Extremal Choice Equilibrium: Existence and Purification with Infinite-Dimensional Externalities

We prove existence and purification results for equilibria in which players choose extreme points of their feasible actions in a class of strategic environments exhibiting a product structure. We assume finite-dimensional action sets and allow for infinite-dimensional externalities. Applied to large games, we obtain existence of Nash equilibrium in pure strategies while allowing a continuum of groups and general dependence of payoffs on average actions across groups, without resorting to saturated measure spaces. Applied to games of incomplete information, we obtain a new purification result for Bayes-Nash equilibria that permits substantial correlation across types, without assuming conditional independence given the realization of a finite environmental state. We highlight our results in examples of industrial organization, auctions, and voting.

Economies and Games with Many Agents

Ph.DDOCTOR OF PHILOSOPH

Universal Regular Conditional Distributions

We introduce a general framework for approximating regular conditional distributions (RCDs). Our approximations of these RCDs are implemented by a new class of geometric deep learning models with inputs in $\mathbb{R}^d$ and outputs in the Wasserstein-$1$ space $\mathcal{P}_1(\mathbb{R}^D)$. We find that the models built using our framework can approximate any continuous functions from $\mathbb{R}^d$ to $\mathcal{P}_1(\mathbb{R}^D)$ uniformly on compacts, and quantitative rates are obtained. We identify two methods for avoiding the "curse of dimensionality"; i.e.: the number of parameters determining the approximating neural network depends only polynomially on the involved dimension and the approximation error. The first solution describes functions in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$ which can be efficiently approximated on any compact subset of $\mathbb{R}^d$. Conversely, the second approach describes sets in $\mathbb{R}^d$, on which any function in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$ can be efficiently approximated. Our framework is used to obtain an affirmative answer to the open conjecture of Bishop (1994); namely: mixture density networks are universal regular conditional distributions. The predictive performance of the proposed models is evaluated against comparable learning models on various probabilistic predictions tasks in the context of ELMs, model uncertainty, and heteroscedastic regression. All the results are obtained for more general input and output spaces and thus apply to geometric deep learning contexts.Comment: Keywords: Universal Regular Conditional Distributions, Geometric Deep Learning, Measure-Valued Neural Networks, Conditional Expectation, Uncertainty Quantification. Additional Information: 27 Pages + 22 Page Appendix, 7 Table

An Analytic Approach to the Structure and Composition of General Learning Problems

Gowers presents, in his 2000 essay "The Two Cultures of Mathematics", two kinds of mathematicians he calls the theory-builders and problem-solvers. Of course both kinds of research are important; theory building may directly lead to solutions to problems, and by studying individual problems one uncovers the general structures of problems themselves. However, referencing a remark of Atiyah, Gowers observes that because so much research is produced, the results that can be ``organised coherently and explained economically'' will be the ones that last. Unlike mathematics, the field of machine learning abounds in problem-solvers --- this is wonderful as it leads to a large number of problems being solved --- but it is with regard to the point of Gowers that we are motivated to develop an appropriately general analytic framework to study machine learning problems themselves. To do this we first locate and develop the appropriate analytic objects to study. Chapter 2 recalls some concepts and definitions from the theory of topological vector spaces. In particular, the families of radiant and co-radiant sets and dualities. In Chapter 4 we will need generalisations of a variety of existing results on these families, and these are presented in Chapter 3. Classically a machine learning problem involves four quantities: an outcome space, a family of predictions (or model), a loss function, and a probability distribution. If the loss function is sufficiently general we can combine it with the set of predictions to form a set of real functions, which under very general assumptions, turns out to be closed, convex, and in particular, co-radiant. With the machinery of the previous two chapters in place, in Chapter 4 we lay out the foundations for an analytic theory of the classical machine learning problem, including a general analysis of link functions, by which we may rewrite almost any loss function as a scoring rule; a discussion of scoring rules and their properisation; and using the co-radiant results from Chapter 3 in particular, a theory of prediction aggregation. Chapters 5 and 6 develop results inspired by and related to adversarial learning. Chapter 5 develops a theory of boosted density estimation with strong convergence guarantees, where density updates are computed by training a classifier, and Chapter 6 uses the theory of optimal transport to formulate a robust Bayes minimisation problem, in which we develop a universal theory of regularisation and deliver new strong results for the problem of adversarial learning

Research in the general area of non-linear dynamical systems Final report, 8 Jun. 1965 - 8 Jun. 1967

Nonlinear dynamical systems research on systems stability, invariance principles, Liapunov functions, and Volterra and functional integral equation

ESSAYS ON GAME THEORY

Ph.DDOCTOR OF PHILOSOPH

Algebraic Topology for Data Scientists

This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been exposed to it, let alone data scientists, computer scientists, and analysts. I have three goals in writing this book. The first is to bring people up to speed who are missing a lot of the necessary background. I will describe the topics in point-set topology, abstract algebra, and homology theory needed for a good understanding of TDA. The second is to explain TDA and some current applications and techniques. Finally, I would like to answer some questions about more advanced topics such as cohomology, homotopy, obstruction theory, and Steenrod squares, and what they can tell us about data. It is hoped that readers will acquire the tools to start to think about these topics and where they might fit in.Comment: 322 pages, 69 figures, 5 table

Essays on the history of dynamic economic analysis

The subsequent three studies in the history of economic analysis ranges over a wide area of subjects. Part one, raises the following questions: To what extent is axiomatic general equilibrium analysis a rational reconstruction of ?Scottish Political Economy? as defined by the writings of David Hume and Adam Smith? How much is gained and how much lost by the axiomatic transformation of the invisible-hand proposition? What are the implications of negative results like the Sonnenschein-Mantel-Debreu demonstrations for the Scottish point of view? Did it reach deadlock, or is there still hope for the dominant trajectory in the history of economics? In contrast to the rich historical literature on the invisible-hand proposition, the present study does not level any paradigmatic criticism at neo-Walrasian analysis. Rather, by focalizing the most important results against the backdrop of Scottish Political Economy, it may inform theory choice within the neo-Walrasian paradigm. Part two translates F.A. Hayek?s informal capital theory into a dynamic equilibrium model. The focus is restricted to Hayek?s largely unrecognized contribution in Utility Analysis and Interest, being restated in The Pure Theory of Capital. The underlying premise is that Hayek adopts infant versions of modern analytical tools during his time at the London School of Economics such that a rational reconstruction of his capital theory by established neoclassical tools is admissible. The major result is that UAI and PTC contain a generalization of the Ramsey-Cass-Koopmans model. In concrete, Hayek provides the solution to an infinite-horizon deterministic social planner optimization problem in a one-sector economy such that the rate of pure time preference encapsulated in the discount factor increases in prospective utility. The endogeneity of myopia is due to intertemporal complementarities and accounted for by the modified Uzawa aggregator. A partial alliance with Frank Knight is established. Part three addresses Ludwig von Mises?s business cycle theory at maturity, as advanced in his opus magnum Human Action. In this work, Mises embeds the business cycle theory which he initially developed in Theorie des Geldes und der Umlaufmittel into the broad context of his methodological convictions. Whereas the initial outline of his cycle theory strongly relies on Böhm-Bawerk?s capital theory, its mature version is built upon a significantly altered framework of real analysis. The paper describes and evaluates the impact of Mises?s praxeology on his conceptualization of real analysis; it provides a simple model to depict and clarify Mises?s outline; it draws implications for his business cycle theory and its core prediction that ?any money-induced traverse by necessity reverses?; it argues that Mises?s core prediction ultimately depends on his barren analytical device; it concludes that Mises?s mature business cycle theory is a regression.Die folgenden drei theoriegeschichtlichen Studien umfassen eine allgemeine Auseinandersetzung mit der Entwicklung dynamischer Analyse in der Wirtschaftswissenschaft. Teil 1 stellt folgende Fragen: Inwieweit ist die axiomatische allgemeine Gleichgewichtsanalyse eine rationale Rekonstruktion der Schottischen Nationalökonomie, definiert durch die werke von David Hume und Adam Smith? War die axiomatische Transformation der Proposition der Unsichtbaren Hand ein Erfolg? Was folgt aus negativen Resultaten wie den Sonnenschein-Mantel-Debreu-Theoreme für die Schottische Nationalökonomie. Teil 2 übersetzt F.A. Hayek's informelle Kapitaltheorie in ein dynamisches Gleichgewichtsmodell. Der fokus beschränkt sich dabei auf seine Beiträge in "Utility Analysis and Interest" sowie "The Pure Theory of Capital". Das Hauptergebnis lautet, dass Hayek eine Verallgemeinerung des Ramsey-Cass-Koopmans -Modells formulierte. Er untersucht dabei das Optimierungproblem eines sozialen Planers über einen unendlichen Horizont bei endogenem Diskontfaktor. Die Rate der reinen Zeitpräferenzrate steigt im erwarteten Nutzen. Es werden Existenz, Eindeutigkeit und Stabilität des Hayek-Systems untersucht und theoriegeschichtlich eingebunden. Teil 3 untersucht die Weiterentwicklung der Österreichischen Konjunkturtheorie durch Ludwig von Mises in dessen Hauptwerk "Human Action". Es werden der Einfluss seiner Erkenntnistheorie auf seine kapitaltheoretischen Argumente untersucht. Im Ergebnis kommt die Arbeit zu dem Schluss, dass die Weiterentwicklung der Österreichischen Konjunkturtheorie einen Rückschritt darstellt, insbesondere relativ zu den Arbeiten von Hayek

Obstacle type problems in Orlicz-Sobolev spaces

Tese de doutoramento, Matemática (Física Matemática e Mecânica dos Meios Contínuos), Universidade de Lisboa, Faculdade de Ciências, 2013This thesis consists of four chapters. In the first chapter we study the regularity of solutions for a class of elliptic problems in Orlicz-Sobolev spaces. In particular, we see that bounded weak solutions of Au := div a(x; jruj)ru _ = f(x); x 2 ; where Ω ⊂ Rn is a bounded domain, for an appropriate a and f are C1 α regular. Using Lewy-Stampacchia inequalities for one obstacle problem we derive C1 α regularity results (both locally and up to the boundary) for the solution of a quasilinear obstacle problem. In the second chapter we prove Lewy-Stampacchia inequalities in abstract form for two obstacles problem and for N-membranes problem. Applying those inequalities we derive C1 α regularity results (both locally and up to the boundary) for A(x)-obstacle problem with two obstacles and for N-membranes problem. As another application of Lewy-Stampacchia inequalities, we study a quasivariational problem related to a stochastic switching game. We prove, that the problem admits at least a maximal and a minimal solution. In the third chapter we extend the regularity of the free boundary of the obstacle problem to a class of heterogeneous quasilinear degenerate elliptic operators (including p(x)-Laplacian). We prove that the free boundary is a porous set and hence has Lebesgue measure zero. We also show that the (n - 1)-dimensional Hausdorff measure of the free boundary is finite (for p(x) > 2), which yields, in particular, that up to a negligible singular set, the free boundary is the union of at most a countable family of C1 hypersurfaces. Finally, in the chapter four of the thesis, after homogenizing the Dirichlet problem for A(x)-Laplacian in Orlicz-Sobolev spaces, we study the homogenization of the A(x)-obstacle problem, then prove convergence of the coincidence sets.Fundação para a Ciência e a Tecnologia (FCT, SFRH/BD/40819/2007