On Two-Player Scalar Discounted Linear Quadratic Games

Abstract

This paper studies feedback Nash equilibria (FNE) in scalar two-player discounted linear-quadratic games. By explicitly discounting the objective function, we reveal the existence of finite-cost equilibria that do not necessarily stabilize the system, motivating a distinction between stable FNE and FNE. We provide a sufficient condition ensuring that every FNE is stabilizing, together with an analytical characterization of the existence, uniqueness, and multiplicity of FNE. For the symmetric setting, closed-form expressions are derived together with a complete characterization of multiplicity in terms of the game parameters. The local attractive and repelling properties of a FNE under the best-response dynamics are characterized and related to multiplicity in the symmetric setting. Analytical and numerical results further reveal an anti-coordination structure in the value functions when multiple FNE exist. Finally, the Community Cost is investigated, showing how the ordering of FNE depends on the relative weighting of the players' objectives.

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