具有竞争关系的首通随机游走的策略几何
Strategic geometry of competing first-passage random walks
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中文总结 AI 辅助
该研究探讨竞争型首通随机游走博弈的策略几何，通过数学分析推导有限路径、特定图结构等场景下的最优策略与均衡，证明其与连续布朗运动博弈的收敛性。
中文摘要 AI 辅助
两名竞争者为独立的匀速随机游走选择起始顶点，每个位置由首位到访者获得，我们研究由此产生的首通位置博弈的空间几何。在每条有限路径上，最优策略恰好是支撑在中心顶点上的分布；证明结合了反射边界调和障碍与乘积链退出概率的参数一致聚合估计。经扩散尺度变换后，完整的双起始收益景观一致收敛于独立反射布朗运动之间的博弈，其唯一均衡集中在中点，具有显式立方稳定性。除路径外，将每个团（clique，阶数为k）的顶点附加两个叶子，得到3k个顶点的图，在此图上每个精确最优策略在所有k个团顶点上随机化；当k=2时，这是一个6个顶点的树，无纯策略均衡。对端点的连续最佳响应是唯一确定的。首通随机排序表示将有限博弈与最大彩票关联，但不将其等同于非策略性涂色、确定性Voronoi分配或吸收陷阱放置。参数一致的陈述源于分析论证或符号多项式恒等式；已识别的有限例外和数值 enclosure 具有可复现的证书。
英文摘要
Two competitors choose starting vertices for independent, constant-speed random walks, and each site is acquired by its first visitor. We study the spatial geometry of the resulting first-passage location game. On every finite path the optimal strategies are exactly the distributions supported on the central vertices. The proof combines reflecting-boundary harmonic barriers with a parameter-uniform aggregate estimate for product-chain exit probabilities. After diffusive rescaling, the complete two-start payoff landscape converges uniformly to the game between independent reflected Brownian motions. Its unique equilibrium concentrates at the midpoint, with explicit cubic stability. Beyond paths, attaching two leaves to every vertex of a clique of order k produces a 3k-vertex graph on which every exact optimal strategy randomizes over all k clique vertices; for k=2, this is a six-vertex tree with no pure equilibrium. The continuum best response to an endpoint is uniquely determined. A first-passage random-ranking representation relates the finite game to maximal lotteries without identifying it with nonstrategic painting, deterministic Voronoi allocation, or absorbing-trap placement. Parameter-uniform statements follow from analytic arguments or symbolic polynomial identities; identified finite exceptions and numerical enclosures have reproducible certificates.
