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Approximate Strong Equilibrium in Job Scheduling Games
, 2009
"... A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is predominantly used in the analysis of multiagent systems. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit sta ..."
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A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is predominantly used in the analysis of multiagent systems. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit stability against coalitional deviations, in that the benefits from a joint deviation are bounded. In this sense, NE approximates strong equilibrium. Coalition formation is a key issue in multiagent systems. We provide a framework for quantifying the stability and the performance of various assignment policies and solution concepts in the face of coalitional deviations. Within this framework we evaluate a given configuration according to three measures: (i) IRmin: the maximal number α, such that there exists a coalition in which the minimal improvement ratio among the coalition members is α, (ii) IRmax: the maximal number α, such that there exists a coalition in which the maximal improvement ratio among the coalition members is α, and (iii) DRmax: the maximal possible damage ratio of an agent outside the coalition. We analyze these measures in job scheduling games on identical machines. In particular, we provide upper and lower bounds for the above three measures for both NE and the wellknown assignment rule Longest Processing Time (LPT). Our results indicate that LPT performs better than a general NE. However, LPT is not the best possible approximation. In particular, we present a polynomial time approximation scheme (PTAS) for the makespan minimization problem which provides a schedule with IRmin of 1 + ε for any given ɛ. With respect to computational complexity, we show that given an NE on m ≥ 3 identical machines or m ≥ 2 unrelated machines, it is NPhard to determine whether a given coalition can deviate such that every member decreases its cost.
The Price of Anarchy on Uniformly Related Machines Revisited
 SAGT 2008. LNCS
, 2008
"... Recent interest in Nash equilibria led to a study of the price of anarchy (POA) and the strong price of anarchy (SPOA) for scheduling problems. The two measures express the worst case ratio between the cost of an equilibrium (a pure Nash equilibrium, and a strong equilibrium, respectively) to the co ..."
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Recent interest in Nash equilibria led to a study of the price of anarchy (POA) and the strong price of anarchy (SPOA) for scheduling problems. The two measures express the worst case ratio between the cost of an equilibrium (a pure Nash equilibrium, and a strong equilibrium, respectively) to the cost of a social optimum. The atomic players are the jobs, and the delay of a job is the completion time of the machine running it, also called the load of this machine. The social goal is to minimize the maximum delay of any job, while the selfish goal of each job is to minimize its own delay, that is, the delay of the machine running it. We consider scheduling on uniformly related machines. While previous studies either consider identical speed machines or an arbitrary number of speeds, focusing on the number of machines as a parameter, we consider the situation in which the number of different speeds is small. We reveal a linear dependence between the number of speeds and the POA. For a set of machine of at most p speeds, the POA turns out to be exactly p + 1. The growth of the POA for large numbers of related machines is therefore a direct result of the large number of potential speeds. We further consider a well known structure of processors, where all machines are of the same speed except for one possibly faster machine. We investigate the POA as a function of both the speed ratio between the fastest machine and the number of slow machines.
On the Inefficiency of Equilibria in Linear Bottleneck Congestion Games
"... Abstract. We study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maxi ..."
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Abstract. We study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weightdependent) latency that he experiences on any of these facilities. We derive upper and (asymptotically) matching lower bounds on the (strong) price of anarchy of linear bottleneck congestion games for a natural load balancing social cost objective (i.e., minimize the maximum latency of a facility). We restrict our studies to linear latency functions. Linear bottleneck congestion games still constitute a rich class of games and generalize, for example, load balancing games with identical or uniformly related machines with or without restricted assignments. 1
Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy
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Partition Equilibrium Always Exists in Resource Selection Games ⋆
"... Abstract. We consider the existence of Partition Equilibrium in Resource Selection Games. Superstrong equilibrium, where no subset of players has an incentive to change their strategies collectively, does not always exist in such games. We show, however, that partition equilibrium (introduced in [4 ..."
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Abstract. We consider the existence of Partition Equilibrium in Resource Selection Games. Superstrong equilibrium, where no subset of players has an incentive to change their strategies collectively, does not always exist in such games. We show, however, that partition equilibrium (introduced in [4] to model coalitions arising in a social context) always exists in general resource selection games, as well as how to compute it efficiently. In a partition equilibrium, the set of players has a fixed partition into coalitions, and the only deviations considered are by coalitions that are sets in this partition. Our algorithm to compute a partition equilibrium in any resource selection game (i.e., load balancing game) settles the open question from [4] about existence of partition equilibrium in general resource selection games. Moreover, we show how to always find a partition equilibrium which is also a Nash equilibrium. This implies that in resource selection games, we do not need to sacrifice the stability of individual players when forming solutions stable against coalitional deviations. In addition, while superstrong equilibrium may not exist in resource selection games, we show that its existence can be decided efficiently, and how to find one if it exists. 1
Collusion in Atomic Splittable Routing Games
"... We investigate how collusion affects the social cost in atomic splittable routing games. Suppose that players form coalitions and each coalition behaves as if it were a single player controlling all the flows of its participants. It may be tempting to conjecture that the social cost would be lower ..."
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We investigate how collusion affects the social cost in atomic splittable routing games. Suppose that players form coalitions and each coalition behaves as if it were a single player controlling all the flows of its participants. It may be tempting to conjecture that the social cost would be lower after collusion, since there would be more coordination among the players. We construct examples to show that this conjecture is not true. Even in very simple singlesourcesingledestination networks, the social cost of the postcollusion equilibrium can be higher than that of the precollusion equilibrium. This counterintuitive phenomenon of collusion prompts us to ask the question: under what conditions would the social cost of the postcollusion equilibrium be bounded by the social cost of the precollusion equilibrium? We show that if (i) the network is “welldesigned ” (satisfying a natural condition), and (ii) the delay functions are affine, then collusion is always beneficial for the social cost in the Nash equilibria. On the other hand, if either of the above conditions is unsatisfied, collusion can worsen the social cost. Our main technique is a novel flowaugmenting algorithm to build Nash equilibria. Our positive result for collusion is obtained by applying this algorithm simultaneously to two different flow value profiles of players and observing the difference in the derivatives of their social costs. Moreover, for a nontrivial subclass of selfish routing games, this algorithm finds the exact Nash equilibrium in polynomial time.
A An extended abstract of this paper
"... Abstract Bottleneck congestion games properly model the properties of many realworld network routing applications. They are known to possess strong equilibriaa strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pu ..."
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Abstract Bottleneck congestion games properly model the properties of many realworld network routing applications. They are known to possess strong equilibriaa strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or singlecommodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and singlecommodity games is strongly NPhard. In addition, we establish a variety of hardness results and lower bounds regarding the
Load Rebalancing Games in Dynamic Systems with Migration Costs
, 2013
"... We consider the following dynamic load balancing game: Given an initial assignment of jobs to identical parallel machines, the system is modified; specifically, some machines are added or removed. Each job’s cost is the load on the machine it is assigned to; thus, when machines are added, jobs have ..."
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We consider the following dynamic load balancing game: Given an initial assignment of jobs to identical parallel machines, the system is modified; specifically, some machines are added or removed. Each job’s cost is the load on the machine it is assigned to; thus, when machines are added, jobs have an incentive to migrate to the new unloaded machines. When machines are removed, the jobs assigned to them must be reassigned. Consequently, other jobs might also benefit from migrations. In our jobextension penalty model, for a given extension parameter δ ≥ 0, if the machine on which a job is assigned to in the modified schedule is different from its initial machine, then the job’s processing time is extended by δ. We provide answers to the basic questions arising in this model. Namely, the existence and calculation of a Nash Equilibrium and a Strong Nash Equilibrium, and their inefficiency compared to an optimal schedule. Our results show that the existence of jobmigration penalties might lead to poor stable schedules; however, if the modification is a result of a sequence of improvement steps or, better, if the sequence of improvement steps can be supervised in some way (by forcing the jobs to play in a specific order) then any stable modified schedule approximates well an optimal one. Our work adds two realistic considerations to the study of job scheduling games: the analysis of the common situation in which systems are upgraded or suffer from failures, and the practical fact according to which job migrations are associated with a cost. 1