Two-player linear-quadratic games — where each player controls a linear dynamical system and minimizes a quadratic cost that depends on both players' actions — are the simplest non-trivial model of strategic interaction in control systems. The feedback Nash equilibrium (where each player's strategy is optimal given the other's) is characterized by coupled Riccati equations. In continuous time, these are well-understood. In discrete time with discounting, less so.
Cavalagli, Bemporad, and Zanon (arXiv:2603.13122) provide a complete analysis of the scalar (one-dimensional state) discrete-time case with discounting. They derive closed-form expressions for when feedback Nash equilibria exist, when they're unique, and what happens when they're not unique.
The anti-coordination pattern is the key finding. When multiple equilibria exist, they exhibit a systematic structure: as one player becomes more aggressive (higher gain), the other becomes more passive (lower gain), and vice versa. The equilibria come in pairs that mirror each other across the symmetric strategy. This isn't an artifact of specific parameter choices — it's a structural property of the coupled best-response equations in the scalar setting.
The classification of local saddle properties determines which equilibria are reachable by iterative best-response algorithms (where players alternate updating their strategies in response to each other). Some equilibria are saddle points of the coupled equations — attracting in one direction, repelling in another. The saddle structure determines whether the iterative process converges or oscillates, and the anti-coordination pattern determines which equilibrium it converges to when it does.