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：我们研究无向图上二元公共品博弈中计算均衡的复杂性。在此类博弈中，玩家对应图中的顶点，面临是否采取行动的二元选择。每个玩家的决策仅取决于图中采取行动的邻居数量，并由每个玩家的二元行为模式编码。我们证明，递减模式（即玩家仅在相邻玩家数量达到阈值时才愿意行动）的博弈始终存在纯纳什均衡，且从任何初始策略配置出发，通过多项式有界的最优反应序列即可到达该均衡。对于形如$10^k10^*$的非单调模式（即玩家希望单独行动或与$k+1$个邻居共同行动），我们证明判断纯纳什均衡是否存在是$\mathsf{NP}$-难问题。我们进一步研究允许不同强度边连接的模型推广：具有整数权重$w$的边表现为$w$条平行边。在该模型中，尽管递减模式仍存在纯纳什均衡，但计算该均衡的任务是$\mathsf{PLS}$-完全的。