广义自私装箱
Generalized selfish bin packing
摘要：
标准装箱是将一组正大小不大于1的物品划分为最少数量的子集（称为箱）的问题，每个子集的总大小最多为1。在装箱博弈中，一个物品具有正的权重，并且给定一个有效的装箱或分割，每个物品都有一个与之相关的成本或收益，我们研究了一类装箱博弈，其中一个物品的收益是它的权重和与它一起装箱的物品的总权重之比，也就是说，成本分摊是线性地基于物品的权重的.我们研究了几种类型的纯纳什均衡：标准纳什均衡，强均衡，严格帕累托最优均衡，弱帕累托最优均衡。我们证明了这类博弈的所有这些类型的均衡。我们研究的（渐近）价格的无政府状态和稳定性（PoA和PoS）的问题就这四种类型的均衡，一般的重量和单位重量的两种情况下。我们表明，虽然一般的权重的情况下是密切相关的著名的第一次拟合算法，所有四个PoA值等于1.7，这是不正确的单位权重。特别是，我们表明，所有这些都严格低于1.7，强PoA等于约1.691（另一个众所周知的数字装箱），而严格的帕累托最优PoA低得多。我们表明，所有的PoS值都等于1，除了那些强平衡，这是等于1.7的一般重量，约为1.611824的单位重量。这个最后的值是不知道的（渐近）近似比的任何众所周知的算法装箱。最后，我们研究收敛到均衡。
英文摘要：
Standard bin packing is the problem of partitioning a set of items with positive sizes no larger than 1 into a minimum number of subsets (called bins) each having a total size of at most 1. In bin packing games, an item has a positive weight, and given a valid packing or partition of the items, each item has a cost or a payoff associated with it. We study a class of bin packing games where the payoff of an item is the ratio between its weight and the total weight of items packed with it, that is, the cost sharing is based linearly on the weights of items. We study several types of pure Nash equilibria: standard Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and weakly Pareto optimal equilibria. We show that any game of this class admits all these types of equilibria. We study the (asymptotic) prices of anarchy and stability (PoA and PoS) of the problem with respect to these four types of equilibria, for the two cases of general weights and of unit weights. We show that while the case of general weights is strongly related to the well-known First Fit algorithm, and all the four PoA values are equal to 1.7, this is not true for unit weights. In particular, we show that all of them are strictly below 1.7, the strong PoA is equal to approximately 1.691 (another well-known number in bin packing) while the strictly Pareto optimal PoA is much lower. We show that all the PoS values are equal to 1, except for those of strong equilibria, which is equal to 1.7 for general weights, and to approximately 1.611824 for unit weights. This last value is not known to be the (asymptotic) approximation ratio of any well-known algorithm for bin packing. Finally, we study convergence to equilibria.
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