Explicit Sequential Equilibria in LQ Deep Structured Games and Weighted Mean-Field Games

We investigate a class of nonzero-sum dynamic stochastic games where players have linear dynamics and quadratic costs, coupled in both dynamics and cost through a linear regression (weighted average) and a quadratic regression (weighted covariance matrix) of states and actions; the linear regression of states is called the deep state. We study collaborative and non-collaborative games under three information structures: perfect sharing, deep state sharing, and no sharing, for three weight types: positive, homogeneous, and asymptotically vanishing. For perfect and deep state sharing structures, we propose a new technique using gauge transformation to solve the players' best-response equations and identify sufficient conditions for a unique subgame perfect Nash equilibrium. The equilibrium is linear in the local state and deep state, with gains obtained from a novel non-standard Riccati equation whose dimension is independent of the number of players, making the solution scalable. Under no sharing with an asymptotically large number of players, we propose an asymptotic population-size-dependent equilibrium and an asymptotic population-size-independent equilibrium (the sequential weighted mean-field equilibrium), and establish their convergence to the infinite-population limits. The main results are extended to infinite-horizon costs and generalized to multiple linear regressions and heterogeneous sub-populations. A numerical example illustrates the difference between the two approximate equilibria. To our knowledge, this is the first paper to propose a unified framework yielding the exact closed-form solution for an arbitrary number of players.