The Qualitative Properties of Symmetric Open‐Loop Nash Equilibria in the State‐Control Dynamic System in Differential Games

Abstract

The local stability, steady state comparative statics, and local comparative dynamics of symmetric open‐loop Nash equilibria in the state‐control dynamic system for a seemingly ubiquitous class of discounted infinite horizon differential games are investigated. It is shown that most of the useful qualitative results occur because the same small number of assumptions is being made about the mathematical structure of the integrand and/or state equations. Applications of the results to exhaustible resource extraction and capital accumulation differential games are provided.

Distributed adaptive generalized Nash equilibrium seeking algorithm with event‐triggered communication

“…In (8), with the estimated aggregation 𝜂 i , ∇ x i J i (x i , 𝜂 i ) is the gradient of cost function J i (x i , 𝜂 i ) with respect to x i . The optimal Lagrange multiplier 𝜆 i is estimated by…”

“…It is calculated that x * = [42.4,47.3,52.2,57.1,62.0] T . The parameters in the designed GNE seeking algorithm (8) are k i = 2, c i1 = c i2 = 1, 𝛼 = 5, and 𝛽 = 10 for all i ∈ {1, … , 5}. Choose 𝜁 1 = [0.4, 0.8, 0.4, 0.51, 0.8] T and 𝜁 2 = [0.2, 0.15, 0.2, 0.15, 0.2] T in event-triggered condition (7) for each agent.…”

“…Choose 𝜁 1 = [0.4, 0.8, 0.4, 0.51, 0.8] T and 𝜁 2 = [0.2, 0.15, 0.2, 0.15, 0.2] T in event-triggered condition (7) for each agent. Figure 2 shows the evolution of all agents' strategies updated by the designed adaptive GNE seeking algorithm (8) with event-triggered communication scheme (7). In Figure 2, dash lines depict GNE x * .…”

“…With the development of networked systems, the design of distributed algorithms for multi‐agent systems has attracted more and more attentions, including distributed optimization and game theory. In the last decade, the (generalized) Nash equilibrium seeking problem has become one of the hot topics in the control and optimization of multi‐agent systems (see the work of Friharf et al [1], Wang et al [2], Han et al [3], Liu et al [4], Wu et al [5], Fang et al [6], Shao et al [7], and Hsueh and Ling [8] and the references therein). In this paper, we focus on a class of aggregative games with shared resources, which has been widely applied in many engineering scenarios, such as smart grids [9], sensor networks [10] and mechanical systems [11].…”

Distributed Nash equilibrium seeking for noncooperative games in nonlinear multi‐agent systems: An event‐triggered neuro‐adaptive approach

“…Game theory has been extensively studied during the past decades, due to its widespread applications in signal processing [1], economic dispatch [2], mobile sensor networks [3], and many other fields [4][5][6][7][8][9]. Among the various topics in the researches on games, distributed Nash equilibrium seeking in noncooperative games has received considerable attention [10][11][12].…”