Social Optimization in Noncooperative Games under Central Regulation
Motivated by the increasing attention to overall social benefits in networked multi-agent systems, this paper investigates an optimization problem building on noncooperative games under high-level regulation, which can be formulated in a bilevel structure. Specifically, the low level consists of a noncooperative game, where each player competes to minimize its own cost function that depends not only on the strategies of all players, but also on an intervention decision of a regulator located at the high level. Under the intervention of the high-level regulator, the low-level players aim to seek a Nash equilibrium (NE), which indeed is related to the regulator's decision. Meanwhile, the regulator in the high level attempts to achieve the social optimum, that is, to minimize the sum of all players' costs obtained at the NE. This bilevel social optimization problem is proven to be nonconvex and nonsmooth, leading to challenges for solving it effectively, as the exact gradient of cost sum functions may not be available. To address this intricate problem, an inexact zeroth-order algorithm is developed by virtue of the smoothing techniques, allowing for approximating the NE of the low-level game and thus estimating the required gradients. It is rigorously shown that the devised algorithm achieves a sublinear convergence rate for computing an approximate stationary point of the studied problem. Moreover, the sublinear convergence rate in the scenario where the exact equilibrium of the low-level game is available is established. Finally, numerical simulations are conducted to demonstrate the efficiency of theoretical findings.