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

2025-03-19

AI Summary

This paper investigates the equivalence between feedback Nash equilibria (FBNE) and open-loop Nash equilibria (OLNE) in zero-sum dynamic games, moving beyond classical linear-quadratic assumptions to address nonconvex-nonconcave objective functions and nonlinear dynamics. Leveraging variational analysis, nonlinear optimization, and dynamic game theory, we establish— for the first time—the equivalence of local FBNE and OLNE under first- and second-order necessary and sufficient optimality conditions, extending the analysis to cases with control constraints and strict complementarity. We prove that any local FBNE trajectory necessarily satisfies the first- and second-order conditions for a local OLNE; conversely, under standard constraint qualifications and second-order sufficient conditions, any local OLNE also constitutes a local FBNE. This result characterizes the fundamental limits of informational advantage conferred by feedback strategies and provides a novel analytical framework for equilibrium analysis in nonlinear dynamic games.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

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.

Problem

• Explores connections between local Nash equilibria in zero-sum games.
• Compares feedback and open-loop information structures in dynamic games.
• Analyzes conditions under which local equilibria yield identical trajectories.

Innovation

• Compares local Nash equilibria in feedback and open-loop structures
• Analyzes nonconvex-nonconcave objectives and nonlinear dynamics
• Links first-order and second-order optimality conditions between equilibria