We consider zero-sum games in which players move between adjacent states, where in each pair of adjacent states one state dominates the other. The states in our game can represent positional advantages in physical conflict such as high ground or camouflage, or product characteristics that lend an advantage over competing sellers in a duopoly. We study the equilibria of the game as a function of the topological and geometric properties of the underlying graph. Our main result characterizes the expected payoff of both players starting from any initial position, under the assumption that the graph does not contain certain types of small cycles. This characterization leverages the block-cut tree of the graph, a construction that describes the topology of the biconnected components of the graph. We identify three natural types of (on-path) pure equilibria, and characterize when these equilibria exist under the above assumptions. On the geometric side, we show that strongly connected outerplanar graphs with undirected girth at least 4 always support some of these types of on-path pure equilibria. Finally, we show that a data structure describing all pure equilibria can be efficiently computed for these games.

翻译：我们研究零和博弈，其中参与者在相邻状态间移动，且每对相邻状态中一个状态支配另一个状态。博弈中的状态可表示物理冲突中的位置优势（如高地或伪装），或双寡头市场中赋予卖方相对于竞争者的优势的产品特性。我们研究博弈均衡与底层图拓扑及几何性质的关系。主要结果刻画了从任意初始位置出发时双方参与者的期望收益，前提是图中不包含特定类型的小环。该刻画利用了图的块割树结构——一种描述图双连通分量拓扑的构造。我们识别出三种自然的（路径上）纯策略均衡类型，并刻画了在上述假设下这些均衡何时存在。在几何层面，我们证明无向围长至少为4的强连通外平面图始终支持其中某些类型的路径上纯策略均衡。最后，我们表明描述所有纯策略均衡的数据结构可在这些博弈中被高效计算。