Social Optimization in Noncooperative Games under Central Regulation

TL;DR

This paper addresses achieving social optimum in a bilevel setting where a regulator's intervention θ shapes a low-level noncooperative game. It introduces a gradient-free, zeroth-order algorithm based on randomized smoothing and Moreau smoothing to handle nonconvex, nonsmooth objectives and inexact computing of the low-level NE. The method attains sublinear convergence to an approximate stationary point, with explicit rates in both inexact and exact NE scenarios. Numerical experiments on an electric-vehicle charging paradigm illustrate effectiveness and practical viability for distributed social optimization.

Abstract

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.

Figures (4)

• Figure 1: Sketch of the problem setup. Each of the $N$ players in the low level aims to minimize its personal cost function $f_i$ given intervention decision $\theta$ of the high-level regulator. Then, the regulator in the high level utilizes required data received from all players to choose an optimal intervention decision to achieve social optimum. Here, $x_{-i}:=\mathop{\mathrm{col}}\limits\{x_i\}_{i\in[N]\backslash\{i\}}$ and $x(\theta):=\mathop{\mathrm{col}}\limits\{x_i(\theta)\}_{i\in[N]}$ represents the NE of the low-level game.
• Figure 2: The trajectory of $|\theta_k-\theta^*|$.
• Figure 3: The trajectory of $|\nabla\hat{\mathbf{F}}(\theta_k)|$.
• Figure 4: Trajectories of players' objective values $f_i(x_{\varepsilon_k}(\theta_k),\theta_k)$, $i\in[10]$ and the regulator's objective value $\sum_{i=1}^{10}f_i(x_{\varepsilon_k}(\theta_k),\theta_k)$.

Theorems & Definitions (34)

• Definition 1
• Example 1
• Remark 1
• Remark 2
• Lemma 1
• Remark 3
• Lemma 2
• proof
• Remark 4
• Remark 5