TY - GEN

T1 - The price of anarchy on uniformly related machines revisited

AU - Epstein, Leah

AU - Van Stee, Rob

PY - 2008

Y1 - 2008

N2 - 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. We consider scheduling on uniformly related machines. Here 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. 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 machines 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 of the fastest machine and the number of slow machines, and give tight bounds for nearly all cases.

AB - 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. We consider scheduling on uniformly related machines. Here 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. 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 machines 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 of the fastest machine and the number of slow machines, and give tight bounds for nearly all cases.

UR - http://www.scopus.com/inward/record.url?scp=44449128942&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-79309-0_6

DO - 10.1007/978-3-540-79309-0_6

M3 - Conference contribution

AN - SCOPUS:44449128942

SN - 3540793089

SN - 9783540793083

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 46

EP - 57

BT - Algorithmic Game Theory - First International Symposium, SAGT 2008, Proceedings

T2 - 1st International Symposium on Algorithmic Game Theory, SAGT 2008

Y2 - 30 April 2008 through 2 May 2008

ER -