  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 mean field games (MFG)-like scenario where a large
number of agents have to make a choice among a set of different potential
target destinations. Each individual both influences and is influenced by the
group's decision, as well as the mean trajectory of all the agents. The model
can be interpreted as a stylized version of opinion crystallization in an
election for example. The agents' biases are dictated first by their initial
spatial position and, in a subsequent generalization of the model, by a
combination of initial position and a priori individual preference. The agents
have linear dynamics and are coupled through a modified form of quadratic cost.
Fixed point based finite population equilibrium conditions are identified and
associated existence conditions are established. In general multiple equilibria
may exist and the agents need to know all initial conditions to compute them
precisely. However, as the number of agents increases sufficiently, we show
that 1) the computed fixed point equilibria qualify as epsilon Nash equilibria,
2) agents no longer require all initial conditions to compute the equilibria
but rather can do so based on a representative probability distribution of
these conditions now viewed as random variables. Numerical results are
reported.
