Linear-quadratic two-person differential game: Nash game versus stackelberg game, local information versus global information

¹ Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China
² Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
³ Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

^* Corresponding author: xwfeng@sdu.edu.cn

Received: 1 May 2023
Accepted: 15 April 2024

Abstract

In this paper, we present a unified framework to study a variety of two-person dynamic decision problems, including stochastic (zero-sum, non-zero-sum) Nash game, Stackelberg game with global information. For these games, the solvability of these problems is discussed via progressive formulations respectively: the abstract quadratic functional, Hamiltonian system for open-loop, and Riccati equation for closed-loop (feedback) representation. Based on the unified framework, time consistency/inconsistency property of related equilibrium is studied. Then we introduce a new type of game, Stackelberg game with local information. For this, the classical best-response machinery adopted for global information is no longer workable. As resolution, a repeated game approach is employed to construct the equilibrium strategies via a backward- and forward-procedure. Moreover, connection of local information pattern to time-inconsistency is also revealed. Finally, relations among zero-sum Nash game, zero-sum Stackelberg game with global information and local information are also identified.