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，玩家对应于图中的顶点，面临着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，并由每个玩家独有的二元模式编码。我们证明，具有递减模式（即玩家仅当其相邻玩家数量不超过某个阈值时才愿意采取行动）的博弈始终存在纯纳什均衡，并且从任意初始配置出发，通过一个多项式有界的最佳响应序列即可达到该均衡。对于形如 $10^k10^*$ 的非单调模式（即玩家希望独自行动或与 $k+1$ 名邻居一同行动），我们证明判断纯纳什均衡是否存在是 $\mathsf{NP}$-难的。我们进一步研究了允许不同强度联系的模型推广：一条具有整数权重 $w$ 的边相当于 $w$ 条平行边。尽管在此模型中，递减模式仍然存在纯纳什均衡，但我们证明求解该均衡是 $\mathsf{PLS}$-完全的。