Approximate strong equilibrium in job scheduling games

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) IR_min: the maximal number α such that there exists a coalition in which the minimal improvement ratio among the coalition members is α,(ii) IR_max: the maximal number α, such that there exists a coalition in which the maximal improvement ratio among the coalition members is α, and (iii) DR_max: 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 well-known 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 IR_min of 1 + ε for any given ε. With respect to computational complexity, we show that given an NE on m ≥ 3 identical machines or m ≥ 2 unrelated machines, it is NP-hard to determine whether a given coalition can deviate such that every member decreases its cost.