Optimal Relative Performance Criteria in Mean Field Contribution Games

We consider mean-field contribution games, where players in a team choose some effort levels at each time period and the aggregate reward for the team depends on the aggregate cumulative performance of all the players. Each player aims to maximize the expected reward of her own share subject to her cost of effort. To reduce the free-rider issue, we propose some relative performance criteria (RPC), based on which the reward is redistributed to each player. We are interested in those RPCs that implement the optimal solution for the corresponding centralized problem, and we call such RPC an optimal one. That is, the expected payoff of each player under the equilibrium associated with an optimal RPC is as large as the value induced by the corresponding problem where players completely cooperate. We first analyze a one-period model with homogeneous players and obtain natural RPCs of different forms. Then, we generalize these results to a multiperiod model in discrete time. Next, we investigate a two-layer mean-field game. The top layer is an interteam game (team-wise competition), in which the reward of a team is impacted by the relative achievement of the team with respect to other teams; the bottom layer is an intrateam contribution game where an RPC is implemented for reward redistribution among team members. We establish the existence of equilibria for the two-layer game and characterize the intrateam optimal RPC. Finally, we extend the (one-layer) results of optimal RPCs to the continuous-time setup as well as to the case with heterogenous players.