Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.14449

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

Marko Maljkovic·Gustav Nilsson·Nikolas Geroliminis
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

eess.SYcs.SY

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marko Maljkovic, Gustav Nilsson, Nikolas Geroliminis. 2024-02-22. On decentralized computation of the leader's strategy in bi-level games. https://arxiv.org/abs/2402.14449

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Co-Investment with Payoff-Sharing Mechanism for Cooperative Decision-Making in Network Design Games

Network-based systems are inherently interconnected, with the design and performance of subnetworks being interdependent. However, the decisions of self-interested operators may lead to suboptimal outcomes for users and the overall system. This paper explores cooperative mechanisms that can simultaneously benefit both operators and users. We address this challenge using a game-theoretical framework that integrates both non-cooperative and cooperative game theory. In the non-cooperative stage, we propose a network design game in which subnetwork decision-makers strategically design local infrastructures. In the cooperative stage, co-investment with payoff-sharing mechanism is developed to enlarge collective benefits and fairly distribute them. To demonstrate the effectiveness of our framework, we conduct case studies on the Sioux Falls network and real-world public transport networks in Zurich and Winterthur, Switzerland. Our evaluation considers impacts on environmental sustainability, social welfare, and economic efficiency. The proposed framework provides a foundation for improving interdependent networked systems by enabling strategic cooperation among self-interested operators.

eess.SY↗
arXiv2026

Generalizable Optimal Control with Transformers: One Policy Across Diverse Systems

Classical optimal control designs a separate controller for each plant. Even for the Linear Quadratic Regulator (LQR), every new model must be identified and its Riccati equation re-solved. We ask whether a single learned policy can instead serve an entire family of systems, and we show that one transformer can. We train the policy to imitate optimal LQR state feedback across a collection of heterogeneous Multiple-Input, Multiple-Output (MIMO) Linear Time-Invariant (LTI) systems that differ in their state and input dimensions and in their cost objectives. A shared representation lets the same parameters control every member of the family. It combines system-wise standardization, zero-padding and masking across dimensions, and an explicit encoding of the cost matrices. At run time, the policy maps a short window of recent states and the specified cost to a control action. It uses no plant matrices and identifies the dynamics implicitly from the state history. We evaluate on $28$ simulated systems over $9{,}675$ closed-loop rollouts, and no unstable rollout was observed in any of them. On the systems seen during training, it attains a median relative sub-optimality of $0.022\%$, even under parameter perturbations of up to $\pm10\%$. It transfers to unseen systems with lightweight fine-tuning, reaching a median sub-optimality of $0.19\%$. These results support transformers as generalizable near-optimal controllers for structured families of linear systems.

eess.SY↗
arXiv2026

Two-Timescale Asymptotic Simulations of Hybrid Inclusions with Applications to Stochastic Hybrid Optimization

Convergence properties of model-free two-timescale asymptotic simulations of singularly perturbed hybrid inclusions are developed. A hybrid inclusion combines constrained differential and difference inclusions to capture continuous (flow) and discrete (jump) dynamics, respectively. Sufficient conditions are established under which sequences of iterates and step sizes constitute a two-timescale asymptotic simulation of such a system, with limiting behavior characterized via weakly invariant and internally chain-transitive sets of an associated boundary layer and reduced system. To illustrate the applicability of these results, conditions are given under which a two-timescale stochastic approximation of a hybrid optimization algorithm asymptotically recovers the behavior of its deterministic counterpart.

eess.SY↗