In selfish bin packing, each item is regarded as a selfish player, who aims to minimize the cost-share by choosing a bin it can fit in. To have a least number of bins used, cost-sharing rules play an important role. The currently best known cost sharing rule has a \emph{price of anarchy} ($PoA$) larger than 1.45, while a general lower bound 4/3 on $PoA$ applies to any cost-sharing rule under which no items have the incentive to move unilaterally to an empty bin. In this paper, we propose a novel and simple rule with a $PoA$ matching the lower bound of $4/3$, thus completely resolving this game. The new rule always admits a Nash equilibrium and its \emph{price of stability} ($PoS$) is one. Furthermore, the well-known bin packing algorithm $BFD$ (Best-Fit Decreasing) is shown to achieve a strong equilibrium, implying that a stable packing with an asymptotic approximation ratio of $11/9$ can be produced in polynomial time. As an extension of the designing framework, we further study a variant of the selfish scheduling game, and design a best coordination mechanism achieving $PoS=1$ and $PoA=4/3$ as well.

翻译：在自私装箱问题中，每个物品被视为一个自私的参与者，其目标是通过选择可容纳自身的箱子来最小化成本分摊。为最小化使用的箱子数量，成本分摊规则起着关键作用。当前已知的最佳成本分摊规则的无政府状态代价（$PoA$）大于1.45，而任何成本分摊规则若使物品无动机单方面迁移至空箱，则其$PoA$的一般下界为4/3。本文提出了一种新颖且简单的规则，其$PoA$达到下界4/3，从而完全解决了该博弈问题。该新规则始终允许纳什均衡存在，且其稳定代价（$PoS$）为1。此外，著名的装箱算法$BFD$（最佳适配递减）被证明能实现强均衡，这意味着可在多项式时间内生成渐近近似比为$11/9$的稳定装箱。作为设计框架的延伸，我们进一步研究了自私调度博弈的变体，并设计了一种最优协调机制，同样实现了$PoS=1$和$PoA=4/3$。