  We consider a dynamic collective choice problem where a large number of
players are cooperatively choosing between multiple destinations while being
influenced by the behavior of the group. For example, in a robotic swarm
exploring a new environment, a robot might have to choose between multiple
sites to visit, but at the same time it should remain close to the group to
achieve some coordinated tasks. We show that to find a social optimum for our
problem, one needs to solve a set of Linear Quadratic Regulator problems, whose
number increases exponentially with the size of the population. Alternatively,
we develop via the Mean Field Games methodology a set of decentralized
strategies that are independent of the size of the population. When the number
of agents is sufficiently large, these strategies qualify as approximately
socially optimal. To compute the approximate social optimum, each player needs
to know its own state and the statistical distributions of the players' initial
states and problem parameters. Finally, we give a numerical example where the
cooperative and noncooperative cases have opposite behaviors. Whereas in the
former the size of the majority increases with the social effect, in the
latter, the existence of a majority is disadvantaged.
