Teamwise Mean Field Competitions

TL;DR

This paper introduces a novel mean field game framework for large team competitions based on rank and Poisson processes, providing explicit solutions and economic insights into team size and performance dynamics.

Contribution

It develops new mean field game models for team competitions with rank-based rewards, including explicit equilibrium solutions and analysis of team size decisions.

Findings

Explicit equilibrium controls are derived for homogeneous parameters.

Team size and performance impacts are analytically examined.

Numerical examples illustrate parameter effects and decision-making comparisons.

Abstract

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 special cases of 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.