Approximate strong equilibria in job scheduling games with two uniformly related machines

Leah Epstein, Michal Feldman, Tami Tamir, Łukasz Witkowski, Marcin Witkowski

Research output: Contribution to journalArticlepeer-review


We consider approximate strong equilibria (SE) in strategic job scheduling games with two uniformly related machines. Jobs are assigned to machines, and each job wishes to minimize its cost, given by the completion time of the machine it is assigned to. Finding a Nash equilibrium (NE) in this game is simple. However, NE-configurations are not stable against coordinated deviations of several jobs. Various measures can be used to evaluate how well an NE-configuration approximates SE. A schedule is said to be an α-SE if there is no coalitional deviation such that every member of the coalition reduces its cost by a factor greater than α. We show that any pure NE on two related machines of speed ratio s is a s2+s-1s2≤54- SE, and provide a matching lower bound. In addition, we show that the LPT (Longest Processing Time) algorithm provides a better approximation ratio than a general NE, in particular, any LPT schedule is a 1.1011-SE. This is in contrast to the LS (List Scheduling) greedy algorithm for which the improvement ratio of coalitional deviations can be arbitrarily large. In addition, we design a fully polynomial time approximation scheme (FPTAS), which computes an NE that is a (1+ε)-SE. We also provide bounds for two other measures of approximate SE, considering the supremum possible improvement of a deviating job and the maximal increase in the cost of non-coalition members, and show that checking whether a specific schedule is an SE is co-NP-complete, which motivates the study of approximate strong equilibria. Finally, we consider multiple machines. We show that any pure NE on m related machines is a 2-SE. We give improved results for identical machines, and in particular, we show that any pure NE is a 1.32-SE.

Original languageEnglish
Pages (from-to)1843-1858
Number of pages16
JournalDiscrete Applied Mathematics
Issue number13-14
StatePublished - Sep 2013

Bibliographical note

Funding Information:
The authors thank an anonymous referee who assisted in simplifying Section 5 , and all the referees of this paper for many valuable comments. The second author is partially supported by the Israel Science Foundation (grant number 1219/09 ), the Leon Recanati Fund of the Jerusalem school of business administration and the Google Inter-university center for Electronic Markets and Auctions .


  • Approximate strong
  • Games
  • Nash equilibrium
  • Scheduling

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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