Distributed Nash equilibrium seeking with limited cost function knowledge via a consensus-based gradient-free method

This article considers a distributed Nash equilibrium seeking problem, where the players only have partial access to other players' actions, such as their neighbors' actions. Thus, the players are supposed to communicate with each other to estimate other players' actions. To solve the problem, a leader-following consensus gradient-free distributed Nash equilibrium seeking algorithm is proposed. This algorithm utilizes only the measurements of the player' local cost function without the knowledge of its explicit expression or the requirement on its smoothness. Hence, the algorithm is gradient-free during the entire updating process. Moreover, the analysis on the convergence of the Nash equilibrium is studied for the algorithm with both diminishing and constant step-sizes, respectively. Specifically, in the case of diminishing step-size, it is shown that the players' actions converge to the Nash equilibrium almost surely, while in the case of fixed step-size, the convergence to the neighborhood of the Nash equilibrium is achieved. The performance of the proposed algorithm is verified through numerical simulations.Building and Construction Authority (BCA)Ministry of Education (MOE)National Research Foundation (NRF)This work was supported in part by Singapore Ministry of Education Academic Research Fund under Grant Tier 1 RG180/17(2017-T1-002- 158), and in part by the National Research Foundation, Prime Minister’s Office, Singapore under the Energy Innovation Research Programme (EIRP) for Building Energy Efficiency Grant Call under Grant NRF2013EWT-EIRP004-051, administered by the Building and Construction Authority