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Following recent interest in the study of computer science problems in a game theoretic setting, we consider the well known bin packing problem where the items are controlled by selfish agents. Each agent is charged with a cost according to the fraction of the used bin space its item requires. That is, the cost of the bin is split among the agents, proportionally to their sizes. Thus, the selfish agents prefer their items to be packed in a bin that is as full as possible. The social goal is to minimize the number of the bins used. The social cost in this case is therefore the number of bins used in the packing.A pure Nash equilibrium is a packing where no agent can obtain a smaller cost by unilaterally moving his item to a different bin, while other items remain in their original positions. A Strong Nash equilibrium is a packing where there exists no subset of agents, all agents in which can profit from jointly moving their items to different bins. We say that all agents in a subset profit from moving their items to different bins if all of them have a strictly smaller cost as a result of moving, while the other items remain in their positions.We measure the quality of the equilibria using the standard measures PoA and PoS that are defined as the worst case worst/best asymptotic ratio between the social cost of a (pure) Nash equilibrium and the cost of an optimal packing, respectively. We also consider the recently introduced measures SPoA and SPoS, that are defined similarly to the PoA and the PoS, but consider only Strong Nash equilibria.We give nearly tight lower and upper bounds of 1.6416 and 1.6428, respectively, on the PoA of the bin packing game, improving upon previous result by Bilò. We study the Strong Nash equilibria of the bin packing game, and show that a packing is a Strong Nash equilibrium iff it is produced by the Subset Sum algorithm for bin Algorithmica (2011) 60: 368-394 369 packing. This characterization implies that the SPoA of the bin packing game equals the approximation ratio of the Subset Sum algorithm, for which an almost tight bound is known. Moreover, the fact that any lower bound instance for the Subset Sum algorithm can be converted by a small modification of the item sizes to a lower bound instance on the SPoS, implies that in the bin packing game SPoA = SPoS. Finally, we address the issue of complexity of computing a Strong Nash packing and show that no polynomial time algorithm exists for finding Strong Nash equilibria, unless P = NP.

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem.Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008, pages 368-380) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the bin-packing problem is precisely the approximation ratio of this heuristic.In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the bin packing problem where item sizes lie on the interval (0, α] for some α ≤ 1, yielding tight bounds for the Strong Price of Anarchy for all α ≤ 1. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all α ≤ 1.

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We consider a scheduling problem on m machines, where each job is controlled by a selfish agent. Each agent is only interested in minimizing its own cost, defined as the total load of the machine that its job is assigned to. We consider the objective of maximizing the minimum load (the value of the 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). These measures are unbounded already for two uniformly related machines [11], and therefore we focus on identical machines. We show that the POS is 1, and derive tight bounds on the pure POA for m ≤ 7 and on the overall pure POA, showing that its value is exactly 1.7. To achieve the upper bound of 1.7, we make an unusual use of weighting functions. Finally, we show that the mixed POA grows exponentially with m for this problem.In addition, we consider a similar setting of selfish jobs with a different objective of minimizing the maximum ratio between the loads of any pair of machines in the schedule. We show that under this objective the POS is 1 and the pure POA is 2, for any m ≥ 2.