A Gradient-Free Distributed Nash Equilibrium Seeking Method with Uncoordinated Step-Sizes

In this paper, we study a Nash equilibrium (NE) seeking problem in a multi-player non-cooperative game over a directed communication graph. Specifically, the players' costs are functions of all players' actions, but only part of which are directly accessible. Moreover, we assume the explicit form/expression/model information of the cost function is unknown, but its value can be measured by the local player. To solve this problem, a non-model based distributed NE seeking algorithm is proposed, which requires no gradient information but the measurements of player's local cost function. A leader-following consensus technique is adopted with a row-stochastic adjacency matrix, which simplifies the implementation and increases the application range of the algorithm as compared to the doubly-stochastic matrix. Moreover, the algorithm is able to work with uncoordinated step-sizes, allowing the players to choose their own preferred step-sizes, which makes the algorithm more distributive. The convergence of the proposed algorithm is rigorously studied for both scenarios of diminishing and constant step-sizes, respectively. It is shown that players' actions converge to the exact NE almost surely for the case of diminishing step-size, and to an approximated NE with a gap depending on the step-size selection for the case of constant step-size. Numerical examples are provided to verify the algorithm's effectiveness.