Mean field games (MFG) and mean field control (MFC) problems have been introduced to study large populations of strategic players. They correspond respectively to non-cooperative or cooperative scenarios, where the aim is to find the Nash equilibrium and social optimum. These frameworks provide approximate solutions to situations with a finite number of players and have found a wide range of applications, from economics to biology and machine learning. In this paper, we study how the players can pass from a non-cooperative to a cooperative regime, and vice versa. The first direction is reminiscent of mechanism design, in which the game's definition is modified so that non-cooperative players reach an outcome similar to a cooperative scenario. The second direction studies how players that are initially cooperative gradually deviate from a social optimum to reach a Nash equilibrium when they decide to optimize their individual cost similar to the free rider phenomenon. To formalize these connections, we introduce two new classes of games which lie between MFG and MFC: $\lambda$-interpolated mean field games, in which the cost of an individual player is a $\lambda$-interpolation of the MFG and the MFC costs, and $p$-partial mean field games, in which a proportion $p$ of the population deviates from the social optimum by playing the game non-cooperatively. We conclude the paper by providing an algorithm for myopic players to learn a $p$-partial mean field equilibrium, and we illustrate it on a stylized model.

翻译：平均场博弈（MFG）与平均场控制（MFC）问题被引入用于研究大规模战略博弈群体，分别对应于非合作与合作场景，其目标在于寻找纳什均衡与社会最优。这些框架为有限玩家情境提供了近似解，并在经济学、生物学及机器学习等领域具有广泛应用。本文研究了玩家如何从非合作机制过渡到合作机制，反之亦然。第一方向类似于机制设计，通过调整博弈定义使非合作玩家达到类似合作场景的结果；第二方向则研究初始合作的玩家如何逐步偏离社会最优，在个体或类似搭便车现象中优化自身成本时逼近纳什均衡。为形式化这些关联，我们引入两类介于MFG与MFC之间的新博弈：λ-插值平均场博弈，其中个体玩家的成本为MFG与MFC成本的λ-插值；以及p-部分平均场博弈，其中比例为p的群体通过非合作方式偏离社会最优。最后，我们为近视型玩家提供了学习p-部分平均场均衡的算法，并通过一个简化模型进行验证。