Lecture 15. Ranking Payoff Distributions: Stochastic Dominance. FirstOrder Stochastic Dominance: higher distribution


 Lawrence Wood
 2 years ago
 Views:
Transcription
1 Lecture 15 Ranking Payoff Distributions: Stochastic Dominance FirstOrder Stochastic Dominance: higher distribution Definition 6.D.1: The distribution F( ) firstorder stochastically dominates G( ) if for every nondecreasing function u : we have u(x)df(x) u(x)dg(x) Proposition 6.D.1: The distribution of monetary payoffs F( ) first order stochastically dominates the distribution G( ) if and only if F(x) G(x) for every x. Proof: First show that stochastic dominance implies that F(x) G(x) for every x. Use proof by contradiction. Assume that F( ) stochastically dominates G( ) but that for some value of x denoted Define the nondecreasing function u(x), where u(x) = 1 for all and u(x) = 0 otherwise. We know and But if contradiction. then u(x)dg(x) > u(x)df(x), a
2 And the other direction: Assume F(x) G(x) for all x, and show that stochastic dominance follows. u(x)df(x) = = u(x)dg(x) + u(x)(df(x)  dg(x)) = = u(x)dg(x) + u(x)d(f(x)  G(x)) Let H(x) = F(x)  G(x), so we need to know if u(x)dh(x) 0 for all nondecreasing functions u(x). To do this, we use integration by parts: but H(o) = 0 and limx H(x) = 0 so that The second term is negative if H(x) 0 everywhere, which is true under the maintained assumption.
3 Second Order Stochastic Dominance: riskier distribution Definition 6.D.1: For any two distributions F( ) and G( ) with the same mean, F( ) secondorder stochastically dominates (or is less risky than) G( ) if for every nondecreasing concave function + u : we have u(x)df(x) u(x)dg(x) Other definition: the variable y is a meanpreserving spread of x, if y = x + z where zdh(z) = 0. Proposition 6.D.2: Consider two distributions F( ) and G( ) with the same mean. Then the following statements are equivalent: (1) F( ) secondorder stochastically dominates G( ) (2) G( ) is a meanpreserving spread of F( ) (3) for all x
4 Demonstration that (2) implies (1). If G( ) is a mean preserving spread of F( ), then u(x)dg(x) = u(x + z)dh(z)df(x) but since zdh(z) = 0 (and xdh(z) = x) u(x)df(x) = u( (x + z)dh(z))df(x) by Jensen s inequality the concavity of u( ) implies that u(x)df(x) > u(x)dg(x)
5 Rabin Critique: (taken from Risk Aversion by M. Rabin and R. Thaler, J. Econ. Perspectives, Winter 2001) Suppose we know that Johnny is a riskaverse expected utility maximizer, and that he will always turn down the gamble of losing $10 or gaining $11. What else can we say about Johnny? Specifically, can we say anything about bets Johnny will be willing to accept in which there is a 50 percent chance of losing $100 and a 50 percent chance of winning some amount $Y? Answer: Johnny will reject the bet no matter what Y is. The logic behind this result is that within the expected utility framework, turning down a moderate stakes gamble means that the marginal utility of money must diminish very quickly. Suppose that you have initial wealth of W, and you reject a lose $10/gain $11 gamble because of diminishing marginal utility of wealth. Then it must be that U(W + 11)  U(W) U(W)  U(W  10). Hence, on average you value each of the dollars between W and W + 11 by at most 10/11 as much as you, on average, value each of the dollars between W  10 and W. By concavity, this implies that you value the dollar W + 11 at most 10/11 as much as you value the dollar W Iterating this observation, if you have the same aversion to the lose $10/gain $11 bet at wealth level W + 21, then you value dollar W = W + 32 by at most 10/11 as you value dollar W = W + 11, which means you value dollar W + 32 by at most 10/11 x 10/11 5/6 as much as dollar W You will th value the W dollar by at most 40 percent as much as th dollar W  10, and the W dollar by at most 2 percent as
6 much as dollar W In words, rejecting the lose $10/gain $11 gamble implies a 10 percent decline in marginal utility for each $21 in additional lifetime wealth, meaning that the marginal utility plummets for substantial changes in lifetime wealth. You care less than 2 percent as much about an additional dollar when you are $900 wealthier than you are now. This rate of deterioration for the value of money is absurdly high, and hence leads to absurd risk aversion. Rubinstein Response: (from Rubinstein Lecture Notes in Microeconomic Theory) Nevertheless, in the economic literature it is usually assumed that a decision maker s preferences over wealth changes are induced from his preferences with regard to final wealth levels. Formally, when starting with wealth w, denote by the decision maker s preferences over lotteries in which the prizes are interpreted as changes in wealth. By the doctrine of
7 consequentialism all relations are derived from the same preference relation,, defined over the final wealth levels by p q iff w + p w + q (where w + p is the lottery that awards a prize w + x with probability p(x)). If is represented by a vnm utility function u, this doctrine implies that for all w, the function v w(x) = u(w + x) is a vnm utility function representing the preferences.
8 Propsect Theory to the Rescue: Theory motivated by experimental evidence that people evaluate wealth relative to a reference level. Two Key Features: Loss Aversion: the displeasure from a monetary loss is greater than the pleasure from a samesized gain (losses resonate more than gains). Diminishing sensitivity: The marginal change in perceived wellbeing is greater for changes that are close to one s reference level than for changes that are far away. Under loss aversion the value function abruptly changes slope at the reference level. People are significantly riskaverse for even small amounts of money. Example: the Rabin example above: people dislike losing $10 much more than they like gaining $11, and hence prefer their status quo to a 50/50 bet of losing $10 or gaining $11. There is a kink in the utility function at the reference level. Diminishing sensitivity implies that a person s utility function becomes less steep as her wealth gets further away from her reference level. For losses relative to the reference level, we have a striking implication: while people are risk averse over gains, they are often risk loving over losses.
9 Kahneman and Tversky (again): Consider the following two distributions: F : $0 with prob. 3/4 and $6000 with prob. 1/4 G : $0 with prob. 2/4, $4000 with prob. 1/4, and $2000 with prob. 1/4. K&T found that 70% of subjects report that they would prefer F to G. This is consistent with diminishing sensitivity. But F is a meanpreserving spread of G, so 70% of responses are inconsistent with the assumption that utility functions are concave.
The Cumulative Distribution and Stochastic Dominance
The Cumulative Distribution and Stochastic Dominance L A T E X file: StochasticDominance Daniel A. Graham, September 1, 2011 A decision problem under uncertainty is frequently cast as the problem of choosing
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationDecision & Risk Analysis Lecture 6. Risk and Utility
Risk and Utility Risk  Introduction Payoff Game 1 $14.50 0.5 0.5 $30  $1 EMV 30*0.5+(1)*0.5= 14.5 Game 2 Which game will you play? Which game is risky? $50.00 Figure 13.1 0.5 0.5 $2,000  $1,900 EMV
More informationECO 463. ExpectedUtility
ECO 463 ExpectedUtility Provide brief explanations as well as your answers. 1. Frank buys a lottery ticket from Sam which promises a 7% chance of winning 700 dollars. Frank is riskneutral and the price
More informationC2922 Economics Utility Functions
C2922 Economics Utility Functions T.C. Johnson October 30, 2007 1 Introduction Utility refers to the perceived value of a good and utility theory spans mathematics, economics and psychology. For example,
More informationLecture 13: Risk Aversion and Expected Utility
Lecture 13: Risk Aversion and Expected Utility Uncertainty over monetary outcomes Let x denote a monetary outcome. C is a subset of the real line, i.e. [a, b]. A lottery L is a cumulative distribution
More informationFund Manager s Portfolio Choice
Fund Manager s Portfolio Choice Zhiqing Zhang Advised by: Gu Wang September 5, 2014 Abstract Fund manager is allowed to invest the fund s assets and his personal wealth in two separate risky assets, modeled
More informationAsset Pricing. Chapter IV. Measuring Risk and Risk Aversion. June 20, 2006
Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Measuring Risk Aversion Utility function Indifference Curves U(Y) tangent lines U(Y + h) U[0.5(Y + h) + 0.5(Y h)] 0.5U(Y + h) + 0.5U(Y h) U(Y
More informationLecture 10  Risk and Insurance
Lecture 10  Risk and Insurance 14.03 Spring 2003 1 Risk Aversion and Insurance: Introduction To have a passably usable model of choice, we need to be able to say something about how risk affects choice
More informationRisk and Insurance. Vani Borooah University of Ulster
Risk and Insurance Vani Borooah University of Ulster Gambles An action with more than one possible outcome, such that with each outcome there is an associated probability of that outcome occurring. If
More informationWe show that prospect theory offers a rich theory of casino gambling, one that captures several features
MANAGEMENT SCIENCE Vol. 58, No. 1, January 2012, pp. 35 51 ISSN 00251909 (print) ISSN 15265501 (online) http://dx.doi.org/10.1287/mnsc.1110.1435 2012 INFORMS A Model of Casino Gambling Nicholas Barberis
More informationApplied Economics For Managers Recitation 5 Tuesday July 6th 2004
Applied Economics For Managers Recitation 5 Tuesday July 6th 2004 Outline 1 Uncertainty and asset prices 2 Informational efficiency  rational expectations, random walks 3 Asymmetric information  lemons,
More information.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationThe Values of Relative Risk Aversion Degrees
The Values of Relative Risk Aversion Degrees Johanna Etner CERSES CNRS and University of Paris Descartes 45 rue des SaintsPeres, F75006 Paris, France johanna.etner@parisdescartes.fr Abstract This article
More informationRisk and Uncertainty. Vani K Borooah University of Ulster
Risk and Uncertainty Vani K Borooah University of Ulster Basic Concepts Gamble: An action with more than one possible outcome, such that with each outcome there is an associated probability of that outcome
More informationIs There A Plausible Theory for Risky Decisions?
Is There A Plausible Theory for Risky Decisions? James C. Cox 1, Vjollca Sadiraj 1, Bodo Vogt 2, and Utteeyo Dasgupta 3 Experimental Economics Center Working Paper 200705 Georgia State University June
More informationUncertainty. BEE2017 Microeconomics
Uncertainty BEE2017 Microeconomics Uncertainty: The share prices of Amazon and the difficulty of investment decisions Contingent consumption 1. What consumption or wealth will you get in each possible
More informationNoBetting Pareto Dominance
NoBetting Pareto Dominance Itzhak Gilboa, Larry Samuelson and David Schmeidler HEC Paris/Tel Aviv Yale Interdisciplinary Center Herzlyia/Tel Aviv/Ohio State May, 2014 I. Introduction I.1 Trade Suppose
More informationRegret and Rejoicing Effects on Mixed Insurance *
Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers
More information1 Uncertainty and Preferences
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we
More informationPDF hosted at the Radboud Repository of the Radboud University Nijmegen
PDF hosted at the Radboud Repository of the Radboud University Nijmegen The following full text is a publisher's version. For additional information about this publication click this link. http://hdl.handle.net/2066/29354
More informationHealth Economics. University of Linz & Demand and supply of health insurance. Gerald J. Pruckner. Lecture Notes, Summer Term 2010
Health Economics Demand and supply of health insurance University of Linz & Gerald J. Pruckner Lecture Notes, Summer Term 2010 Gerald J. Pruckner Health insurance 1 / 25 Introduction Insurance plays a
More informationChoice Under Uncertainty
Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange
More informationLecture 11 Uncertainty
Lecture 11 Uncertainty 1. Contingent Claims and the StatePreference Model 1) Contingent Commodities and Contingent Claims Using the simple twogood model we have developed throughout this course, think
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationIntroduction to Game Theory IIIii. Payoffs: Probability and Expected Utility
Introduction to Game Theory IIIii Payoffs: Probability and Expected Utility Lecture Summary 1. Introduction 2. Probability Theory 3. Expected Values and Expected Utility. 1. Introduction We continue further
More informationThe Effect of Ambiguity Aversion on Insurance and Selfprotection
The Effect of Ambiguity Aversion on Insurance and Selfprotection David Alary Toulouse School of Economics (LERNA) Christian Gollier Toulouse School of Economics (LERNA and IDEI) Nicolas Treich Toulouse
More informationChoice under Uncertainty
Choice under Uncertainty Jonathan Levin October 2006 1 Introduction Virtually every decision is made in the face of uncertainty. While we often rely on models of certain information as you ve seen in the
More informationA Model of Casino Gambling
A Model of Casino Gambling Nicholas Barberis Yale University March 2009 Abstract Casino gambling is a hugely popular activity around the world, but there are still very few models of why people go to casinos
More informationA Model of Casino Gambling
A Model of Casino Gambling Nicholas Barberis Yale University June 2011 Abstract We show that prospect theory offers a rich theory of casino gambling, one that captures several features of actual gambling
More informationEcon 132 C. Health Insurance: U.S., Risk Pooling, Risk Aversion, Moral Hazard, Rand Study 7
Econ 132 C. Health Insurance: U.S., Risk Pooling, Risk Aversion, Moral Hazard, Rand Study 7 C2. Health Insurance: Risk Pooling Health insurance works by pooling individuals together to reduce the variability
More informationFind an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.
374 Chapter 8 The Mathematics of Likelihood 8.3 Expected Value Find an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.
More informationManagerial Economics
Managerial Economics Unit 9: Risk Analysis Rudolf WinterEbmer Johannes Kepler University Linz Winter Term 2012 Managerial Economics: Unit 9  Risk Analysis 1 / 1 Objectives Explain how managers should
More informationIntermediate Micro. Expected Utility
Intermediate Micro Expected Utility Workhorse model of intermediate micro Utility maximization problem Consumers Max U(x,y) subject to the budget constraint, I=P x x + P y y Health Economics Spring 2015
More informationIndividual Preferences, Monetary Gambles, and Stock Market Participation: A Case for Narrow Framing
Individual Preferences, Monetary Gambles, and Stock Market Participation: A Case for Narrow Framing By NICHOLAS BARBERIS, MING HUANG, AND RICHARD H. THALER* We argue that narrow framing, whereby an agent
More informationProspect Theory Ayelet Gneezy & Nicholas Epley
Prospect Theory Ayelet Gneezy & Nicholas Epley Word Count: 2,486 Definition Prospect Theory is a psychological account that describes how people make decisions under conditions of uncertainty. These may
More informationA Portfolio Model of Insurance Demand. April 2005. Kalamazoo, MI 49008 East Lansing, MI 48824
A Portfolio Model of Insurance Demand April 2005 Donald J. Meyer Jack Meyer Department of Economics Department of Economics Western Michigan University Michigan State University Kalamazoo, MI 49008 East
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationChapter 5 Uncertainty and Consumer Behavior
Chapter 5 Uncertainty and Consumer Behavior Questions for Review 1. What does it mean to say that a person is risk averse? Why are some people likely to be risk averse while others are risk lovers? A riskaverse
More informationLecture Note 14: Uncertainty, Expected Utility Theory and the Market for Risk
Lecture Note 14: Uncertainty, Expected Utility Theory and the Market for Risk David Autor, Massachusetts Institute of Technology 14.03/14.003, Microeconomic Theory and Public Policy, Fall 2010 1 Risk Aversion
More informationEconomics 2020a / HBS 4010 / HKS API111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4
Economics 00a / HBS 4010 / HKS API111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with
More informationNational Sun YatSen University CSE Course: Information Theory. Gambling And Entropy
Gambling And Entropy 1 Outline There is a strong relationship between the growth rate of investment in a horse race and the entropy of the horse race. The value of side information is related to the mutual
More informationLecture notes on Moral Hazard, i.e. the Hidden Action PrincipleAgent Model
Lecture notes on Moral Hazard, i.e. the Hidden Action PrincipleAgent Model Allan CollardWexler April 19, 2012 CoWritten with John Asker and Vasiliki Skreta 1 Reading for next week: Make Versus Buy in
More informationDemand and supply of health insurance. Folland et al Chapter 8
Demand and supply of health Folland et al Chapter 8 Chris Auld Economics 317 February 9, 2011 What is insurance? From an individual s perspective, insurance transfers wealth from good states of the world
More informationSlot Machine Stopping Decisions: Evidence for Prospect Theory Preferences?
Slot Machine Stopping Decisions: Evidence for Prospect Theory Preferences? Jaimie W. Lien 1 Tsinghua University School of Economics and Management Version: August 29 th, 2011 Abstract This paper uses a
More informationBasic Utility Theory for Portfolio Selection
Basic Utility Theory for Portfolio Selection In economics and finance, the most popular approach to the problem of choice under uncertainty is the expected utility (EU) hypothesis. The reason for this
More informationChoice Under Uncertainty Insurance Diversification & Risk Sharing AIG. Uncertainty
Uncertainty Table of Contents 1 Choice Under Uncertainty Budget Constraint Preferences 2 Insurance Choice Framework Expected Utility Theory 3 Diversification & Risk Sharing 4 AIG States of Nature and Contingent
More informationA Model of Casino Gambling
A Model of Casino Gambling Nicholas Barberis Yale University February 2009 Abstract Casino gambling is a hugely popular activity around the world, but there are still very few models of why people go to
More informationWe never talked directly about the next two questions, but THINK about them they are related to everything we ve talked about during the past week:
ECO 220 Intermediate Microeconomics Professor Mike Rizzo Third COLLECTED Problem Set SOLUTIONS This is an assignment that WILL be collected and graded. Please feel free to talk about the assignment with
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationLecture 8 The Subjective Theory of Betting on Theories
Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational
More informationThe St. Petersburg Paradox despite riskseeking preferences: An experimental study
The St. Petersburg Paradox despite riskseeking preferences: An experimental study James C. Cox a, Eike B. Kroll b, *, Vjollca Sadiraj a and Bodo Vogt b a Georgia State University, Andrew Young School
More informationProspect Theory and the Demand for Insurance
Prospect Theory and the emand for Insurance avid L. Eckles and Jacqueline Volkman Wise ecember 211 Abstract We examine the effect of prospect theory preferences on the demand for insurance to determine
More informationMTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationInsurance Demand under Prospect Theory: A Graphical Analysis. by Ulrich Schmidt
Insurance Demand under Prospect Theory: A Graphical Analysis by Ulrich Schmidt No. 1764 March 2012 Kiel Institute for the World Economy, Hindenburgufer 66, 24105 Kiel, Germany Kiel Working Paper No. 1764
More informationMicroeconomics Tutoral 5: Choice under Uncertainty and Revealed Preferences October, 21th
Université Paris 1 Panthéon Sorbonne Microeconomics: PSME  ngelo Secchi 1 Master I, cademic year 20152016 Microeconomics tutorials supervised by Elisa Baku 2 and ntoine Marsaudon 3. Microeconomics Tutoral
More informationThe "Dutch Book" argument, tracing back to independent work by. F.Ramsey (1926) and B.deFinetti (1937), offers prudential grounds for
The "Dutch Book" argument, tracing back to independent work by F.Ramsey (1926) and B.deFinetti (1937), offers prudential grounds for action in conformity with personal probability. Under several structural
More informationCompulsory insurance and voluntary selfinsurance: substitutes or complements? A matter of risk attitudes
Compulsory insurance and voluntary selfinsurance: substitutes or complements? A matter of risk attitudes François Pannequin, Ecole Normale Supérieure de Cachan and Centre d Economie de la Sorbonne pannequin@ecogest.enscachan.fr
More informationEconomics 206 Problem Set 1 Winter 2007 Vincent Crawford
Economics 206 Problem Set 1 Winter 2007 Vincent Crawford This problem set, which is optional, covers the material in the first half of the course, roughly in the order in which topics are discussed in
More informationThe effect of exchange rates on (Statistical) decisions. Teddy Seidenfeld Mark J. Schervish Joseph B. (Jay) Kadane Carnegie Mellon University
The effect of exchange rates on (Statistical) decisions Philosophy of Science, 80 (2013): 504532 Teddy Seidenfeld Mark J. Schervish Joseph B. (Jay) Kadane Carnegie Mellon University 1 Part 1: What do
More informationYou Are What You Bet: Eliciting Risk Attitudes from Horse Races
You Are What You Bet: Eliciting Risk Attitudes from Horse Races PierreAndré Chiappori, Amit Gandhi, Bernard Salanié and Francois Salanié March 14, 2008 What Do We Know About Risk Preferences? Not that
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationDecision making in the presence of uncertainty II
CS 274 Knowledge representation Lecture 23 Decision making in the presence of uncertainty II Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Informationgathering actions Many actions and their
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationHow to Gamble If You Must
How to Gamble If You Must Kyle Siegrist Department of Mathematical Sciences University of Alabama in Huntsville Abstract In red and black, a player bets, at even stakes, on a sequence of independent games
More informationReview Horse Race Gambling and Side Information Dependent horse races and the entropy rate. Gambling. Besma Smida. ES250: Lecture 9.
Gambling Besma Smida ES250: Lecture 9 Fall 200809 B. Smida (ES250) Gambling Fall 200809 1 / 23 Today s outline Review of Huffman Code and Arithmetic Coding Horse Race Gambling and Side Information Dependent
More informationHomework Assignment #1: Answer Key
Econ 497 Economics of the Financial Crisis Professor Ickes Spring 2012 Homework Assignment #1: Answer Key 1. Consider a firm that has future payoff.supposethefirm is unlevered, call the firm and its shares
More informationMoney and Capital in an OLG Model
Money and Capital in an OLG Model D. Andolfatto June 2011 Environment Time is discrete and the horizon is infinite ( =1 2 ) At the beginning of time, there is an initial old population that lives (participates)
More informationRisk Aversion and ExpectedUtility Theory: A Calibration Theorem
Risk Aversion and ExpectedUtility Theory: A Calibration Theorem Matthew Rabin Department of Economics University of California Berkeley First draft distributed: October 13 1997 Current draft: May 29 1999
More informationInsurance. Michael Peters. December 27, 2013
Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall
More informationIndifference Curves and the Marginal Rate of Substitution
Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationAuctioning Keywords in Online Search
Auctioning Keywords in Online Search Jianqing Chen The Uniersity of Calgary iachen@ucalgary.ca De Liu Uniersity of Kentucky de.liu@uky.edu Andrew B. Whinston Uniersity of Texas at Austin abw@uts.cc.utexas.edu
More informationIs it possible to beat the lottery system?
Is it possible to beat the lottery system? Michael Lydeamore The University of Adelaide Postgraduate Seminar, 2014 The story One day, while sitting at home (working hard)... The story Michael Lydeamore
More informationFinancial Markets. Itay Goldstein. Wharton School, University of Pennsylvania
Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationDiscrete Random Variables; Expectation Spring 2014 Jeremy Orloff and Jonathan Bloom
Discrete Random Variables; Expectation 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom This image is in the public domain. http://www.mathsisfun.com/data/quincunx.html http://www.youtube.com/watch?v=9xubhhm4vbm
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationInformatics 2D Reasoning and Agents Semester 2, 201516
Informatics 2D Reasoning and Agents Semester 2, 201516 Alex Lascarides alex@inf.ed.ac.uk Lecture 29 Decision Making Under Uncertainty 24th March 2016 Informatics UoE Informatics 2D 1 Where are we? Last
More informationLecture Note on Auctions
Lecture Note on Auctions Takashi Kunimoto Department of Economics McGill University First Version: December 26 This Version: September 26, 28 Abstract. There has been a tremendous growth in both the number
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More information2. Information Economics
2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many
More informationOptimization under uncertainty: modeling and solution methods
Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino email: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte
More informationPortfolio Allocation and Asset Demand with MeanVariance Preferences
Portfolio Allocation and Asset Demand with MeanVariance Preferences Thomas Eichner a and Andreas Wagener b a) Department of Economics, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany.
More informationEconomics 2020a / HBS 4010 / HKS API111 Fall 2011 Practice Problems for Lectures 1 to 11
Economics 2020a / HBS 4010 / HKS API111 Fall 2011 Practice Problems for Lectures 1 to 11 LECTURE 1: BUDGETS AND REVEALED PREFERENCE 1.1. Quantity Discounts and the Budget Constraint Suppose that a consumer
More informationSweating the Small Stu :
Sweating the Small Stu : Demand for Low Deductibles in Homeowners Insurance Justin Sydnor This Draft: July, 2006 Abstract How much do individuals pay to insure against moderate losses? To address this
More informationRational Expectations at the Racetrack : Testing Expected Utility Theory Using Betting Market Equilibrium (VERY PRELIMINARY DRAFT)
Rational Expectations at the Racetrack : Testing Expected Utility Theory Using Betting Market Equilibrium (VERY PRELIMINARY DRAFT) Amit Gandhi University of Chicago Abstract We present an adaptation of
More informationTesting Efficiency in the Major League of Baseball Sports Betting Market.
Testing Efficiency in the Major League of Baseball Sports Betting Market. Jelle Lock 328626, Erasmus University July 1, 2013 Abstract This paper describes how for a range of betting tactics the sports
More informationHow to build a probabilityfree casino
How to build a probabilityfree casino Adam Chalcraft CCR La Jolla dachalc@ccrwest.org Chris Freiling Cal State San Bernardino cfreilin@csusb.edu Randall Dougherty CCR La Jolla rdough@ccrwest.org Jason
More informationChoice under Uncertainty
Choice under Uncertainty Theoretical Concepts/Techniques Expected Utility Theory Representing choice under uncertainty in statecontingent space Utility Functions and Attitudes towards Risk Risk Neutrality
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationOn Compulsory PerClaim Deductibles in Automobile Insurance
The Geneva Papers on Risk and Insurance Theory, 28: 25 32, 2003 c 2003 The Geneva Association On Compulsory PerClaim Deductibles in Automobile Insurance CHUSHIU LI Department of Economics, Feng Chia
More informationReadings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem
Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,
More informationA Note on Proebsting s Paradox
A Note on Proebsting s Paradox Leonid Pekelis March 8, 2012 Abstract Proebsting s Paradox is twostage bet where the naive Kelly gambler (wealth growth rate maximizing) can be manipulated in some disconcertingly
More informationCost Minimization and the Cost Function
Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is
More information