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In this paper, we investigate a class of nonzero-sum dynamic stochasticgames, where players have linear dynamics and quadratic cost functions. Theplayers are coupled in both dynamics and cost through a linear regression(weighted average) as well as a quadratic regression (weighted covariancematrix) of the states and actions, where the linear regression of states iscalled deep state. We study collaborative and non-collaborative games underthree information structures: perfect sharing, deep state sharing, and nosharing for three different types of weights: positive, homogeneous andasymptotically vanishing weights. For perfect and deep state sharinginformation structures, we propose a transformation-based technique to solvefor the best-response equations of players and identify a few sufficientconditions under which a unique subgame perfect Nash equilibrium exists. Theequilibrium is linear in the local state and deep state, and the correspondinggains are obtained by solving a novel non-standard Riccati equation (whosedimension is independent of the number of players, thus making the solutionscalable). When the information structure is no-sharing and the number ofplayers is asymptotically large, one approximate population-size-dependentequilibrium and one approximate population-size-independent equilibrium (alsocalled sequential weighted mean-field equilibrium) are proposed, and theirconvergence to the infinite-population limits are established. In addition, themain results are extended to infinite-horizon cost function, and generalized tomultiple orthogonal linear regressions and heterogeneous sub-populations. Anumerical example is provided to demonstrate the difference between the twoproposed approximate equilibria.

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