  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.
