Dario Paccagnan

Dario Paccagnan, Basilio Gentile, Francesca Parise et al.

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibrium. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.

Dario Paccagnan, Francesca Parise, John Lygeros

Several works have recently suggested to model the problem of coordinating the charging needs of a fleet of electric vehicles as a game, and have proposed distributed algorithms to coordinate the vehicles towards a Nash equilibrium of such game. However, Nash equilibria have been shown to posses desirable system-level properties only in simplified cases. In this work, we use the concept of price of anarchy to analyze the inefficiency of Nash equilibria when compared to the social optimum solution. More precisely, we show that i) for linear price functions depending on all the charging instants, the price of anarchy converges to one as the population of vehicles grows; ii) for price functions that depend only on the instantaneous demand, the price of anarchy converges to one if the price function takes the form of a positive pure monomial; iii) for general classes of price functions, the asymptotic price of anarchy can be bounded. For finite populations, we additionaly provide a bound on the price of anarchy as a function of the number vehicles in the system. We support the theoretical findings by means of numerical simulations.

Dario Paccagnan, Marco C. Campi

Variational inequalities are modelling tools used to capture a variety of decision-making problems arising in mathematical optimization, operations research, game theory. The scenario approach is a set of techniques developed to tackle stochastic optimization problems, take decisions based on historical data, and quantify their risk. The overarching goal of this manuscript is to bridge these two areas of research, and thus broaden the class of problems amenable to be studied under the lens of the scenario approach. First and foremost, we provide out-of-samples feasibility guarantees for the solution of variational and quasi variational inequality problems. Second, we apply these results to two classes of uncertain games. In the first class, the uncertainty enters in the constraint sets, while in the second class the uncertainty enters in the cost functions. Finally, we exemplify the quality and relevance of our bounds through numerical simulations on a demand-response model.

Rahul Chandan, Dario Paccagnan, Jason R. Marden

Today's multiagent systems have grown too complex to rely on centralized controllers, prompting increasing interest in the design of distributed algorithms. In this respect, game theory has emerged as a valuable tool to complement more traditional techniques. The fundamental idea behind this approach is the assignment of agents' local cost functions, such that their selfish minimization attains, or is provably close to, the global objective. Any algorithm capable of computing an equilibrium of the corresponding game inherits an approximation ratio that is, in the worst case, equal to its price-of-anarchy. Therefore, a successful application of the game design approach hinges on the possibility to quantify and optimize the equilibrium performance. Toward this end, we introduce the notion of generalized smoothness, and show that the resulting efficiency bounds are significantly tighter compared to those obtained using the traditional smoothness approach. Leveraging this newly-introduced notion, we quantify the equilibrium performance for the class of local resource allocation games. Finally, we show how the agents' local decision rules can be designed in order to optimize the efficiency of the corresponding equilibria, by means of a tractable linear program.

Vinod Ramaswamy, Dario Paccagnan, Jason R. Marden

The price of anarchy and price of stability are three well-studied performance metrics that seek to characterize the inefficiency of equilibria in distributed systems. The distinction between these two performance metrics centers on the equilibria that they focus on: the price of anarchy characterizes the quality of the worst-performing equilibria, while the price of stability characterizes the quality of the best-performing equilibria. While much of the literature focuses on these metrics from an analysis perspective, in this work we consider these performance metrics from a design perspective. Specifically, we focus on the setting where a system operator is tasked with designing local utility functions to optimize these performance metrics in a class of games termed covering games. Our main result characterizes a fundamental trade-off between the price of anarchy and price of stability in the form of a fully explicit Pareto frontier. Within this setup, optimizing the price of anarchy comes directly at the expense of the price of stability (and vice versa). Our second results demonstrates how a system-operator could incorporate an additional piece of system-level information into the design of the agents' utility functions to breach these limitations and improve the system's performance. This valuable piece of system-level information pertains to the performance of worst performing agent in the system.

Rahul Chandan, Dario Paccagnan, Jason R. Marden

The design of distributed algorithms is central to the study of multiagent systems control. In this paper, we consider a class of combinatorial cost-minimization problems and propose a framework for designing distributed algorithms with a priori performance guarantees that are near-optimal. We approach this problem from a game-theoretic perspective, assigning agents cost functions such that the equilibrium efficiency (price of anarchy) is optimized. Once agents' cost functions have been specified, any algorithm capable of computing a Nash equilibrium of the system inherits a performance guarantee matching the price of anarchy. Towards this goal, we formulate the problem of computing the price of anarchy as a tractable linear program. We then present a framework for designing agents' local cost functions in order to optimize for the worst-case equilibrium efficiency. Finally, we investigate the implications of our findings when this framework is applied to systems with convex, nondecreasing costs.

Dario Paccagnan, Jason R. Marden

A fundamental challenge in multiagent systems is to design local control algorithms to ensure a desirable collective behaviour. The information available to the agents, gathered either through communication or sensing, naturally restricts the achievable performance. Hence, it is fundamental to identify what piece of information is valuable and can be exploited to design control laws with enhanced performance guarantees. This paper studies the case when such information is uncertain or inaccessible for a class of submodular resource allocation problems termed covering problems. In the first part of this work we pinpoint a fundamental risk-reward tradeoff faced by the system operator when conditioning the control design on a valuable but uncertain piece of information, which we refer to as the cardinality, that represents the maximum number of agents that can simultaneously select any given resource. Building on this analysis, we propose a distributed algorithm that allows agents to learn the cardinality while adjusting their behaviour over time. This algorithm is proved to perform on par or better to the optimal design obtained when the exact cardinality is known a priori.

Daniel Marks, Dario Paccagnan, Mark van der Wilk

Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.

Dario Paccagnan, Daniel Marks, Marco C. Campi et al.

Data-driven methods have become paramount in modern systems and control problems characterized by growing levels of complexity. In safety-critical environments, deploying these methods requires rigorous guarantees, a need that has motivated much recent work at the interface of statistical learning and control. However, many existing approaches achieve this goal at the cost of sacrificing valuable data for testing and calibration, or by constraining the choice of learning algorithm, thus leading to suboptimal performances. In this paper, we describe Pick-to-Learn (P2L) for Systems and Control, a framework that allows any data-driven control method to be equipped with state-of-the-art safety and performance guarantees. P2L enables the use of all available data to jointly synthesize and certify the design, eliminating the need to set aside data for calibration or validation purposes. In presenting a comprehensive version of P2L for systems and control, this paper demonstrates its effectiveness across a range of core problems, including optimal control, reachability analysis, safe synthesis, and robust control. In many of these applications, P2L delivers designs and certificates that outperform commonly employed methods, and shows strong potential for broad applicability in diverse practical settings.

Miriam Fischer, Dario Paccagnan

We study mathematical programs with equilibrium constraints, in which a leader knows their own cost function, but lacks a model of the followers' response. Instead, the leader can only query this response at specific points. While this setting precludes the use of gradient-based methods, existing zeroth-order approaches treat the composed objective entirely as a black box, deploying zeroth-order tools across both the leader and follower. Such approaches are inefficient, as they discard information the leader already possesses about their own cost function. In this work we instead propose to deploy zeroth-order tools only where they are truly needed: to handle the unknown, non-smooth followers' response. Specifically, we first propose PZOS, an algorithm that combines exact partial gradients of the leader's cost with zeroth-order Jacobian estimates of the followers' response in a chain-rule-inspired manner, and establish that it achieves a strictly lower variance bound than the black-box baseline. Second, we introduce the partial Goldstein subdifferential, a stationarity notion tailored to this composite structure, and prove convergence of our algorithm to both standard and partial Goldstein stationary points. Finally, we validate our method on two application domains -- toll optimization in routing games and defense-attack investment in security games -- demonstrating consistent improvements over black-box baselines in convergence speed, objective value, and estimator variance, with robust performance even under few queries per iteration.

Andreas Feik, Nicolas Lanzetti, Saverio Bolognani et al.

In many multiagent settings, such as electric vehicle charging and traffic routing, agents must make decisions in the face of uncertain behavior exhibited by others. Often, this uncertainty arises from multiple sources, such as incomplete information, limited computation, or bounded rationality, ultimately impacting the aggregate behavior. To tackle this challenge, we follow recent work on strategically robust game theory and postulate that agents seek protection directly against deviations around the emergent behavior, as opposed to explicitly modeling all sources of uncertainty. Specifically, we propose that each agent protects itself against the worst-case aggregate behavior within an optimal-transport-based ambiguity set centered at the emergent aggregate population behavior. This leads to a novel equilibrium concept, called strategically robust Wardrop equilibrium, that enables one to interpolate between standard Wardrop equilibria (no robustness) and security strategies (maximum robustness). In the setting of convex aggregative games, we establish the existence of a pure strategically robust Wardrop equilibrium and provide tractable computational tools for computing it. Through an application in electric vehicle charging, we demonstrate that strategically robust Wardrop equilibria lead to better decisions, protecting agents against the uncertain aggregate behavior of the population. Remarkably, we also observe that strategic robustness can lead to lower equilibrium costs for all agents, uncovering a "coordination-via-robustification" effect.

Aamal Hussain, Dan Leonte, Francesco Belardinelli et al.

Beyond specific settings, many multi-agent learning algorithms fail to converge to an equilibrium solution, and instead display complex, non-stationary behaviours such as recurrent or chaotic orbits. In fact, recent literature suggests that such complex behaviours are likely to occur when the number of agents increases. In this paper, we study Q-learning dynamics in network polymatrix games where the network structure is drawn from classical random graph models. In particular, we focus on the Erdos-Renyi model, a well-studied model for social networks, and the Stochastic Block model, which generalizes the above by accounting for community structures within the network. In each setting, we establish sufficient conditions under which the agents' joint strategies converge to a unique equilibrium. We investigate how this condition depends on the exploration rates, payoff matrices and, crucially, the sparsity of the network. Finally, we validate our theoretical findings through numerical simulations and demonstrate that convergence can be reliably achieved in many-agent systems, provided network sparsity is controlled.