This paper studies competitions with rank-based reward among a large number of teams. Within each sizable team, we consider a mean-field contribution game in which each team member contributes to the jump intensity of a common Poisson project process; across all teams, a mean field competition game is formulated on the rank of the completion time, namely the jump time of Poisson project process, and the reward to each team is paid based on its ranking. On the layer of teamwise competition game, three optimization problems are introduced when the team size is determined by: (i) the team manager; (ii) the central planner; (iii) the team members' voting as partnership. We propose a relative performance criteria for each team member to share the team's reward and formulate some mean field games of mean field games, which are new to the literature. In all problems with homogeneous parameters, the equilibrium control of each worker and the equilibrium or optimal team size can be computed in an explicit manner, allowing us to analytically examine the impacts of some model parameters and discuss their economic implications. Two numerical examples are also presented to illustrate the parameter dependence and comparison between different team size decision making.

中文翻译：

---

Teamwise 平均场比赛

本文研究了大量团队之间基于排名奖励的比赛。在每个规模较大的团队中，我们考虑一个平均场贡献博弈，其中每个团队成员都对共同泊松项目过程的跳跃强度做出贡献；在所有团队中，按照完成时间的排名，即泊松项目过程的跳跃时间制定一个平均场比赛游戏，并根据其排名支付给每个团队的奖励。在团队竞赛层面，当团队规模由以下因素决定时，引入三个优化问题：（i）团队经理；(ii) 中央计划者；(iii) 团队成员作为合伙人投票。我们为每个团队成员提出了一个相对的绩效标准来分享团队的奖励并制定一些平均场比赛的平均场比赛，对文学来说是新的。在所有具有同质参数的问题中，每个工人的均衡控制和均衡或最佳团队规模都可以以明确的方式计算，使我们能够分析检查一些模型参数的影响并讨论它们的经济影响。还给出了两个数值例子来说明不同团队规模决策之间的参数依赖性和比较。