From Nash Equilibrium to Social Optimum and Back: A Mean Field Perspective

Mean field games (MFGs) and mean field control (MFC) 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. They approximate finite player problems and have many applications in economics, biology, machine learning. This paper studies how 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 initially cooperative players gradually deviate from social optimum to reach Nash equilibrium when they optimize their individual cost very much in the spirit of the free-rider phenomenon. To formalize these connections, we introduce and theoretically analyze two new classes of games which lie between MFG and MFC: $\lambda$ -interpolated MFGs, in which the cost of an individual player is an interpolation of the MFG and the MFC costs, and p-partial MFGs, in which a proportion of the population deviates from social optimum by behaving non-cooperatively. We conclude by providing an algorithm for myopic players to learn a p-partial mean field equilibrium.