We investigate pure Nash equilibria in generalized graphk-coloring games where we are given an edge-weighted undirected graph together with a set of k colors. Nodes represent players and edges capture their mutual interests. The strategy set of each player consists of k colors. The utility of a player v in a given state or coloring is given by the sum of the weights of edges {v, u} incident to v such that the color chosen by v is different than the one chosen by u, plus the profit gained by using the chosen color. Such games form some of the basic payoff structures in game theory, model lots of real-world scenarios with selfish players and extend or are related to several fundamental class of games. We first show that generalized graph k-coloring games are potential games. In particular, they are convergent and thus Nash Equilibria always exist. We then evaluate their performance by means of the widely used notions of price of anarchy and price of stability and provide tight bounds for two natural and widely used social welfare, i.e., utilitarian and egalitarian social welfare.

中文翻译：

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广义图k-着色游戏

我们研究了广义图k色游戏中的纯Nash均衡，在该博弈中，我们得到了一个边缘加权无向图以及一组k颜色。节点代表参与者，边缘抓住他们的共同利益。每个玩家的策略集由k种颜色组成。的玩家的效用v在给定的状态或着色是由边缘{的权重的总和给定v，ù }入射到v，使得由所选择的颜色v大于由所选择的一个不同ù，再加上使用所选颜色获得的利润。这样的游戏构成了游戏理论中的一些基本收益结构，用自私的玩家为现实世界中的许多场景建模，并扩展了游戏的某些基本类别或与之相关。我们首先证明广义图k着色游戏是潜在游戏。特别是，它们是收敛的，因此纳什均衡始终存在。然后，我们通过广泛使用的无政府状态价格和稳定价格概念来评估其绩效，并为功利主义和均等社会福利这两种自然而广泛使用的社会福利提供了严格的界限。