We study the complexity of computing equilibria in binary public goods games on undirected graphs. In such a game, players correspond to vertices in a graph and face a binary choice of performing an action, or not. Each player's decision depends only on the number of neighbors in the graph who perform the action and is encoded by a per-player binary pattern. We show that games with decreasing patterns (where players only want to act up to a threshold number of adjacent players doing so) always have a pure Nash equilibrium and that one is reached from any starting profile by following a polynomially bounded sequence of best responses. For non-monotonic patterns of the form $10^k10^*$ (where players want to act alone or alongside $k + 1$ neighbors), we show that it is $\mathsf{NP}$-hard to decide whether a pure Nash equilibrium exists. We further investigate a generalization of the model that permits ties of varying strength: an edge with integral weight $w$ behaves as $w$ parallel edges. While, in this model, a pure Nash equilibrium still exists for decreasing patters, we show that the task of computing one is $\mathsf{PLS}$-complete.
翻译:我们研究了在非方向图形上的二进制公益游戏中计算平衡的复杂性。 在这样的游戏中,玩家对应于图表中的顶点, 并面对一个执行动作的二进制选择。 每个玩家的决定只取决于图表中执行动作的邻居人数, 并且由每个玩家的二进制模式编码。 我们显示, 模式不断下降的游戏( 玩家只想要达到一个最接近的玩家的门槛) 总是有一个纯净的纳什平衡, 并且从任何初始配置中, 可以通过一个组合式组合式组合式组合式的最佳响应序列达到一个平衡。 对于表格 $10k10 $( 玩家想要单独或与$k+$1$邻居一起行动)的非单调模式来说, 我们显示, $\ mathf{NP} 很难决定是否存在纯纳什平衡。 我们进一步调查允许不同强度的模型的概括化: 与整体重量的边缘值为$w$的边缘值, 从任何初始配置中可以表现为$w$的平行边缘。 对于表格来说, 10+$ $ $xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx