Potential games in volatile environments

This papers studies the co-evolution of networks and play in the context of finite population potential games. Action revision, link creation and link destruction are combined in a continuous-time Markov process. I derive the unique invariant distribution of this process in closed form, as well as the marginal distribution over action profiles and the conditional distribution over networks. It is shown that the equilibrium interaction topology is an inhomogeneous random graph. Furthermore, a characterization of the set of stochastically stable states is provided, generalizing existing results to models with endogenous interaction structures.

Stochastic mirror descent dynamics and their convergence in monotone variational inequalities

We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences that respectively pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around this average.Comment: 23 pages; updated proofs in Section 3 and Section

Constrained Interactions and Social Coordination

We consider a co-evolutionary model of social coordination and network formation whereagents may decide on an action in a 2 x 2- coordination game and on whom to establish costly links to. We find that a payoff dominant convention is selected for a wider parameter range when agents may only support a limited number of links as compared to a scenario where agents are not constrained in their linking choice. The main reason behind this result is that constrained interactions create a tradeoff between the interactions an agent has and those he would rather have. Further, we discuss convex linking costs and provide suffcient conditions for the payoff dominant convention to be selected in mxm coordination games.

On a General class of stochastic co-evolutionary dynamics

This paper presents a unified framework to study the co-evolution of networks and play, using the language of evolutionary game theory. We show by examples that the set-up is rich enough to encompass many recent models discussed by the literature. We completely characterize the invariant distribution of such processes and show how to calculate stochastically stable states by means of a treecharacterization algorithm. Moreover, specializing the process a bit further allows us to completely characterize the generated random graph ensemble. This new result demonstrates a new and rather general relation between random graph theory and evolutionary models with endogenous interaction structures.

ON THE CONVERGENCE OF GRADIENT-LIKE FLOWS WITH NOISY GRADIENT INPUT

In view of solving convex optimization problems with noisy gradient input, we analyze the asymptotic behavior of gradient-like flows under stochastic disturbances. Specifically, we focus on the widely studied class of mirror descent schemes for convex programs with compact feasible regions, and we examine the dynamics' convergence and concentration properties in the presence of noise. In the vanishing noise limit, we show that the dynamics converge to the solution set of the underlying problem (a.s.). Otherwise, when the noise is persistent, we show that the dynamics are concentrated around interior solutions in the long run, and they converge to boundary solutions that are sufficiently "sharp". Finally, we show that a suitably rectified variant of the method converges irrespective of the magnitude of the noise (or the structure of the underlying convex program), and we derive an explicit estimate for its rate of convergence.Comment: 36 pages, 5 figures; revised proof structure, added numerical case study in Section

Random block coordinate methods for inconsistent convex optimisation problems

We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal gradient method, we establish global convergence of the last iterate as well optimal $O(1/k)$ and $O(1/k^{2})$ complexity rates in the convex and strongly convex case, respectively, $k$ being the iteration count. Motivated by the increased complexity in the control of distribution level electric power systems, we test the performance of our method on a second-order cone relaxation of an AC-OPF problem. Distributed control is achieved via the distributed locational marginal prices (DLMPs), which are obtained \revise{as} dual variables in our optimisation framework.Comment: Changed title and revised manuscrip

Hessian barrier algorithms for linearly constrained optimization problems

In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent (MD), and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a non-degeneracy condition, the algorithm converges to the problem's set of critical points; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is $\mathcal{O}(1/k^\rho)$ for some $\rho\in(0,1]$ that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard non-convex test functions and large-scale traffic assignment problems.Comment: 27 pages, 6 figure

Evolutionary dynamics and rationality

Es werden drei Modelle der dynamischen evolutionären Spieltheorie betrachtet. Von besonderem Interesse ist der Zusammenhang zwischen den Fixpunkten dieser dynamischen Systeme und dem Nash Gleichgewicht. Ein weiterer Teil der Arbeit beschäftigt sich mit der potentiellen Rolle evolutionärer Dynamiken in Spielen in extensiver Form, wenn mehrere Teilspielperfekte Gleichgewichte existieren.We discuss three models of dynamic evolutionary game theory. In particular we are interested in the relation between the fixed points of these dynamical systems and the Nash equilibrium of the underlying game. Another part of the work focuses on the potential role of evolutionary dynamics in games of extensive form when there are several subgame perfect equilibria

Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games

Co-evolutionary dynamics of networks and play

Diese Dissertation präsentiert drei inhaltlich zusammenhängende Artikel zu Ko-evolutionären Dynamiken von Netzwerken und Spielen. Ein Ko-evolutionärer Prozess besteht aus drei elementaren Ereignissen- Revision von gewählten Aktionen, Kreation eines links, Zerstörung eines links- welche zusammengefasst eine aggregierte Spieldynamik generieren. Unser Augenmerk liegt in der Charakterisierung des asymptotischen Verhaltens derartiger Prozesse. Kapitel 2, "On a general class of stochastic co-evolutionary dynamics", beschreibt den abstrakten mathematischen Rahmen eines Ko-evolutorischen Modells. Die Spieler sind charakterisiert durch eine Nutzenfunktionen auf einem gemeinsamen Aktionenraum und verwenden probabilistische Verhaltensregeln in den eingangs genannten Ereignissen. Zulässige Verhaltensregeln erfüllen eine Irreduzibilitätsannahme sowie ein Prinzip der großen Abweichungen. Neben diesen technischen Annahmen werden substantielle Annahmen an das Verhalten der Spieler auf ein Minimum reduziert. Dies generiert eine wohl definierte Markov-Kette, dessen langfristiges Verhalten durch Graphen-theoretische Methoden nach Freidlin und Wentzell [Random perturbations of dynamical systems, Springer, 1998] studiert werden kann. Wir beschreiben eine allgemeine Methode mit der stochastisch-stabile Zustände identifiziert werden können. Unter weiteren schwachen Annahmen lässt sich das induzierte Zufallsgraphemodell vollständig charakterisieren. Es zeigt sich eine äußerst interessante und unerwartete Beziehung zwischen Ko-evolutorischen Modellen und dem Modell der inhomogenen Zufallsgraphen. Kapitel 4 präsentiert den Artikel "Potential games played in volatile environments". Dort diskutieren wir ein Ko-evolutorisches Modell in der Klasse von Potentialspielen und Logit-Verhaltensregeln. Die invariante Verteilung sowie das generierte Zufallsgraphenmodell sind vollständig bestimmbar. Weiteres werden einige Statistiken des Zufallsgraphemodells in geschlossener Form präsentiert, wie etwa die ``degree distribution'' des Zufallsgraphen. Kapitel 5 erweitert dieses Modell durch Heterogenität in den Präferenzen der Spieler. Wir definieren eine neue Klasse von Spielen, ``structured Bayesian interaction games'', aufbauend auf einer jungen Literatur der evolutionären Spieltheorie die sich mit ``Bayesian population games'' beschäftigt. Kapitel 3 stellt eine Verbindung zwischen den Markov-Ketten von Kapitel 2, und den Markov-Prozessen der Kapitel 4 und 5 her. Ein abschließendes Kapitel resümiert die Ergebnisse der Dissertation und gibt Ausblicke für zukünftige Forschungsvorhaben.This dissertation presents three interrelated papers on the co-evolution of networks and play. The general structure of these models combines three elementary events- action adjustment, link creation and link destruction- to one stochastic game dynamics, and focuses on the asymptotic properties of these processes. Chapter 2 presents the general mathematical framework of a co-evolutionary model. The players have arbitrary utility functions, defined on a common set of actions, and employ probabilistic behavioral rules in the above mentioned events. Admissible rules satisfy irreducibility and a large deviations assumption. Beside these technical assumptions, we try to avoid making substantial behavioral assumptions. This generates a well-defined Markov chain, whose long-run properties can be studied analytically by making use of tree-characterization methods due to Freidlin and Wentzell [Random perturbations of dynamical systems, Springer, 1998]. We provide a general technique to compute stochastically stable states in such co-evolutionary models, by defining suitable cost-functions. Making some mild additional assumptions on the structure of the behavioral rules, we demonstrate an interesting and unforeseen connection between the derived ensemble of networks and inhomogeneous random graphs. The models presented in chapters 4 and 5 particularize the general framework to the class of potential games and behavioral rules of the logit-response form. Under these assumptions, chapter 4 gives a full description of the induced ensemble of networks, and provides additionally closed-form expression for statistics of this ensemble, such as the degree-distribution. The model presented in chapter 5 is more general by allowing the players to have idiosyncratic preferences. This leads us to the definition of structured Bayesian interaction games, following a recent literature of evolutionary game theory studying Bayesian population games. Chapter 3 establishes a connection between the Markov chain constructed in the general framework of chapter 2 and the continuous-time Markov processes considered in the models of chapters 4 and 5. Chapter 6 recapitulates the results of the thesis and gives an outlook for future research projects