More Information is Not Always Better: Connections between Zero-Sum Local Nash Equilibria in Feedback and Open-Loop Information Patterns

Non-cooperative dynamic game theory provides a principled approach to modeling sequential decision-making among multiple noncommunicative agents. A key focus has been on finding Nash equilibria in two-agent zero-sum dynamic games under various information structures. A well-known result states that in linear-quadratic games, unique Nash equilibria under feedback and open-loop information structures yield identical trajectories. Motivated by two key perspectives -- (i) many real-world problems extend beyond linear-quadratic settings and lack unique equilibria, making only local Nash equilibria computable, and (ii) local open-loop Nash equilibria (OLNE) are easier to compute than local feedback Nash equilibria (FBNE) -- it is natural to ask whether a similar result holds for local equilibria in zero-sum games. To this end, we establish that for a broad class of zero-sum games with potentially nonconvex-nonconcave objectives and nonlinear dynamics: (i) the state/control trajectory of a local FBNE satisfies local OLNE first-order optimality conditions, and vice versa, (ii) a local FBNE trajectory satisfies local OLNE second-order necessary conditions, (iii) a local FBNE trajectory satisfying feedback sufficiency conditions also constitutes a local OLNE, and (iv) with additional hard constraints on agents' actuations, a local FBNE where strict complementarity holds also satisfies local OLNE first-order optimality conditions, and vice versa.