Distributed adaptive Nash equilibrium seeking for games of heterogeneous high-order players over a directed graph

This paper investigates the distributed Nash equilibrium (NE) seeking problem for non-cooperative games involving heterogeneous high-order players over a directed graph. We first address the NE seeking problem of heterogeneous high-order integrators by employing a state transformation method, which reduces the high-order integrators to single integrators. A distributed NE seeking strategy is proposed, incorporating position estimators, gradient descent and state feedback. Furthermore, for the case of heterogeneous high-order players with parametric uncertainties, we propose another distributed strategy based on the model reference adaptive NE seeking idea, where a linear high-order virtual player is designed for each player to track. Both strategies converge to NE asymptotically, avoiding the need to interact with high-order derivative information or rely on shared gains. Finally, numerical simulations are performed to verify the effectiveness of the proposed algorithms.