  This paper investigates the social optimum for a dynamic linear quadratic
collective choice problem where a group of agents choose among multiple
alternatives or destinations. The agents' common objective is to minimize the
average cost of the entire population. A naive approach to finding a social
optimum for this problem involves solving a number of linear quadratic
regulator (LQR) problems that increases exponentially with the population size.
By exploiting the problem's symmetries, we first show that one can equivalently
solve a number of LQR problems equal to the number of destinations, followed by
an optimal transport problem parameterized by the fraction of agents choosing
each destination. Then, we further reduce the complexity of the solution search
by defining an appropriate system of limiting equations, whose solution is used
to obtain a strategy shown to be asymptotically optimal as the number of agents
becomes large. The model includes a congestion effect captured by a negative
quadratic term in the social cost function, which may cause agents to escape to
infinity in finite time. Hence, we identify sufficient conditions, independent
of the population size, for the existence of the social optimum. Lastly, we
investigate the behavior of the model through numerical simulations in
different scenarios.
