The Power of One Secret Agent

I am a job. In job-scheduling applications, my friends and I are assigned to machines that can process us. In the last decade, thanks to our strong employee committee, and the rise of algorithmic game theory, we are getting more and more freedom regarding our assignment. Each of us acts to minimize his own cost, rather than to optimize a global objective. My goal is different. I am a secret agent operated by the system. I do my best to lead my fellow jobs to an outcome with a high social cost. My naive friends keep doing the best they can, each of them performs his best-response move whenever he gets the opportunity to do so. Luckily, I am a charismatic guy. I can determine the order according to which the naive jobs perform their best-response moves. In this paper, I analyze my power, formalized as the Price of a Traitor (PoT), in cost-sharing scheduling games - in which we need to cover the cost of the machines that process us. Starting from an initial Nash Equilibrium (NE) profile, I join the instance and hurt its stability. A sequence of best-response moves is performed until I vanish, leaving the naive jobs in a new NE. For an initial NE assignment, S_0, the PoT measures the ratio between the social cost of a worst NE I can lead the jobs to, starting from S_0, and the social cost of S_0. The PoT of a game is the maximal such ratio among all game instances and initial NE assignments. My analysis distinguishes between instances with unit- and arbitrary-cost machines, and instances with unit- and arbitrary-length jobs. I give exact bounds on the PoT for each setting, in general and in symmetric games. While it turns out that in most settings my power is really impressive, my task is computationally hard (and also hard to approximate)

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 alpha, such that there exists a coalition in which the minimal improvement ratio among the coalition members is alpha, (ii) IR_max: the maximal number alpha, such that there exists a coalition in which the maximal improvement ratio among the coalition members is alpha, 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+epsilon for any given epsilon. 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

Congestion Games with Multisets of Resources and Applications in Synthesis

In classical congestion games, players\u27 strategies are subsets of resources. We introduce and study multiset congestion games, where players\u27 strategies are multisets of resources. Thus, in each strategy a player may need to use each resource a different number of times, and his cost for using the resource depends on the load that he and the other players generate on the resource. Beyond the theoretical interest in examining the effect of a repeated use of resources, our study enables better understanding of non-cooperative systems and environments whose behavior is not covered by previously studied models. Indeed, congestion games with multiset-strategies arise, for example, in production planing and network formation with tasks that are more involved than reachability. We study in detail the application of synthesis from component libraries: different users synthesize systems by gluing together components from a component library. A component may be used in several systems and may be used several times in a system. The performance of a component and hence the system\u27s quality depends on the load on it. Our results reveal how the richer setting of multisets congestion games affects the stability and equilibrium efficiency compared to standard congestion games. In particular, while we present very simple instances with no pure Nash equilibrium and prove tighter and simpler lower bounds for equilibrium inefficiency, we are also able to show that some of the positive results known for affine and weighted congestion games apply to the richer setting of multisets