Adaptive Nash Equilibrium Seeking Strategies for Games with Second-order and Mixed-order players

Inspired by the observation that the implementations of many existing Nash equilibrium seeking algorithms require centralized information, which is hard to be obtained, this paper considers the design of adaptive laws for Nash equilibrium seeking in two types of games. First, games in which the players are modeled as second-order integrators are considered, followed by games with both first-order integrator-type players and second-order integrator-type players. The proposed Nash equilibrium seeking algorithms are designed by including a regulation term (for second-order players) and an optimization term in the control inputs in which their associated gains governed by adaptive laws. Lyapunov functions are constructed to show stabilities of the Nash equilibrium under the proposed methods. It is proven that with the adaptive laws, the control gains involved in the proposed algorithms are effective without utilizing any centralized information. In the last, simulation examples are given to numerically illustrate the proposed algorithms.