We study a Stackelberg game to examine how two agents determine to cooperate while competing with each other. Each selects an arrival time to a destination, the earlier one fetching a higher reward. There is, however, an inherent penalty in arriving too early as well as a risk in traveling alone. This gives rise to the possibility of the agents cooperating by traveling together while competing for the reward. In our prior work [1] we studied this problem as a sequential game among a set of $N$ competing agents in continuous time, and defined the formation of a group traveling together as arriving at exactly the same time. In the present study, we relax this definition to allow arrival times within a small window, and study a 2-agent game in both continuous and discrete time, referred to as the flock formation game. We derive and examine the properties of the subgame perfect equilibrium (SPE) of this game.

翻译：本研究通过斯塔克尔伯格博弈分析两个智能体如何在竞争关系中达成协作。每个智能体需选择抵达目的地的时刻，先到者将获得更高回报。然而，过早抵达存在固有惩罚，且单独行进伴随风险。这促使智能体可能通过结伴行进实现合作，同时保持对回报的竞争。在前期研究[1]中，我们将该问题建模为连续时间内$N$个竞争智能体的序贯博弈，并将结伴行进定义为严格同时抵达。本研究中，我们放宽该定义以允许抵达时刻存在微小时间窗，并在连续与离散时间框架下研究双智能体博弈（称为集群形成博弈）。我们推导并分析了该博弈子博弈完美均衡（SPE）的性质。