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We consider a class of dynamic collective choice models with social interactions, whereby a large number of nonuniform 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 noncooperative 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. [ABSTRACT FROM AUTHOR]

We study large population leader-follower stochastic multi-agent systems where the agents have linear stochastic dynamics and are coupled via their quadratic cost functions. The cost of each leader is based on a trade-off between moving toward a certain reference trajectory which is unknown to the followers and staying near their own centroid. On the other hand, followers react by tracking a convex combination of their own centroid and the centroid of the leaders. We approach this large population dynamic game problem by use of so-called Mean Field (MF) linear-quadratic-Gaussian (LQG) stochastic control theory. In this model, followers are adaptive in the sense that they use a likelihood ratio estimator (on a sample population of the leaders' trajectories) to identify the member of a given finite class of models which is generating the reference trajectory of the leaders. Under appropriate conditions, it is shown that the true reference trajectory model is identified by each follower in finite time with probability one as the leaders' population goes to infinity. Furthermore, we show that the resulting sets of mean field control laws for both leaders and adaptive followers possess an almost sure \varepsilonN-Nash equilibrium property for a system with population N where \varepsilonN goes to zero as N goes to infinity. Numerical experiments are presented illustrating the results. [ABSTRACT FROM AUTHOR]