Francesca Parise

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.

Francesca Parise, Asuman Ozdaglar

We provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with multidimensional and possibly constrained strategy sets. We delineate the relations between these conditions and characterize classes of networks that satisfy each of these conditions.

Francesca Parise, Basilio Gentile, John Lygeros

We consider the framework of average aggregative games, where the cost function of each agent depends on his own strategy and on the average population strategy. We focus on the case in which the agents are coupled not only via their cost functions, but also via constraints coupling their strategies. We propose a distributed algorithm that achieves an almost Nash equilibrium by requiring only local communications of the agents, as specified by a sparse communication network. The proof of convergence of the algorithm relies on the auxiliary class of network aggregative games and exploits a novel result of parametric convergence of variational inequalities, which is applicable beyond the context of games. We apply our theoretical findings to a multi-market Cournot game with transportation costs and maximum market capacity.

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.

Francesca Parise, Maria Elena Valcher, John Lygeros

A fundamental question in systems biology is what combinations of mean and variance of the species present in a stochastic biochemical reaction network are attainable by perturbing the system with an external signal. To address this question, we show that the moments evolution in any generic network can be either approximated or, under suitable assumptions, computed exactly as the solution of a switched affine system. Motivated by this application, we propose a new method to approximate the reachable set of switched affine systems. A remarkable feature of our approach is that it allows one to easily compute projections of the reachable set for pairs of moments of interest, without requiring the computation of the full reachable set, which can be prohibitive for large networks. As a second contribution, we also show how to select the external signal in order to maximize the probability of reaching a target set. To illustrate the method we study a renown model of controlled gene expression and we derive estimates of the reachable set, for the protein mean and variance, that are more accurate than those available in the literature and consistent with experimental data.

Shuhao Yan, Francesca Parise, Eilyan Bitar

We study sample average approximations (SAA) of chance constrained programs. SAA methods typically approximate the actual distribution in the chance constraint using an empirical distribution constructed from random samples assumed to be independent and identically distributed according to the actual distribution. In this paper, we consider a nonstationary variant of this problem, where the random samples are assumed to be independently drawn in a sequential fashion from an unknown and possibly time-varying distribution. This nonstationarity may be driven by changing environmental conditions present in many real-world applications. To account for the potential nonstationarity in the data generation process, we propose a novel robust SAA method exploiting information about the Wasserstein distance between the sequence of data-generating distributions and the actual chance constraint distribution. As a key result, we obtain distribution-free estimates of the sample size required to ensure that the robust SAA method will yield solutions that are feasible for the chance constraint under the actual distribution with high confidence.

Muhammed O. Sayin, Francesca Parise, Asuman Ozdaglar

We present a novel variant of fictitious play dynamics combining classical fictitious play with Q-learning for stochastic games and analyze its convergence properties in two-player zero-sum stochastic games. Our dynamics involves players forming beliefs on the opponent strategy and their own continuation payoff (Q-function), and playing a greedy best response by using the estimated continuation payoffs. Players update their beliefs from observations of opponent actions. A key property of the learning dynamics is that update of the beliefs on Q-functions occurs at a slower timescale than update of the beliefs on strategies. We show both in the model-based and model-free cases (without knowledge of player payoff functions and state transition probabilities), the beliefs on strategies converge to a stationary mixed Nash equilibrium of the zero-sum stochastic game.

Marco Avella-Medina, Francesca Parise, Michael T. Schaub et al.

As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known -- a premise not necessarily fulfilled for large, uncertain networks. Accordingly, centrality measures may fail to faithfully extract the importance of nodes in the presence of uncertainty. To mitigate these problems, we suggest a statistical approach based on graphon theory: we introduce formal definitions of centrality measures for graphons and establish their connections to classical graph centrality measures. A key advantage of this approach is that centrality measures defined at the modeling level of graphons are inherently robust to stochastic variations of specific graph realizations. Using the theory of linear integral operators, we define degree, eigenvector, Katz and PageRank centrality functions for graphons and establish concentration inequalities demonstrating that graphon centrality functions arise naturally as limits of their counterparts defined on sequences of graphs of increasing size. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score.

Francesca Parise, Sergio Grammatico, Basilio Gentile et al.

We consider network aggregative games to model and study multi-agent populations in which each rational agent is influenced by the aggregate behavior of its neighbors, as specified by an underlying network. Specifically, we examine systems where each agent minimizes a quadratic cost function, that depends on its own strategy and on a convex combination of the strategies of its neighbors, and is subject to personalized convex constraints. We analyze the best response dynamics and we propose alternative distributed algorithms to steer the strategies of the rational agents to a Nash equilibrium configuration. The convergence of these schemes is guaranteed under different sufficient conditions, depending on the matrices defining the cost and on the network. Additionally, we propose an extension to the network aggregative game setting that allows for multiple rounds of communications among the agents, and we illustrate how it can be combined with consensus theory to recover a solution to the mean field control problem in a distributed fashion, that is, without requiring the presence of a central coordinator. Finally, we apply our theoretical findings to study a novel multi-dimensional, convex-constrained model of opinion dynamics and a hierarchical demand-response scheme for energy management in smart buildings, extending literature results.

Sergio Grammatico, Francesca Parise, Marcello Colombino et al.

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.