A Dynamic Game Model of Collective Choice in Multiagent Systems

Inspired by successful biological collective decision mechanisms such as honey bees searching for a new colony or the collective navigation of fish schools, we consider a scenario where a large number of agents engaged in a dynamic game have to make a choice among a finite set of different potential target destinations. Each individual both influences and is influenced by the group's decision, as represented by the mean trajectory of all agents. The model can be interpreted as a stylized version of opinion crystallization in an election for example. In the most general formulation, agents' biases are dictated by a combination of initial position, individual dynamics parameters, and a priori individual preference. Agents are assumed linear and coupled through a modified form of quadratic cost, whereby the terminal cost captures the discrete choice component of the problem. Following the mean field games methodology, we identify sufficient conditions under which allocations of destination choices over agents lead to self-replication of the overall mean trajectory under the best response by the agents. Importantly, we establish that when the number of agents increases sufficiently, 1) the best response strategies to the self-replicating mean trajectories qualify as epsilon-Nash equilibria of the population game; and 2) these epsilon-Nash strategies can be computed solely based on the knowledge of the joint probability distribution of the initial conditions, dynamics parameters, and destination preferences, now viewed as random variables. Our results are illustrated through numerical simulations.