WINE 2009: Internet and Network Economics pp 232-243

# Maximizing the Minimum Load: The Cost of Selfishness

• Leah Epstein
• Elena Kleiman
• Rob van Stee
Conference paper

DOI: 10.1007/978-3-642-10841-9_22

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5929)
Cite this paper as:
Epstein L., Kleiman E., van Stee R. (2009) Maximizing the Minimum Load: The Cost of Selfishness. In: Leonardi S. (eds) Internet and Network Economics. WINE 2009. Lecture Notes in Computer Science, vol 5929. Springer, Berlin, Heidelberg

## Abstract

We consider a scheduling problem where each job is controlled by a selfish agent, who is only interested in minimizing its own cost, which is defined as the total load on the machine that its job is assigned to. We consider the objective of maximizing the minimum load (cover) over the machines. Unlike the regular makespan minimization problem, which was extensively studied in a game theoretic context, this problem has not been considered in this setting before.

We study the price of anarchy (poa) and the price of stability (pos). We show that on related machines, both these values are unbounded. We then focus on identical machines. We show that the $$\textsc{pos}$$ is 1, and we derive tight bounds on the $$\textsc{poa}$$ for m ≤ 6 and nearly tight bounds for general m. In particular, we show that the $$\textsc{poa}$$ is at least 1.691 for larger m and at most 1.7. Hence, surprisingly, the $$\textsc{poa}$$ is less than the $$\textsc{poa}$$ for the makespan problem, which is 2. To achieve the upper bound of 1.7, we make an unusual use of weighting functions. Finally, in contrast we show that the mixed $$\textsc{poa}$$ grows exponentially with m for this problem, although it is only Θ(logm/loglogm) for the makespan.