Dynamic Collective Choice: Social Optima

We consider a dynamic collective choice problem where a large number of agents 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 some group to achieve coordinated tasks. Finding a social optimum for our problem reduces to solving a set of linear quadratic regulator problems, whose number, however, increases exponentially with the size of the population. Alternatively, we develop via the mean field games methodology a set of decentralized strategies characterized via the fixed points of a suitable operator. In the homogeneous parameter agents case, the procedure reduces to solving a vector fixed point equation of size
$l$
equal to the number of destinations, followed by each agent comparing only
$l$
regulator costs, independently of the number of agents. When the latter is sufficiently large, the strategies qualify as approximately socially optimal. To compute the approximate social optimum, each agent only needs to know its own state and the statistical distributions of the agents’ initial states and problem parameters. A numerical example illustrates the benefits of cooperative strategies compared to noncooperative ones in achieving an adequate splitting of agents among multiple destinations that each require collective attention.