Nash equilibrium seeking in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e89" altimg="si2.svg"><mml:mi>N</mml:mi></mml:math>-coalition games via a gradient-free method

This paper studies an N -coalition non-cooperative game problem, where the players in same coalition cooperatively minimize sum of their local cost functions under a directed communication graph , while collectively acting as virtual player to play with other coalitions. Moreover, it is assumed that have no access explicit functional form but only function value costs. To solve discrete-time gradient-free Nash equilibrium seeking strategy, based on gradient tracking method, proposed. Specifically, estimator developed locally Gaussian smoothing estimate partial gradients, and tracker constructed trace average gradients among within coalition. With sufficiently small constant step-size, we show all players’ actions approximately converge at geometric rate strongly monotone mapping condition. Numerical simulations are conducted verify effectiveness proposed algorithm.