Distributed Nash Equilibrium Searching via Fixed-Time Consensus-Based Algorithms

In this paper, distributed algorithms are designed to search the Nash equilibrium (NE) for an $N$-player game in continuous-time. The agents are not assumed to have direct access of other agents' states, and instead, they estimate other agents' states by communicating with their neighbours. Advanced consensus algorithms are implemented for such purposes, and consequently the game is decentralised into $N$ subsystems interacting over a communication network. It is proved that, for any communication network with a connected graph, a fixed-time consensus is achieved, independent of the initial conditions, based on which an NE can be obtained asymptotically by the gradient descent term for the fixed-time consensus-based algorithm. Then, the results are extended to a fixed-time NE seeking with modifications of the gradient terms, where both the consensus and the optimisation can be obtained in fixed time, and the upper bound of the settling time is established by the Lyapunov theory. A simulation example is presented to verify the effectiveness of the theoretical development, where some comparisons with other works are studied to demonstrate the advantages of the proposed algorithms.