We introduce and initiate the study of a natural class of repeated two-player matrix games, called Repeated-Until-Collision (RUC) games. In each round, both players simultaneously pick an action from a common action set $\{1, 2, \dots, n\}$. Depending on their chosen actions, they derive payoffs given by $n \times n$ matrices $A$ and $B$, respectively. If their actions collide (i.e., they pick the same action), the game ends, otherwise, it proceeds to the next round. Both players want to maximize their total payoff until the game ends. RUC games can be interpreted as pursuit-evasion games or repeated hide-and-seek games. They also generalize hand cricket, a popular game among children in India. We show that under mild assumptions on the payoff matrices, every RUC game admits a Nash equilibrium (NE). Moreover, we show the existence of a stationary NE, where each player chooses their action according to a probability distribution over the action set that does not change across rounds. Remarkably, we show that all NE are effectively the same as the stationary NE, thus showing that RUC games admit an almost unique NE. Lastly, we also show how to compute (approximate) NE for RUC games.

翻译：本文引入并系统研究了一类自然的重复双人矩阵博弈模型——持续至碰撞（Repeated-Until-Collision, RUC）博弈。在每一轮中，双方玩家同时从公共行动集$\{1, 2, \dots, n\}$中选择一个行动。根据所选行动，他们分别获得由$n \times n$矩阵$A$和$B$定义的收益。若双方行动发生碰撞（即选择相同行动），则博弈终止；否则博弈进入下一轮。双方玩家均希望最大化博弈结束前的总收益。RUC博弈可被解释为追逃博弈或重复捉迷藏博弈，同时也可推广至印度儿童中流行的“手板球”游戏。我们证明：在收益矩阵的温和假设下，每个RUC博弈均存在纳什均衡（NE）。此外，我们证明了平稳纳什均衡的存在性：在此均衡中，每位玩家按照行动集上的固定概率分布选择行动，且该分布不随轮次改变。值得关注的是，我们证明所有纳什均衡本质上与平稳纳什均衡等价，从而表明RUC博弈具有几乎唯一的纳什均衡。最后，我们还给出了RUC博弈（近似）纳什均衡的计算方法。