We consider selfish colorful bin packing games in which a set of items, each one controlled by a selfish player, are to be packed into a minimum number of unit capacity bins. Each item has one of \(m\ge 2\) colors and no items of the same color may be adjacent in a bin. All bins have the same unitary cost which is shared among the items it contains, so that players are interested in selecting a bin of minimum shared cost. We adopt two standard cost sharing functions, i.e., the egalitarian and the proportional ones. Although, under both cost functions, these games do not converge in general to a (pure) Nash equilibrium, we show that Nash equilibria are guaranteed to exist. We also provide a complete characterization of the efficiency of Nash equilibria under both cost functions for general games, by showing that the prices of anarchy and stability are unbounded when \(m\ge 3\), while they are equal to 3 when \(m=2\). We finally focus on the subcase of games with uniform sizes (i.e., all items have the same size). We show a tight characterization of the efficiency of Nash equilibria and design an algorithm which returns Nash equilibria with best achievable performance. All of our bounds on the price of anarchy and stability hold with respect to both their absolute and asymptotic version.

中文翻译：

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自私的多彩垃圾箱游戏

我们考虑自私的多彩垃圾箱游戏，其中将一组由自私玩家控制的物品包装到最小数量的单位容量垃圾箱中。每个项目都有一个\（m \ ge 2 \）颜色，并且同一颜色的任何物品都不得在垃圾箱中相邻。所有垃圾箱都具有相同的单位成本，该成本在包含的物品之间共享，因此玩家有兴趣选择最低共享成本的垃圾箱。我们采用两种标准的成本分摊功能，即均等功能和比例功能。尽管在这两个成本函数下，这些博弈一般不会收敛到（纯）纳什均衡，但我们证明纳什均衡必定存在。我们还提供了纳什均衡的下普通游戏既经济功能的效率一个完整的表征，通过表明无政府状态和稳定的价格时，无界\（M \ GE 3 \） ，而它们等于3时\（ m = 2 \）。最后，我们将重点放在大小一致的游戏的子情况下（即所有项目都具有相同的大小）。我们对Nash均衡的效率进行了严格的刻画，并设计了一种算法，该算法可返回具有最佳可实现性能的Nash均衡。关于无政府状态和稳定性的所有局限性都取决于其绝对值和渐近值。