k when s is a leaf). Clearly, sine those edges arrive prior to the edges onneting the opies to the binary tree and by the disussion above, the assignment of those edges to the nodes loser to the root is a valid outome of the greedy algorithm. We now examine lients orresponding to edges outside the opies. Consider lient orresponding to an edge e = R s,, s) onneting the root R s, of opy 8

19 to a leaf s of the binary tree. R s, has lateny funtion f Rs, x) = α s x/5 and already two lients, while s has lateny funtion f s x) = α s x and no lients. So, the inrease in the total ost will be equal to α s for both hoies of lient. Similary, onsider lient orresponding to an edge e = R s,, s) onneting the root R s, of opy I s, k to a leaf s of the binary tree. R s, has lateny funtion f Rs, x) = α s x/5 and already two lients, while s has lateny funtion f s x) = α s x and one lient. So, the inrease in the total ost will be equal to α s for both hoies of lient. Now, onsider lient orresponding to an edge e = R s, s) onneting the root R s of opy Ik s to a non-leaf s of the binary tree. R s has lateny funtion f Rs x) = α s x/5 and already two lients, while s has lateny funtion f s x) = α s x and no lients. So, the inrease in the total ost will be equal to α s for both hoies of lient. Until now, we have argued about lients onneting opies of instane I k α, t) to the binary tree. We proeed to handle the lients onneting nodes of the binary tree to their parents. Consider lient 4 orresponding to an edge e = ls), s) onneting the left hild ls) to its parent s, for the ase where ls) is a leaf of the binary tree. ls) has lateny funtion f ls) x) = α s x/5 and already two lients, while s has lateny funtion f s x) = α s x and one lient. So, the inrease in the total ost will be equal to α s for both hoies of lient 4. Similarly, onsider lient 5 orresponding to an edge e = rs), s) onneting the right hild rs) to its parent s, for the ase where rs) is a leaf of the binary tree. Both rs) and s have already two lients and the same lateny funtions. Therefore, the inrease in the total ost will be equal to 5α s for both hoies of lient 5. Moreover, onsider lient 6 orresponding to an edge e = ls), s) onneting the left hild ls) to its parent s, for the ase where ls) is not a leaf of the binary tree. ls) has lateny funtion f ls) x) = α s x/7 and already three lients, while s has lateny funtion f s x) = α s x and one lient. So, the inrease in the total ost will be equal to α s for both hoies of lient 6. Finally, onsider lient 7 orresponding to an edge e = rs), s) onneting the right hild rs) to its parent s, for the ase where rs) is not a leaf of the binary tree. rs) has lateny funtion f rs) x) = 5α s /7 and already three lients, while s has lateny funtion f s x) = α s x and already two lients. Therefore, the inrease in the total ost will be equal to 5α s for both hoies of lient 7. In order to bound the optimal ost it suffies to onsider the assignment of eah lient to the server whih is loser to the leaves. In this assignment, eah non-root server is assigned I s, k one lient. Therefore, the ost of the servers that form Ik R is /5 + optir k ), the ost of eah non-root node s is α s, the ost of the servers that form Ik s is α s/5+optik s ), while the ost of the servers that form I s, k and I s, k is α s /5 + opti s, k ) and α s/5 + opti s, k ), respetively. Denote by opt and gr the optimal ost and the ost of the greedy assignment of the onstrution. We have that opt k 5 + optir k ) k 5 + k = + i= s S i i= s S i α s + 5 i= s S i = 8/7) k 5, ) 6αs 5 + optis k ) + 6αs 5 + 4α s 5 s S k α s ) + s S k s S k ) 9αs 5 + optis, k ) + optis, k ) 9αs 5 + 4α s 5 + α ) s 5 9