We consider a class of dynamic collective choice models with social interactions,
whereby a large number of non-uniform agents have to individually settle on one
of multiple discrete alternative choices, with the relevance of their would-be
choices continuously impacted by noise and the unfolding group behavior.
This class of problems is modeled here as a so-called Min-LQG game,
i.e., a linear quadratic Gaussian dynamic and non-cooperative game,
with an additional combinatorial aspect in that it includes a final
choice-related minimization in its terminal cost.
The presence of this minimization term is key to enforcing some specific discrete choice by each individual agent.
The theory of mean field games is invoked to generate a class of
decentralized agent feedback control strategies, which are then shown
to converge to an exact Nash equilibrium of the game as the number
of players increases to infinity.
A key building block in our approach is an explicit solution to the
problem of computing the best response of a generic agent to some
arbitrarily posited smooth mean field trajectory.
Ultimately, an agent is shown to face a continuously revised
discrete choice problem, where greedy choices dictated by current
conditions must be constantly balanced against the risk of the future
process noise upsetting the wisdom of such decisions.
We show that any Nash equilibrium of the game is defined by
an a priori computable probability matrix which describes the distribution of the players' choices over the alternatives.
The results are illustrated through simulations.
Published November 2018 , 26 pages