Nash and Wardrop equilibria in aggregative games with coupling constraints

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.Comment: IEEE Trans. on Automatic Control (Accepted without changes). The first three authors contributed equall

Spectral algorithm for non-destructive damage localisation: Application to an ancient masonry arch model

Structural monitoring and vibration-based damage identification methods are fundamental tools for condition assessment and early-stage damage identification, especially when dealing with the conservation of historical constructions and the maintenance of strategic civil structures. However, although the substantial advances in the field, several issues must still be addressed to broaden the application range of such tools and to assert their reliability. This study deals with the experimental validation of a novel method for non-destructive damage identification purposes. This method is based on the use of spectral output signals and has been recently validated by the authors through a numerical simulation. After a brief insight into the basic principles of the proposed approach, the spectral-based technique is applied to identify the experimental damage induced on a masonry arch through statically increasing loading. Once the direct and cross spectral density functions of the nodal response processes are estimated, the system's output power spectrum matrix is built and decomposed in eigenvalues and eigenvectors. The present study points out how the extracted spectral eigenparameters contribute to the damage analysis allowing to detect the occurrence of damage and to locate the target points where the cracks appear during the experimental tests. The sensitivity of the spectral formulation to the level of noise in the modal data is investigated and discussed. As a final evaluation criterion, the results from the spectrum-driven method are compared with the ones obtained from existing non-model based damage identification methods.info:eu-repo/semantics/publishedVersio

Approximate solution to the reactive power flow and its application to voltage stability in microgrids

the work presents two main contributions. First, for sufficiently-high nsource voltages, we guarantee the existence of a high-voltage solution for the reactive power flow equations and provide its approximate analytical expression. We derive a bound on the approximation error and study its asymptotic behavior for large source voltages. Second, we apply our previous result to to a recently proposed droop control for voltage stabilizatio

Equilibria in aggregative games

This thesis studies equilibrium problems in aggregative games. A game describes the interaction among selfish rational agents, each of them choosing his strategy to optimize his own cost function, which depends also on the strategies of the other agents. In particular, the thesis focuses on aggregative games, where the cost of each agent is a sole function of his strategy and of the average agents’ strategy. Not only such class of games can model a wide spectrum of applications, ranging from traffic or transmission networks to electricity or commodity markets, but it also lends itself to an elegant mathematical analysis. The first part of the thesis investigates the relation between Nash and Wardrop equilibria, which are two classical concepts in game theory. Thanks to the powerful framework of variational inequalities, we derive bounds on the distance between the two equilibria and use them to show that the agents’ strategies at the Nash equilibrium converge to those at the Wardrop equilibrium, when the number of agents grows to infinity. Moreover, we propose novel sufficient conditions to guarantee uniqueness of the Nash equilibrium for a specific aggregative game, which is often used in applications. The second part of the thesis is dedicated to the design of algorithms that converge to Nash equilibrium and to Wardrop equilibrium in presence of constraints coupling the agents’ strategies. Due to privacy issues and to the large number of agents at hand in real-life applications, centralized solutions might not be desirable. Hence, we first propose two parallel algorithms, where a central operator gathers and broadcasts aggregate information to coordinate the computations carried out by the agents. Then we design a distributed algorithm that only relies on local communications among the agents. We test the proposed algorithms in three case studies, where we also numerically verify the results of the first part of the thesis. The last part of the thesis introduces the novel concept of equilibrium with inertia. Both classical Nash and Wardrop equilibria assume that each agent has the flexibility to change his strategy whenever this leads to an improvement. In some applications, however, this hypothesis is not realistic. We show that introducing an inertial coefficient which penalizes action switches leads to a richer set of equilibria, which is however in general not convex. Since classical algorithms for Nash and Wardrop equilibria cannot be used in presence of the inertial coefficients, we propose natural agents dynamics and guarantee their convergence to an equilibrium with inertial coefficients

A Distributed Algorithm For Almost-Nash Equilibria of Average Aggregative Games With Coupling Constraints

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 a shared constraint coupling their strategies. We propose a distributed algorithm that achieves an ε -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. We apply our theoretical findings to a multimarket Cournot game with transportation costs and maximum market capacity. © 2019 IEEE.ISSN:2325-587

Mean Field Modeling of Large-Scale Energy Systems

This work proposes mean field game-type models for two instances of large- scale energy systems, namely plug-in electric vehicles and thermostatically controlled loads. Theoretical and numerical analysis show that both systems possess an equilibrium configuration which is optimal for the individuals and beneficial for the overall population

The Nash Equilibrium With Inertia in Population Games

In the traditional game-theoretic set up, where agents select actions and experience corresponding utilities, a Nash equilibrium is a configuration where no agent can improve their utility by unilaterally switching to a different action. In this article, we introduce the novel notion of inertial Nash equilibrium to account for the fact that in many practical situations switching action does not come for free. Specifically, we consider a population game and introduce the coefficients c(ij) describing the cost an agent incurs by switching from action i to action j. We define an inertial Nash equilibrium as a distribution over the action space where no agent benefits in switching to a different action, while taking into account the cost of such switch. First, we show that the set of inertial Nash equilibria contains all the Nash equilibria, is in general nonconvex, and can be characterized as a solution to a variational inequality. We then argue that classical algorithms for computing Nash equilibria cannot be used in the presence of switching costs. Finally, we propose a better-response dynamics algorithm and prove its convergence to an inertial Nash equilibrium. We apply our results to study the taxi drivers' distribution in Hong Kong.ISSN:0018-9286ISSN:1558-252