From Nash Equilibrium to Social Optimum and vice versa: a Mean Field Perspective

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.