Nash Social Welfare in Selfish and Online Load Balancing
Vittorio Bilò, Gianpiero Monaco, Luca Moscardelli, Cosimo Vinci
NNash Social Welfare in Selfish and Online Load Balancing (cid:63)
Vittorio Bil , Gianpiero Monaco , Luca Moscardelli , and Cosimo Vinci University of Salento, Italy University of L’Aquila, Italy University of Chieti-Pescara, Italy Gran Sasso Science Institute, Italy
Abstract.
In load balancing problems there is a set of clients, each wishing to select a resource froma set of permissible ones, in order to execute a certain task. Each resource has a latency function,which depends on its workload, and a client’s cost is the completion time of her chosen resource. Twofundamental variants of load balancing problems are selfish load balancing (aka. load balancing games ),where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and onlineload balancing , where clients appear online and have to be irrevocably assigned to a resource without anyknowledge about future requests. We revisit both selfish and online load balancing under the objectiveof minimizing the
Nash Social Welfare , i.e., the geometric mean of the clients’ costs. To the best ofour knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash SocialWelfare has not been considered so far as a benchmarking quality measure in load balancing problems.We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitiveratio of the greedy algorithm under very general latency functions, including polynomial ones. For thisparticular class, we also prove that the greedy strategy is optimal as it matches the performance of anypossible online algorithm.
Keywords:
Congestion games · Nash social welfare · Pure Nash equilibrium · Price of anarchy · Onlinealgorithms.
In load balancing problems there is a set of clients, each wishing to select a resource from a set of permissibleones, in order to execute a certain task. Each resource has a latency function, which depends on its workload,and a client’s cost is the completion time of her chosen resource. These problems stand at the foundations ofthe Theory of Computing and have been studied under a variety of objective functions, such as the maximumclient’s cost (aka. the makespan) [40,41,42,48] and the average weighted client’s cost (see [26] for an excellentsurvey).Two extensively studied variants of load balancing problems are selfish load balancing [61] (aka. loadbalancing games ) and online load balancing [40]. Selfish load balancing, where clients are non-cooperativeselfish players aimed at minimizing their own cost solely, constitutes a notable subclass of weighted congestiongames [53] and, as such, enjoys some nice theoretical properties. For instance, they always admit pure NashEquilibria [43]. Moreover, under the assumption that all tasks have unitary weight ( unweighted congestiongames ), any best-response dynamics converges to a pure Nash Equilibrium in polynomial time [1]. In onlineload balancing, instead, clients appear online and have to be irrevocably assigned to a resource without anyknowledge about future requests.Interpreting the set of clients of a load balancing problem as a society and adopting the terminologyof welfare economics, the makespan and the average weighted client’s cost objective functions get called,respectively, the egalitarian and the utilitarian social function. In the case of unweighted tasks, the egalitarianfunction is defined as max i x i , and the utilitarian one is defined as n (cid:80) i x i , where n is the number of clientsand x = ( x , x , ... ) is the vector encoding the clients’ costs. Another interesting social function is the NashSocial Welfare (NSW) [51], which is defined as ( (cid:81) i x i ) n , i.e., as the geometric mean of the clients’ costs.These definitions naturally extend to the more general case of weighted tasks (see Section 2). (cid:63) This work was partially supported by the Italian MIUR PRIN 2017 Project ALGADIMAR Algorithms, Games,and Digital Markets. a r X i v : . [ c s . G T ] J u l V. Bil et al.
The NSW is a celebrated welfare measure in many settings, such as Fisher markets [11,19] and fairdivision [4,20,17,23,24,29,39], as it satisfies a set of interesting properties (most of the aforementioned papersfocus on fairness properties such as envy-freeness and maximin share, and Pareto optimality) and achievesa balanced compromise between the equity of the egalitarian social welfare function and the efficiency of theutilitarian one. We notice that when x i >
0, for any i = 1 , . . . , n , this balance holds regardless of whetherthe objective is maximizing or minimizing the NSW. The case where each x i can be either a positive ornegative value has been considered in [4]. In the context of congestion games we do not take into accountenvy-freeness and maximin share, however, it is easy to see that an outcome that minimizes the NSW isPareto optimal. Another interesting motivation for considering the NSW in load balancing comes from thefollowing observation. An alternative reasonable way to define a client’s cost can come by taking the ratiobetween the completion time of her chosen resource and the completion time she could obtain when beingthe only client in the system (i.e., when she is the unique user of the fastest resource). This definition avoidssituations where the cost of a specific client determines almost completely the value of the social welfare.This happens, for instance, when there is a client i owing a highly time-consuming task. Here, both theutilitarian and the egalitarian social welfare end up depending on the cost of i , thus almost neglecting theother clients’ costs. In this setting, the NSW is the proper metric to use. More generally, the NSW is theonly correct mean to use when averaging normalized results, that is, results that are presented as ratios toreference values [34]. It is important to emphasize the scale-freeness of the NSW in load balancing problems,that is, the NSW is a robust social welfare function as its analysis is not affected by this change in thedefinition of a client’s cost. The literature concerning the efficiency of Nash equilibria in selfish load balancingis highly tied with that of its superclass of congestion games. In the following, we first focus on results forthe mostly studied case of the utilitarian social welfare . In this setting, it is assumed that all clients selectingthe same resource experience the same cost.The efficiency of pure Nash equilibria in congestion games has been first considered in [6] and [27],where it has been independently shown that the price of anarchy is 5 / √ / / . / egalitarian social welfare . The study of the price of anarchy was initiated in[47], where weighted congestion games of m parallel links with linear latency functions are considered. Theprice of anarchy for the egalitarian social welfare is Θ ( log m log log m ). The lower bound was shown in [47] and theupper bound in [30]. For load balancing games, the price of anarchy is Θ ( log n log log n ) where n is the number of ash Social Welfare in Load Balancing 3 players [36], while for unweighted congestion games is Θ ( √ n ) [27]. [55] proves that the price of anarchy ofnon-atomic congestion games with general non-decreasing latency function is Ω ( n ). Online Load Balancing.
The performance of greedy load balancing with respect to the utilitarian socialwelfare and under affine latency functions has been studied in [7,22,59]. [7] considers a more general modelwhere each client has a load vector denoting her impact on each resource (i.e., how much her assignment to aresource will increase its load) and the objective is to minimize the L p norm of the load of the resources. Theirresults, together with [22], imply a competitive ratio of the greedy algorithm equal to 3 + 2 √ ≈ . / √ ≈ . √ m identical machines,[40] shows that the greedy algorithm achieves a competitive ratio of exactly 2 − m and this bound is proventhe best possible one for m = 2 , . m and no algorithm can achieve a competitive ratio bettern than 1 .
88 [58]. For relatedmachines, [5,8] show a tight bound of log m , while [21] considers the case of unrelated machines with theobjective of minimizing the norm of the machines loads. We revisit both selfish and online load balancing under the objective of minimizing the NSW. To the best ofour knowledge, this is the first work adopting the NSW as a benchmarking quality measure in load balancingproblems. We analyze the price of anarchy [47] of pure Nash equilibria (the loss in optimality due to selfishbehavior) and the competitive ratio of online algorithms (the loss in optimality due to lack of information)under very general latency functions. These questions have been widely addressed under the utilitarian andegalitarian functions, but never under the NSW.We notice that by adopting the NSW as new metric, we are not going to modify the set of Nash equilibriabut only the social values. The main difference between the NSW and the classical notion of utilitarian socialwelfare consists in the fact that, while in the latter the players’ costs are summed, in the former they aremultiplied. This may lead to think that, by turning the costs into their logarithms, a classical utilitariananalysis can be easily adapted to deal with the NSW. Actually, this is not the case. In fact, on the one hand,using this idea for bounding a performance ratio (e.g., the price of anarchy or the competitive ratio), oneobtains a bound on the ratio between two logarithms (each one having the product of the players’ costsas argument). On the other hand, we are interested in bounding the ratio between the argument of theselogarithms, and there is no direct correlation between these two ratios (notice that logarithm of the latterratio is equal to the difference between the corresponding utilitarian social costs, and therefore it is notrelated to the former one). Thus, the analysis of the NSW requires different proof arguments. In order tohave another evidence of this fact, it is worth noticing that the results obtained for the NSW substantiallydiffer from the ones holding for the utilitarian social function, not only from a quantitative point of view,but also from a qualitative one. In fact, while it is well known (see [22]) that for the utilitarian social welfarethe simpler combinatorial structure of load balancing games does not improve the price of anarchy of generalcongestion games, our Theorem 10 (deferred to the appendix) and Corollary 1 show that, for the NSW, evenfor the case of linear latency functions, the price of anarchy drops from n to 2.All upper bounds shown in this paper are quite general, given that they hold for any non-decreasing andpositive latency function. Moreover, the provided matching lower bounds hold for latency functions verifying V. Bil et al. mild assumptions; it is worth to remark that they are satisfied by the well studied class of polynomial latencyfunctions and by many other ones.In particular, Theorem 1 provides an upper bound to the price of anarchy for the case of weighted loadbalancing games, while Theorem 2 gives a matching lower bound. Similarly, we focus on unweighted games(a special case of weighted ones) by providing tight bounds that, in general, are lower than the ones thatcan be obtained for weighted games (see Subsection 3.2). However, Corollaries 1 (or 2) and 3 show that,when considering polynomial latency functions of degree p , the two analyses (for weighted games and forunweighted ones) give the same tight bound of 2 p . Furthermore, when considering weighted games, the tightbound of 2 p holds even for symmetric games (Corollary 1) and for games with identical resources (Corollary2). We also provide a tight analysis holding for non-atomic games (see Subsection 3.3); for the case ofpolynomial latency functions of degree p , Corollary 4 shows that the price of anarchy is (cid:16) e e (cid:17) p (cid:39) (1 . p .For the online setting, we analyze the greedy algorithm that assigns every client to a resource minimizingthe total cost of the instance revealed up to the time of its appearance. We provide a tight analysis of thecompetitive ratio of the greedy algorithm, and we show that, when considering polynomial latency functionsof degree p , there exists no online algorithm achieving a competitive ratio better than the one of the greedyalgorithm, that is equal to 4 p (see Section 4). In Table 1, we consider the case of polynomial latency functions,and we compare the performance under the NSW with that under the utilitarian social welfare studied insome previous works.The rest of the paper is structured as follows. Section 2 introduces the model. Sections 3 and 4 are devotedto the performance analysis of the price of anarchy and of the competitive ratio, under the selfish and theonline setting, respectively. Finally, in Section 5 we give some conclusive remarks and state some interestingopen problems. Due to lack of space, some proofs are sketched or omitted, and are left to the appendix. NSW USWWeighted 2 p ( Φ p ) p +1 ∼ Θ (cid:16) p log( p ) (cid:17) p +1 , [2]Unweighted 2 p ( k +1) p +1 − k p +1 ( k +2) p ( k +1) p +1 − ( k +2) p +( k +1) p − k p +1 ∼ Θ (cid:16) p log( p ) (cid:17) p +1 , [2]Non-atomic (cid:16) e e (cid:17) p (cid:16) − p ( p + 1) − ( p +1) /p (cid:17) − ∼ Θ (cid:16) p log( p ) (cid:17) , [54]Online 4 p (2 / ( p +1) − − ( p +1) ∼ Θ ( p ) p +1 , [21] Table 1.
Tight bounds on the performance of load balancing with polynomial latency functions of maximum degree p ,under the NSW and the utilitarian social welfare (USW). Φ p denotes the unique solution of equation x p +1 = ( x + 1) p ,and k := (cid:98) Φ p (cid:99) . We observe that the performance under the NSW case is definitely better (even asymptotically) thanthat under the USW case, except for the non-atomic setting. Given k ∈ N , let [ k ] := { , , . . . , k } . A class C of functions is called ordinate-scaling if, for any f ∈ C and α ≥
0, the function g such that g ( x ) = αf ( x ) for any x ≥
0, belongs to C ; abscissa-scaling if, for any f ∈ C and α ≥
0, the function g such that g ( x ) = f ( αx ) for any x ≥
0, belongs to C ; all-constant-including ifit contains all the constant functions (i.e., all functions f such that f ( x ) = c for some c > unbounded-including if all the latency functions f , except for the constant ones, verify lim x →∞ f ( x ) = ∞ . Let P ( p )denote the class of polynomial latencies of maximum degree p , i.e., the class of functions f ( x ) = (cid:80) pd =0 α d x d ,with α d ≥ d ∈ [ p ] ∪ { } and α d > d ∈ [ p ] ∪ { } . A function f is quasi-log-convex if x ln( f ( x )) is convex.We first deal with selfish load balancing , by defining load balancing games , and then we turn our attentionto the online setting. ash Social Welfare in Load Balancing 5 A weighted (atomic) load balancing game , or load balancing game forbrevity, is a tuple LB = ( N, R, ( (cid:96) j ) j ∈ R , ( w i ) i ∈ N , ( Σ i ) i ∈ N ) , where N is a set of n ≥ R is a finite set of resources, (cid:96) j : R > → R > is the (non-decreasing and positive) latency functionof resource j ∈ R , and, for each i ∈ N , w i > i and Σ i ⊆ R (with Σ i (cid:54) = ∅ ) is herset of strategies (or admissible resources). For notational simplicity, we assume that each latency function (cid:96) verifies (cid:96) (0) = 0.An unweighted load balancing game is a weighted load balancing game with unitary weights. A symmetricweighted load balancing game is a congestion game in which each player can select all the resources, i.e., Σ i = R for any i ∈ N .Given a class C of latency functions, let ULB ( C ) be the class of unweighted load balancing games, WLB ( C )be the class of weighted load balancing games, and SWLB ( C ) be the class of weighted symmetric loadbalancing games, all having latency functions in the class C . We say that resources are identical if all of themhave the same latency function. Non-Atomic Load balancing Games.
The counterpart of the class of atomic load balancing games is thatof non-atomic load balancing games [10,52,62]: these games are a good approximation for atomic ones whenplayers become infinitely many and the contribution of each player to social welfare becomes infinitesimallysmall. A non-atomic load balancing game is a tuple
NLB = (
N, R, ( (cid:96) j ) j ∈ R , ( r i ) i ∈ N , ( Σ i ) i ∈ N ), where N is a setof n ≥ types of players, R is a finite set of resources, (cid:96) j : R > → R > is the (non-decreasing and positive)latency function of resource j ∈ R ; moreover, given i ∈ N , r i ∈ R ≥ is the amount of players of type i and Σ i ⊆ R is the set of strategies of every player of type i .Given a class C of latency functions, let NLB ( C ) be the class of non-atomic load balancing games, and SNLB ( C ) be the class of symmetric non-atomic load balancing games, all having latency functions in theclass C . Strategy Profiles and Cost Functions.
In atomic load balancing games, a strategy profile is an n -tuple σ = ( σ , . . . , σ n ), where σ i ∈ Σ i is the resource chosen by each player i ∈ N in σ . Given a strategy profile σ , let k j ( σ ) := (cid:80) i ∈ N : σ i = j w i be the congestion of resource j ∈ R in σ , and let cost i ( σ ) := (cid:96) σ i ( k σ i ( σ )) bethe cost of player i ∈ N in σ .In non-atomic load balancing games, a strategy profile is an n -tuple ∆ = ( ∆ , . . . , ∆ n ), where ∆ i : Σ i → R ≥ is a function denoting, for each resource j ∈ Σ i , the amount ∆ i ( j ) of players of type i selecting resource j , so that (cid:80) j ∈ Σ i ∆ i ( j ) = r i . Observe that ∆ i ( j ) = 0 if j / ∈ Σ i . For a strategy profile ∆ , the congestion ofresource j ∈ R in ∆ , denoted as k j ( ∆ ) := (cid:80) i ∈ N ∆ i ( j ), is the total amount of players using resource j in ∆ and its cost is given by cost j ( ∆ ) = (cid:96) j ( k j ( ∆ )). The cost of a player of type i selecting a resource j ∈ Σ i isequal to cost j ( ∆ ) and each player aims at minimizing it. Nash Social Welfare.
In atomic load balancing games, the
Nash Social Welfare (NSW) of a strat-egy profile σ is defined as: NSW ( σ ) := (cid:0)(cid:81) i ∈ N cost i ( σ ) w i (cid:1) (cid:80) i ∈ N wi . Using the previous definition, for un-weighted games we get
NSW ( σ ) = (cid:0)(cid:81) i ∈ N cost i ( σ ) (cid:1) n . Given a strategy profile σ , let R ( σ ) := { j ∈ R : k j ( σ ) > } . For weighted load balancing games we get: NSW ( σ ) = (cid:0)(cid:81) i ∈ N cost i ( σ ) (cid:1) (cid:80) i ∈ N wi = (cid:16)(cid:81) j ∈ R ( σ ) (cid:96) j ( k j ( σ )) k j ( σ ) (cid:17) (cid:80) i ∈ N wi = (cid:16)(cid:81) j ∈ R ( σ ) (cid:96) j ( k j ( σ )) k j ( σ ) (cid:17) (cid:80) j ∈ R ( σ ) kj ( σ ) .Let SP ( LB ) be the set of strategy profiles of an atomic load balancing game LB . An optimal strategyprofile σ ∗ ( LB ) of a load balancing game LB is a strategy profile σ ∗ ∈ arg min σ ∈ SP ( LB ) NSW ( σ ), i.e., a strategyprofile minimizing the NSW.Analogously, for the non-atomic setting, we have NSW ( ∆ ) = (cid:16)(cid:81) j ∈ R ( ∆ ) cost j ( ∆ ) k j ( ∆ ) (cid:17) (cid:80) j ∈ R ( ∆ ) kj ( ∆ ) , where R ( ∆ ) := { j ∈ R : k j ( ∆ ) > } . Let SP ( NLB ) be the set of strategy profiles of a non-atomic loadbalancing game
NLB . An optimal strategy profile ∆ ∗ ( NLB ) of a load balancing game
NLB is a strategyprofile ∆ ∗ ∈ arg min ∆ ∈ SP ( NLB ) NSW ( ∆ ), i.e., a strategy profile minimizing the NSW. Pure Nash Equilibria and their Efficiency.
In the atomic setting, for a given strategy profile σ , let( σ − i , σ (cid:48) i ) := ( σ , σ , . . . , σ i − , σ (cid:48) i , σ i +1 , . . . , σ n ), i.e., a strategy profile equal to σ , except for strategy σ (cid:48) i . A pure Nash equilibrium is a strategy profile σ such that cost i ( σ ) ≤ cost i ( σ − i , σ (cid:48) i ) for any σ (cid:48) i ∈ Σ i and i ∈ N ,i.e., a strategy profile in which no player can improve her cost by unilateral deviations. Let PNE ( LB ) be the V. Bil et al. set of pure Nash equilibria of a load balancing game LB . The Nash price of anarchy of LB is defined as: NPoA ( LB ) = sup σ ∈ PNE ( LB ) NSW ( σ ) NSW ( σ ∗ ( LB )) Given a class G of load balancing games, the Nash price of anarchy of G is defined as NPoA ( G ) = sup LB ∈G NPoA ( LB ). In the non-atomic setting, a pure Nash equilibrium isa strategy profile ∆ such that, for any player type i ∈ N , resources j, j (cid:48) ∈ Σ i such that ∆ i ( j ) > cost j ( ∆ ) ≤ cost j (cid:48) ( ∆ ) holds, that is, an outcome of the game in which no player can improve her situationby unilaterally deviating to another strategy. The Nash price of anarchy of a non-atomic game
NLB (denotedas
NPoA ( NLB )) is defined as in the atomic setting, and again, given a class G of non-atomic load balancinggames, the Nash price of anarchy of G is defined as NPoA ( G ) = sup NLB ∈G NPoA ( NLB ). We now introduce online load balancing. There is a natural correspondence between a load balancing gameand an instance of the online load balancing problem. When dealing with the online setting, as usual inthe literature, we adopt a different nomenclature. In particular, an instance I of the online load balancingproblem is a tuple I = ( N, R, ( (cid:96) j ) j ∈ R , ( w i ) i ∈ N , ( Σ i ) i ∈ N ) , where N = [ n ] is a set of n ≥ clients , R is a finiteset of resources, (cid:96) j : R > → R > is the (non-decreasing and positive) latency function of resource j ∈ R , and,for each i ∈ N , w i > i and Σ i ⊆ R (with Σ i (cid:54) = ∅ ) is her set of admissible resources .Furthermore, in the online setting an assignment of clients to resources is called state: A state is an n -tuple σ = ( σ , . . . , σ n ), where σ i ∈ Σ i ⊆ R is the resource assigned to player i ∈ N in σ . As in load balancinggames, given a class of latency latency functions C , let WLB ( C ) denote class of load balancing instances withlatency functions in C .The NSW of a state and the optimal state are defined analogously to the selfish load balancing setting. The online setting. In online load balancing , clients appear in online fashion, in consecutive steps ; when aclient appears, an irrevocable decision has to be taken in order to assign it to a resource. We assume w.l.o.g.that clients appear in increasing order, i.e., client i ∈ [ n ] appears before client j ∈ [ n ] if and only if i < j .More formally, for any i ∈ [ n ], an online algorithm has to assign client i to a resource being admissible for itwithout the knowledge of the future clients i + 1 , i + 2 , . . . ; the assignment of client i decided by the algorithmat step i cannot be modified at later steps.Notice that at each step i > i to the instance of step i − Competitive Ratio.
Following the standard performance measure in competitive analysis, we evaluate theperformance of an online algorithm in terms of its competitiveness (or competitive ratio ).An online algorithm A is c -competitive on instance I if the following holds: Let σ and σ ∗ be the statecomputed by algorithm A and the optimal state for I , respectively. Then, NSW ( σ ) ≤ c · NSW ( σ ∗ ). Thecompetitive ratio CR A ( I ) of algorithm A on instance I is the smallest c such that A is c -competitive on I [18].Given a class I of load balancing instances, the competitive ratio CR A ( I ) of Algorithm A on I is simplygiven by the maximum competitive ratio of A over all instances I ∈ I ,i.e., CR A ( I ) = sup I ∈I CR A ( I ). Greedy algorithm.
A natural algorithm proposed in [7] for this problem is to assign each client to theresource yielding the minimum increase to the social welfare (ties are broken arbitrarily). This results to greedyassignments . Therefore, given an instance of online load balancing, an assignment of clients to resources iscalled a greedy assignment if the assignment of a client to a resource minimizes the total cost of the instancerevealed up to the time of its appearance.
In this section we focus on selfish load balancing. In particular, in Subsection 3.1 we deal with the analysisof the price of anarchy in weighted load balancing games, in Subsection 3.2 we consider the subclass ofunweighted load balancing games, while in Subsection 3.3 we analyze the price of anarchy of non-atomicload balancing games.
NPoA for Weighted Load Balancing Games
We first provide an upper bound to the Nash price of anarchy of weighted load balancing games. ash Social Welfare in Load Balancing 7
Theorem 1.
Let C be a class of latency functions. The Nash price of anarchy ofweighted load balancing games with latency functions in C is NPoA ( WLB ( C )) ≤ sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o . Proof.
Let LB ∈ WLB ( C ) be a weighted load balancing game with latency functions in C , and let σ and σ ∗ be a worst case pure Nash equilibrium and an optimal strategy profile of LB , respectively. Let k j denote k j ( σ ) and o j denote k j ( σ ∗ ).Since σ is a pure Nash equilibrium, we have that cost i ( σ ) ≤ cost i ( σ − i , σ ∗ i ). Thus, we get (cid:81) i ∈ N cost i ( σ ) w i ≤ (cid:81) i ∈ N cost i ( σ − i , σ ∗ i ) w i . Since cost i ( σ ) = (cid:96) σ i ( k σ i ) and cost i ( σ − i , σ ∗ i ) ≤ (cid:96) σ ∗ i ( k σ ∗ i + w i ),it holds that (cid:81) i ∈ N cost i ( σ ) w i = (cid:81) i ∈ N (cid:96) σ i ( k σ i ) w i = (cid:81) j ∈ R ( σ ) (cid:96) j ( k j ) (cid:80) i : j = σi w i = (cid:81) j ∈ R ( σ ) (cid:96) j ( k j ) k j and (cid:81) i ∈ N cost i ( σ − i , σ ∗ i ) w i ≤ (cid:81) i ∈ N (cid:96) σ ∗ i ( k σ ∗ i + w i ) w i ≤ (cid:81) i ∈ N (cid:96) σ ∗ i ( k σ ∗ i + o σ ∗ i ) w i = (cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( k j + o j ) (cid:80) i : j = σ ∗ i w i = (cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( k j + o j ) o j . By putting together the above inequalities we get (cid:89) j ∈ R ( σ ) (cid:96) j ( k j ) k j = (cid:89) i ∈ N cost i ( σ ) w i ≤ (cid:89) i ∈ N cost i ( σ − i , σ ∗ i ) w i ≤ (cid:89) j ∈ R ( σ ∗ ) (cid:96) j ( k j + o j ) o j . (1)By exploiting the properties of the logarithmic function and by using (1), we obtain ln ( NPoA ( LB )) = ln (cid:16)(cid:81) j ∈ R ( σ ) (cid:96) j ( k j ) k j (cid:17) (cid:80) i ∈ N wi (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( o j ) o j (cid:17) (cid:80) i ∈ N wi ≤ ln (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( k j + o j ) o j (cid:17) (cid:80) i ∈ N wi (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( o j ) o j (cid:17) (cid:80) i ∈ N wi = (cid:80) j ∈ R ( σ ∗ ) o j (ln( (cid:96) j ( k j + o j )) − ln( (cid:96) j ( o j ))) (cid:80) i ∈ N w i , (2) Since (cid:80) i ∈ N w i = (cid:80) j ∈ R k j = (cid:80) j ∈ R o j , we have that (2) is upper bounded by the optimal solution of thefollowing optimization problem on some new linear variables ( α j ) j ∈ R (as (2) is the solution obtained bysetting α = 1 for each j ∈ R ): max (cid:80) j ∈ R ( σ ∗ ) α j o j (ln( (cid:96) j ( k j + o j )) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R α j k j (3)s.t. (cid:88) j ∈ R α j k j = (cid:88) j ∈ R α j o j , α j ≥ ∀ j ∈ R. Fact 1
The maximum value of the optimization problem considered in (3) is at most sup k ≥ o > ,o >k ≥ ,f ,f ∈C ( o − k ) o (ln( f ( k + o )) − ln( f ( o )))+( k − o ) o (ln( f ( k + o )) − ln( f ( o ))) k o − k o . By Fact 1, and by continuing from (2), we have that the upper bound provided in Fact 1 ishigher or equal than ln(
NPoA ( LB )). Thus, by exponentiating such inequality, we get NPoA ( LB ) ≤ sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o . Hence, by the arbitrariness of LB ∈ WLB ( C ), theclaim follows. (cid:117)(cid:116) In the following theorem we show that the upper bound derived in Theorem 1 is tight under mild assumptionson the latency functions.
Theorem 2.
Let C be a class of latency functions.(i) If C is abscissa-scaling and ordinate-scaling, then NPoA ( WLB ( C )) ≥ sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o . (ii) If C is abscissa-scaling, ordinate-scaling, and unbounded-including, the previous inequality holds evenfor symmetric weighted load balancing games. V. Bil et al.
Proof (Sketch of the proof ).
We show part (ii) of the claim only (the proof of part (i) resorts tosimilar arguments and is left to the appendix). Let us assume that C is abscissa-scaling, ordinate-scaling, and unbounded-including. In order to prove part (ii), we equivalently show that for any M < sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o there exists a game LB ∈ WLB ( C ) suchthat NPoA ( LB ) > M .Let f , f ∈ C , k , k , o , o ≥ k ≥ o > , o > k ≥
0, and a sufficiently small (cid:15) > (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o > M + (cid:15). Let f, g ∈ C be such that f ( x ) := f ( o x )and g ( x ) := f ( o x ), and let k := k /o and h := k /o . Since (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o = (cid:16) f ( k +1) f (1) (cid:17) − hk − h (cid:16) g ( h +1) g (1) (cid:17) k − k − h we have that (cid:18) f ( k + 1) f (1) (cid:19) − hk − h (cid:18) g ( h + 1) g (1) (cid:19) k − k − h > M + (cid:15) , for some f, g ∈ C , k ≥ , and h < . (4)Observe that f and g can be chosen in such a way that they are non-constant functions. Indeed, if one ofthem is constant, it is sufficient replacing it with an arbitrary non-constant function, so that (4) holds as well.Since C is unbounded-including and f, g are non-constant, we have that lim x →∞ f ( x ) = lim x →∞ g ( x ) = ∞ .We consider the case h > h = 0 is analogue and is left to the appendix). Given twointegers m ≥ s ≥
1, let LB ( m, s ) be a symmetric weighted load balancing game where the resources arepartitioned into 2 m groups R , R , R . . . , R m . Each group R j has s j − resources and the latency functionof each resource r ∈ R j is defined as (cid:96) r ( x ) := α j ˆ f j ( β j x ) withˆ f j := (cid:40) f if j ≤ m − g if j ≥ m , β j := (cid:40)(cid:0) sk (cid:1) j − if j ≤ m − (cid:0) sh (cid:1) j − m (cid:0) sk (cid:1) m − if m ≤ j ≤ m , (5) α j := (cid:16) f ( k ) f ( k +1) (cid:17) j − if j ≤ m − (cid:16) g ( h ) g ( h +1) (cid:17) j − m (cid:16) f ( k ) g ( h +1) (cid:17) (cid:16) f ( k ) f ( k +1) (cid:17) m − if m ≤ j ≤ m − g ( h ) g (1) (cid:16) g ( h ) g ( h +1) (cid:17) m − (cid:16) f ( k ) g ( h +1) (cid:17) (cid:16) f ( k ) f ( k +1) (cid:17) m − if j = 2 m . (6)The set of players N is partitioned into 2 m − N , N , . . . , N m − , and each group N j has s j playershaving weight w j := 1 /β j +1 . Let σ be the strategy profile in which, for any j ∈ [2 m − R j is selected by exactly s players of group N j (see Figure 1.a). One can show that, for any integer m ≥
3, there exists a sufficiently large s m such that σ is a pure Nash equilibrium of the game LB ( m, s m )(see the appendix for a complete proof).Now, let σ ∗ be the strategy profile of LB ( m, s m ) in which, for any j ∈ [2 m − R j +1 is selected by exactly one player of group N j (see Figure 1.b). By exploiting the definitions of α j , β j ,ˆ f j , w j , and N j , and by choosing a sufficiently large m , one can show that the following inequalities hold (seethe appendix for a complete proof): NSW ( σ ) NSW ( σ ∗ ) ≥ lim m →∞ (cid:18) (cid:81) m − j =1 ( α j ˆ f j ( β j s m w j ) ) | Nj | wj (cid:81) mj =2 ( α j ˆ f j ( β j w j − ) ) | Nj − | wj − (cid:19) (cid:80) m − j =1 | Nj | wj − (cid:15) = (cid:16) f ( k +1) f (1) (cid:17) − hk − h (cid:16) g ( h +1) g (1) (cid:17) k − k − h − (cid:15) > M + (cid:15) − (cid:15) = M, thus showing part (ii) of the claim. (cid:117)(cid:116) When considering functions belonging to the class P ( p ) of polynomials of maximum degree p , the followingtechnical lemma holds. Lemma 1. sup k ≥ o > ,o >k ≥ ,f ,f ∈P ( p ) (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o = 2 p . Given Lemma 1, and since the class of polynomial latency functions is ordinate-scaling, abscissa-scaling,and unbounded-including, the following corollary of Theorems 1 and 2 establishes the exact Nash price ofanarchy for polynomial latency functions. ash Social Welfare in Load Balancing 9 b b b R ( s resources) b bb bb b s b b b R ( s resources) b bb bb b R m − ( s m − resources) bbb b b b R m ( s m − resources) b b b (a) b bb bb b R b bb bb b b bb bb b b bb bb b b bb bb b b bb bb b b bb bb b b b b R ( s resources) b b b R ( s resources) R m − ( s m − resources) R m ( s m − resources) b b b (b) R bbb bbb bbb bbb bbb bbb bbb b b b b b b b b b Fig. 1.
The LB used in the proof of Theorem 2. Columns represent resources and squares represent players (number j inside a square means that the player belongs to group N j ). (a): a Nash equilibrium σ ; (b): the strategy profile σ ∗ . Corollary 1.
The Nash price of anarchy of weighted load balancing games with polynomial latency functions(even for symmetric games) of maximum degree p is NPoA ( WLB ( C )) = 2 p . When considering identical resources with polynomial latency functions, the price of anarchy does not de-crease, as shown in the following corollary of Theorem 2.
Corollary 2.
The Nash price of anarchy of weighted load balancing games with polynomial latency functionsof maximum degree p and identical resources is at least p . NPoA for Unweighted Load Balancing Games
We first provide an upper bound to the Nash price of anarchy of unweighted load balancing games.
Theorem 3.
Let C be a class of latency functions. The Nash price of anarchy of unweighted load balancinggames with latency functions in C is NPoA ( ULB ( C )) ≤ sup f ∈C ,k ∈ N ,o ∈ [ k ] (cid:16) f ( k +1) f ( o ) (cid:17) ok . We show that the upper bound derived in Theorem 3 is tight if the considered latency functions areordinate-scaling (the proof is deferred to the appendix). The following result for polynomial latency functionsholds.
Corollary 3.
The Nash price of anarchy of unweighted load balancing games with polynomial latency func-tions of maximum degree p is NPoA ( ULB ( C )) = 2 p . NPoA for Non-Atomic Load Balancing Games
We first provide an upper bound to the Nash price of anarchy of non-atomic load balancing games.
Theorem 4.
Let C be a class of latency functions. The Nash price of anarchy of non-atomic load balancinggames with latency functions in C is NPoA ( NLB ( C )) ≤ sup f ∈C ,k ≥ o> (cid:16) f ( k ) f ( o ) (cid:17) ok . We show that the upper bound derived in Theorem 4 is tight the considered latency functions areall-constant-including (the proof is deferred to the appendix). The following result for polynomial latencyfunctions holds.
Corollary 4.
The Nash price of anarchy of non-atomic load balancing games with polynomial latency func-tions of maximum degree p (even for symmetric games) is NPoA ( NLB ( P ( p ))) = NPoA ( SNLB ( P ( p ))) = (cid:16) e e (cid:17) p (cid:39) (1 . p . We first provide an upper bound on the competitive ratio of the greedy algorithm.
Theorem 5.
Let C be a class of quasi-log-convex functions. The competitive ratio of the greedy al-gorithm G applied to load balancing instances with latency functions in C is CR G ( WLB ( C )) ≤ sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) k o f ( k ) k f ( o ) o (cid:17) o − k o k − o k (cid:16) f ( k + o ) k o f ( k ) k f ( o ) o (cid:17) k − o o k − o k , where we set f (0) := 1 . We show that, when considering the greedy algorithm, the upper bound derived in Theorem 5 is tight if theconsidered latency functions are abscissa-scaling and ordinate-scaling (the proof is deferred to the appendix).The following result for polynomial latency functions holds (the proof is deferred to the appendix).
Corollary 5.
The competitive ratio of the greedy algorithm applied to weighted load balancing instances withpolynomial latency functions of maximum degree p is CR G ( WLB ( C )) = 4 p . We show that, when considering polynomial latency functions, the upper bound of Corollary 5 is tightfor any online algorithm, i.e., we are able to provide a matching lower bound to the online load balancingproblem (the proof is deferred to the appendix).
To the best of our knowledge, this is the first work that adopts the NSW as a benchmarking quality measurein load balancing problems. Several open problems deserve further investigation.First of all, our paper mostly focuses on evaluating the performance of selfish and online load balancing.Concerning complexity issues, it is worth noticing that, on the one hand, when considering unweightedplayers, an optimal configuration with respect to the NSW can be trivially computed in polynomial time byexploiting the same techniques developed in [25,50] for the utilitarian social welfare ([25,50] use, in turn, anapproach similar to the one adopted in [31] for the computation of a Nash equilibrium); on the other hand,when considering weighted players, a simple reduction from the NP-complete problem
PARTITION showsthat the problem becomes NP-hard. Therefore, an interesting open problem is that of providing polynomialtime approximation algorithms for the weighted case (we notice that Corollary 5 provides a 4 p -approximationalgorithm for weighted load balancing instances with polynomial latency functions of maximum degree p ).Moreover, a natural extension of our results consists in considering other families of congestion games,being more general than the one of load balancing games, such as the family of matroid congestion games[1,44].Finally, it would be interesting to apply the NSW measure to other classes of games, whose performances,in the literature, have only been analysed with respect to the utilitarian and/or egalitarian social welfarefunctions. References
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A.1 Proof of Fact 1
First of all, by exploiting the structure of the optimization problem, we can introduce the normalizationconstraint (cid:80) j ∈ R α j k j = (cid:80) j ∈ R α j o j = 1 without affecting the optimal value of the problem. By introducingsuch normalization constraint, the optimization problem becomes the following linear program:max (cid:88) j ∈ R ( σ ∗ ) α j o j (ln( (cid:96) j ( k j + o j )) − ln( (cid:96) j ( o j ))) (7)s.t. (cid:88) j ∈ R α j k j = 1 , (cid:88) j ∈ R α j o j = 1 , α j ≥ ∀ j ∈ R. By standard arguments of linear programming, we have that an optimal solution of (7) is given by a vertex ofthe polyhedral region defined by the linear constraints of (7), and such vertex can be obtained by nullifyingat least | R | − α and α , such that α ≥ α ≥
0. If both variables α and α are positive, we havethat they are univocally determined by the constraints α k + α k = 1 and α o + α o = 1, so that α = o − k k o − k o > , α = k − o k o − k o > , α j = 0 ∀ j ≥ . (8)By symmetry, we can assume w.l.o.g. that k o − k o >
0, so that k > o ≥ o > k ≥ α and α is null, and assume w.l.o.g. that α = 0. In thiscase, we necessarily get k = o > α = 1 /o , and the value of the objective function becomesln( f (2 o )) − ln( f ( o )). Anyway, we obtain the same value of the objective function by using in (7) the valuesof α and α considered in (8), and by setting k = o > o > k ≥
0. We also observe that, if o = 0and α , α >
0, the value of the objective function is ln( f ( k + o )) − ln( f ( o )) ≤ ln( f (2 o )) − ln( f ( o )),i.e., at most equal to the value of the objective function in which one of the two variables among α and α is null. Thus, we may omit the case o = 0.We conclude that, by considering the objective function of (7) with the values α and α defined in (8),and by considering the supremum of the objective function over k ≥ o > o > k ≥
0, we obtain theupper bound of the claim.
A.2 Proof of Theorem 2
First of all, we deal with part (ii) of the claim: Let us assume that C is abscissa-scaling, ordinate-scaling, and unbounded-including. In order to prove part (ii), we equivalently show that for any M < sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o there exists a game LB ∈ WLB ( C ) suchthat NPoA ( LB ) > M .Let f , f ∈ C , k , k , o , o ≥ k ≥ o > , o > k ≥
0, and a sufficiently small (cid:15) > (cid:16) f ( k + o ) f ( o ) (cid:17) ( o − k o k o − k o (cid:16) f ( k + o ) f ( o ) (cid:17) ( k − o o k o − k o > M + (cid:15). Let f, g ∈ C be such that f ( x ) := f ( o x ) and g ( x ) := f ( o x ), and let k := k /o and h := k /o . Since (cid:18) f ( k + o ) f ( o ) (cid:19) ( o − k o k o − k o (cid:18) f ( k + o ) f ( o ) (cid:19) ( k − o o k o − k o = (cid:18) f ( k + 1) f (1) (cid:19) − hk − h (cid:18) g ( h + 1) g (1) (cid:19) k − k − h we have that (cid:18) f ( k + 1) f (1) (cid:19) − hk − h (cid:18) g ( h + 1) g (1) (cid:19) k − k − h > M + (cid:15) , for some f, g ∈ C , k ≥ , and h < . (9)Observe that f and g can be chosen in such a way that they are non-constant functions. Indeed, if one ofthem is constant, it is sufficient replacing it with an arbitrary non-constant function, so that (9) holds as well.Since C is unbounded-including and f, g are non-constant, we have that lim x →∞ f ( x ) = lim x →∞ g ( x ) = ∞ . First of all, we assume that h >
0. Given two integers m ≥ s ≥
1, let LB ( m, s ) be a symmetricweighted load balancing game where the resources are partitioned into 2 m groups R , R , R . . . , R m . Eachgroup R j has s j − resources and the latency function of each resource r ∈ R j is defined as (cid:96) r ( x ) := α j ˆ f j ( β j x )with ˆ f j := (cid:40) f if j ≤ m − g if j ≥ m , β j := (cid:40)(cid:0) sk (cid:1) j − if j ≤ m − (cid:0) sh (cid:1) j − m (cid:0) sk (cid:1) m − if m ≤ j ≤ m , (10) α j := (cid:16) f ( k ) f ( k +1) (cid:17) j − if j ≤ m − (cid:16) g ( h ) g ( h +1) (cid:17) j − m (cid:16) f ( k ) g ( h +1) (cid:17) (cid:16) f ( k ) f ( k +1) (cid:17) m − if m ≤ j ≤ m − g ( h ) g (1) (cid:16) g ( h ) g ( h +1) (cid:17) m − (cid:16) f ( k ) g ( h +1) (cid:17) (cid:16) f ( k ) f ( k +1) (cid:17) m − if j = 2 m . (11)The set of players N is partitioned into 2 m − N , N , . . . , N m − , and each group N j has s j playershaving weight w j := 1 /β j +1 . Let σ be the strategy profile in which, for any j ∈ [2 m − R j is selected by exactly s players of group N j (see Figure 1.a). Observe that, by construction of α j , β j , w j , the following properties hold: α j f ( k ) = α j +1 f ( k + 1) if j ≤ m − α j f ( k ) = α j +1 g ( h + 1) if j = m − α j g ( h ) = α j +1 g ( h + 1) if m ≤ j ≤ m − α j g ( h ) = α j +1 g (1) if j = 2 m − , β j w j s = k, w j | N j | = k j if j ≤ m − β j w j s = h, w j | N j | = h j +1 − m k m − if m ≤ j ≤ m − β j +1 w j = 1 if j ≤ m − We now show that, by choosing a sufficiently large s , the strategy profile σ is a pure Nash equilibrium of LB ( m, s ). Let j ∈ [2 m − t ∈ [2 m ], and i be an arbitrary player selecting a resource r j of group R j in thestrategy profile σ , and assume that she deviates to a resource r t of group R t . We have three cases: • t = j + 1: First of all, assume that j ≤ m −
2. By using (12), we get cost i ( σ ) = (cid:96) r j ( k r j ( σ )) = α j ˆ f j ( β j sw j ) = α j f ( k ) = α j +1 f ( k + 1) = α j +1 f ( β j +1 sw j +1 + β j +1 w j ) = α j +1 ˆ f j +1 ( β j +1 ( sw j +1 + w j )) = (cid:96) r h ( k r h ( σ − i , { r t } )) = cost i ( σ − i , { r t } ). The cases j = m − m ≤ j ≤ m −
2, and j = 2 m − cost i ( σ ) = α j ˆ f j ( β j sw j ) = α j +1 ˆ f j +1 ( β j +1 ( sw j +1 + w j )) = cost i ( σ − i , { r t } ), where we set w m := 0. • t ≤ j : From the previous case, we have that if one player is playing a resource at some level l , and deviatesto some resource at level l + 1, her cost does not change. Thus, we necessarily have that the cost of eachresource in strategy profile σ is a non-increasing function of the level l ∈ [2 m ] which it belongs to. Thus,since t ≤ j , we necessarily have that cost i ( σ ) ≤ cost i ( σ − i , { r t } ). • t > j + 1 : If we consider the asymptotic behaviour of cost i ( σ ) and cost i ( σ − i , { r t } ) with respect to pa-rameter s , we get cost i ( σ ) = α j ˆ f j ( β j sw j ) = α j ˆ f j ( Θ ( s j − · s · s − j )) = Θ (1), thus cost i ( σ ) does notdepend on s ; furthermore, we get cost i ( σ − i , { r t } ) ≥ α j ˆ f j ( β t w j +1 ) = α j ˆ f j ( Θ ( s t − s − j )) ≥ α j ˆ f j ( Θ ( s )),thus, since lim x →∞ ˆ f ( x ) = ∞ , we have that cost i ( σ − i , { r t } ) can be arbitrarily large as s increases. Weconclude that, by taking a sufficiently large s , we get cost i ( σ ) ≤ cost i ( σ − i , { r t } ) for any j and t > j + 1.The previous case-analysis shows that player i does not improve her cost after deviating in favour of anyresource r t at level t , for any t ∈ [2 m ], and thus that σ is a pure Nash equilibrium of LB ( m, s ). For anyinteger m ≥
3, let s m be a sufficiently large integer such that (according to the previous case-analysis) σ isa pure Nash equilibrium of the game LB ( m, s m ).Now, let σ ∗ be the strategy profile of LB ( m, s m ) in which, for any j ∈ [2 m − R j +1 is selected by exactly one player of group N j (see Figure 1.b). By exploiting the definitions of α j , β j ,ˆ f j , w j , and N j , we have that: NPoA ( LB ( m, s m )) ≥ NSW ( σ ) NSW ( σ ∗ )= (cid:81) m − j =1 (cid:16) α j ˆ f j ( β j s m w j ) (cid:17) | N j | w j (cid:81) mj =2 (cid:16) α j ˆ f j ( β j w j − ) (cid:17) | N j − | w j − (cid:80) m − j =1 | Nj | wj ash Social Welfare in Load Balancing 15= (cid:16)(cid:81) m − j =1 ( α j f ( k )) | N j | w j (cid:17) (cid:16)(cid:81) m − j = m ( α j g ( h )) | N j | w j (cid:17)(cid:16)(cid:81) m − j =2 ( α j f (1)) | N j − | w j − (cid:17) (cid:16)(cid:81) mj = m ( α j g (1)) | N j − | w j − (cid:17) (cid:80) m − j =1 | Nj | wj (13)= (cid:16)(cid:81) m − j =1 ( α j f ( k )) k j (cid:17) (cid:16)(cid:81) m − j = m ( α j g ( h )) h j +1 − m k m − (cid:17)(cid:16)(cid:81) m − j =2 ( α j f (1)) k j − (cid:17) (cid:16)(cid:81) mj = m ( α j g (1)) h j − m k m − (cid:17) (cid:80) m − j =1 | Nj | wj = (cid:16)(cid:81) m − j =1 ( α j +1 f ( k + 1)) k j (cid:17) (cid:16)(cid:81) m − j = m − ( α j +1 g ( h + 1)) h j +1 − m k m − (cid:17) ( α m g (1)) h m k m − (cid:16)(cid:81) m − j =2 ( α j f (1)) k j − (cid:17) (cid:16)(cid:81) mj = m ( α j g (1)) h j − m k m − (cid:17) (cid:80) m − j =1 | Nj | wj (14)= (cid:16)(cid:81) m − j =1 ( α j +1 f ( k + 1)) k j (cid:17) (cid:16)(cid:81) m − j = m − ( α j +1 g ( h + 1)) h j +1 − m k m − (cid:17) ( α m g (1)) h m k m − (cid:16)(cid:81) m − j =1 ( α j +1 f (1)) k j (cid:17) (cid:16)(cid:81) m − j = m − ( α j +1 g (1)) h j +1 − m k m − (cid:17) (cid:80) m − j =1 | Nj | wj = (cid:16)(cid:81) m − j =1 ( α j +1 f ( k + 1)) k j (cid:17) (cid:16)(cid:81) m − j = m − ( α j +1 g ( h + 1)) h j +1 − m k m − (cid:17)(cid:16)(cid:81) m − j =1 ( α j +1 f (1)) k j (cid:17) (cid:16)(cid:81) m − j = m − ( α j +1 g (1)) h j +1 − m k m − (cid:17) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − = (cid:32)(cid:32) m − (cid:89) j =1 (cid:18) f ( k + 1) f (1) (cid:19) k j (cid:33) (cid:32) m − (cid:89) j = m − (cid:18) g ( h + 1) g (1) (cid:19) h j +1 − m k m − (cid:33)(cid:33) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − = (cid:32)(cid:18) f ( k + 1) f (1) (cid:19) (cid:80) m − j =1 k j (cid:18) g ( h + 1) g (1) (cid:19) (cid:80) m − j = m − h j +1 − m k m − (cid:33) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − , (15) where (13) and (14) come from (12). We have two cases: k > k = 1. If k >
1, by continuing from (15)and by considering a sufficiently large m , we get (cid:32)(cid:18) f ( k + 1) f (1) (cid:19) (cid:80) m − j =1 k j (cid:18) g ( h + 1) g (1) (cid:19) (cid:80) m − j = m − h j +1 − m k m − (cid:33) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − = (cid:18) f ( k + 1) f (1) (cid:19) km − − kk − (cid:18) g ( h + 1) g (1) (cid:19) k m − ( − hm − h ) km − − kk − km − (cid:18) − hm +11 − h (cid:19) = (cid:18) f ( k + 1) f (1) (cid:19) km − − kk − km − − kk − km − (cid:18) − hm +11 − h (cid:19) (cid:18) g ( h + 1) g (1) (cid:19) km − ( − hm − h ) km − − kk − km − (cid:18) − hm +11 − h (cid:19) = (cid:18) f ( k + 1) f (1) (cid:19) − h − hm +11 − h − hm +1 + km − (cid:18) k − km − − k (cid:19) (cid:18) g ( h + 1) g (1) (cid:19) km − (cid:18) k − km − − k (cid:19)(cid:18) − hm − hm +1 (cid:19) − h − hm +1 + km − (cid:18) k − km − − k (cid:19) ≥ lim m →∞ (cid:18) f ( k + 1) f (1) (cid:19) − h − hm +11 − h − hm +1 + km − (cid:18) k − km − − k (cid:19) (cid:18) g ( h + 1) g (1) (cid:19) km − (cid:18) k − km − − k (cid:19)(cid:18) − hm − hm +1 (cid:19) − h − hm +1 + km − (cid:18) k − km − − k (cid:19) − (cid:15) (16)= (cid:18) f ( k + 1) f (1) (cid:19) − h (1 − h )+( k − (cid:18) g ( h + 1) g (1) (cid:19) k − − h )+( k − − (cid:15) (17)= (cid:18) f ( k + 1) f (1) (cid:19) − hk − h (cid:18) g ( h + 1) g (1) (cid:19) k − k − h − (cid:15)>M + (cid:15) − (cid:15) (18)= M, (19)where (16) holds if m is sufficiently large, (17) comes from the fact that k > h <
1, and (18) comesfrom (9). If k = 1, by continuing from (15), we get: (cid:32)(cid:18) f ( k + 1) f (1) (cid:19) (cid:80) m − j =1 k j (cid:18) g ( h + 1) g (1) (cid:19) (cid:80) m − j = m − h j +1 − m k m − (cid:33) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − = (cid:18) f ( k + 1) f (1) (cid:19) m − m −
2+ 1 − hm +11 − h (cid:18) g ( h + 1) g (1) (cid:19) − hm − hm −
2+ 1 − hm +11 − h ≥ lim m →∞ (cid:18) f ( k + 1) f (1) (cid:19) m − m −
2+ 1 − hm +11 − h (cid:18) g ( h + 1) g (1) (cid:19) − hm − hm −
2+ 1 − hm +11 − h − (cid:15) (20)= (cid:18) f ( k + 1) f (1) (cid:19) (cid:18) g ( h + 1) g (1) (cid:19) − (cid:15) = (cid:18) f ( k + 1) f (1) (cid:19) − hk − h (cid:18) g ( h + 1) g (1) (cid:19) k − k − h − (cid:15) (21) >M + (cid:15) − (cid:15) (22)= M, (23)where (20) holds if m is sufficiently large, (21) comes from the fact that k = 1 and h <
1, and (22) comesfrom (9). By (19) and (23), we have that, for a sufficiently large m , NPoA ( LB ( m, s m )) ≥ M , thus showingpart (ii) of the claim.If h = 0, we consider a load balancing game defined as LB ( m, s m ), but restricted to the resources ofgroups R , . . . , R m and to the players of groups N , . . . , N m − . By using the same proof arguments as thoseused for h >
0, one can show the claim as well.We now show part (i). Assume that C is abscissa-scaling and ordinate-scaling. Analogously to the proofof part (ii), we have that (9) holds. Moreover, let LB (cid:48) ( m, s ) be a weighted load balancing game equal to game LB ( m, s ) defined in the proof of part (ii), except for the strategy set of each player: for any j ∈ [2 m − N j is Σ j := R j ∪ R j +1 . Let σ and σ ∗ be the strategy profiles defined asin game LB ( m, s ). By considering the case h = j + 1 analyzed in the proof of part (ii) of the claim, it alsoholds that σ is a pure Nash equilibrium of LB (cid:48) ( m, s ) for any s ≥
1. Therefore, if we take a sufficiently large m , an arbitrary s ≥
1, and by applying to game LB (cid:48) ( m, s ) the same inequalities as in (19) and (23), part (i)follows. A.3 Proof of Lemma 1
We have that sup k ≥ o > ,o >k ≥ ,f ,f ∈P ( p ) (cid:18) f ( k + o ) f ( o ) (cid:19) ( o − k o k o − k o (cid:18) f ( k + o ) f ( o ) (cid:19) ( k − o o k o − k o = sup k ≥ o > ,o >k ≥ ,α ,...,α p , ≥ β ,...,β p ≥ (cid:18) (cid:80) pd =0 α d ( k + o ) d (cid:80) pd =0 α d o d (cid:19) ( o − k o k o − k o (cid:18) (cid:80) pd =0 β d ( k + o ) d (cid:80) pd =0 β d o d (cid:19) ( k − o o k o − k o = sup k ≥ o > ,o >k ≥ (cid:18) max d ∈ [ p ] ∪{ } ( k + o ) d o d (cid:19) ( o − k o k o − k o (cid:18) max d ∈ [ p ] ∪{ } ( k + o ) d o d (cid:19) ( k − o o k o − k o = sup k ≥ o > ,o >k ≥ (cid:18)(cid:18) k + o o (cid:19) p (cid:19) ( o − k o k o − k o (cid:18)(cid:18) k + o o (cid:19) p (cid:19) ( k − o o k o − k o = sup k ≥ , ≤ h< (cid:16) ( k + 1) − hk − h ( h + 1) k − k − h (cid:17) p , (24) ash Social Welfare in Load Balancing 17 where (24) can be obtained by setting k := k /o and h := k /o . Now, we show that the maximumvalue of function F ( k, h ) := ( k + 1) − hk − h ( h + 1) k − k − h over k ≥ ≤ h < F ( k, h )) = − hk − h ln( k + 1) + k − k − h ln( h + 1) ≤ ln (cid:16) − hk − h ( k + 1) + k − k − h ( h + 1) (cid:17) , where the last inequality holdssince ln( F ( k, h )) is defined as convex combination of ln( k + 1) and ln( h + 1), and because of the concavityof the natural logarithm. Thus, we get F ( k, h ) ≤ − hk − h ( k + 1) + k − k − h ( h + 1) = ( k − h ) + ( k − h ) k − h = 2 . (25)Finally, since F ( k, h ) = 2 for k = 1 and h = 0, and because of (25), we have that the maximum of F ( k, h )over k ≥ ≤ h < p . A.4 Proof of Corollary 2
Let (cid:15) >
0. Let LB (cid:48) ( m ) be the load balancing game defined as the game LB (cid:48) ( m, s ) considered in the proof ofpart (i) of Theorem 2, with s = 2, k = 1, h = 0, and f, g defined as f ( x ) = g ( x ) = x p . One can easily observethat LB (cid:48) ( m ) is a game with identical resources. Furthermore, because of the proof of Theorem 2, there existsa sufficiently large integer m such that NPoA ( LB (cid:48) ( m )) > p − (cid:15) , and the claim follows by the arbitrarinessof (cid:15) > B Missing Proofs of Subsection 3.2
B.1 Proof of Theorem 3
Let LB ∈ ULB ( C ) be an unweighted load balancing game with latency functions in C , and let σ and σ ∗ bea worst-case pure Nash equilibrium and an optimal strategy profile of LB , respectively. Let k j denote k j ( σ )and o j denote k j ( σ ∗ ). As in Theorem 1, we get (cid:89) j ∈ R ( σ ) (cid:96) j ( k j ) k j ≤ (cid:89) j ∈ R ( σ ∗ ) (cid:96) j ( k j + 1) o j . (26)By exploiting the properties of the logarithmic function, we getln ( NPoA ( LB )) = ln (cid:16)(cid:81) j ∈ R ( σ ) (cid:96) j ( k j ) k j (cid:17) n (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( o j ) o j (cid:17) n ≤ ln (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( k j + 1) o j (cid:17) n (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( o j ) o j (cid:17) n (27)= (cid:80) j ∈ R ( σ ∗ ) o j (ln( (cid:96) j ( k j + 1)) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R k j , where (27) comes from (26). Now, let R + := { j ∈ R ( σ ∗ ) : k j ≥ o j } . We have that (cid:80) j ∈ R ( σ ∗ ) o j (ln( (cid:96) j ( k j + 1)) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R k j ≤ (cid:80) j ∈ R ( σ ∗ ) o j (ln( (cid:96) j ( k j + 1)) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R ( σ ∗ ) k j ≤ (cid:80) j ∈ R + o j (ln( (cid:96) j ( k j + 1)) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R + k j (28) ≤ max j ∈ R + o j (ln( (cid:96) j ( k j + 1)) − ln( (cid:96) j ( o j ))) k j ≤ sup f ∈C ,k ∈ N ,o ∈ [ k ] o (ln( f ( k + 1)) − ln( f ( o ))) k , where (28) holds because for any j ∈ R ( σ ∗ ) \ R + , it holds that o j (ln( (cid:96) j ( k j + 1)) − ln( (cid:96) j ( o j ))) ≤
0. Therefore,we conclude that ln (
NPoA ( LB )) ≤ sup f ∈C ,k ∈ N ,o ∈ [ k ] o (ln( f ( k + 1)) − ln( f ( o ))) k , and by exponentiating the previous inequality we get the claim. B.2 Tightness of the Upper Bound in Theorem 3Theorem 6.
Let C be a class of latency functions. If C is ordinate-scaling, then NPoA ( ULB ( C )) ≥ sup f ∈C ,k ∈ N ,o ∈ [ k ] (cid:16) f ( k +1) f ( o ) (cid:17) ok . Proof.
In order to prove the theorem, we equivalently show that, for any
M < sup f ∈C ,k ∈ N ,o ∈ [ k ] (cid:16) f ( k +1) f ( o ) (cid:17) ok ,there exists a game LB ∈ ULB ( C ) such that NPoA ( LB ) > M .Fix an arbitrary M < sup f ∈C ,k ∈ N ,o ∈ [ k ] (cid:16) f ( k +1) f ( o ) (cid:17) ok . Let f ∈ C , k ∈ N , o ∈ [ k ], and a sufficiently small (cid:15) > (cid:18) f ( k + 1) f ( o ) (cid:19) ok > M + (cid:15). (29)Given an integer m >
0, let LB ( m ) be an unweighted load balancing game with ( k − o + 1) m + o resources,partitioned into m groups R , R , . . . , R m such that R j := { r j, , r j, , . . . , r j,k − o } for any j ∈ [ m − R m := { r m, , r m, , . . . , r m,k } . Each resource r j,h has latency function (cid:96) r j,h ( x ) := α j,h f ( x ), with α j,h := (cid:16) f ( k ) f ( k +1) (cid:17) j − if h = 0 f ( k ) f (1) (cid:16) f ( k ) f ( k +1) (cid:17) j − otherwise.We have n := mk players split into m groups N , N , . . . , N m of k players each. For j ∈ [ m − Σ j of players of group N j is R j ∪ { r j +1 , } , and the set of strategies Σ m of players in N m is R m .Let σ be the strategy profile such that, for any j ∈ [ m ], all k players of group N j select resource r j, , sothat each resource r j, has congestion k , and all the remaining resources have null congestion (see Figure 2.a).We show that σ is a pure Nash equilibrium. Given an arbitrary player i of group N j with j ∈ [ m ], such playerhas a cost equal to (cid:96) r j, ( k ) = α j, f ( k ) = (cid:16) f ( k ) f ( k +1) (cid:17) j − f ( k ) when playing strategy σ i . If j ∈ [ m − i unilaterally deviates to strategy r j +1 , , her cost is (cid:96) r j +1 , ( k + 1) = α j +1 , f ( k + 1) = (cid:16) f ( k ) f ( k +1) (cid:17) j f ( k + 1) = (cid:16) f ( k ) f ( k +1) (cid:17) j − f ( k ) = (cid:96) r j, ( k ), thus her cost does not improve. Analogously, if j ∈ [ m ], and player i unilaterallydeviates to any strategy r j,h with h (cid:54) = 0, her cost is (cid:96) r j,h (1) = α j,h f (1) = f ( k ) f (1) (cid:16) f ( k ) f ( k +1) (cid:17) j − f (1) = (cid:96) r j, ( k ),thus her cost does not improve as well. We conclude that σ is a pure Nash equilibrium.Now, let σ ∗ be a strategy profile defined as follows: (i) for any j ∈ [ m − o players of group N j selectresource r j +1 , , and each of the k − o remaining players of N j selects a distinct resource of R j \ { r j, } , (ii)all the k players of group N m select a distinct resource of E m \ { r m, } . Thus, in σ ∗ , any resource of type r j, with j > o , resource r , has null congestion, and the remaining resources have unitarycongestion (see Figure 2.b). By some algebraic manipulation, it holds that NSW ( σ ) NSW ( σ ∗ ) = ash Social Welfare in Load Balancing 19 r , r , r , r ,k − o b b b R b bb bb b k r , r , r , r ,k − o b b b R b bb bb b r m − , r m − , r m − ,k − o b b b R m − b bb bb b r m, r m, r m, r m,k b b b R m b bb bb b mmmmm-1m-1m-1m-1 b b bb b bb b b (a) r , r , r , r ,k − o b b b R o r , r , r , r ,k − o b b b R b bb bb b r m − , r m − , r m − ,k − o b b b R m − b bb bb b r m, r m, r m, r m,k b b b R m mmmm-2m-2 m-1 b b bb b bb b b (b) b b b b bb bb b m-1m-1m-1 Fig. 2.
The LB used in the proof of Theorem 6. Columns represent resources and squares represent players (number j inside a square means that the player belongs to group N j ). (a): The Nash equilibrium σ ; (b): The strategy profile σ ∗ . = (cid:81) mj =1 (cid:96) j, ( k ) k (cid:81) m − j =1 (cid:16) (cid:96) r j +1 , ( o ) o (cid:81) r ∈ R j \{ r j, } (cid:96) r (1) (cid:17) (cid:81) r ∈ R m \{ r m, } (cid:96) r (1) km = (cid:81) mj =1 (cid:18)(cid:16) f ( k ) f ( k +1) (cid:17) j − f ( k ) (cid:19) k (cid:81) m − j =1 (cid:34)(cid:18)(cid:16) f ( k ) f ( k +1) (cid:17) j f ( o ) (cid:19) o (cid:18) f ( k ) f (1) (cid:16) f ( k ) f ( k +1) (cid:17) j − f (1) (cid:19) k − o (cid:35) (cid:18) f ( k ) f (1) (cid:16) f ( k ) f ( k +1) (cid:17) m − f (1) (cid:19) k km = (cid:81) mj =1 (cid:18)(cid:16) f ( k ) f ( k +1) (cid:17) j f ( k + 1) (cid:19) k (cid:81) m − j =1 (cid:34)(cid:18)(cid:16) f ( k ) f ( k +1) (cid:17) j f ( o ) (cid:19) o (cid:18)(cid:16) f ( k ) f ( k +1) (cid:17) j f ( k + 1) (cid:19) k − o (cid:35) (cid:16)(cid:16) f ( k ) f ( k +1) (cid:17) m f ( k + 1) (cid:17) k km = (cid:18)(cid:81) mj =1 (cid:16) f ( k ) f ( k +1) (cid:17) kj (cid:19) f ( k + 1) km (cid:18)(cid:81) m − j =1 (cid:16) f ( k ) f ( k +1) (cid:17) kj (cid:19) f ( o ) o ( m − f ( k + 1) ( k − o )( m − (cid:16) f ( k ) f ( k +1) (cid:17) km f ( k + 1) k km = (cid:18) f ( k + 1) km f ( o ) o ( m − f ( k + 1) ( k − o )( m − f ( k + 1) k (cid:19) km = (cid:18) f ( k + 1) o ( m − f ( o ) o ( m − (cid:19) km = (cid:18) f ( k + 1) f ( o ) (cid:19) o ( m − km . (30)By using (29) and (30), and by choosing a sufficiently large m , we get NPoA ( LB ( m )) ≥ NSW ( σ ) NSW ( σ ∗ ) = (cid:18) f ( k + 1) f ( o ) (cid:19) o ( m − km ≥ lim m →∞ (cid:18) f ( k + 1) f ( o ) (cid:19) o ( m − km − (cid:15) = (cid:18) f ( k + 1) f ( o ) (cid:19) ok − (cid:15)> M + (cid:15) − (cid:15) = M, thus showing the claim. (cid:117)(cid:116) B.3 Proof of Corollary 3
The claim follows from the following lemma.
Lemma 2. sup f ∈P ( p ) ,k ∈ N ,o ∈ [ k ] (cid:16) f ( k +1) f ( o ) (cid:17) ok = 2 p . Proof.
We have that sup f ∈P ( p ) ,k ∈ N ,o ∈ [ k ] (cid:18) f ( k + 1) f ( o ) (cid:19) ok = sup α ,α ,...,α p ≥ ,k ∈ N ,o ∈ [ k ] (cid:18) (cid:80) pd =0 α d ( k + 1) d (cid:80) pd =0 α d o d (cid:19) ok = sup k ∈ N ,o ∈ [ k ] (cid:18) max d ∈ [ p ] ∪{ } ( k + 1) d o d (cid:19) ok = sup k ∈ N ,o ∈ [ k ] (cid:18)(cid:18) k + 1 o (cid:19) p (cid:19) ok = (cid:32) sup k ∈ N ,o ∈ [ k ] (cid:18) k + 1 o (cid:19) ok (cid:33) p = 2 p , (31)where (31) holds for the following reasons: First of all, we have that ( k +1 o ) ok = 2 if o = k = 1, thus showingthat 2 ≤ sup k ∈ N ,o ∈ [ k ] (cid:0) k +1 o (cid:1) ok ; furthermore, by setting x := k/o , we obtain ( k +1 o ) ok = (cid:0) x + o (cid:1) x ≤ ( x +1) x ≤ x ≥ x + 1 which holds for any x ≥ (cid:117)(cid:116) C Missing Proofs of Subsection 3.3
C.1 Proof of Theorem 4
Let
NLB ∈ NLB ( C ) be a non-atomic load balancing game with latency functions in C , and let ∆ and ∆ ∗ be a worst-case pure Nash equilibrium and an optimal strategy profile of NLB , respectively. Let k j denote k j ( ∆ ) and o j denote k j ( ∆ ∗ ).For any player type i and pair ( j, j ∗ ) of resources, let α ij,j ∗ be the amount of players of type i selectingresource j in ∆ and resource j ∗ in ∆ ∗ . Clearly, it holds that, for any i ∈ N , (cid:80) j,j ∗ ∈ R α ij,j ∗ = r i .Since ∆ is a pure Nash equilibrium, if there exists i ∈ N such that α ij,j ∗ >
0, we have that cost j ( ∆ ) ≤ cost j ∗ ( ∆ ). For any j, j ∗ ∈ R , let A j,j ∗ = (cid:80) i ∈ N α ij,j ∗ . Clearly, it holds that cost j ( ∆ ) A j,j ∗ ≤ cost j ∗ ( ∆ ) A j,j ∗ . (32)Since, for any j ∈ R ( ∆ ), (cid:80) j ∗ ∈ R A j,j ∗ = k j and, symmetrically, for any j ∗ ∈ R ( ∆ ∗ ), (cid:80) j ∈ R A j,j ∗ = o j , itfollows that (cid:89) j,j ∗ ∈ R cost j ( ∆ ) A j,j ∗ = (cid:89) j ∈ R ( ∆ ) cost j ( ∆ ) k j (33) ash Social Welfare in Load Balancing 21 and (cid:89) j,j ∗ ∈ R cost j ∗ ( ∆ ∗ ) A j,j ∗ = (cid:89) j ∈ R ( ∆ ∗ ) cost j ( ∆ ) o j . (34)By multiplying (32) over all pairs of resources in R and by exploiting (33) and (34), we obtain (cid:89) j ∈ R ( ∆ ) (cid:96) j ( k j ) k j = (cid:89) j ∈ R ( ∆ ) cost j ( ∆ ) k j = (cid:89) j,j ∗ ∈ R cost j ( ∆ ) A j,j ∗ ≤ (cid:89) j,j ∗ ∈ R cost j ∗ ( ∆ ) A j,j ∗ = (cid:89) j ∈ R ( ∆ ∗ ) cost j ( ∆ ) o j = (cid:89) j ∈ R ( ∆ ∗ ) (cid:96) j ( k j ) o j . (35)By exploiting the properties of the logarithmic function, we getln ( NPoA ( LB )) = ln (cid:16)(cid:81) j ∈ R ( ∆ ) (cid:96) j ( k j ) k j (cid:17) (cid:80) i ∈ N ri (cid:16)(cid:81) j ∈ R ( ∆ ∗ ) (cid:96) j ( o j ) o j (cid:17) (cid:80) i ∈ N ri ≤ ln (cid:16)(cid:81) j ∈ R ( ∆ ∗ ) (cid:96) j ( k j ) o j (cid:17) (cid:80) i ∈ N ri (cid:16)(cid:81) j ∈ R ( ∆ ∗ ) (cid:96) j ( o j ) o j (cid:17) (cid:80) i ∈ N ri (36)= (cid:80) j ∈ R ( ∆ ∗ ) o j ln( (cid:96) j ( k j )) − (cid:80) j ∈ R ( ∆ ∗ ) o j ln( (cid:96) j ( o j )) (cid:80) i ∈ N r i = (cid:80) j ∈ R ( ∆ ∗ ) o j (ln( (cid:96) j ( k j )) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R k j , ≤ (cid:80) j ∈ R + o j (ln( (cid:96) j ( k j )) − ln( (cid:96) j ( o j ))) (cid:80) j ∈ R + k j (37) ≤ max j ∈ R + o j (ln( (cid:96) j ( k j )) − ln( (cid:96) j ( o j ))) k j ≤ sup f ∈C ,k ≥ o> o (ln( f ( k )) − ln( f ( o ))) k , where (36) comes from (35), and (37) is obtained by using similar arguments as in Theorem 3 (in particular,see inequalities (28)). Therefore, we conclude thatln ( NPoA ( NLB )) ≤ sup f ∈C ,k ≥ o> o (ln( f ( k )) − ln( f ( o ))) k , and by exponentiating the previous inequality we get the claim. C.2 Tightness of the Upper Bound of Theorem 4Theorem 7.
Let C be a class of latency functions. If C is all-constant-including, then NPoA ( NLB ( C )) = NPoA ( SNLB ( C )) ≥ sup f ∈C ,k ≥ o> (cid:16) f ( k ) f ( o ) (cid:17) ok . Proof.
To show the theorem, we equivalently show that, for any
M < sup f ∈C ,k ≥ o> (cid:16) f ( k ) f ( o ) (cid:17) ok , there exists asymmetric non-atomic load balancing game NLB ∈ SNLB ( C ) such that NPoA ( NLB ) > M . Fix an arbitrary M < sup f ∈C ,k ≥ o> (cid:16) f ( k ) f ( o ) (cid:17) ok . Let f ∈ C and k ≥ o > (cid:16) f ( k ) f ( o ) (cid:17) ok > M . Let NLB be a symmetricnon-atomic load balancing game with a unique player type, say 1, and two resources having latency definedas (cid:96) ( x ) := f ( x ) and (cid:96) ( x ) := f ( k ). Assume that the amount of players of type 1 is r = k . Let ∆ be thestrategy profile in which all players select resource 1, and let ∆ ∗ be the strategy profile in which an amount o of players selects resource 1 and the remaining one (i.e., k − o ) selects resource 2. We trivially have that ∆ is apure Nash equilibrium. Thus, we obtain NPoA ( NLB ) ≥ NSW ( ∆ ) NSW ( ∆ ∗ ) = (cid:16) (cid:96) ( k ) k (cid:96) ( o ) o (cid:96) ( k − o ) k − o (cid:17) k = (cid:16) f ( k ) k f ( o ) o f ( k ) k − o (cid:17) k = (cid:16) f ( k ) f ( o ) (cid:17) ok > M, and the claim follows. (cid:117)(cid:116) C.3 Proof of Corollary 4
We have that
NPoA ( NLB ( P ( p ))) = NPoA ( SNLB ( P ( p ))) (38)= sup f ∈C ,k ≥ o> (cid:18) f ( k ) f ( o ) (cid:19) ok (39)= sup α ,α ,...,α p ≥ ,k ≥ o> (cid:18) (cid:80) pd =0 α d k d (cid:80) pd =0 α d o d (cid:19) ok = sup k ≥ o> (cid:18) max d ∈ [ p ] ∪{ } k d o d (cid:19) ok = max d ∈ [ p ] ∪{ } sup k ≥ o> (cid:18) k d o d (cid:19) ok = max d ∈ [ p ] ∪{ } (cid:32) sup k ≥ o> (cid:18) ko (cid:19) ok (cid:33) d = (cid:32) sup k ≥ o> (cid:18) ko (cid:19) ok (cid:33) p , = (cid:18) sup x> x x (cid:19) p , (40)= (cid:16) e e (cid:17) p , (41)where (38) and (39) come from Theorems 4 and 7 (observe that polynomial latency functions are all-constant-including), (40) can be obtained by setting x := k/o , and (41) comes from the fact that function F ( x ) := x /x is maximized by x = e . D Missing Proofs of Section 4
D.1 Proof of Theorem 5
Let I ∈ WLB ( C ) be a load balancing instance with latency functions in C , and let σ and σ ∗ be the statesreturned by the greedy algorithm and an optimal strategy profile of LB , respectively.Let k j denote k j ( σ ) and o j denote k j ( σ ∗ ). For any i ∈ N and resource j , let ( σ i ) be the partial statein which the first i clients have been assigned according to σ , and let ( σ i − , j ) be the state in whichthe first i − σ and client i is assigned to resource j . By defi-nition of greedy algorithm, we have that σ i ∈ arg min j ∈ R NSW ( σ i − , j ) = arg min j ∈ R (cid:81) l ≤ i cost l ( σ i − ,j ) (cid:81) l ≤ i − cost l ( σ i − ) =arg min j ∈ R (cid:96) j ( k j ( σ i − ,j )) kj ( σ i − ,j ) (cid:96) j ( k j ( σ i − )) kj ( σ i − , where we set (cid:96) j (0) := 1. Thus, we can equivalently define the greedyassignment by saying that each client i is assigned to the resource j minimizing (cid:96) j ( k j ( σ i − ,j )) kj ( σ i − ,j ) (cid:96) j ( k j ( σ i − )) kj ( σ i − , so that (cid:96) σ i ( k σ i ( σ i )) k σi ( σ i ) (cid:96) σ i ( k σ i ( σ i − )) k σi ( σ i − ) ≤ (cid:96) σ ∗ i ( k σ i ( σ i − , σ ∗ i )) k σ ∗ i ( σ i − ,σ ∗ i ) (cid:96) σ ∗ i ( k σ ∗ i ( σ i − )) k σ ∗ i ( σ i − ) . (42) ash Social Welfare in Load Balancing 23 We have that: (cid:89) i ∈ N (cid:96) σ i ( k σ i ( σ i )) k σi ( σ i ) (cid:96) σ i ( k σ i ( σ i − )) k σi ( σ i − ) = (cid:89) j ∈ R ( σ ) (cid:89) i ∈ N : σ i = j (cid:96) j ( k j ( σ i )) k j ( σ i ) (cid:96) j ( k j ( σ i − )) k j ( σ i − ) = (cid:89) j ∈ R ( σ ) (cid:96) j ( k j ( σ n )) k j ( σ n ) (43)= (cid:89) j ∈ R ( σ ) (cid:96) j ( k j ) k j , (44)where (43) is obtained by exploiting telescoping properties. Furthermore, we get (cid:89) i ∈ N (cid:96) σ ∗ i ( k σ ∗ i ( σ i − , σ ∗ i )) k σ ∗ i ( σ i − ,σ ∗ i ) (cid:96) σ ∗ i ( k σ ∗ i ( σ i − )) k σ ∗ i ( σ i − ) = (cid:89) i ∈ N (cid:96) σ ∗ i ( k σ ∗ i ( σ i − ) + w i ) k σ ∗ i ( σ i − )+ w i (cid:96) σ ∗ i ( k σ ∗ i ( σ i − )) k σ ∗ i ( σ i − ) ≤ (cid:89) i ∈ N (cid:96) σ ∗ i ( k σ ∗ i + w i ) k σ ∗ i + w i (cid:96) σ ∗ i ( k σ ∗ i ) k σ ∗ i (45)= (cid:89) j ∈ R ( σ ∗ ) (cid:89) i ∈ N : σ ∗ i = j (cid:96) j ( k j + w i ) k j + w i (cid:96) j ( k j ) k j ≤ (cid:89) j ∈ R ( σ ∗ ) (cid:89) i ∈ N : σ ∗ i = j (cid:96) j ( k j + (cid:80) t ≤ i : σ ∗ t = j w t ) k j + (cid:80) t ≤ i : σ ∗ t = j w t (cid:96) j ( k j + (cid:80) t
Given a quasi-log-convex latency function f , we have that f ( x + z ) x + z f ( x ) ≤ f ( x + y + z ) x + y + z f ( x + y ) x + y for any x, y, z ≥ .Proof. Since the function g such that g ( t ) = t ln( f ( t )) is convex, we have that g ( x + z ) − g ( x ) ≤ g ( x + y + z ) − g ( x + y ) for any x, y, z ≥
0, thus, by exponentiating the previous inequality, the claim follows. (cid:117)(cid:116)
By putting together (42), (44), and (48), we get (cid:89) j ∈ R ( σ ) (cid:96) j ( k j ) k j = (cid:89) i ∈ N (cid:96) σ i ( k σ i ( σ i )) k σi ( σ i ) (cid:96) σ i ( k σ i ( σ i − )) k σi ( σ i − ) ≤ (cid:89) i ∈ N (cid:96) σ ∗ i ( k σ ∗ i ( σ i − , σ ∗ i )) k σ ∗ i ( σ i − ,σ ∗ i ) (cid:96) σ ∗ i ( k σ ∗ i ( σ i − )) k σ ∗ i ( σ i − ) ≤ (cid:89) j ∈ R ( σ ∗ ) (cid:96) j ( k j + o j ) k j + o j (cid:96) j ( k j ) k j . (49)By exploiting the properties of the logarithmic function, we obtainln ( CR G ( I )) = ln (cid:16)(cid:81) j ∈ R ( σ ) (cid:96) j ( k j ) k j (cid:17) (cid:80) i ∈ N wi (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( o j ) o j (cid:17) (cid:80) i ∈ N wi ≤ ln (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( k j + o j ) kj + oj (cid:96) j ( k j ) kj (cid:17) (cid:80) i ∈ N wi (cid:16)(cid:81) j ∈ R ( σ ∗ ) (cid:96) j ( o j ) o j (cid:17) (cid:80) i ∈ N wi (50)= (cid:80) j ∈ R ( σ ∗ ) (( k j + o j ) ln( (cid:96) j ( k j + o j )) − k j ln( (cid:96) j ( k j )) − o j ln( (cid:96) j ( o j ))) (cid:80) i ∈ N w i , (51)where (50) comes from (49). Since (cid:80) i ∈ N w i = (cid:80) j ∈ R k j = (cid:80) j ∈ R o j , we have that (51) is upper bounded bythe optimal solution of the following optimization problem on some new linear variables ( α j ) j ∈ R :max (cid:80) j ∈ R ( σ ∗ ) α j (( k j + o j ) ln( (cid:96) j ( k j + o j )) − k j ln( (cid:96) j ( k j )) − o j ln( (cid:96) j ( o j ))) (cid:80) j ∈ R α j k j s.t. (cid:88) j ∈ R α j k j = (cid:88) j ∈ R α j o j , α j ≥ ∀ j ∈ R. By normalizing the denominator of the objective function, we obtain the following equivalent linear program:max (cid:88) j ∈ R ( σ ∗ ) α j (( k j + o j ) ln( (cid:96) j ( k j + o j )) − k j ln( (cid:96) j ( k j )) − o j ln( (cid:96) j ( o j ))) (52)s.t. (cid:88) j ∈ R α j k j = 1 , (cid:88) j ∈ R α j o j = 1 , α j ≥ ∀ j ∈ R. We have the following fact, whose proof is omitted, since it is similar to that of Fact 1.
Fact 3
The maximum value of the linear program considered in (3) is at most sup k ≥ o > ,o >k ≥ ,f ,f ∈C ( o − k ) F ( f , o , k ) + ( k − o ) F ( f , o , k ) k o − k o , where F ( f, o, k ) := ( k + o ) ln( f ( k + o )) − k ln( f ( k )) − o ln( f ( o )) . By continuing from (51) and by using Fact 3, we getln ( CR G ( I )) ≤ sup k ≥ o > ,o >k ≥ ,f ,f ∈C ( o − k ) F ( f , o , k ) + ( k − o ) F ( f , o , k ) k o − k o . By exponentiating the previous inequality, we get the claim.
D.2 Tightness of the Upper Bound of Theorem 5Theorem 8.
Let C be a class of latency functions and let G be the greedy algorithm. If C is abscissa-scalingand ordinate-scaling, then CR G ( WLB ( C )) ≥ sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) o − k o k − o k (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) k − o o k − o k . (53) Proof.
Let us assume that C is abscissa-scaling and ordinate-scaling. We equivalently show that for any M < sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:16) f ( k + o ) k o f ( k ) k f ( o ) o (cid:17) o − k o k − o k (cid:16) f ( k + o ) k o f ( k ) k f ( o ) o (cid:17) k − o o k − o k there exists an instance I ∈ WLB ( C ) such that NPoA ( I ) > M . ash Social Welfare in Load Balancing 25 Let f , f ∈ C , k , k , o , o ≥ k ≥ o > , o > k ≥
0, and let (cid:15) > (cid:16) f ( k + o ) k o f ( k ) k f ( o ) o (cid:17) o − k o k − o k (cid:16) f ( k + o ) k o f ( k ) k f ( o ) o (cid:17) k − o o k − o k > M + (cid:15). Let f, g ∈ C be such that f ( x ) := f ( o x ) and g ( x ) := f ( o x ), and let k := k /o and h := k /o . Since (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) o − k o k − o k (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) k − o o k − o k = (cid:18) f ( k + 1) k +1 f ( k ) k f (1) (cid:19) − hk − h (cid:18) g ( h + 1) h +1 g ( h ) h g (1) (cid:19) k − k − h we have that (cid:18) f ( k + 1) k +1 f ( k ) k f (1) (cid:19) − hk − h (cid:18) g ( h + 1) h +1 g ( h ) h g (1) (cid:19) k − k − h > M + (cid:15) , for some f, g ∈ C , k ≥ , and h < . (54)First of all, we assume that h >
0. Given an integer m ≥
3, let I ( m ) be a load balancing instance having2 m resources r , r , r . . . , r m and 2 m − j is { r j , r j +1 } .Each resource r j has a latency function defined as (cid:96) j ( x ) := α j ˆ f j ( β j x ), and the weight of each client j isdefined as w j := 1 /β j +1 , where α j , ˆ f j , and β j are defined as follows:ˆ f j := (cid:40) f if j ≤ m − g if j ≥ m , β j := (cid:40)(cid:0) k (cid:1) j − if j ≤ m − (cid:0) h (cid:1) j − m (cid:0) k (cid:1) m − if m ≤ j ≤ m , (55) α j := (cid:16) f ( k ) k +1 f ( k +1) k +1 (cid:17) j − if j ≤ m − (cid:16) g ( h ) h +1 g ( h +1) h +1 (cid:17) j − m (cid:16) f ( k ) g ( h ) h g ( h +1) h +1 (cid:17) (cid:16) f ( k ) k +1 f ( k +1) k +1 (cid:17) m − if m ≤ j ≤ m − g ( h ) g (1) (cid:16) g ( h ) h +1 g ( h +1) h +1 (cid:17) m − (cid:16) f ( k ) g ( h ) h g ( h +1) h +1 (cid:17) (cid:16) f ( k ) k +1 f ( k +1) k +1 (cid:17) m − if j = 2 m . (56)Observe that, by construction of α j , β j , w j , the following properties hold: α j f ( k ) = α j +1 f ( k +1) k +1 f ( k ) k if j ≤ m − α j f ( k ) = α j +1 g ( h +1) h +1 g ( h ) h if j = m − α j g ( h ) = α j +1 g ( h +1) h +1 g ( h ) h if m ≤ j ≤ m − α j g ( h ) = α j +1 g (1) if j = 2 m − , β j w j = k, w j = k j if j ≤ m − β j w j = h, w j = h j +1 − m k m − if m ≤ j ≤ m − β j +1 w j = 1 if j ≤ m − Let σ be the strategy profile in which each client j is assigned to resource r j . We show that σ is a statethat can be possibly returned by the greedy algorithm when clients are processed in reverse order w.r.t.index j . We equivalently show that NSW ( σ j ) NSW ( σ j +1 ) ≤ NSW ( σ j +1 ,r j +1 ) NSW ( σ j +1 ) for any j ≤ m −
1, where σ j denotes thepartial assignment in which each client t ≥ j is assigned to resource r t , and ( σ j +1 , r j +1 ) denotes the partialassignment in which each client t ≥ j + 1 is assigned to resource r t and client j is assigned to resource r j +1 .Let j ∈ [2 m − j ≤ m −
2. By using (57), we get
NSW ( σ j ) NSW ( σ j +1 ) = (cid:96) r j ( k r j ( σ )) k rj ( σ ) = (cid:16) α j ˆ f j ( β j w j ) (cid:17) w j = ( α j f ( k )) w j = (cid:32) α j +1 f ( k + 1) k +1 f ( k ) k (cid:33) w j = α w j j +1 f ( k + 1) kw j + w j f ( k ) kw j = α w j j +1 f ( k + 1) w j +1 + w j f ( k ) w j +1 = ( α j +1 f ( k + 1)) w j +1 + w j ( α j +1 f ( k )) w j +1 = ( α j +1 f ( β j +1 ( w j +1 + w j ))) w j +1 + w j ( α j +1 f ( β j +1 w j +1 )) w j +1 = (cid:16) α j +1 ˆ f j +1 ( β j +1 ( w j +1 + w j )) (cid:17) w j +1 + w j (cid:16) α j +1 ˆ f j +1 ( β j +1 w j +1 ) (cid:17) w j +1 = (cid:96) r j +1 ( k r j +1 ( σ j +1 , r j +1 )) k rj +1 ( σ j +1 ,r j +1 ) (cid:96) r j +1 ( k r j +1 ( σ j +1 )) k rj +1 ( σ j +1 ) = NSW ( σ j +1 , r j +1 ) NSW ( σ j +1 ) . The cases j = m − m ≤ j ≤ m −
2, and j = 2 m − NSW ( σ j ) NSW ( σ j +1 ) = (cid:16) α j ˆ f j ( β j w j ) (cid:17) w j = (cid:16) α j +1 ˆ f j +1 ( β j +1 ( w j +1 + w j )) (cid:17) w j +1 + w j (cid:16) α j +1 ˆ f j +1 ( β j +1 w j +1 ) (cid:17) w j +1 = NSW ( σ j +1 , r j +1 ) NSW ( σ j +1 ) , (58)where we set (cid:16) α m ˆ f m ( β m w m ) (cid:17) w m := 1 and w m := 0. Now, let σ ∗ be the strategy profile of I ( m ) inwhich each client j ∈ [ m −
1] is assigned to resource r j +1 . By exploiting the definitions of α j , β j , ˆ f j , and w j ,and by considering a sufficiently large m , we have that: NPoA ( I ( m )) ≥ NSW ( σ ) NSW ( σ ∗ )= (cid:81) m − j =1 (cid:16) α j ˆ f j ( β j w j ) (cid:17) w j (cid:81) mj =2 (cid:16) α j ˆ f j ( β j w j − ) (cid:17) w j − (cid:80) m − j =1 wj = (cid:81) m − j =1 (cid:18) ( α j +1 ˆ f j +1 ( β j +1 ( w j +1 + w j ) )) wj +1+ wj ( α j +1 ˆ f j +1 ( β j +1 w j +1 ) ) wj +1 (cid:19)(cid:81) mj =2 (cid:16) α j ˆ f j ( β j w j − ) (cid:17) w j − (cid:80) m − j =1 wj (59)= (cid:81) m − j =1 (cid:18) ( α j +1 ˆ f j +1 ( β j +1 ( w j +1 + w j ) )) wj +1+ wj ( α j +1 ˆ f j +1 ( β j +1 w j +1 ) ) wj +1 (cid:19)(cid:81) m − j =1 (cid:16) α j +1 ˆ f j +1 ( β j +1 w j ) (cid:17) w j (cid:80) m − j =1 wj = m − (cid:89) j =1 (cid:16) α j +1 ˆ f j +1 ( β j +1 ( w j +1 + w j )) (cid:17) w j +1 + w j (cid:16) α j +1 ˆ f j +1 ( β j +1 w j +1 ) (cid:17) w j +1 (cid:16) α j +1 ˆ f j +1 ( β j +1 w j ) (cid:17) w j (cid:80) m − j =1 wj = (cid:32) m − (cid:89) j =1 (cid:32) f ( k + 1) k j +1 + k j f ( k ) k j +1 f (1) k j (cid:33) m − (cid:89) j = m − (cid:32) g ( h + 1) h j +2 − m k m − + h j +1 − m k m − g ( h ) h j +2 − m k m − g (1) h j +1 − m k m − (cid:33)(cid:33) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − = (cid:32) m − (cid:89) j =1 (cid:18) f ( k + 1) k +1 f ( k ) k f (1) (cid:19) k j m − (cid:89) j = m − (cid:18) g ( h + 1) h +1 g ( h ) h g (1) (cid:19) h j +1 − m k m − (cid:33) (cid:80) m − j =1 kj + (cid:80) m − j = m − hj +1 − mkm − ≥ (cid:18) f ( k + 1) k +1 f ( k ) k f (1) (cid:19) − hk − h (cid:18) g ( h + 1) h +1 g ( h ) h g (1) (cid:19) k − k − h − (cid:15) (60) >M + (cid:15) − (cid:15) (61)= M, (62) where (59) comes from (58), (60) can be shown by using similar arguments as in the proof of Theorem 2 (seesteps (17) and (21)), and (61) comes from (54). By (62), the claim follows.If h = 0, we consider a load balancing instance defined as I ( m ), but restricted to resources r , r , . . . , r m and to players in [ m − h >
0, one can show theclaim as well. (cid:117)(cid:116)
D.3 Proof of Corollary 5
The proof follows from the following lemma.
Lemma 3. sup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) o − k o k − o k (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) k − o o k − o k = 4 p . ash Social Welfare in Load Balancing 27 Proof.
We have thatsup k ≥ o > ,o >k ≥ ,f ,f ∈C (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) o − k o k − o k (cid:18) f ( k + o ) k + o f ( k ) k f ( o ) o (cid:19) k − o o k − o k = sup k ≥ o > ,o >k ≥ ,α ,...,α p ≥ ,β ,...,β p ≥ (cid:32) (cid:0)(cid:80) pd =0 α d ( k + o ) d (cid:1) k + o (cid:0)(cid:80) pd =0 α d k d (cid:1) k (cid:0)(cid:80) pd =0 α d o d (cid:1) o (cid:33) o − k o k − o k (cid:32) (cid:0)(cid:80) pd =0 β d ( k + o ) d (cid:1) k + o (cid:0)(cid:80) pd =0 β d k d (cid:1) k (cid:0)(cid:80) pd =0 β d o d (cid:1) o (cid:33) k − o o k − o k = sup k ≥ o > ,o >k ≥ ,α ,...,α p ,β ,...,β p ≥ (cid:32)(cid:18) (cid:80) pd =0 α d ( k + o ) d (cid:80) pd =0 α d k d (cid:19) k (cid:18) (cid:80) pd =0 α d ( k + o ) d (cid:80) pd =0 α d o d (cid:19) o (cid:33) o − k o k − o k · (cid:32)(cid:18) (cid:80) pd =0 β d ( k + o ) d (cid:80) pd =0 β d k d (cid:19) k (cid:18) (cid:80) pd =0 β d ( k + o ) d (cid:80) pd =0 β d o d (cid:19) o (cid:33) k − o o k − o k = sup k ≥ o > ,o >k ≥ (cid:32)(cid:18) max d ∈ [ p ] ∪{ } ( k + o ) d k d (cid:19) k (cid:18) max d ∈ [ p ] ∪{ } ( k + o ) d o d (cid:19) o (cid:33) o − k o k − o k · (cid:32)(cid:18) max d ∈ [ p ] ∪{ } ( k + o ) d k d (cid:19) k (cid:18) max d ∈ [ p ] ∪{ } ( k + o ) d o d (cid:19) o (cid:33) k − o o k − o k = sup k ≥ o > ,o >k ≥ (cid:32)(cid:18) ( k + o ) p k p (cid:19) k (cid:18) ( k + o ) p o p (cid:19) o (cid:33) o − k o k − o k (cid:32)(cid:18) ( k + o ) p k p (cid:19) k (cid:18) ( k + o ) p o p (cid:19) o (cid:33) k − o o k − o k = sup k ≥ , ≤ h< (cid:18) ( k + 1) k +1 k k (cid:19) − hk − h (cid:18) ( h + 1) h +1 h h (cid:19) k − k − h p , (63)where (63) can be obtained by setting k := k /o and h := k /o . Now, we show that the maxi-mum value of function F ( k, h ) := (cid:16) ( k +1) k +1 k k (cid:17) − hk − h (cid:16) ( h +1) h +1 h h (cid:17) k − k − h over k ≥ ≤ h < F ( k, h )) = − hk − h (( k + 1) ln( k + 1) − k ln( k )) + k − k − h (( h + 1) ln( h + 1) − h ln( h )) ≤ (cid:16) − hk − h ( k + 1) + k − k − h ( h + 1) (cid:17) ln (cid:16) − hk − h ( k + 1) + k − k − h ( h + 1) (cid:17) , where the second last inequality holds becauseof the concavity of the function g defined as g ( x ) := ( x + 1) ln( x + 1) − x ln( x ) and since ln( F ( k, h )) is definedas convex combination of g ( k ) and g ( h ). Thus, we get F ( k, h ) ≤ (cid:18) − hk − h ( k + 1) + k − k − h ( h + 1) (cid:19) − hk − h ( k +1)+ k − k − h ( h +1) = (cid:18) ( k − h ) + ( k − h ) k − h (cid:19) ( k − h )+( k − h ) k − h = 2 = 4 . (64)Finally, since F ( k, h ) = 4 for k = 1 and h = 0, and because of (64), we have that the maximum of F ( k, h )over k ≥ ≤ h < p . (cid:117)(cid:116) D.4 Tightness of the Upper Bound of Corollary 5 w.r.t. any Online Algorithm.Theorem 9.
The competitive ratio of any online algorithm A applied to load balancing instances with poly-nomial latencies of maximum degree p is at least CR A ( P ( p )) ≥ p , even for instances with identical resources.Proof. We equivalently show that, for any online algorithm A and (cid:15) >
0, there exists a load balancinginstance I such that CR A ( I ) ≥ p − (cid:15) . We construct an instance similar to that defined in Theorem 17 of [22]. Given an integer m ≥ w >
0, let I ( m ) be a load balancing instance with identicalpolynomial latency functions of type (cid:96) ( x ) = x p , and recursively defined as follows: – If m = 0, I ( m ) has no clients and there is a unique resource denoted as fundamental resource of I (0). – If m ≥
1, then: (i) I ( m ) contains a sub-instance equivalent to I ( i −
1) for any i ∈ [ m ]; (ii) I ( m ) has afurther resource r denoted as fundamental resource of I ( m ); (iii) there are further m clients such that,for any i ∈ [ m ], the i -th client has weight w i := 2 i − and can select among r and the fundamentalresource r ( i ) of the sub-instance of type I ( i −
1) included in I ( m ); (iv) for any client i ∈ [ m ], r and r ( i )are respectively denoted as first and second resource of the i -th client included in I ( m ).Let σ and σ ∗ be the states of I ( m ) in which each client is assigned to her first and second resource, respectively.We have that σ is a state that can be returned by any online algorithm if clients are processed accordingto the following partial ordering: (i) given two clients i and i having their first resource in sub-instancesof type I ( m ) and I ( m ) respectively, if m < m then client i is processed before client i ; (ii) the clientsdefined in the same sub-instance are processed in increasing order with respect to their weights. This fact istrue since each time the greedy algorithm processes some client i according to the partial ordering definedabove, the congestions of the first and the second resource of that client are equal. Thus, since the latencyfunctions are equal too, any online algorithm cannot distinguish between the two resources selectable byeach client, and by symmetry both choices can potentially lead to the same worst-case competitive ratio.We have the following fact: Fact 4
Given two integers m ≥ and i ∈ [ m − ∪ { } such that j ≥ i , the number N ( m, i ) of sub-instancesof I ( m ) equivalent to I ( j ) for some j ≥ i is N ( m, i ) = 2 m − i .Proof. We show the claim by induction on h ( i ) := m − i ≥
0. If h ( i ) = 0 the unique sub-instance equivalentto I ( j ) for some j ≥ i is the entire instance I ( m ), thus N ( m, i ) = 1 = 2 h ( i ) = 2 m − i and the base stepholds. Now, assume that the claim holds for any h ( i ) ≥
0. Observe that we can associate in a one-to-onecorrespondence each sub-instance that is equivalent to I ( j ) for some j ≥ i , with a sub-instance equivalentto I ( i − N ( m, i ) = N ( m, i − − N ( m, i ) ⇒ N ( m, i −
1) = 2 N ( m, i ). Thus, we have that N ( m, i −
1) = 2 N ( m, i ) = 2 · h ( i ) = 2 m − i +1 = 2 h ( i )+1 , and the inductive step holds. (cid:117)(cid:116) Let N ( m, i ) be defined as in Fact 4 and let R ( i ) be the set of fundamental resources for sub-instances oftype I ( i ). Observe that, for any i ∈ [ m ] and resource r such that i clients select r as first resource, r is thefundamental resource of a sub-instance of type I ( i ), i.e., r ∈ R ( i ). Thus, by exploiting Fact 4, we get NSW ( σ ) = (cid:89) i ∈ [ m ] (cid:89) r ∈ R ( i ) (cid:96) ( k r ( σ )) k r ( σ ) (cid:80) r ∈ R kr ( σ ) = (cid:89) i ∈ [ m ] (cid:96) i (cid:88) j =1 w j ( (cid:80) ij =1 w j ) | R ( i ) | (cid:80) i ∈ [ m ] ( (cid:80) ij =1 wj ) | R ( i ) | = (cid:89) i ∈ [ m ] (cid:96) i (cid:88) j =1 j − ( (cid:80) ij =1 j − ) ( N ( m,i ) − N ( m,i +1)) (cid:80) i ∈ [ m ] ( (cid:80) ij =1 2 j − ) ( N ( m,i ) − N ( m,i +1)) = (cid:89) i ∈ [ m ] (cid:0) i − (cid:1) p ( i − ) m − i − (cid:80) i ∈ [ m ] ( i − ) m − i − (65)and NSW ( σ ∗ ) = (cid:89) i ∈ [ m ] (cid:89) r ∈ R ( i ) (cid:89) j ∈ [ i ] (cid:96) ( k r ( j ) ( σ ∗ )) k r ( j ) ( σ ∗ ) (cid:80) r ∈ R kr ( σ ) ash Social Welfare in Load Balancing 29 = (cid:89) i ∈ [ m ] (cid:89) j ∈ [ i ] (cid:96) ( w j ) w j ( N ( m,i ) − N ( m,i +1)) (cid:80) i ∈ [ m ] ( i − ) m − i − = (cid:89) i ∈ [ m ] (cid:89) j ∈ [ i ] (cid:0) j − (cid:1) p j − m − i − (cid:80) i ∈ [ m ] ( i − ) m − i − = (cid:89) i ∈ [ m ] p ( (cid:80) i − j =0 j j ) m − i − (cid:80) i ∈ [ m ] ( i − ) m − i − = (cid:89) i ∈ [ m ] p ( i i − ( i − )) m − i − (cid:80) i ∈ [ m ] ( i − ) m − i − (66)Let (cid:15) >
0. By (65) and (66), and by taking a sufficiently large integer m >
1, we get CR A ( I ) ≥ NSW ( σ ) NSW ( σ ∗ )= (cid:81) i ∈ [ m ] (cid:0) i − (cid:1) p ( i − ) m − i − (cid:81) i ∈ [ m ] p ( i i − i − m − i − (cid:80) i ∈ [ m ] ( i − ) m − i − = (cid:81) i ∈ [ m ] (cid:0) i − (cid:1) p ( i − ) − i − (cid:81) i ∈ [ m ] p ( i i − i − − i − (cid:80) i ∈ [ m ] ( i − ) − i − = (cid:81) i ∈ [ m ] (cid:0) i (cid:1) p ( i − ) − i − (cid:81) i ∈ [ m ] p ( i i − i − − i − (cid:80) i ∈ [ m ] ( i − ) − i − (cid:89) i ∈ [ m ] (cid:18) i − i (cid:19) p ( i − ) − i − (cid:80) i ∈ [ m ] ( i − ) − i − = (2 p ) (cid:80) i ∈ [ m ] ( − i − i − − − i ) (cid:80) i ∈ [ m ](1 / − − i − (cid:89) i ∈ [ m ] (cid:18) i − i (cid:19) p ( / − − i − ) (cid:80) i ∈ [ m ] ( / − − i − ) (67)We have the following fact: Fact 5 lim m →∞ (cid:89) i ∈ [ m ] (cid:18) i − i (cid:19) p ( / − − i − ) (cid:80) i ∈ [ m ] ( / − − i − ) = 1 . Proof.
Set α i := p ln (cid:16) i − i (cid:17) and β i := (cid:0) / − − i − (cid:1) . We will equivalently show that lim m →∞ (cid:80) mi =1 α i β i (cid:80) mi =1 β i = 0,since, by exponentiating this equality, we get the claim. Set a m := (cid:80) mi =1 α i β i and b m := (cid:80) mi =1 β i . Wehave that sequence ( b m ) m ≥ is positive, increasing, and unbounded. Thus, by the Stolz-Cesaro Theorem,we have that lim m →∞ a m b m = lim m →∞ a m +1 − a m b m +1 − b m . We conclude that lim m →∞ (cid:80) mi =1 α i β i (cid:80) mi =1 β i = lim m →∞ a m b m =lim m →∞ a m +1 − a m b m +1 − b m = lim m →∞ α m β m β m = lim m →∞ p ln (cid:0) m − m (cid:1) = 0, and the claim follows. (cid:117)(cid:116) By continuing from (67), we get= (2 p ) (cid:80) i ∈ [ m ] ( − i − i − − − i ) (cid:80) i ∈ [ m ](1 / − − i − (cid:89) i ∈ [ m ] (cid:18) i − i (cid:19) p ( / − − i − ) (cid:80) i ∈ [ m ] ( / − − i − ) ≥ lim m →∞ (2 p ) (cid:80) i ∈ [ m ] ( − i − i − − − i ) (cid:80) i ∈ [ m ](1 / − − i − (cid:89) i ∈ [ m ] (cid:18) i − i (cid:19) p ( / − − i − ) (cid:80) i ∈ [ m ] ( / − − i − ) − (cid:15) = lim m →∞ (2 p ) (cid:80) i ∈ [ m ] ( − i − i − − − i ) (cid:80) i ∈ [ m ](1 / − − i − − (cid:15) (68)= lim m →∞ (2 p ) − m − m + m +21 − m − / ( m +2 − m − ) − (cid:15) = (2 p ) (cid:18) lim m →∞ − m − m + m +21 − m − / ( m +2 − m − ) (cid:19) − (cid:15) = (2 p )( lim m →∞ m / m ) ) − (cid:15) = (2 p ) − (cid:15) = 4 p − (cid:15), where (68) comes from Fact 5. We conclude that there exists a load balancing instance I such that CR A ( I ) ≥ p − (cid:15) , thus, for the arbitrariness of (cid:15) , the claim follows. E Lower bound for Linear Congestion Games
Unweighted congestion games are a further generalization of unweighted load balancing games. The differenceis that the strategy set of each player i ∈ N is a collection Σ i ⊆ R \ {∅} , i.e., a strategy is a non-emptysubset of R . Furthermore, given a strategy profile σ = ( σ , . . . , σ n ) (with σ i ∈ Σ i ), the cost of each player i ∈ N is cost i ( σ ) := (cid:80) j ∈ σ i (cid:96) j ( k j ( σ )), where k j ( σ ) := | i ∈ N : j ∈ σ i | is the congestion of resource j instrategy profile σ . In the following theorem, we show that, even for linear latency functions, the Nash priceof anarchy of unweighted congestion games with linear latency functions is non-constant in the number ofplayers, differently from the case of load balancing games. This fact exhibits a substantial difference withrespect to the case of the price of anarchy when the considered social function is the sum of the players’costs. Indeed, in such case, the price of anarchy for linear congestion games is finite, and the price of anarchyof load balancing games is as high as that of general linear congestion games. Theorem 10.
The Nash price of anarchy of linear congestion games is at least n − o (1) , where n is thenumber of players (and o (1) is an infinitesimal w.r.t. to n ).Proof. We show that, for any (cid:15) ∈ (0 , / CG with linear latency functionsand n ≥ NPoA ( CG ) ≥ (cid:100) n(cid:15) (cid:101) − (cid:100) n(cid:15) (cid:101) n , (69)and this fact will imply the claim, as (cid:100) n(cid:15) (cid:101) − (cid:100) n(cid:15) (cid:101) n ∈ Θ ( n − (cid:15) ) for any fixed (cid:15) ∈ (0 , / (cid:15) ∈ (0 , / n ≥
2, and m := (cid:100) n(cid:15) (cid:101) . Let CG ( n, (cid:15) ) be an unweighted congestion game with n players defined as follows: Theset of resources is organized into three groups R , R , R , with R j := { r j, , . . . , r j,n − m } for any j ∈ [2], and R := { r , , . . . , r ,m } . The latency function of each resource r j,h is (cid:96) r j,h ( x ) := α j x , where α = m +1, α = 1,and α = m . There are two groups of players N , N , with N := { i , , . . . , i ,n − m } and N := { i , , . . . , i ,m } .Each player i ,h ∈ N has two strategies S ,h and S ∗ ,h defined as S ,h := { r ,h } and S ∗ ,h := { r ,h } , and eachplayer i ,h ∈ N has two strategies S ,h and S ∗ ,h defined as S ,h := R and S ∗ ,h := { r ,h } . Let σ (resp. σ ∗ )be the strategy profile such that each player i t,h plays strategy S t,h (resp. S ∗ t,h ), for any t ∈ [2]. One caneasily show that cost i ( σ ) = cost i ( σ − i , σ ∗ i ) for any player i , thus σ is a pure Nash equilibrium. We have that: NPoA ( CG ( n, (cid:15) )) ≥ NSW ( σ ) NSW ( σ ∗ )= (cid:32)(cid:32) (cid:89) i ∈ N cost i ( σ ) cost i ( σ ∗ ) (cid:33) (cid:32) (cid:89) i ∈ N cost i ( σ ) cost i ( σ ∗ ) (cid:33)(cid:33) n = (cid:32)(cid:32) (cid:89) i ∈ N cost i ( σ − i , σ ∗ i ) cost i ( σ ∗ ) (cid:33) (cid:32) (cid:89) i ∈ N cost i ( σ − i , σ ∗ i ) cost i ( σ ∗ ) (cid:33)(cid:33) n ash Social Welfare in Load Balancing 31 = (cid:32)(cid:18) α ( m + 1) α (cid:19) n − m (cid:18) α α (cid:19) m (cid:33) n = ( m + 1) n − mn ≥ (cid:100) n(cid:15) (cid:101) − (cid:100) n(cid:15) (cid:101) n , (70)thus (69) holds, and the claim follows.(70)thus (69) holds, and the claim follows.