On decentralized computation of the leader's strategy in bi-level games

TL;DR

The paper addresses computing local Stackelberg equilibria in broad bi-level games with private constraints by proposing a decentralized, first-order method based on projected gradient descent with Armijo stepsize. It relies on the Implicit Function Theorem to obtain differentiable Jacobians of follower equilibria with respect to the leader’s action, enabling distributed gradient-based updates to the leader’s strategy. A surrogate best-response formulation ensures the KKT Jacobians are invertible, providing explicit expressions for the Jacobians and guaranteeing convergence to an l-SE under standard regularity assumptions. For quadratic aggregative games with polytopic follower constraints, it introduces a decentralized ADMM-based warm-start to obtain interior NE efficiently, and validates the approach with smart-mobility case studies showing effective handling of general convex constraints and initialization issues. The work advances privacy-preserving, scalable computation of leader strategies in hierarchical decision-making, with practical impact in energy management and transportation optimization.

Abstract

Motivated by the omnipresence of hierarchical structures in many real-world applications, this study delves into the intricate realm of bi-level games, with a specific focus on exploring local Stackelberg equilibria as a solution concept. While existing literature offers various methods tailored to specific game structures featuring one leader and multiple followers, a comprehensive framework providing formal convergence guarantees to a local Stackelberg equilibrium appears to be lacking. Drawing inspiration from sensitivity results for nonlinear programs and guided by the imperative to maintain scalability and preserve agent privacy, we propose a decentralized approach based on the projected gradient descent with the Armijo stepsize rule. The main challenge here lies in assuring the existence and well-posedness of Jacobians that describe the leader's decision's influence on the achieved equilibrium of the followers. By meticulous tracking of the Implicit Function Theorem requirements at each iteration, we establish formal convergence guarantees to a local Stackelberg equilibrium for a broad class of bi-level games. Building on our prior work on quadratic aggregative Stackelberg games, we also introduce a decentralized warm-start procedure based on the consensus alternating direction method of multipliers addressing the previously reported initialization issues. Finally, we provide empirical validation through two case studies in smart mobility, showcasing the effectiveness of our general method in handling general convex constraints, and the effectiveness of its extension in tackling initialization issues.

Figures (5)

• Figure 1: Schematic sketch of the problem setup. Each of the $N$ followers aims to optimize the personal objective $J_i$ under the parametrized local constraints $x_i\in\mathcal{X}_i(x_i,\pi)$. The followers communicate with the leader through the communication hub that is used as a medium to collect the locally computed Jacobians $\textbf{D}_{\pi_t}x^{*}_i$ in every update step of the leader's action.
• Figure 2: Ilustrative example of the lower-level NE evolution when the projected gradient descent algorithm is initialized by $\pi_0^a$, i.e., such that the NE is on the boundary of the feasible set $\mathcal{X}_i$ (red), and by $\pi_0^b$, i.e., such that the NE is in the interior (orange).
• Figure 3: Illustration of the problem setup with ride-hailing companies $\mathcal{I}=\{I_1, I_2, I_3\}$ operating in a region with charging stations $\mathcal{M}=\{M_1, M_2, M_3, M_4\}$. The central authority $L$ chooses the electricity price $\pi\in\mathcal{P}\subseteq\mathds{R}^4$ so as to respect the discount budget $B_i$ of each company $i\in\mathcal{I}$.
• Figure 4: The plots show the evolution of the total vehicle accumulation at the charging stations $\sigma(x)$, the price of charging $\pi_j$ at the station $M_j$, the leader's objective $J_L(x^*(\pi),\pi)$, and the portion of the budgets used at each iteration.
• Figure 5: Evolution of the central authority's objective for different initializations of the upper-level iterative loop.

Theorems & Definitions (12)

• Definition 1: Nash Equilibrium
• Definition 2: Quadratic Aggregative Games
• Proposition 1
• Definition 3: Local Stackelberg Equilibrium
• Lemma 1: Proposition 2.3.3 of Bertsekas/99
• Theorem 1: Theorem 1.B1 of Implicit
• Lemma 2
• Theorem 2
• Theorem 3
• Theorem 4: Internal v-NE feasibility check