This work proposes a dynamic and adversarial resource allocation problem in a graph environment, which is referred to as the dynamic Defender-Attacker Blotto (dDAB) game. A team of defender robots is tasked to ensure numerical advantage at every node in the graph against a team of attacker robots. The engagement is formulated as a discrete-time dynamic game, where the two teams reallocate their robots in sequence and each robot can move at most one hop at each time step. The game terminates with the attacker's victory if any node has more attacker robots than defender robots. Our goal is to identify the necessary and sufficient number of defender robots to guarantee defense. Through a reachability analysis, we first solve the problem for the case where the attacker team stays as a single group. The results are then generalized to the case where the attacker team can freely split and merge into subteams. Crucially, our analysis indicates that there is no incentive for the attacker team to split, which significantly reduces the search space for the attacker's winning strategies and also enables us to design defender counter-strategies using superposition. We also present an efficient numerical algorithm to identify the necessary and sufficient number of defender robots to defend a given graph. Finally, we present illustrative examples to verify the efficacy of the proposed framework.

翻译：本文提出了一种图环境下的动态对抗资源分配问题，称为动态防御者-攻击者Blotto（dDAB）博弈。一组防御机器人需确保图结构中每个节点对攻击机器人形成数量优势。该对抗过程被建模为离散时间动态博弈，双方按序重新分配机器人，每台机器人每时间步最多移动一个节点。若任一节点上攻击机器人数量超过防御机器人，则攻击方获胜。目标是确定保证防御所需的防御机器人必要且充分的数量。通过可达性分析，我们首先解决了攻击方保持单一集群情况下的问题，随后将结论推广至攻击方可自由分拆合并为子集群的情形。关键分析表明，攻击方没有分拆动机，这显著缩小了攻击方获胜策略的搜索空间，并使我们能够利用叠加原理设计防御方反制策略。我们还提出了一种高效数值算法，用于确定给定图结构所需防御机器人的必要且充分数量。最后，通过算例验证了所提框架的有效性。