Approximate Strong Equilibrium in Job Scheduling Games
A Nash Equilibriun (NE) is a strategy profile that is resilient to unilateral deviations, and is predominantly used in analysis of competitive games. 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 (SE) .
We provide a framework for quantifying the stability and the performance of various assignment policies and solution concept in the face of coalitional deviations. Within this framework we evaluate a given configuration according to three measurements: (i) IR min : the maximal number α, such that there exists a coalition in which the minimum improvement ratio among the coalition members is α (ii) IR max : the maximum improvement ratio among the coalition’s members. (iii) DR max : the maximum possible damage ratio of an agent outside the coalition.
This framework can be used to study the proximity between different solution concepts, as well as to study the existence of approximate SE in settings that do not possess any such equilibrium. We analyze these measurements in job scheduling games on identical machines. In particular, we provide upper and lower bounds for the above three measurements for both NE and the well-known assignment rule Longest Processing Time (LPT) (which is known to yield a NE). Most of our bounds are tight for any number of machines, while some are tight only for three machines. We show that both NE and LPT configurations yield small constant bounds for IR min and DR max . As for IR max , it can be arbitrarily large for NE configurations, while a small bound is guaranteed for LPT configurations. For all three measurements, LPT performs strictly better than NE.
With respect to computational complexity aspects, we show that given a NE on m ≥ 3 identical machines and a coalition, it is NP-hard to determine whether the coalition can deviate such that every member decreases its cost. For the unrelated machines settings, the above hardness result holds already for m ≥ 2 machines.
KeywordsNash Equilibrium Congestion Game Identical Machine Improvement Ratio Strong Equilibrium
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