Stability and Bayesian Consistency in Two-Sided Marketsi Thank Navin
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Lectures on Nonparametric Bayesian Statistics
Lectures on Nonparametric Bayesian Statistics Notes for the course by Bas Kleijn, Aad van der Vaart, Harry van Zanten (Text partly extracted from a forthcoming book by S. Ghosal and A. van der Vaart) version 4-12-2012 UNDER CONSTRUCTION 1 Introduction Why adopt the nonparametric Bayesian approach for inference? The answer lies in the si- multaneous preference for nonparametric modeling and desire to follow a Bayesian proce- dure. Nonparametric (and semiparametric) models can avoid the arbitrary and possibly un- verifiable assumptions inherent in parametric models. Bayesian procedures may be desired for the conceptual simplicity of the Bayesian paradigm, easy interpretability of Bayesian quantities or philosopohical reasons. 1.1 Motivation Bayesian nonparametrics is the study of Bayesian inference methods for nonparametric and semiparametric models. In the Bayesian nonparametric paradigm a prior distribution is assigned to all unknown quantities (parameters) involved in the modeling, whether finite or infinite dimensional. Inference is made from the “posterior distribution”, the conditional distribution of all parameters given the data. A model completely specifies the conditional distribution of all observable given all unobserved quantities, or parameters, while a prior distribution specifies the distribution of all unobservables. From this point of view, random effects and latent variables also qualify as parameters, and distributions of these quantities, often considered as part of the model itself from the classical point of view, are considered part of the prior. The posterior distribution involves an inversion of the order of conditioning. Existence of a regular version of the posterior is guaranteed under mild conditions on the relevant spaces (see Section 2). -
Equilibrium Refinements
Equilibrium Refinements Mihai Manea MIT Sequential Equilibrium I In many games information is imperfect and the only subgame is the original game. subgame perfect equilibrium = Nash equilibrium I Play starting at an information set can be analyzed as a separate subgame if we specify players’ beliefs about at which node they are. I Based on the beliefs, we can test whether continuation strategies form a Nash equilibrium. I Sequential equilibrium (Kreps and Wilson 1982): way to derive plausible beliefs at every information set. Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 2 / 38 An Example with Incomplete Information Spence’s (1973) job market signaling game I The worker knows her ability (productivity) and chooses a level of education. I Education is more costly for low ability types. I Firm observes the worker’s education, but not her ability. I The firm decides what wage to offer her. In the spirit of subgame perfection, the optimal wage should depend on the firm’s beliefs about the worker’s ability given the observed education. An equilibrium needs to specify contingent actions and beliefs. Beliefs should follow Bayes’ rule on the equilibrium path. What about off-path beliefs? Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 3 / 38 An Example with Imperfect Information Courtesy of The MIT Press. Used with permission. Figure: (L; A) is a subgame perfect equilibrium. Is it plausible that 2 plays A? Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 4 / 38 Assessments and Sequential Rationality Focus on extensive-form games of perfect recall with finitely many nodes. An assessment is a pair (σ; µ) I σ: (behavior) strategy profile I µ = (µ(h) 2 ∆(h))h2H: system of beliefs ui(σjh; µ(h)): i’s payoff when play begins at a node in h randomly selected according to µ(h), and subsequent play specified by σ. -
Lecture Notes
GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies .................................. -
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University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/58404 This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. Library Declaration and Deposit Agreement 1. STUDENT DETAILS Please complete the following: Full name: …………………………………………………………………………………………….Gui Pedro Araújo de Mendonça University ID number: ………………………………………………………………………………0759252 2. THESIS DEPOSIT 2.1 I understand that under my registration at the University, I am required to deposit my thesis with the University in BOTH hard copy and in digital format. The digital version should normally be saved as a single pdf file. 2.2 The hard copy will be housed in the University Library. The digital version will be deposited in the University’s Institutional Repository (WRAP). Unless otherwise indicated (see 2.3 below) this will be made openly accessible on the Internet and will be supplied to the British Library to be made available online via its Electronic Theses Online Service (EThOS) service. [At present, theses submitted for a Master’s degree by Research (MA, MSc, LLM, MS or MMedSci) are not being deposited in WRAP and not being made available via EthOS. This may change in future.] 2.3 In exceptional circumstances, the Chair of the Board of Graduate Studies may grant permission for an embargo to be placed on public access to the hard copy thesis for a limited period. -
Characterizing Solution Concepts in Terms of Common Knowledge Of
Characterizing Solution Concepts in Terms of Common Knowledge of Rationality Joseph Y. Halpern∗ Computer Science Department Cornell University, U.S.A. e-mail: [email protected] Yoram Moses† Department of Electrical Engineering Technion—Israel Institute of Technology 32000 Haifa, Israel email: [email protected] March 15, 2018 Abstract Characterizations of Nash equilibrium, correlated equilibrium, and rationaliz- ability in terms of common knowledge of rationality are well known (Aumann 1987; arXiv:1605.01236v1 [cs.GT] 4 May 2016 Brandenburger and Dekel 1987). Analogous characterizations of sequential equi- librium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using results of Halpern (2009, 2013). ∗Supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, by ONR un- der grant N00014-02-1-0455, by the DoD Multidisciplinary University Research Initiative (MURI) pro- gram administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grants F49620-02-1-0101 and FA9550-05-1-0055. †The Israel Pollak academic chair at the Technion; work supported in part by Israel Science Foun- dation under grant 1520/11. 1 Introduction Arguably, the major goal of epistemic game theory is to characterize solution concepts epistemically. Characterizations of the solution concepts that are most commonly used in strategic-form games, namely, Nash equilibrium, correlated equilibrium, and rational- izability, in terms of common knowledge of rationality are well known (Aumann 1987; Brandenburger and Dekel 1987). We show how to get analogous characterizations of sequential equilibrium (Kreps and Wilson 1982), (trembling hand) perfect equilibrium (Selten 1975), and quasi-perfect equilibrium (van Damme 1984) for arbitrary n-player games, using results of Halpern (2009, 2013). -
Robust Numerical Bayesian Unit Root Test for Model
ROBUST NUMERICAL BAYESIAN UNIT ROOT TEST FOR MODEL UNCERTAINTY by XUEDONG WU (Under the Direction of Jeffrey H. Dorfman) ABSTRACT Unit root testing is an important procedure when performing time series analysis, since all the succeeding inferences should be performed based on a stationary series. Thus, it is crucial to test the stationarity of the time series in hand accurately and efficiently. One issue among the existing popular unit root testing methods is that they all require certain assumptions about the model specification, whether on the functional form or the stochastic term distribution; then all the analyses are performed based on the pre- determined model. However, various circumstances such as data incompleteness, variable selection, and distribution misspecification which may lead to an inappropriate model specification that will produce an erroneous conclusion since the test result depends on the particular model considered. This dissertation focuses on confronting this issue by proposing a new numerical Bayesian unit root test incorporating model averaging which can take model uncertainty as well as variable transformation into account. The first chapter introduces a broad literature review of all the building blocks need for the development of the new methods, including traditional frequentist unit root tests, Bayesian unit root tests, and Bayesian model averaging. Following chapter II elaborates the mathematical derivation of the proposed methods, Monte Carlo simulation study and results, as well as testing conclusions on the benchmark Nelson and Plosser (1982) macroeconomic time series. Chapter III applies the method to investigate the effect of data frequency on unit root test results particularly for financial data. -
Bayesian Treatments to Panel Data Models with an Application To
Bayesian Treatments to Panel Data Models with an Application to Models of Productivity Junrong Liu Department of Economics/Rice University Robin Sickles Department of Economics/Rice University E. G. Tsionas University of Athens This Draft November 25, 2013 Abstract This paper considers two models for uncovering information about technical change in large heterogeneous panels. The first is a panel data model with nonparametric time effects. Second, we consider a panel data model with common factors whose number is unknown and their effects are firm-specific. This paper proposes a Bayesian approach to estimate the two models. Bayesian inference techniques organized around MCMC are applied to implement the models. Monte Carlo experiments are performed to examine the finite-sample performance of this approach, which dominates a variety of estimators that rely on parametric assumptions. In order to illustrate the new method, the Bayesian approach has been applied to the analysis of efficiency trends in the U.S. largest banks using a dataset based on the Call Report data from FDIC over the period from 1990 to 2009. Keywords: Panel data, time-varying heterogeneity, Bayesian econometrics, banking studies, productivity JEL Classification: C23; C11; G21; D24 1 1. Introduction In this paper, we consider two panel data models with unobserved heterogeneous time-varying effects, one with individual effects treated as random functions of time, the other with common factors whose number is unknown and their effects are firm-specific. This paper has two distinctive features and can be considered as a generalization of traditional panel data models. Firstly, the individual effects that are assumed to be heterogeneous across units as well as to be time varying are treated nonparametrically, following the spirit of the model from Bai (2009) and Kneip et al. -
Acebayes: an R Package for Bayesian Optimal Design of Experiments Via Approximate Coordinate Exchange
acebayes: An R Package for Bayesian Optimal Design of Experiments via Approximate Coordinate Exchange Antony M. Overstall David C. Woods Maria Adamou University of Southampton University of Southampton University of Southampton Abstract We describe the R package acebayes and demonstrate its use to find Bayesian optimal experimental designs. A decision-theoretic approach is adopted, with the optimal design maximising an expected utility. Finding Bayesian optimal designs for realistic problems is challenging, as the expected utility is typically intractable and the design space may be high-dimensional. The package implements the approximate coordinate exchange algo- rithm to optimise (an approximation to) the expected utility via a sequence of conditional one-dimensional optimisation steps. At each step, a Gaussian process regression model is used to approximate, and subsequently optimise, the expected utility as the function of a single design coordinate (the value taken by one controllable variable for one run of the experiment). In addition to functions for bespoke design problems with user-defined utility functions, acebayes provides new functions tailored to finding designs for common generalised linear and nonlinear models. The package provides a step-change in the com- plexity of problems that can be addressed, enabling designs to be found for much larger numbers of variables and runs than previously possible. We provide tutorials on the ap- plication of the methodology for four illustrative examples of varying complexity where designs are found for the goals of parameter estimation, model selection and prediction. These examples demonstrate previously unseen functionality of acebayes. Keywords: A-optimality, computer experiments, D-optimality, decision-theoretic design, Gaus- sian process regression, generalised linear models, high-dimensional design, model selection, nonlinear models, prediction, pseudo-Bayesian design. -
TESIS DOCTORAL Three Essays on Game Theory
UNIVERSIDAD CARLOS III DE MADRID TESIS DOCTORAL Three Essays on Game Theory Autor: José Carlos González Pimienta Directores: Luis Carlos Corchón Francesco De Sinopoli DEPARTAMENTO DE ECONOMÍA Getafe, Julio del 2007 Three Essays on Game Theory Carlos Gonzalez´ Pimienta To my parents Contents List of Figures iii Acknowledgments 1 Chapter 1. Introduction 3 Chapter 2. Conditions for Equivalence Between Sequentiality and Subgame Perfection 5 2.1. Introduction 5 2.2. Notation and Terminology 7 2.3. Definitions 9 2.4. Results 12 2.5. Examples 24 2.6. Appendix: Notation and Terminology 26 Chapter 3. Undominated (and) Perfect Equilibria in Poisson Games 29 3.1. Introduction 29 3.2. Preliminaries 31 3.3. Dominated Strategies 34 3.4. Perfection 42 3.5. Undominated Perfect Equilibria 51 Chapter 4. Generic Determinacy of Nash Equilibrium in Network Formation Games 57 4.1. Introduction 57 4.2. Preliminaries 59 i ii CONTENTS 4.3. An Example 62 4.4. The Result 64 4.5. Remarks 66 4.6. Appendix: Proof of Theorem 4.1 70 Bibliography 73 List of Figures 2.1 Notation and terminology of finite extensive games with perfect recall 8 2.2 Extensive form where no information set is avoidable. 11 2.3 Extensive form where no information set is avoidable in its minimal subform. 12 2.4 Example of the use of the algorithm contained in the proof of Proposition 2.1 to generate a game where SPE(Γ) = SQE(Γ). 14 6 2.5 Selten’s horse. An example of the use of the algorithm contained in the proof of proposition 2.1 to generate a game where SPE(Γ) = SQE(Γ). -
Arxiv:1503.04493V1 [Math.ST] 16 Mar 2015 a Nsbaia Tpictnwiepr Fti Okwsfina Was Work This of Part Hospitality
Semiparametric Bernstein-von Mises Theorem: Second Order Studies Yun Yang∗, Guang Cheng† and David B. Dunson‡ March 17, 2015 Abstract The major goal of this paper is to study the second order frequentist properties of the marginal posterior distribution of the parametric component in semiparametric Bayesian mod- els, in particular, a second order semiparametric Bernstein-von Mises (BvM) Theorem. Our first contribution is to discover an interesting interference phenomenon between Bayesian esti- mation and frequentist inferential accuracy: more accurate Bayesian estimation on the nuisance function leads to higher frequentist inferential accuracy on the parametric component. As the second contribution, we propose a new class of dependent priors under which Bayesian inference procedures for the parametric component are not only efficient but also adaptive (w.r.t. the smoothness of nonparametric component) up to the second order frequentist validity. However, commonly used independent priors may even fail to produce a desirable root-n contraction rate for the parametric component in this adaptive case unless some stringent assumption is imposed. Three important classes of semiparametric models are examined, and extensive simulations are also provided. Key words: Bernstein-von Mises theorem; second order asymptotics; semiparametric model. 1. Introduction A semiparametric model is indexed by a Euclidean parameter of interest θ Θ Rp and an ∈ ⊂ infinite-dimensional nuisance function η belonging to a Banach space . For example, in the Cox H proportional hazards model, θ is a regression covariate vector corresponding to the log hazard ratio, while η is a cumulative hazard function. By introducing a joint prior Π on the product space Θ , we can make Bayesian inferences for the parameter of interest, e.g., credible set, ×H through MCMC sampling from the marginal posterior distribution. -
Sequential Preference Revelation in Incomplete Information Settings”
Online Appendix for “Sequential preference revelation in incomplete information settings” † James Schummer∗ and Rodrigo A. Velez May 6, 2019 1 Proof of Theorem 2 Theorem 2 is implied by the following result. Theorem. Let f be a strategy-proof, non-bossy SCF, and fix a reporting order Λ ∆(Π). Suppose that at least one of the following conditions holds. 2 1. The prior has Cartesian support (µ Cartesian) and Λ is deterministic. 2 M 2. The prior has symmetric Cartesian support (µ symm Cartesian) and f is weakly anonymous. 2 M − Then equilibria are preserved under deviations to truthful behavior: for any N (σ,β) SE Γ (Λ, f ), ,µ , 2 h U i (i) for each i N, τi is sequentially rational for i with respect to σ i and βi ; 2 − (ii) for each S N , there is a belief system such that β 0 ⊆ N ((σ S ,τS ),β 0) SE β 0(Λ, f ), ,µ . − 2 h U i Proof. Let f be strategy-proof and non-bossy, Λ ∆(Π), and µ Cartesian. Since the conditions of the two theorems are the same,2 we refer to2 arguments M ∗Department of Managerial Economics and Decision Sciences, Kellogg School of Manage- ment, Northwestern University, Evanston IL 60208; [email protected]. †Department of Economics, Texas A&M University, College Station, TX 77843; [email protected]. 1 made in the proof of Theorem 1. In particular all numbered equations refer- enced below appear in the paper. N Fix an equilibrium (σ,β) SE Γ (Λ, f ), ,µ . -
Evolutionary Game Theory: a Renaissance
games Review Evolutionary Game Theory: A Renaissance Jonathan Newton ID Institute of Economic Research, Kyoto University, Kyoto 606-8501, Japan; [email protected] Received: 23 April 2018; Accepted: 15 May 2018; Published: 24 May 2018 Abstract: Economic agents are not always rational or farsighted and can make decisions according to simple behavioral rules that vary according to situation and can be studied using the tools of evolutionary game theory. Furthermore, such behavioral rules are themselves subject to evolutionary forces. Paying particular attention to the work of young researchers, this essay surveys the progress made over the last decade towards understanding these phenomena, and discusses open research topics of importance to economics and the broader social sciences. Keywords: evolution; game theory; dynamics; agency; assortativity; culture; distributed systems JEL Classification: C73 Table of contents 1 Introduction p. 2 Shadow of Nash Equilibrium Renaissance Structure of the Survey · · 2 Agency—Who Makes Decisions? p. 3 Methodology Implications Evolution of Collective Agency Links between Individual & · · · Collective Agency 3 Assortativity—With Whom Does Interaction Occur? p. 10 Assortativity & Preferences Evolution of Assortativity Generalized Matching Conditional · · · Dissociation Network Formation · 4 Evolution of Behavior p. 17 Traits Conventions—Culture in Society Culture in Individuals Culture in Individuals · · · & Society 5 Economic Applications p. 29 Macroeconomics, Market Selection & Finance Industrial Organization · 6 The Evolutionary Nash Program p. 30 Recontracting & Nash Demand Games TU Matching NTU Matching Bargaining Solutions & · · · Coordination Games 7 Behavioral Dynamics p. 36 Reinforcement Learning Imitation Best Experienced Payoff Dynamics Best & Better Response · · · Continuous Strategy Sets Completely Uncoupled · · 8 General Methodology p. 44 Perturbed Dynamics Further Stability Results Further Convergence Results Distributed · · · control Software and Simulations · 9 Empirics p.