This paper considers a reach-avoid game on a rectangular domain with two defenders and one attacker. The attacker aims to reach a specified edge of the game domain boundary, while the defenders strive to prevent that by capturing the attacker. First, we are concerned with the barrier, which is the boundary of the reach-avoid set, splitting the state space into two disjoint parts: (1) defender dominance region (DDR) and (2) attacker dominance region (ADR). For the initial states lying in the DDR, there exists a strategy for the defenders to intercept the attacker regardless of the attacker's best effort, while for the initial states lying in the ADR, the attacker can always find a successful attack strategy. We propose an attack region method to construct the barrier analytically by employing Voronoi diagram and Apollonius circle for two kinds of speed ratios. Then, by taking practical payoff functions into considerations, we present optimal strategies for the players when their initial states lie in their winning regions, and show that the ADR is divided into several parts corresponding to different strategies for the players. Numerical approaches, which suffer from inherent inaccuracy, have already been utilized for multiplayer reach-avoid games, but computational complexity complicates solving such games and consequently hinders efficient on-line applications. However, this method can obtain the exact formulation of the barrier and is applicable for real-time updates.

中文翻译：

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两名防守者和一名攻击者的避免触及游戏：一种分析方法


本文考虑在矩形区域上进行的回避游戏，其中有两名防守者和一名攻击者。攻击者的目标是到达游戏域边界的指定边缘，而防御者则努力通过捕获攻击者来阻止这种情况。首先，我们关注障碍，它是避免触及集的边界，将状态空间分为两个不相交的部分：（1）防御者优势区域（DDR）和（2）攻击者优势区域（ADR）。对于位于 DDR 的初始状态，无论攻击者如何尽力，防御者都存在拦截攻击者的策略，而对于位于 ADR 的初始状态，攻击者总能找到成功的攻击策略。我们提出了一种攻击区域方法，通过使用 Voronoi 图和 Apollonius 圆对两种速度比来解析构造障碍。然后，考虑到实际的收益函数，我们提出了当玩家的初始状态位于获胜区域时的最优策略，并表明 ADR 被分为与玩家的不同策略相对应的几个部分。数值方法存在固有的不准确性，已被用于多人回避游戏，但计算复杂性使解决此类游戏变得复杂，从而阻碍了高效的在线应用。然而，该方法可以获得障碍的精确公式并且适用于实时更新。