Distributed Selfish Load Balancing on Networks

Transcription

1 Distributed Selfish Load Balancing on Networks Petra Berenbrink Martin Hoefer Thomas Sauerwald Abstract We study distributed load balancing in networks with selfish agents In the simplest model considered here, there are n identical machines represented by verticen a network and m n selfish agents that unilaterally decide to move from one vetex to another if thimproves their experienced load We present several protocols for concurrent migration that satisfy desirable properties such as being based only on local information and computation and the absence of global coordination or cooperation of agents Our main contribution is to show rapid convergence of the resulting migration process to states that satisfy different stability or balance criteria In particular, the convergence time to a Nash equilibrium is only logarithmic in m and polynomial in n, where the polynomial depends on the graph structure In addition, we show reduced convergence times to approximate Nash equilibria Finally, we extend our results to networks of machines with different speeds or to agents that have different weights and show similar results for convergence to approximate and exact Nash equilibria Introduction Load balancing is an essential requirement in large networks to ensure efficient utilization of resources and satisfactory performance of the system In many large computer networks load balancing becomes a challenge because of the absence of global information and coordination When there is only local information available about the load situation and even the existence of machines, a centralized approach to load balancing inappropriate or even impossible Instead, one then needs to develop protocols that respect the informational and computational restrictions of the scenario In addition, the protocols should guarantee rapid convergence to balanced states Some distributed algorithmic approaches for load balancing have been proposed in algorithmic game theory, see, eg, [5, 5, 2, 8] In this context tasks are considered as selfish agents that act unilaterally and migrate concurrently between machines without global coordination Such an approach has two main advantages over protocols that use more centralized optimization Firstly, concurrent migration and unilateral decision making of multiple agents controlling the tasks reduce the coordination overhead and may still allow for rapid convergence ie, sublinear in the number of agents Secondly, such task agents have an incentive to follow the protocol This an advantage in modern computer networks that are influenced by a variety of economic incentives and developments In these networks centralized coordination is often absent and user actions are made in a selfish manner While these properties make protocols for concurrent selfish load balancing desirable, their existence and convergence properties are not well-understood in many load balancing contexts A preliminary version appeared at the 22nd ACM-SIAM Symposium on Discrete Algorithms SODA [8] Dept of Computer Science, Simon Fraser University, Canada petra@cssfuedu Supported by DFG grant Ho 383/3- Dept of Computer Science, RWTH Aachen University, Germany mhoefer@csrwth-aachende Max-Planck Institute for Computer Science, Saarbrücken, Germany sauerwal@mpi-infmpgde

2 INTRODUCTION 2 In this paper, we present protocols for selfish load balancing in a discrete network balancing model There are n identical machines which represent verticen an arbitrary, undirected graph G = V, E, and m tasks that are initially assigned arbitrarily to the machines Our protocols proceed in a round-based fashion In each round, every task picks a neighboring machine at random and decides probabilistically whether or not to migrate to that machine Hence, the tasks only need local information, ie, they have to know the load of machine they are currently assigned to and the load of the neighboring machinen the graph The main challenge in the design of concurrent protocols such as ours to carefully choose appropriate migration probabilitien order to guarantee rapid convergence while avoiding oscillation effects Our scenario represents a significant extension of the existing literature on selfish load balancing, as concurrent protocols have been considered essentially only for complete graphs [4, 7, 2, 5, 5, 6] In our model there are several concepts of a state in which the assignment of tasks stable or balanced The standard notion of stability is the Nash equilibrium NE, in which no player can unilaterally improve by moving to a neighboring machine Note that throughout this paper we restrict to pure NE and states that do not involve randomization A weaker and more rapidly achievable condition is that of ε-approximate Nash equilibrium ε-apx NE, where no player can decrease his personal load by less than a factor of ε However, in an NE the load difference between the least loaded and most loaded machines can still be in the order of the network diameter in our model In this paper, we consider protocols and study the convergence times to approximate and exact NE We extend our results to networks of machines with different speeds or to agents that have different weights and show similar results for convergence to approximate and exact NE Contribution and Techniques We propose and analyze concurrent probabilistic protocols to obtain approximate and exact NE for identical machines or machines with speeds, and for weighted tasks on identical machines Identical machines We present a protocol that reaches an exact NE after a number of rounds that depends only logarithmically on m Let be the maximum degree of the graph and µ 2 be the second smallest eigenvalue of the Laplace matrix of G The dynamics reach a NE in expected time O /µ 2 ln m + ln n + E /µ 2 For m δn 4 and δ >, the first part of this convergence time is the time needed to reach a 2/+δ-apx NE The second part is the time needed so that no agent wants to unilaterally deviate to a neighboring machine Therefore, the convergence time is only logarithmic in the number of agents m, but the dependency on n is polynomial and connected to the structure of the graph Obviously, in general a polynomial dependence on n cannot be avoided eg, for paths Machines with speeds In this case, each machine i V has a speed N,, and thus processes tasks at a different rate Let S := and δ >, then using our protocol a set of m δ 8 n 3 S agents converges to an 2/+δ-apx NE in expected time Olnm polyn polys max on any graph, where s max is the maximum speed An exact NE is reached after additional time Opolyn polys max, so in total also in time Olnm polyn polys max Weighted tasks Finally, we also consider the case that each task l has a weight w l N, w l In this case, the expected convergence time to a NE is O W 3 w max, where W is the sum of all weights and w max is the maximum weight

3 INTRODUCTION 3 Outline of the paper Due to technical reasons we present our resultn a different order than the one stated above After presenting the necessary definitions and preliminarien Section 2, we first derive in Section 3 the results on load balancing with speeds Theorems 3 and 32, as thintroduces the main technical framework The results on networks with identical machines are discussed in Section 4 Then we enhance the general approach for machines with speeds to show convergence to apx NE Theorems 4 and 42 Finally, Section 5 contains the extension to weighted tasks Theorem 5 and is mainly based on our previous approach 2 Related Work Most closely related to our paper is [5], where the case of identical machinen a complete graph is studied There, a protocol that is equivalent to ours shown to arrive at a NE in time Olog log m+ polyn Note that for complete graphs the NE and optimal allocations are identical An extension of this model to weighted tasks studied in [6] There the authors consider a similar protocol for the complete graph They show that the expected time to reach an approximate NE is Omn wmax 3 Here we extend the results of both papers significantly by studying dynamics on general graphs and machines that have speeds This also requires to use different techniques that allow to capture the connections between convergence time and graph structure Our paper relates to a general stream of works for selfish load balancing on complete graphs There is a variety of issues that have been considered, starting with seminal papers on algorithms and dynamics to reach NE [4, 6] More directly related are concurrent protocols for selfish load balancing in different contexts that allow convergence results similar to ours Whereas some papers consider protocols that use some form of global information [5] or coordinated migration [20], others consider infinitesimal or splittable tasks [9, 4] or work without rationality assumptions [7, 2] The machine modeln these cases range from identical, uniformly related linear with speeds to unrelated machines The latter also contains the case when there are access restrictions of certain agents to certain machines In contrast, in our model players migrate over a network and can access machines only depending on their current location This a fundamental difference to all previous related work in this area For an overview of work on selfish load balancing see, eg, [3] A slightly different approach for discrete selfish load balancing on networks are finite congestion games [29], for which the convergence times of sequential best-response dynamics to exact and approximate NE have been extensively studied [0, 3, 30] A general approach for a concurrent better-response protocol is [], which inspired by similar results for non-atomic congestion games [8] In this protocol, agents pick strategies only by imitation of other agents, and there is rapid convergence even for general delay functions, but only to an approximate equilibrium concept where all agents experience a similar cost While this represents a generally applicable approach, the obtained approximate equilibrium might be far from any apx NE A different line of research are no-regret and similar payoff-based learning dynamics [23, 24, 25, 26], but they usually converge quickly only in the history of play and/or to classes of mixed NE In contrast, we present protocols that reach pure exact and approximate NE rapidly Our protocol is also related to a vast amount of literature on non-selfish load balancing over networks, where results usually concern the case of identical machines and unweighted tasks Often there are additional restrictions on the graph structure such as regular graphs, expander graphs, tori, etc A central measure of balance is the discrepancy, ie, the difference between most and least loaded machine in the network In expectation, our protocols mimic continuous diffusion, which has been studied initially in [2, 9] and later, eg, in [27] This work established the connection between convergence, discrepancy, and eigenvalues of graph matrices Closer to our paper are discrete diffusion processes prominently studied in [28], where the authorntroduce a general

4 2 NOTATION AND PRELIMINARIES 4 technique to bound the load deviations between an idealized and the actual processes Recently, randomized extensions of the algorithm in [28] have been considered, eg, [3, 2] However, either machines have to communicate with their neighbors to determine the number of tasks that should move [28, 2], or the tasks perform independent random walks [3] In the first case, machines have a strong control over their tasks, while in the second case tasks may jump from an underloaded to an overloaded machine, which clearly is undesirable in a game-theoretic context 2 Notation and Preliminaries We consider an arbitrary, undirected and connected graph G = V, E with n = V vertices representing machines The degree of a vertex i V is degi denotes the maximum degree of any vertex in V For two vertice, j, degi, j = max{degi, degj} is the maximum degree of i and j There are m taskn the system, which are initially assigned arbitrarily to the n machines We denote by x a state of the system, ie, a fixed assignment of tasks to the machines For any machine i V, we denote by x i the set of tasks are that assigned to machine i in x We consider a probabilistic migration process, in which the state is a random variable in each time step In particular, let X t be the state at the end of step t Similarly, Xi t is the subset of tasks assigned to machine i V, and it is a random variable for every t due to the probabilistic nature of our migration protocols Each task l has a weight w l N and, unless stated otherwise, we assume uniform tasks with w l = Each vertex i V is a machine with a speed N, We define S = n i= Note that we can also handle rational speeds by normalization to integers By s max and s min we denote the maximal and minimal speed of a machine, respectively If s max = s min, we have a network of identical machines and assume wlog = for all i V In the case of uniform tasks the load of a machine is defined as the sum of tasks assigned to it, divided by the speed of the machine In the case of weighted tasks the load is the sum of the weights of these jobs divided by the speed In particular, by W x i we denote the weight on machine i in state x, ie, the sum of weights of all tasks that are located on i Similarly, W Xi t is the weight at the end of step t Let W := W x i For a state x the load of vertex i is denoted by Lx i and equals Lx i = W x i / Each task on machine i experiences a disutility of Lx i in state x Naturally, for our process at time t, we obtain the random variable LXi t A task is a selfish agent that strives to minimize the experienced load The task is only aware of the load of the machine it is currently located at and is able to inspect the load on one of the neighboring machines We consider a round-based process In each iteration every task uses a protocol to decide to which of the neighbouring machinet potentially migrates The Laplace L matrix is a standard matrix associated to undirected graphs, which is based on adjacency and degree information eg, [] Formally, L is defined as the n n-matrix where L i,j is equal to degi if j = i, if {i, j} EG and 0 otherwise The eigenvalues of the Laplace matrix are known to encode valuable structural information for dynamic load balancing processes, see, eg [22, 7] A state x is an ε-approximate Nash equilibrium ε-apx NE for any 0 ε if no task can decrease the experience load by more than factor of ε machine i and every neighboring machine j ε W x i W x j + For ε = 0 we call such a state an exact Nash equilibrium NE In such a state we have for every

5 3 UNIFORM TASKS AND RELATED MACHINES 5 Our game can be expressed as an atomic congestion game, and thus the following function due to Rosenthal [29] is a potential function for our game with uniform tasks: Φ x = W x i k= k = W x i W x i + 2 Whenever a single player makes a unilateral assignment change, the change in the potential function equals the load change experienced by the player Thus, the local optima of the potential function are exactly the NE of the game, and the potential function measures the progress to NE from a players point of view We will also use the quadratic potential function that is standard in the load balancing literature Φ 0 x = W x i 2 It measures progress to a completely balanced state from a global point of view This function will be helpful when we prove convergence to apx NE This leads to the following general definition Definition 2 For r {0, }, define Φ r x := W x i W x i + r For our migration process, we often consider the change of the potential, which is Φ r X t := Φ r X t Φ r X t We will also use a normalized formulation of Φ 0 x denoted by Ψ 0 x Definition 22 The function Ψ 0 x is defined as The potential change of Ψ 0 is defined as Note that for identical machinet holds Ψ 0 x = Φ 0 x m2 S Ψ 0 X t := Ψ 0 X t Ψ 0 X t = Φ 0 X t Ψ 0 x = Φ 0 x 2m m n + n m2 n 2 = W x i m n 2 As a standard convention, the term with high probability means with probability at least n c for some constant c > 0 Our main results are upper bounds on the expected time for our protocols to reach an apx NE We point out that one can get corresponding upper bounds which hold with high probability at the cost of a multiplicative increase by Olog n

6 3 UNIFORM TASKS AND RELATED MACHINES 6 for each task l in parallel do Let i = il the current machine of task l Choose a neighboring machine j uar if LXi t LXj t > / then Move task l from resource i to j with probability end if end for degi degi, j LXi t LXj t α + W X t i Figure : Protocol I for uniform tasks and machines with speeds We set α := 4s max 3 Uniform Tasks and Related Machines In this section, we first present our results on general networks and machines with speeds, as our analysis of this case introduces our main approach Protocol I in Figure allows tasks to move to neighboring machines with a smaller load In more detail, in each round every player randomly chooses a neighboring machine If the anticipated load of the other machine is smaller, the player moves to it with a probability that depends on several parameters: the degree of the actual vertices, the speeds of both machines and their load difference Note that these are all local parameters The factor degi/ degi, j = degi/max{degi, degj} is crucial to prevent that too many tasks from low degree vertices move to neighbors with a much larger degree Alternatively, we can use / in our analysis but this would require that all tasks know the maximum degree Our first result in this section shows that, for sufficiently large m, we reach an apx NE in time logarithmic in m The second result bounds the time to reach a NE Theorem 3 Let m δ 8 diamg n S for some δ > and ε = / + δ Then Protocol I reaches a ε-approximate Nash equilibrium in expected time O lnm diamg 2 S smax, s min which is Olnm polyn polys max on any graph Theorem 32 If m 8 diamg n S, then Protocol I reaches a Nash equilibrium in expected time O lnm diamg 2 S smax + diamg 2 n 2 3 S s 3 max s min on any graph Otherwise, Protocol I reaches a Nash equilibrium in expected time OdiamG 2 n 2 3 S s 3 max In both cases, the time in Olnm polyn polys max on any graph In expectation, the protocol behaves like a a continuous diffusion process To avoid oscillation we need α 4s max This, however, implies that even though tasks can have a large incentive for migration eg, if they move from a full and slow to an empty and fast machine, they never migrate

7 3 UNIFORM TASKS AND RELATED MACHINES 7 with a probability of more than /4s max Thus, to reach an apx NE where all players have a small incentive to migrate, it might take Ωs max many rounds for the last players to move Thus, a convergence time that polynomially depends on s max is unavoidable, given the way our protocol is defined In Section 3 we prove some fundamental bounds on the potential change in one step of Protocol I Using these insights we show Theorem 3 and Theorem 32 in Section 32 3 Potential Function Analysis First we introduce some additional definitions Note that throughout the paper we first estimate the expected potential decrease occuring in a single round of the process In particular, we usually condition on the event that X t is some arbitrary but fixed state x Definition 33 For any i, j V with {i, j} E and any given state x, the expected flow along this edge in a single round of our protocol starting from x is Lx i Lx j if Lx i Lx j > α degi,j + f i,j x := 0 otherwise Note that f i,j x is always non-negative Further, let } Ẽx := {i, j E : Lx i Lx j > sj, be the set of edges over which tasks have an incentive to move when the system in state x Finally, for any r {0, } we define Λ r i,jx := 2α 2 degi, j + f i,j x + r r We usually use the shorthand Ẽ instead of Ẽx Our aim is to prove that in one round the system makes progress towards a NE, ie, to show that E [ Φ r X t X t = x ] > 0 Let us first consider the potential change when the number of tasks transferred over any edge is exactly its expected number Hence, we define Φ r X t Xi t = x := W x W x + r E [ W X t i Xt = x ] 2 r E [ W Xi t Xt = x ] Note that the expected load at machine i at step t given the load vector x at step t equals E [ W Xi t X t = x ] = W x i + f j,i x f i,j x j Ni The following lemma generalizes [7, Lemma 2] to the setting with speeds Lemma 34 For any round t N it holds that Φ r X t X t = x f i,j x Λ r i,jx i,j Ẽx

8 3 UNIFORM TASKS AND RELATED MACHINES 8 Proof Let X t = x be fixed For reasons of simplification we sometimes omit the conditioning in this proof We order the edgen Ẽx increasingly according to f i,jx Let Ẽx = {e, e 2,, e } be the edgen that order We activate the edgen Ẽx sequentially and Ẽ x bound the potential change after the activation of each edge For any k Ẽx, let Xt,k i be the set of tasks assigned to vertex i after the k-th activation of an edge in Ẽx in round t Moreover, for any k Ẽx, and Φ r X t,k = Φ r X t,k Φ r X t,k Φ r X t = Ẽ k= Φ r X t,k Let us now fix k with e k = i, j and consider Φ r X t,k Note that f i,j x > 0 We have Similarly, Consequently, Φ r X t,k = W Xt,k i W Xt,k i = W Xt,k i W Xt,k i W X t,k j W x j + degj f i,j x W X t,k i W x i degi f i,j x W t,k i + r + W X W X t,k i + r W X W X t,k i + r + W X t,k j W X t,k t,k j W X t,k j + r j + r t,k j W X t,k f i,j x W X t,k i f i,j x + r W Xt,k j + f i,j x W X t,k j + f i,j x + r W X t,k i = f i,j x 2 W X t,k W xi degi f i,j x f i,j x 2 fi,j x f i,j x 2 f i,j x f i,jx W xi W x j j j + r f i,jx f i,jx + r r W x j + degj f i,j x + r r 2di, j f i,j x + + rsi rsj, 3

9 3 UNIFORM TASKS AND RELATED MACHINES 9 where until this point, we have not used the particular definition of f i,j x In the next step we use the definition of f i,j x to obtain f i,j x 2 αdi, j + = f i,j x 2α 2 di, j and summing up over all edges yields the claim f i,j x 2di, j f i,j x + f i,j x + r r, + + rsi rsj The previous lemma lower bounds the potential improvement given that the expectations are exactly attained To relate this to the real potential change, we have to upper bound the variance of the loads Lemma 35 For any step t and any state x, Proof Note that Var [ W X t i Xt = x ] = i,j Ẽ Var [ W Xi t Xt = x ] = f i,j x + Var [ Ci t ] At i, where Ci t and At i are the random variables conditioned on Xt = x defined as follows Ci t is the random variable that counts the number of tasks that come to link i from another link in x, and A t i is the random variable that counts the number of tasks that abandon link i in x Note that linearity of the variance for independent variables and the fact that Var [ Y ] = Var [Y ] yields Var [ Ci t ] At i = Var [ ] C t i + Var [ ] A t i = Var [ ] C t i + Var [ ] A t i Observe that Ci t = j : j,i Ẽ Zt j,i, where Zt j,i 0 is the random variable that counts the number tasks that move from j to i from x We have Zj,i t Bin W x j, αdi, j Lx j Lx i + W xj Similarly, A t i Bin W x i, αdi, j j : i,j Ẽ Lx i Lx j + W xi } Since {Z j,i t : j Ni is a set of independent random variables, we have Var [ Ci t ] = j : j,i Ẽ j : j,i Ẽ Var [ Bin W x j, αdi, j Lx j Lx i +, ] αdi, j Lx j Lx i + W xj

10 3 UNIFORM TASKS AND RELATED MACHINES 0 where the last inequality follows from Var [Binn, p] = np p np Similarly, Var [ A t ] i = Var Bin W x i, αdi, j Lx i Lx j + W xi Hence, j : i,j Ẽ Var [ W Xi t Xt = x ] j : i,j Ẽ αdi, j Lx i Lx j + j : j,i Ẽ αdi, j Lx j Lx i + + j : i,j Ẽ αdi, j Lx i Lx j + Since every edge in Ẽ occurs exactly twice in above sum, once with weight / and once with weight /, we obtain Var [ W X t i Xt = x ] i,j Ẽ = i,j Ẽ + Lx i Lx j αdi, j αdi, j Lx i Lx j + = i,j Ẽ f i,j x + With Lemma 35 at hand, we will now provide a bound on the real potential change, ie, we include the deviation which occurs since the actual number of transferred tasks may differ from their expected values Lemma 36 For any step t and any state x, E [ Φ r X t X t = x ] f i,j x Λ ri,jx si sj i,j Ẽx Proof We obtain E [ Φ r X t X t = x ] = Φ r x E [ Φ r X t X t = x ] = W x i W x i + r r E [ W Xi t Xt = x ] E [ W Xi t2 X t = x ] = W x i W x i + r r E [ W Xi t Xt = x ] E [ W Xi t Xt = x ] 2 Var [ W Xi t Xt = x ]

11 3 UNIFORM TASKS AND RELATED MACHINES = Φ r X t X t = x f i,j x Λ ri,jx si sj, i,j Ẽ Var [ W X t i Xt = x ] where the last inequality follows by applying Lemma 34 and Lemma 35 The next technical lemma builds on Lemma 36 and shows that every edge i, j Ẽx with a sufficiently large load difference contributes positively to E [ Φ r X t X t = x ] Lemma 37 Set α := 4s max 4 Then the following two statements hold for any state x: If r =, then for any i, j Ẽx with Lx i Lx j +, we have Λ i,jx + 2s max 2 If r = 0, then for any i, j Ẽx with Lx i Lx j +, we have Λ 0 i,jx + 2 Proof We first start with the claim for r = We obtain that Λ i,jx = 2α 2 di, j + f i,j x + r r by definition = 2α 2 di, j Lx i Lx j + αdi, j by assumption 2s max = s max 2s max since s max + + = s max

12 3 UNIFORM TASKS AND RELATED MACHINES 2 Similarly, consider now the second claim where r = 0 Then Λ 0 i,jx = 2α 2 di, j + f i,j x + r r by definition = 2α 2 di, j Lx i Lx j αdi, j 2 + by assumption 2s max = s max 2s max since s max Convergence to Approximate and Exact Nash Equilibria In this section, we finally prove our main theorems We use Lemma 34 for the migration in a single round and consider the potential drop wrt an ideal potential value if exactly the expected loads are realized Then, using Lemma 36, we bound the difference between ideal and realized potential values by analyzing the variance of the migration process This yields bounds on the expected drop of the potential in one round and is the main ingredient to prove the theorems Let us define for any given state x the maximum load difference as L x := max Lx i m/s In order to prove our first theorem, we use function Φ 0 and its normalized version Ψ 0 For any given state x, we observe a simple relation between the potential value Φ 0 x and the maximum load difference L x Note that Φ 0 x = W x i 2 / is minimized when all W x i = m/s Thimplies Φ 0 x m 2 /S 2 = m 2 /S The following observation considers the normalized version of Ψ 0 x Observation 38 For any state x of the system it holds that Proof Clearly, Φ 0 x m2 S = = L x 2 Ψ 0 x = Φ 0 x m2 S S L x 2 m/s + W x i m/s 2 m/s m2 S m/s W x i m/s = 2 m/s W x i m/s 2 s 2 i + = W x i m/s 2 + W x i m/s 2 W x i m/s 2 m2 S

13 3 UNIFORM TASKS AND RELATED MACHINES 3 and L x 2 Moreover, max W x i 2 m/s W x i m/s 2 = max s 2 i W x i m/s 2 W x i m/s 2 = W x i / m/s 2 S L x 2 Lemma 39 Let γ := 32 S diamg 2 s max /s min Let x be a state such that L x 8 diamg n Then E [ Ψ 0 X t X t = x ] Ψ 0 x γ Proof In this proof, we consider the progress of Φ 0 X t for an arbitrary but fixed assignment X t = x Consider now the protocol before the execution of the next step t Let l V be a vertex with Lx l m/s = L x, and assume wlog that Lx > m/s Then there must be another vertex k V with Lx k m/s Thimplies that there is an edge {p, q} E on a path from l to k such that Lx p m/s Lx q m/s + L x/ diamg, which is equivalent to By Lemma 36, E [ Φ 0 X t X t = x ] Lx p Lx q L x/ diamg i,j Ẽ f i,j x Λ i,j x si sj Let us now define for any edge i, j Ẽ, i,j Φ 0 X t X t = x := f i,j x Λ 0i,jx si sj, so that E [ Φ 0 X t X t = x ] i,j ΦX t X t = x i,j Ẽ Let us now group Ẽ into three disjoint groups where we omit the argument x throughout: { Ẽ := i, j Ẽ : Lx i Lx j + } p, q { Ẽ 2 := i, j Ẽ : Lx i Lx j < + } p, q Ẽ 3 := p, q

14 3 UNIFORM TASKS AND RELATED MACHINES 4 The intuition behind this grouping is as follows Ẽ contains all edges with a load difference large enough so that, in expectation, the edge yields a decrease of the potential Ẽ 2 contains all edges with a small load difference Due to the variance, these edges could increase the potential However, since the load difference is small, their total impact on the potential is limited Finally, Ẽ3 consists of the edge p, q which has a large load difference and thereby decreases the potential significantly in expectation Group : Lemma 37 with r = 0 shows that i,j Φ 0 X t X t = x = f i,j x Λ 0i,jx si sj 0 i,j Ẽ i,j Ẽ Group 2: In this case we consider the value of i,j Ẽ2 i,jφ 0 X t X t = x Let i, j be an edge in Ẽ2 Then, plugging in the definition of Λ 0 i,j x and f i,jx yields i,j Φ 0 X t X t = x = f i,j x Λ 0 i,jx f i,j x + Lx i Lx j = α di, j + 2α 2 di, j + f i,j x Lx i Lx j s α di, j j Lx i Lx j = α di, j Lx i Lx j Lx i Lx j α } {{ } α di, j 0 Lx i Lx j α di, j > + α 2 α, since we assumed that all speeds are larger than one Summing up over all edgen Ẽ2, we obtain i,j Ẽ2 i,j Φ 0 X t X t = x 2 E α Group 3: Consider now the edge p, q, ie, the set Ẽ3: p,q Φ 0 X t X t = x = f p,q x Λ 0p,qx sp sq Lx p Lx q α dp, q s p + s q Lx p Lx q α dp, q s p + s q = 2 2 α L x diamg 2 L x/ diamg α dp, q L s p + x 2 diamg s q L x 2 2diamG 2 α dp, q s p + s q, Lx p Lx q 2

15 3 UNIFORM TASKS AND RELATED MACHINES 5 where the second last inequality holds since Lx p Lx q L x/ diamg and α 2 The last inequality holds since by assumption, L x 8 diamg n Combining the contribution of all three groups yields E [ Φ 0 X t X t = x ] i,j Φ 0 X t X t = x + i,j Φ 0 X t X t = x + p,q Φ 0 X t X t = x {i,j} Ẽ {i,j} Ẽ2 0 2 E α + L x 2 2 diamg 2 α dp, q s p + s q L x 2 4 diamg 2 α dp, q s p + s q, where the last inequality uses L x 8 diamg n Now we continue with the normalized version of Φ 0 x, ie, with Ψ 0 x By Observation 38, Ψ 0 x S L x 2 Since Ψ 0 X t = Φ 0 X t and α = 4s max we obtain as long as L x > 8 diamg n that E [ Ψ 0 X t X t = x ] L x 2 4 diamg 2 α dp, q s p + s q Ψ 0 x 4 S diamg 2 α s p + s q Ψ 0 x 32 S diamg 2 smax s min With γ := 32 S diamg 2 s max /s min we have E [ Ψ 0 X t X t = x ] γ Ψ 0x, or equivalently E [ Ψ 0 X t X t = x ] Ψ 0 x γ Lemma 30 Let γ := 32 S diamg 2 s max /s min and τ := γ 2 ln n + lnψ 0 X 0 Then, with probability at least n, there exists a round t [, τ] such that L X t < 8 diamg n Proof Let us define an auxiliary random variable Ψ 0 by Ψ 0 X 0 := Ψ 0 X 0, and for any round t, { Ψ 0 X t Ψ 0 X t if [L X t 8 diamg n ] [ Ψ 0 X t > 0] = 0 otherwise Then, for any t, it follows by Lemma 39, ] E [ Ψ0 X t X t = x γ Ψ 0 X t

16 3 UNIFORM TASKS AND RELATED MACHINES 6 We have for τ = γ 2 ln n + ln Ψ 0 X 0, ] E [ Ψ0 X t = ] E [ Ψ0 X t X t = x P [X t = x] x γ Ψ 0 X t P [X t = x] x τ γ Ψ 0 X 0 n 2 Hence by Markov nequality, We consider two cases Pr [ Ψ0 X τ n ] n Case : For all time-steps t [0,, τ], Ψ 0 X t = Ψ 0 X t Then by Observation 38, L X τ Ψ 0 X t = Ψ 0 X t n /2 Case 2: There exists a step t [,, τ] such that Ψ 0 X t Ψ 0 X t Let t be the smallest time step with that property Hence, Ψ 0 X t Ψ 0 X t, but Ψ 0 X t = Ψ 0 X t If Ψ 0 X t = 0, then L X t Ψ 0 X t = Ψ 0 X t = 0 If Ψ 0 X t 0, then by definition of Ψ 0 X t, Ψ0 X t Ψ 0 X t Ψ0 X t 0 L X t < 8 diamg n In all cases we have shown that there exists a step [0, τ] so that L X t < 8 diamg n This completes the proof of the lemma 32 Proof of Theorem 3 First note that for any state x we have Ψ 0 x m 2 /s min m 2 We now show that for m 8δ n 3, any state x with L x 8 diamg n 8 n 3 indeed a / + δ-apx NE For every i V we have W x i / m/s L x 8 n 3 Consider now any pair i, j with {i, j} E Then W x i / 8 n 3 + m/s and similarly W x j + / max { /, m/s 8n 3 + / } We are looking for the smallest possible ε [0, such that ε W X T i / W X T j + / Plugging in our bounds from above, a simple calculation yields the result m 8δ n 3 From Lemma 30, it follows now that with probability at least n 4 after τ = 32S diamg 2 s max /s min 4 ln n + ln Ψ 0 X 0 = O lnm diamg 2 S smax s min steps, we reach a / + δ-apx NE in the time-interval [0,,, τ] Hence after expected O lnm diamg 2 S smax s min rounds, we have reached a / + δ-apx NE This proves Theorem 3

17 3 UNIFORM TASKS AND RELATED MACHINES Proof of Theorem 32 Now we show that there is still a small potential drop even if L is already relatively small First we show some results that will be needed in our later proofs Observation 3 Let x be a state that is not a Nash equilibrium Then there exist two neighbouring vertice, j such that Lx i Lx j > If all speeds are integers, then we also have Lx i Lx j + Proof The fact that two neighbouring vertice, j exist with Lx i Lx j > follows directly from the definition of a NE Also, W x i W x j > W x i > W x j + 32 Let us now fix any, and sum of weighted tasks W x j Our goal is to lower bound the smallest possible W x i such that the strict inequality above is fulfilled We proceed by a case distinction First we assume / W x j + is an integer Then, and therefore, W x i W x j + +, Lx i Lx j = W x i W x j + + Now we assume that W x j + is not an integer Hence, si W x i W x j + We now use the fact that a b a b b to conclude that for any pair of integers a, b where a is not a multiple of b W x i W x j + + and Lx i Lx j = W x i W x j + Again, we define a normalized version of Φ For every state x, we define m 2 Ψ x = Φ x S + n m S n The next Lemma shows a relation between L and Ψ 4s min Lemma 32 Let x be an arbitrary state With the definition above we have 0 Ψ x S L x 2 + n L x + n

18 3 UNIFORM TASKS AND RELATED MACHINES 8 Proof We first show the left inequality, ie, we show a lower bound on Φ x In order to bound the minimum, we characterize the NE with minimum potential The loads are equilibrated to m/s on all machines when the weight is W x i = m /S for every machine i Thus, in every state x of the game we have a weight W x i = m /S + δ i, where δ i [ m S Using thinsight, we bound msi S + δ 2 m i S + + δ i Φ x = = m 2 S 2 = m2 S + nm S = m2 S + nm S + 2mδ i S + 2m S + + δ2 i + m S + δ i δ i + m2 S + nm S n 4s min, δ i δ i +, m m S δ 2 i + δ i ] and δ i = 0 because the function yy + /4 for any y This proves the lower bound in the lemma To show the upper bound, we note that Lx i m/s L x and calculate Φ x m2 S nm S + n 4s min = Φ 0 x + Lx i m2 S nm S + n S L x 2 + Lx i m + n S 4s min S L x 2 + n L x + n, 4s min where the first inequality follows from Observation 38 This finishes the proof of the lemma Lemma 33 Assume that at step t the system is not in a Nash equilibrium We have E [ Ψ X t+ X t = x ] Ψ x 8 s 3 max Proof Note that Observation 3 provides the minimum decrease required to apply Lemma 37 for Φ x Plugging in this bound for Λ i,j in Lemma 36 yields an expected additive decrease of at least /8 s 3 max per round, as long as x is not a NE Lemma 34 Assume that at step τ we have L X τ 8 diamg n Let T be the additional number of steps such that X τ+t is a Nash equilibrium Then E [T ] 560 diamg 2 n 2 3 S s 3 max Proof For reasons of simplification let us assume τ = 0 Lemma 32 implies that for every state x with L x 8 diamg n Ψ x S 8 diamg n 2 + n 8 diamg n + n S 70 diamg 2 n 2 2

19 3 UNIFORM TASKS AND RELATED MACHINES 9 Hence we can assume that E [ Ψ X 0 ] 70 S diamg 2 n 2 2 The rest of the proof is done by a standard martingale argument Let T be the end of the first time step after the system reaches a NE Let t T be the minimum of t and T and let V = /8 s 3 max We define Z t = Ψ X t + t V Observe that T is a stopping time for Z t t0 {Z t } t T is a supermartingale since by Lemma 33 with X t = x, E [Z t+ Z t = z] = E [ Ψ X t+ + V t + Ψ x + t V = z ] E [ Ψ X t+ Ψ x = z t V ] + t + V z tv V + t + V = z Hence E [Z t+ ] = z E [Z t+ Z t = z] Pr [Z t = z] z z Pr [Z t = z] = E [Z t ] We obtain Therefore, V E[T ] E[ΨX T ] + V E[T ] = E[Z T ] E[Z 0 ] 70 S diamg 2 n 2 2 E[T ] 70 S diamg 2 n 2 2 /V 70 S diamg 2 n s 3 max = 560 diamg 2 n 2 3 S s 3 max Now we are ready to show the theorem From Lemma 30 we know that there is an integer such that T γ 2 ln n + ln Ψ 0 X 0 Pr [ t T : L X t 8 diamg n ] /4 Let T 2 be the additional number of steps until X t+t 2 which implies Hence, for is a NE From Lemma 34 we get E [T 2 ] 560 diamg 2 n 2 3 S s 3 max, Pr [ T diamg 2 n 2 3 S s 3 max] /4 T = T diamg 2 n 2 3 S s 3 max, the probability that X T is not a NE is at most 2/4 Now divide the time into intervals of length T The expected number of these time intervals at most + /2 + /2 2 + /2 3 2 In addition, after c log 2 n intervals the state X t is a NE with probability at least /n c This proves Theorem 32

20 4 UNIFORM TASKS AND IDENTICAL MACHINES 20 4 Uniform Tasks and Identical Machines In this section we consider the case of identical machines We use Protocol I with s max = and α = 3 For uniform tasks and identical machines, a task assigned to i prefers machine j if and only if W Xi t W Xj t > With µ 2 being the second smallest eigenvalue of the Laplace matrix, we show the following result Theorem 4 Let m δn 5 for some δ > With α = 3, Protocol I reaches a 2/+δ-approximate Nash equilibrium in expected time O /µ 2 ln m Theorem 42 With α = 3, Protocol I reaches a Nash equilibrium in expected time 4 Proof of Theorem 4 O /µ 2 ln m + ln n + E /µ 2 For a state x, we define f i,j x an Section 3, ie, as the expected load that is sent from i to j in one round of Protocol I The next lemma bounds the potential change for the edges with a load difference of at most two Lemma 43 Let µ 2 be the second smallest eigenvalue of the Laplace matrix and let α = 3 Assume Ψ 0 x 2 E /µ 2, then E [ Ψ 0 X t X t = x ] µ 2 Ψ 0 x 36 Proof Consider Rosenthal s potential function, which simplifies to Φ x = W x i W x i + since all = We obtain Φ x = W x i W x i + = = Φ 0 x + m = Ψ 0 x + m2 n + m W x i 2 + W x i Hence, for any given state x functions Φ x, Φ 0 x and Ψ 0 x differ only by additive terms and Φ X t X t = x = Ψ 0 X t X t = x Note that for uniform tasks and with α = 3 we have { W xi W x j 6 degi,j if W x i W x j > f i,j x = 0 otherwise We also have and Λ 0 i,jx = 8 degi, j f i,j x, Ẽx := {i, j E : W x i W x j > },

21 4 UNIFORM TASKS AND IDENTICAL MACHINES 2 where we omit the argument x whenever possible Applying Lemma 36 we get E [ Ψ 0 X t X t = x ] f i,j x Λ 0 i,jx 2 = = i,j Ẽ {i,j} Ẽ {i,j} Ẽ {i,j} Ẽ Now we apply x x 2 /2 for x 2 to obtain W x i W x j 6 di, j W x i W x j 3 di, j 2 W x i W x j 2 9 di, j 8 di, j W x i W x j 6 di, j {i,j} Ẽ W x i W x j 3 di, j E [ Ψ 0 X t X t = x ] = 2 W x i W x j 2 9 di, j W x i W x j 2 8 di, j {i,j} Ẽ {i,j} Ẽ {i,j} Ẽ 2 W x i W x j 2 3 di, j We conclude from Lemma 43 that E [ Ψ 0 X t X t = x ] {i,j} Ẽ W x i W x j 2 8 di, j {i,j} Ẽ W x i W x j 2 8 On the other hand, a classic result from standard continuous diffusion [7, Theorem 4] see also [22, Proof of Theorem ] shows that W x i W x j 2 {i,j} E 8 µ 2 8 Ψ 0x, where we recall that µ 2 is the second smallest eigenvalue of the Laplace matrix By noting that W x i W x j 2 for all e E \ Ẽ, we derive W x i W x j 2 {i,j} Ẽ 8 µ 2 8 Ψ 0x E 8 Hence if µ 2 Ψ 0 x 2 E, we get Thimplies that as long as Ψ 0 x 2 E /µ 2, E [ Ψ 0 X t X t = x ] µ 2 36 Ψ 0x E [ Ψ 0 X t X t = x ] Ψ 0 x E [ Ψ 0 X t X t = x ] = This completes the proof µ 2 Ψ 0 x 36

22 4 UNIFORM TASKS AND IDENTICAL MACHINES 22 An Lemma 30, we prove that: Lemma 44 Let γ := 36 µ 2 and τ := γ 2 ln n + lnφ 0 X 0 Then with probability at least n, there exists a round t [, τ] such that Φ 0 X t < 2 E /µ 2 Proof The proof is the same as Lemma 30, except that the conditions on L X t are replaced by the conditions on Ψ 0 X t Now we are ready to show Theorem 4 Proof of Theorem 4 First we show that for m δn 5, any state x with Ψ 0 x 2 E /µ 2 n 5 is a 2/ + δ-apx NE Note that 2 E /µ 2 4 E 2 diamg n 5, using the fact that for any graph G, µ 2 diamg 2 E [, Lemma 9] For the approximation ratio we note that, given Ψ 0 x n 5, the maximum load must satisfy The minimum load must satisfy W max x = max W x i m/n + n 3 W min x = min W x i m/n n 3 Recall that in an ε-apx NE, for all neighboring vertice, j the following inequality must hold εw x i W x j + Thus, as we have W x i m/n + n 3 and W x j max{0, m/n n 3 } + for all edges i, j E along which players want to move, it suffices for ε to satisfy εm/n + n 3 m/n n 3 By solving this for ε and using m δn 5, we get ε δ /δ + = 2/δ + as desired Using Lemma 44 along with the fact that Ψ 0 X 0 m 2, it follows that with probability at least n that there is a step t [, τ] with Ψ 0 X t 2 E /µ 2 n 5 and τ O /µ 2 ln n + ln m an Lemma 44 Since this holds for an arbitray initial state at step 0 with m tasks, it follows that the expected time to reach such a state is at most O /µ 2 ln n + ln m With m Ωn, the theorem follows 42 Proof of Theorem 42 The proof of Theorem 42 is very similar to the proof of Theorem 32 Using Lemma 44, we can bound the number of steps to reach a state τ with Φ 0 X τ < n 5 Therefore, it remains to consider the number of additional steps to reach a NE from such a state Lemma 45 Assume that at step t the system is not in a Nash equilibrium Then we have E[Ψ 0 X t+ X t = x] = Ψ 0 x 8 Proof If in a step t we are not in a NE, there is an edge {i, j} E with W Xi t W Xt j > Then Equation 42 of Lemma 43 yields E[Ψ 0 X t+ X t = x] Ψ 0 x 8

23 5 WEIGHTED TASKS AND IDENTICAL MACHINES 23 for each task l in parallel do Let i = il the current machine of task l and w l the weight of task l Choose neighbor machine j uar if W Xi t W Xj t > w l then Move task l from machine i to j with probability end if end for degi degi, j W Xt i W X t 2α Wi t j Figure 2: Protocol II for weighted tasks and uniform speeds Lemma 46 Assume that at step τ we have Φ 0 X τ 2 E /µ 2 Let T be the additional number of steps until X τ+t is a Nash equilibrium Then E [T ] 36 E /µ 2 Similarly to Theorem 32, we are now able to prove Theorem 42 by combining Lemma 44 to reach a state τ with Φ 0 X τ n 5 and Lemma 46 to reach from there a NE By Markov s inequality and the union bound, with probability at least /2, the number of total steps needed for this at most 36 2 ln n + lnφ 0 X E µ 2 µ 2 Since for any state x, Φ 0 x m 2, it follows that the total number of expected steps for Protocol I to reach a NE is at most O ln n + ln m + E µ 2 µ 2 5 Weighted Tasks and Identical Machines In this section we assume that every task l [m] has an integral weight w l 5 Protocol and Results We denote the maximal and minimal weight of any task by w max and w min, respectively A task l with weight w l N that is located at machine i in state x prefers machine j over i if and only if W x i > W x j + w l, which is equivalent to W x i W x j > w x Our protocol for weighted tasks is given in Figure 2 Theorem 5 With α = 4w max, Protocol II reaches a Nash equilibrium in expected time O W 3 w max

24 5 WEIGHTED TASKS AND IDENTICAL MACHINES Proof of Theorem 5 Let x be an arbitrary state We define for every task l an indicator variable Hi,j l x {0, } H for happy which is one if task l prefers machine j over its current location i, and zero otherwise H i,j x is defined as the fraction of the total weight of jobs on machine i that prefer machine j over i, ie, w l H i,j x := W x i l x i : H l i,j x= Note that for uniform weights, H i,j x {0, }, ie, either all tasks or no task on i prefer j over i For non-uniform weights however, H i,j x may potentially be as small as w min /W apart from being 0 For any j Ni at least one of the variables H i,j x or H j,i x must be zero With these definitions we get E [ W Xi t X t = x ] = W x i w l Pr [r moves to j] j Ni:W x i >W x j l x i : Hi,j l x= + w l Pr [l moves to i] j Ni:W x i <W X i l x j : Hj,i l x= = W x i + j Ni:W x i >W x j j Ni:W x i <W x j H i,j x W x i W x j 2α di, j H j,i x W x j W x i 2α di, j Again, we denote by Ẽx the set of edges i, j with H i,jx > 0 and omit x whenever possible Then E [ W i X t X t = x ] = W x i j:i,j Ẽ H i,j x W x i W x j 2α di, j + j:j,i Ẽ H j,i x W x j W x i 2α di, j So the expected flow from i to j, {i, j} E, in a single round of the protocol starting from state x is f i,j x := H i,j x W x i W x j 2α di, j Note that at least one of the flows, f i,j x or f j,i x has to be zero Further, observe that i, j Ẽx if and only if f i,j x > 0 Let us first consider the potential change ΦX t X t = x recall, this the change under the assumption that the flow on every edge is exactly its expected value Φ 0 X t X t = x := W x i 2 E [ W X t i ] 2 The following lemma generalizes [7, Lemma 2] to the setting with weights The statement and the proof is similar to Lemma 34 Lemma 52 For any step t N and any state x it holds that Φ 0 X t X t = x αdi, j f i,j x di, j H i,j x i,j Ẽ 4 f i,j x

25 5 WEIGHTED TASKS AND IDENTICAL MACHINES 25 Proof We follow the same proof as for Lemma 34 Recall that in the proof of Lemma 34, we did not use the particular definition of f i,j x Hence we can use the same notations and definitions as in the proof of Lemma 34 to obtain a simplified version of equation 3 for weighted tasks but with uniform speeds, ie, Φ 0 X t,k f i,j x 2 W x i W x j 2 2di, j f i,j x, where e k = i, j represents the k-th activation of an edge in Ẽx Plugging in the definition of f i,j x, we get 4α di, j Φ 0 X t,k f i,j x H i,j x f i,j x α di, j H i,j x Summing over all i, j Ẽ yields the result f i,j x 2 2di, j f i,j x di, j 4 f i,j x Now we bound the sum of the variances of the random variables W X t i conditioned on Xt = x for any x We define w i,j x as the maximum weight of a task that prefers in state x either j over i or i over j We define w i,j x = 0 if there is no such task Lemma 53 For any step t N and state x it holds that, Var [ W Xi t X t = x ] = i,j Ẽ w i,j x H i,j x W x i W x j α di, j Proof We have Var [ W X t X t = x] = Var [ C t i A t i], where the random variable Ci t counts the number of tasks that come to link i from another link in step t, and the variable A t i counts the number of tasks that abandon link i in step t Again we have Var [ Ci t A t [ ] [ ] i] = Var C t i + Var A t i Observe that Ci t = j Ni l x j : H l j,i x= w l Y t l,j, where Y t that l,j Similarly, is the Bernoulli random variable indicating whether task l jumps to j at step t Note Yl,j t Ber α di, j W x j W x i 2 W x j A t i = l x i w l Z t l,i,

26 5 WEIGHTED TASKS AND IDENTICAL MACHINES 26 where Zl,i t is the Bernoulli random variable indicating whether a task l located at i before step t moves to another vertex Again, we have Zl,i t Ber α di, j W x i W x j 2 W x i By independence, Var [ Ci t ] = j Ni j Ni j Ni:H l i,j x= [ wl 2 Var Ber α di, j W x ] j W x i 2 W x j l x j : Hj,i l x= w i,j x w l α di, j W x j W x i, 2 W x j l x j : Hj,i l x= Similarly, Var [ A t ] i = wl 2 Var Ber α l x i i, j j Ni:H l i,j x= W x i W x j 2 W x i wl 2 α di, j W x i W x j 2 W x i l x i j Ni:Hi,j l x= = wl 2 α di, j W x i W x j 2 W x i j Ni l x i :Hi,j l x= w i,j x w l α di, j W x i W x j 2 W x i j Ni l x i :Hi,j l x= This gives Var [ W X t X t = x ] = i,j Ẽ = i,j Ẽ = i,j Ẽ w i,j x 2 w l l x i : Hi,j l x= w i,j x w l l x i : Hi,j l x= w i,j x H i,j x α di, j W x i W x j 2 W x i α di, j W x i W x j W x i α di, j W x i W x j The following lemma is similar to Lemma 36 where we have different speeds but uniform weights Lemma 54 For any step t and any state x it holds that E [ Φ 0 X t X t = x ] w i,jx + H i,j x α i,j Ẽ H i,j x W x i W x j α di, j

27 5 WEIGHTED TASKS AND IDENTICAL MACHINES 27 Proof From the definition of Φ 0 we get Therefore, E [ Φ 0 X t X t = x ] = E [ Φ 0 X t X t = x ] = W x i 2 E [ W X t i 2 X t = x ] = Var [ W X t i X t = x ] + E [ W X t i X t = x ] 2 [ E W X t i X t = x ] 2 Var [ W Xi t X t = x ] We first lower bound the first two sums Using Lemma 52 we get [ E W X t i X t = x ] 2 W x i 2 α di, j f i,j x H i,j x i,j Ẽ = i,j Ẽ = i,j Ẽ H i,j x W x i W x j 2α di, j H i,j x W x i W x j α i,j Ẽ di, j 4 f i,j x α di, j di, j 4 f i,j x H i,j x α H i,j x 2 f i,j x Now we define α Ax := H i,j x 2 f i,j x di, j Then by the above and using Lemma 53 to bound the sum of variances, we arrive at E [ Φ 0 X t X t = x ] H i,j x W x i W x j α α H i,j x 2 f i,j x = i,j Ẽ i,j Ẽ w i,j x H i,j x W x i W x j α di, j Ax w i,j x H i,j x W x i W x j α di, j We lower bound Ax as follows: α Ax = H i,j x 2 f i,j x di, j α = H i,j x 2 H i,j x W x i W x j 2α H i,j x H i,j x w i,j x + α = H i,jx w i,j x + α = w i,j x + w i,jx + H i,j x α

28 6 CONCLUSIONS 28 Hence E [ Φ 0 X t X t = x ] {i,j} Ẽ w i,jx + H i,j x α H i,j x W x i W x j α di, j With this lemma at hand, it is easy to conclude the proof of Theorem 5 Proof of Theorem 5 Assume there are tasks on machine i that prefer machine j over i Then, since H i,j x, w i,j x w max and α 4w max, we have w i,j x + H i,j x α We also have H i,j x w min /W if H i,j x > 0 Lemma 54 implies that w max + 4w max 2 Hence as long as we have not reached a NE, E [ Φ 0 X t X t = x ] 2 wmin W w min 4w max = 8 w min 2 W w max Next observe that for any state x, we have Φ 0 x W 2, where W is the sum of the weights of all tasks Hence an the proof of Theorem 32, we obtain the following upper bound on the expected time to reach a NE: W 2 8 W w max w min 2 8 W 3 w max, where the last inequality holds, since w min This completes the proof of Theorem 5 6 Conclusions In this paper we initiate the study of concurrent protocols for selfish load balancing on general networks Our protocols rely only on local information and computation and yield rapid convergence times to approximate and exact NE for systems with many agents We show a number of generalisations eg, to networks with uniformly related machines of different speeds or agents with weights There are many open problems that arise from our work, such as finding improved convergence times or general lower bounds for concurrent protocols On a more technical side, the generalization of our techniques an interesting open problem, eg, to more general potential functions that work for networks with both speeds and weights [4] Another interesting problem is whether approaches that do not use potential functions eg, [4] can be applied here References [] Heiner Ackermann, Petra Berenbrink, Simon Fischer, and Martin Hoefer Concurrent imitation dynamicn congestion games In Proc 28th Symp Principles of Distributed Computing PODC, pages 63 72, 2009 [2] Heiner Ackermann, Simon Fischer, Martin Hoefer, and Marcel Schöngens Distributed algorithms for QoS load balancing Distributed Computing, 235 6:32 330, 20