A Simpli ed Axiomatic Approach to Ambiguity Aversion

Save this PDF as:

Size: px
Start display at page:

Transcription

1 A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN March 2009 Abstract This paper takes the Anscombe-Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann-Morgenstern independence axiom to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity. Keywords: Ambiguity; Savage axioms; Anscombe-Aumann framework; independence axiom; Ellsberg paradox JEL code: D81 I am grateful to Peter Klibano for encouragement and comments. 1

2 1 Introduction One of the famous problems that highlights the di erence between risk and ambiguity (or uncertainty) is the two-color Ellsberg problem (see Ellsberg, 1961). A decision maker is faced with two urns. The rst urn contains 50 red and 50 yellow balls, and the second contains 100 balls but in an unknown mixture of red and yellow. The decision maker will be paid \$10 if she can draw a red ball and must choose from which urn to draw. The rst urn generates a known payo distribution, so it is risky, but the second urn generates an unknown payo distribution, so it is ambiguous. Subjects systematically avoid the ambiguous urn in favor of the risky urn, thereby creating the need for a model of choice behavior which can accommodate the distinction between risk and ambiguity. Ambiguity aversion is inconsistent with the standard subjective expected utility models, and so those models must be generalized. Two axiomatizations of subjective expected utility stand out in the literature. Savage (1954) assumes that the state space is in nite and that the object of choice is an act, which is a mapping from states of nature to payo s. Anscombe and Aumann (1963) assume a nite state space and that the object of choice is a horse lottery, which is a mapping from states of nature into roulette lotteries, or probability distributions. In this paper I generalize subjective expected utility by assuming Savage s in nite state space, Anscombe and Aumann s formulation of horse and roulette lotteries, and applying Savage s axioms to preferences over horse lotteries instead of acts. In addition, I apply the familiar von Neumann-Morgenstern independence axiom when preferences are restricted to roulette lotteries. This very simple approach yields a preference representation of 2

3 the form 1 Z Z W (h) = w u(x)d(h s (x)) d(s); (1) S X where the state of nature s 2 S determines the objective probability distribution (roulette lottery) h s over payo s x 2 X, is the subjective probability distribution over states of nature, u is a von Neumann-Morgenstern utility function governing attitudes toward risk, and w is another utility function, this time governing attitudes toward ambiguity. The overall preference function W is de ned over horse lotteries h 2 H, which are assignments of objective probability distributions to states of nature. We call preferences that have the representation in (1) second-order expected utility preferences. Other researchers have used two methods to generalize the subjective expected utility model to allow for ambiguity aversion. One approach is based on nonadditive probabilities, and examples can be found in Gilboa (1987), Schmeidler (1989), Gilboa and Schmeidler (1989), Sarin and Wakker (1992), and Maccheroni, Marinacci, and Rustichini (2006). The alternative approach, which has generated considerable recent interest, is based on second-order probabilities, or subjective probability distributions de ned over objective probability distributions. An early example of this research can be found in Hazen (1987), 2 and more recent work can be found in Klibano, Marinacci, and Mukerji (2005), Nau (2006), Strzalecki (2007), Ahn (2008), Chew and Sagi (2008), and Ergin and Gul (2009). All of these papers generate functional forms similar to that in (1), but with di erent interpretations and di erent constructions. The interpretation here is not as a true second-order probability, since the subjective probability distribution is de ned over states of nature and not other (objective) 1 This functional speci cation is also proposed by Kreps and Porteus (1978) for the analysis of dynamic choices under risk. The speci cation here was rst proposed in Neilson (1993). 2 Hazen and Lee (1991) show how Hazen s model accommodates evidence such as the Ellsberg paradox. 3

4 probability distributions. The key to the construction here is that states correspond to objective probability distributions, as in the Anscombe-Aumann approach, and this paper s contribution to the literature is the simplicity of the axiomatic framework. In particular, no new axioms are proposed, only a di erent application of the old axioms. Section 2 sets up the Anscombe-Aumann framework, identi es the axioms, and provides the representation theorem. Section 3 shows that the second-order expected utility representation in expression (1) can easily accommodate ambiguity averse behavior when the function w is concave and the subjective distribution is uniform. Section 4 o ers a brief conclusion. 2 The representation theorem The model adopts the roulette and horse lottery framework of Anscombe and Aumann (1963). The bounded interval X is the payo space, and (X) is the set of all probability distributions over X. Members of (X) are also called roulette lotteries. Let S be the set of states of the world, with generic element s. De ne to be the set of all subsets of S, with generic element E, which is interpreted as an event. Savage (1954) de nes an act as a mapping from S to X, while Anscombe and Aumann de ne a horse lottery as a mapping from S to (X), that is, a mapping assigning a roulette lottery to every state. The resolution of an act is an outcome in the payo space, while the resolution of a horse lottery is a roulette lottery, which is a probability distribution over payo s. Let H denote the set of all horse lotteries. The key to this paper is applying the Savage axioms to horse lotteries instead of acts. To do so, assume that the individual has a preference ordering % de ned over H. In what follows, f; f 0 ; h; h 0 2 H are horse lotteries, ; 0 ; ; 0 2 (X) are roulette 4

5 lotteries, and E; E 0 ; E i 2 are events. Abusing notation when the context is clear, the roulette lottery is also a degenerate horse lottery assigning the same probability distribution to every state in S, that is, h s = for all s 2 S. These degenerate horse lotteries are called constant lotteries. The set E c is the complement of E in S, that is, S n E. A set E is null if h f whenever h s = f s for all s 2 E c, and where is the indi erence relation. It is said that h = f on E if h s = f s for all s 2 E. It is said that h f given E if and only if h 0 f 0 whenever h s = h 0 s for s 2 E, f s = fs 0 for s 2 E, and h 0 s = fs 0 for all s 2 E c. We use the following axioms over %. Axioms A1 - A7 are the Savage axioms, and axiom A8 applies von Neumann and Morgenstern s independence axiom to constant horse lotteries, which are simply probability distributions. A1: (Ordering) is complete and transitive. A2: (Sure-thing principle) If f = f 0 and h = h 0 on E, and f = h and f 0 = h 0 on E c, then f h if and only if f 0 h 0. A3: (Eventwise monotonicity) If E is not null and if f = and h = on E, then f h given E if and only if A4: (Weak comparative probability) Suppose that, f = on E, f = on E c, h = on E 0, and h = on E 0c, and suppose that 0 0, f 0 = 0 on E, f 0 = 0 on E c, h 0 = 0 on E 0, and h 0 = 0 on E 0c. Then f h if and only if f 0 h 0. A5: (Nondegeneracy) for some ; 2 (X). A6: (Small event continuity) If f h, for every 2 (X) there is a nite partition of S such that for every E i in the partition, if f 0 = on E i and f 0 = f on Ei c then f 0 h, and if h 0 = on E i and h 0 = h on Ei c then f h 0. A7: (Uniform monotonicity) For all E 2 and for all 2 h(e), if f given E, then f h given E. If f given E, then h f given E. 5

6 A8: (Independence over risk) 0 if and only if a +(1 a) a 0 +(1 a) for all 2 (X) and all scalars a 2 (0; 1). Axioms A1 - A7 are the standard Savage axioms modi ed so that they govern preferences over horse lotteries instead of preferences over acts. The main di erence between these axioms and Savage s, then, is that here probability distributions in (X) replace outcomes in X. It is worth di erentiating the sure-thing principle A2 from the independence over risk axiom A8. Consider the following application of axiom A2 to roulette lotteries, in which the horse lotteries f; f 0 ; h; h 0 2 H yield the following probability distributions in states E; E c 2, where ; ; ; 2 (X): E E c f f 0 h h 0 The sure-thing principle states that f h if and only if f 0 h 0, that is, preferences only depend on states in which the two horse lotteries being considered have di erent outcomes. This has the same spirit as the independence axiom A8, but with one key di erence. Here the mixture f of and is not a probability mixture, and thus it is not in (X). Consequently, axiom A2 does not imply axiom A8. However, axiom A3 guarantees that preferences on constant horse lotteries are identical to preferences over the corresponding roulette lotteries. Theorem 1 Preferences satisfy A1 - A8 if and only if there exists a unique probability 6

7 measure :! [0; 1], a function u : X! R, and a function w : R! R such that for all f; h 2 H, h f if and only if Z S Z Z Z w u(x)d(h s (x)) d(s) w u(x)d(f s (x)) d(s). X S X Moreover, the function u is unique up to increasing a ne transformations, and for a given speci cation of u the function w is unique up to increasing a ne transformations. Proof. Proof of the "if" direction is standard. For the "only if" direction, by the Savage axioms A1 - A7, there exists a unique probability measure :! [0; 1], and a function v : (X)! R such that the preference ordering % is represented by the functional Z W (h) = v (h s ) d(s). (2) S Furthermore, is unique and v is unique up to increasing a ne transformations. By axiom A8, % restricted to constant horse lotteries can be represented by Z V () = u(x)d(x); (3) X where u is unique up to increasing a ne transformations. Axiom A3 implies that V () and v() must represent the same preferences over roulette lotteries, and so there exists a monotone function w : R! R such that v() = w(v ()): (4) Since v is unique up to increasing a ne transformations, so is w for a given speci - 7

8 cation of u. Substituting expression (3) into (4) and then (4) into (2) yields Z W (h) = S Z w u(x)dh s (x)) d(s): (5) X It is worth pointing out why the axioms do not get us all the way to subjective expected utility (and ambiguity neutrality). The key is that axioms A1 - A7 only link the event space to the roulette lottery space (X), and not all the way to the payo space X. Axiom A8 places structure on the link between the roulette lottery space and the payo space, but not enough to provide that missing link. Consequently, risk attitudes and ambiguity attitudes remain separated. 3 Ambiguity aversion The second-order expected utility speci cation in (1) easily accommodates ambiguity attitudes as revealed in patterns such as the Ellsberg paradox. First consider the two-color paradox described in the introduction, where an individual is paid \$10 for drawing a red ball from one of two urns. States correspond to the number of red balls in the ambiguous urn, and for ease of exposition assume that this is a continuous variable between 0 and 100. Betting on the unambiguous urn corresponds to a constant horse lottery f which yields a 50:50 chance of \$10 in every state. Betting on the ambiguous urn corresponds to a horse lottery h which yields a probability s=100 of winning \$10. Since there are only two payo s, we can normalize the von Neumann-Morgenstern utility function u so that u(10) = 1 and u(0) = 0. If is uniform over states we get W (f) = w 1 2 8

9 and W (h) = Z s w ds: 100 In the Ellsberg example individuals tend to choose f over h, and Jensen s inequality implies that W (f) W (h) if w is concave. Uniform and concave w can also explain the three-color Ellsberg paradox. In this paradox a single urn contains 90 balls, 30 of which are red and the remaining 60 an unknown mixture of yellow and black. When individuals are given a chance to win \$100 if they draw a red ball or \$10 if they draw a yellow ball, they tend to bet on a red ball. When individuals are given a chance to win \$10 if they draw either a red or black ball, or \$10 if they draw either a yellow or black ball, they tend to bet on the yellow and black combination. In a subjective expected utility context the rst choice suggests that fewer than 30 of the balls are yellow, and the second choice suggests that fewer than 30 of the balls are black; hence the paradox. Let s de ne the state according to the number of yellow balls, so that 60 s is the number of black balls, and normalize u in the same way as above. The constant horse lottery f is a bet on drawing a red ball, the horse lottery f 0 is a bet on drawing a red or black ball, the horse lottery h is a bet on drawing a yellow ball, and the constant horse lottery h 0 is a bet on drawing a yellow or black ball. The Ellsberg preferences have f h but h 0 f 0. One can compute 1 W (f) = w 3 W (h) = 1 60 ; Z 60 0 Z 60 W (f 0 ) = W (h 0 ) = w : 3 s w w ds; 90 s ds; 9

10 If w is concave Jensen s inequality implies that W (f) W (h) and W (h 0 ) W (f 0 ). Finally, consider an Ellsberg-like situation in which there are three urns, all of which contain a mixture of 10 red and yellow balls. Urn 1 is unambiguous and contains 50 red and 50 yellow balls. Urn 2 contains an unknown mixture of 100 balls, but with at least 30 red and at least 30 yellow balls. Urn 3 is completely ambiguous, containing a completely unknown mixture of 100 red and yellow balls. An individual can win \$10 for drawing a red ball from one of the urns. Intuition from the two-color Ellsberg paradox suggests that the individual would prefer urn 1 to urn 2 to urn 3. Letting f be the bet on urn 1, g the bet on urn 2, and h the bet on urn 3, and assuming uniform subjective probabilities, one can compute 1 W (f) = w 2 W (g) = 1 40 W (h) = ; Z Z s w 100 w s 100 ds; ds; and concave w implies the above preferences. 4 Conclusion This paper applies old axioms from Savage (1954) and von Neumann and Morgenstern (1944) to an old choice framework developed by Anscombe and Aumann (1963) in which states of the world correspond to objective risks. The axioms lead to secondorder expected utility preferences which consist of a subjective probability measure over states of the world, a utility function governing risk attitudes, and another utility function governing ambiguity attitudes. Concavity of the rst utility function implies 10

11 risk aversion, and concavity of the second is consistent with ambiguity aversion in the Ellsberg paradoxes when the subjective distribution over states is uniform. The second-order expected utility model is also consistent with the Kreps-Porteus (1978) model governing temporal uncertainty. References Ahn, D.S. (2008). "Ambiguity without a state space," Review of Economic Studies 75, Anscombe, F.J. and R.J. Aumann (1963). "A de nition of subjective probability," Annals of Mathematical Statistics 34, Chew, S.H. and J.S. Sagi (2008). "Small worlds: Modeling attitudes toward sources of uncertainty," Journal of Economic Theory 139, Ellsberg, D. (1961). "Risk, ambiguity, and the Savage axioms," Quarterly Journal of Economics 75, Ergin, H. and F. Gul (2009). "A subjective theory of compound lotteries," Journal of Economic Theory, forthcoming. Gilboa, I. (1987). "Subjective utility with purely subjective non-additive probabilities," Journal of Mathematical Economics 16, Gilboa, I. and D. Schmeidler (1989). "Maxmin expected utility with a non-unique prior," Journal of Mathematical Economics 18, Hazen, G.B. (1987). "Subjectively weighted linear utility," Theory and Decision 23, Hazen, G. and J.S. Lee (1991). "Ambiguity aversion in the small and in the large for weighted linear utility," Journal of Risk and Uncertainty 4, Klibano, P., M. Marinacci, and S. Mukerji (2005). "A smooth model of decision making under ambiguity," Econometrica 73, Kreps, D.M. and E.L. Porteus (1978). "Temporal resolution of uncertainty and dynamic choice theory," Econometrica 46, Maccheroni, F., M. Marinacci, and A. Rustichini (2006). "Ambiguity aversion, robustness, and the variational representation of preferences," Econometrica 74,

12 Nau, R.F. (2006). "Uncertainty aversion with second-order utilities and probabilities," Management Science 52, Neilson, W.S. (1993). "Ambiguity aversion: An axiomatic approach using secondorder probabilities," mimeo. Sarin, R.K. and P. Wakker (1992). "A simple axiomatization of nonadditive expected utility," Econometrica 60, Savage, L.J. (1954). Foundations of Statistics, New York: Wiley. Schmeidler, D. (1989). "Subjective probability and expected utility without additivity," Econometrica 57, Strzalecki, T. (2007). "Axiomatic foundations of multiplier preferences," mimeo. von Neumann, J. and O. Morgenstern (1944). Theory of Games and Economic Behavior, Princeton: Princeton University Press. 12

A Simpli ed Axiomatic Approach to Ambiguity Aversion

A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 37996-0550 wneilson@utk.edu May 2010 Abstract This paper takes the

Subjective expected utility theory does not distinguish between attitudes toward uncertainty (ambiguous

MANAGEMENT SCIENCE Vol. 52, No. 1, January 2006, pp. 136 145 issn 0025-1909 eissn 1526-5501 06 5201 0136 informs doi 10.1287/mnsc.1050.0469 2006 INFORMS Uncertainty Aversion with Second-Order Utilities

WORKING PAPER SERIES

Institutional Members: CEPR, NBER and Università Bocconi WORKING PAPER SERIES On the Smooth Ambiguity Model: A Reply Peter Klibanoff, Massimo Marinacci and Sujoy Mukerji Working Paper n. 40 This Version:

Lecture 11 Uncertainty

Lecture 11 Uncertainty 1. Contingent Claims and the State-Preference Model 1) Contingent Commodities and Contingent Claims Using the simple two-good model we have developed throughout this course, think

Schneeweiss: Resolving the Ellsberg Paradox by Assuming that People Evaluate Repetitive Sampling

Schneeweiss: Resolving the Ellsberg Paradox by Assuming that People Evaluate Repetitive Sampling Sonderforschungsbereich 86, Paper 5 (999) Online unter: http://epub.ub.uni-muenchen.de/ Projektpartner Resolving

Scale-Invariant Uncertainty-Averse Preferences and Source-Dependent Constant Relative Risk Aversion

Scale-Invariant Uncertainty-Averse Preferences and Source-Dependent Constant Relative Risk Aversion Costis Skiadas October 2010; this revision: May 7, 2011 Abstract Preferences are defined over consumption

Economics 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

Red Herrings: Some Thoughts on the Meaning of Zero-Probability Events and Mathematical Modeling. Edi Karni*

Red Herrings: Some Thoughts on the Meaning of Zero-Probability Events and Mathematical Modeling By Edi Karni* Abstract: Kicking off the discussion following Savage's presentation at the 1952 Paris colloquium,

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

The Effect of Ambiguity Aversion on Insurance and Self-protection

The Effect of Ambiguity Aversion on Insurance and Self-protection David Alary Toulouse School of Economics (LERNA) Christian Gollier Toulouse School of Economics (LERNA and IDEI) Nicolas Treich Toulouse

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

arxiv:1208.2354v1 [physics.soc-ph] 11 Aug 2012

A Quantum Model for the Ellsberg and Machina Paradoxes arxiv:1208.2354v1 [physics.soc-ph] 11 Aug 2012 Diederik Aerts and Sandro Sozzo Center Leo Apostel for Interdisciplinary Studies Brussels Free University

A SUBJECTIVE SPIN ON ROULETTE WHEELS. by Paolo Ghirardato, Fabio Maccheroni, Massimo Marinacci and Marciano Siniscalchi 1

A SUBJECTIVE SPIN ON ROULETTE WHEELS by Paolo Ghirardato, Fabio Maccheroni, Massimo Marinacci and Marciano Siniscalchi 1 Abstract We provide a simple behavioral definition of subjective mixture of acts

U n iversity o f H ei delberg

U n iversity o f H ei delberg Department of Economics Discussion Paper Series No 592 482482 Benevolent and Malevolent Ellsberg Games Adam Dominiak and Peter Duersch May 2015 Benevolent and Malevolent Ellsberg

Coherent Odds and Subjective Probability

Coherent Odds and Subjective Probability Kim C. Border Uzi Segal y October 11, 2001 1 Introduction Expected utility theory, rst axiomatized by von Neumann and Morgenstern [33], was considered for manyyears

Accident Law and Ambiguity

Accident Law and Ambiguity Surajeet Chakravarty and David Kelsey Department of Economics University of Exeter January 2012 (University of Exeter) Tort and Ambiguity January 2012 1 / 26 Introduction This

Estimating the Relationship between Economic Preferences: A Testing Ground for Unified Theories of Behavior: Appendix

Estimating the Relationship between Economic Preferences: A Testing Ground for Unified Theories of Behavior: Appendix Not for Publication August 13, 2012 1 Appendix A: Details of Measures Collected Discount

Second order beliefs models of choice under imprecise risk: Nonadditive second order beliefs versus nonlinear second order utility

Theoretical Economics 9 (2014), 779 816 1555-7561/20140779 econd order beliefs models of choice under imprecise risk: Nonadditive second order beliefs versus nonlinear second order utility Raphaël Giraud

Uncertainty. 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

Exact Nonparametric Tests for Comparing Means - A Personal Summary

Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola

Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

EXTENSION RULES OR WHAT WOULD THE SAGE DO? January 23, 2013

EXTENSION RULES OR WHAT WOULD THE SAGE DO? EHUD LEHRER AND ROEE TEPER January 23, 2013 Abstract. Quite often, decision makers face choices that involve new aspects and alternatives never considered before.

COGNITIVE DISSONANCE AND CHOICE

COGNITIVE DISSONANCE AND CHOICE Larry G. Epstein Igor Kopylov February 2006 Abstract People like to feel good about past decisions. This paper models selfjusti cation of past decisions. The model is axiomatic:

De Finetti Meets Ellsberg

203s-35 De Finetti Meets Ellsberg Larry G. Epstein, Kyoungwon Seo Série Scientifique Scientific Series Montréal Septembre 203 203 Larry G. Epstein, Kyoungwon Seo. Tous droits réservés. All rights reserved.

Savages Subjective Expected Utility Model

Savages Subjective Expected Utility Model Edi Karni Johns Hopkins University November 9, 2005 1 Savage s subjective expected utility model. In his seminal book, The Foundations of Statistics, Savage (1954)

Betting on Machina s reflection example: An Experiment on Ambiguity

Betting on Machina s reflection example: An Experiment on Ambiguity Olivier L Haridon Greg-Hec & University Paris Sorbonne & Lætitia Placido Greg-Hec & CNRS October 2008 Abstract In a recent paper, Machina

EconS 503 - Advanced Microeconomics II Handout on Cheap Talk

EconS 53 - Advanced Microeconomics II Handout on Cheap Talk. Cheap talk with Stockbrokers (From Tadelis, Ch. 8, Exercise 8.) A stockbroker can give his client one of three recommendations regarding a certain

CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

Economics Discussion Paper Series EDP-1005. Agreeing to spin the subjective roulette wheel: Bargaining with subjective mixtures

Economics Discussion Paper Series EDP-1005 Agreeing to spin the subjective roulette wheel: Bargaining with subjective mixtures Craig Webb March 2010 Economics School of Social Sciences The University of

Risk, Ambiguity, and the Rank-Dependence Axioms

American Economic Review 2009, 99:1, 385 392 http://www.aeaweb.org/articles.php?doi=10.1257/aer.99.1.385 Risk, Ambiguity, and the Rank-Dependence Axioms By Mark J. Machina* Choice problems in the spirit

Information Gaps for Risk and Ambiguity

Information Gaps for Risk and Ambiguity Russell Golman, George Loewenstein, and Nikolos Gurney May 6, 2015 Abstract We apply a model of preferences for information to the domain of decision making under

2. CHOICE UNDER UNCERTAINTY. Simple and Compound Lotteries

2 CHOICE UNDER UNCERTAINTY Ref: MWG Chapter 6 Subjective Expected Utility Theory Elements of decision under uncertainty Under uncertainty, the DM is forced, in effect, to gamble A right decision consists

Directed search and job rotation

MPRA Munich Personal RePEc Archive Directed search and job rotation Fei Li and Can Tian University of Pennsylvania 4. October 2011 Online at http://mpra.ub.uni-muenchen.de/33875/ MPRA Paper No. 33875,

ISSN 1518-3548. Working Paper Series

ISSN 1518-3548 Working Paper Series 180 A Class of Incomplete and Ambiguity Averse Preferences Leandro Nascimento and Gil Riella December, 2008 ISSN 1518-3548 CGC 00.038.166/0001-05 Working Paper Series

Uncertainty, Learning and Ambiguity in Economic Models on Climate Policy: Some Classical Results and New Directions *

Page 1 14/09/2007 Uncertainty, Learning and Ambiguity in Economic Models on Climate Policy: Some Classical Results and New Directions * Andreas Lange AREC, University of Maryland, USA - alange@arec.umd.edu

Choice 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

Aggregating Tastes, Beliefs, and Attitudes under Uncertainty

Aggregating Tastes, Beliefs, and Attitudes under Uncertainty Eric Danan Thibault Gajdos Brian Hill Jean-Marc Tallon July 20, 2014 Abstract We provide possibility results on the aggregation of beliefs and

Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

Optimal insurance contracts with adverse selection and comonotonic background risk

Optimal insurance contracts with adverse selection and comonotonic background risk Alary D. Bien F. TSE (LERNA) University Paris Dauphine Abstract In this note, we consider an adverse selection problem

Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication

Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

Pricing Cloud Computing: Inelasticity and Demand Discovery

Pricing Cloud Computing: Inelasticity and Demand Discovery June 7, 203 Abstract The recent growth of the cloud computing market has convinced many businesses and policy makers that cloud-based technologies

14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

Calibrated Uncertainty

Calibrated Uncertainty Faruk Gul and Wolfgang Pesendorfer Princeton University July 2014 Abstract We define a binary relation (qualitative uncertainty assessment (QUA)) to describe a decision-maker s subjective

A Comparison of Option Pricing Models

A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might

Robust Asset Allocation Under Model Ambiguity

Robust Asset Allocation Under Model Ambiguity Sandrine Tobelem (London School of Economics - Statistics Department) (Joint work with Pauline Barrieu) Bachelier Conference - Toronto - June 2010 1 Introduction

1 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

Risk Aversion. Expected value as a criterion for making decisions makes sense provided that C H A P T E R 2. 2.1 Risk Attitude

C H A P T E R 2 Risk Aversion Expected value as a criterion for making decisions makes sense provided that the stakes at risk in the decision are small enough to \play the long run averages." The range

Sample Problems. Practice Problems

Lecture Notes Linear Systems - Elimination page Sample Problems. Solve each of the following system of linear equations. 3x 2y = 26 x + 5y = 8 a) b) 2x + 5y = 27 5y = 3x + 8 c) ( 2y x = 50 2 x = y + 25

Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers

Since the early days of probability theory, there has been a distinction

Journal of Economic Perspectives Volume 22, Number 3 Summer 2008 Pages 173 188 Probability and Uncertainty in Economic Modeling Itzhak Gilboa, Andrew W. Postlewaite, and David Schmeidler Since the early

Voluntary Voting: Costs and Bene ts

Voluntary Voting: Costs and Bene ts Vijay Krishna y and John Morgan z November 7, 2008 Abstract We study strategic voting in a Condorcet type model in which voters have identical preferences but di erential

Oligopoly. Chapter 10. 10.1 Overview

Chapter 10 Oligopoly 10.1 Overview Oligopoly is the study of interactions between multiple rms. Because the actions of any one rm may depend on the actions of others, oligopoly is the rst topic which requires

Towards an evolutionary cooperative game theory

Towards an evolutionary cooperative game theory Andre Casajus and Harald Wiese March 2010 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March 2010 1 / 32 Two game theories

Similarity-Based Mistakes in Choice

INSTITUTO TECNOLOGICO AUTONOMO DE MEXICO CENTRO DE INVESTIGACIÓN ECONÓMICA Working Papers Series Similarity-Based Mistakes in Choice Fernando Payro Centro de Investigación Económica, ITAM Levent Ülkü Centro

Games Played in a Contracting Environment

Games Played in a Contracting Environment V. Bhaskar Department of Economics University College London Gower Street London WC1 6BT February 2008 Abstract We analyze normal form games where a player has

.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

Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

Optimal Auctions. Jonathan Levin 1. Winter 2009. Economics 285 Market Design. 1 These slides are based on Paul Milgrom s.

Optimal Auctions Jonathan Levin 1 Economics 285 Market Design Winter 29 1 These slides are based on Paul Milgrom s. onathan Levin Optimal Auctions Winter 29 1 / 25 Optimal Auctions What auction rules lead

Third-degree Price Discrimination, Entry and Welfare

Third-degree Price Discrimination, Entry and Welfare Sílvia Ferreira Jorge DEGEI, Universidade de Aveiro, Portugal Cesaltina Pacheco Pires CEFAGE UE, Departamento de Gestão, Universidade de Évora, Portugal

SUBJECTIVE AND OTHER PROBABILITIES

SUBJECTIVE AND OTHER PROBABILITIES IGOR KOPYLOV Personalistic views hold that probability measures the confidence that a particular individual has in the truth of a particular proposition. Savage (1972,

Subjective Probability

Subjective Probability Nabil I. Al-Najjar and Luciano De Castro Northwestern University March 2010 Abstract We provide an overview of the idea of subjective probability and its foundational role in decision

Financial Economics Lecture notes. Alberto Bisin Dept. of Economics NYU

Financial Economics Lecture notes Alberto Bisin Dept. of Economics NYU September 25, 2010 Contents Preface ix 1 Introduction 1 2 Two-period economies 3 2.1 Arrow-Debreu economies..................... 3

A Dynamic Mechanism and Surplus Extraction Under Ambiguity 1 Subir Bose University of Leicester, Leicester, UK

A Dynamic Mechanism and Surplus Extraction Under Ambiguity 1 Subir Bose University of Leicester, Leicester, UK Arup Daripa Birkbeck College, London University Bloomsbury, London, UK Final Revision: January

Working Paper Series

RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos Hervés-Beloso, Emma Moreno- García and

The Annals of Mathematical Statistics, Vol. 34, No. 1. (Mar., 1963), pp. 199-205.

A Definition of Subjective Probability F. J. Anscombe; R. J. Aumann The Annals of Mathematical Statistics, Vol. 34, No. 1. (Mar., 1963), pp. 199-205. Stable URL: http://links.jstor.org/sici?sici=0003-4851%28196303%2934%3a1%3c199%3aadosp%3e2.0.co%3b2-8

A Welfare Criterion for Models with Distorted Beliefs

A Welfare Criterion for Models with Distorted Beliefs Markus Brunnermeier, Alp Simsek, Wei Xiong August 2012 Markus Brunnermeier, Alp Simsek, Wei Xiong () Welfare Criterion for Distorted Beliefs August

Statistics 100A Homework 4 Solutions

Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.

The Binomial Distribution

The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing

Analyzing the Demand for Deductible Insurance

Journal of Risk and Uncertainty, 18:3 3 1999 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State

«The Effect of Ambiguity Aversion on Self-insurance and Self-protection»

JEEA Rennes 22 & 23 Octobre 2009 «The Effect of Ambiguity Aversion on Self-insurance and Self-protection» Nicolas Treich, Toulouse School of Economics (LERNA-INRA) E-mail: ntreich@toulouse.inra.fr Motivation

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION

Basic 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

Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

C2922 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,

Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania

Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically

10 Dynamic Games of Incomple Information

10 Dynamic Games of Incomple Information DRAW SELTEN S HORSE In game above there are no subgames. Hence, the set of subgame perfect equilibria and the set of Nash equilibria coincide. However, the sort

The 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 Saints-Peres, F-75006 Paris, France johanna.etner@parisdescartes.fr Abstract This article

Interlinkages between Payment and Securities. Settlement Systems

Interlinkages between Payment and Securities Settlement Systems David C. Mills, Jr. y Federal Reserve Board Samia Y. Husain Washington University in Saint Louis September 4, 2009 Abstract Payments systems

5. Probability Calculus

5. Probability Calculus So far we have concentrated on descriptive statistics (deskriptiivinen eli kuvaileva tilastotiede), that is methods for organizing and summarizing data. As was already indicated

DEPARTMENT OF ECONOMICS EXPLAINING THE ANOMALIES OF THE EXPONENTIAL DISCOUNTED UTILITY MODEL

DEPARTMENT OF ECONOMICS EXPLAINING THE ANOMALIES OF THE EXPONENTIAL DISCOUNTED UTILITY MODEL Ali al-nowaihi, University of Leicester, UK Sanjit Dhami, University of Leicester, UK Working Paper No. 07/09

Regret 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

Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

Quality differentiation and entry choice between online and offline markets

Quality differentiation and entry choice between online and offline markets Yijuan Chen Australian National University Xiangting u Renmin University of China Sanxi Li Renmin University of China ANU Working

No-Betting Pareto Dominance

No-Betting 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

Dynamics and Stability of Political Systems

Dynamics and Stability of Political Systems Daron Acemoglu MIT February 11, 2009 Daron Acemoglu (MIT) Marshall Lectures 2 February 11, 2009 1 / 52 Introduction Motivation Towards a general framework? One

SONDERFORSCHUNGSBEREICH 504

SONDERFORSCHUNGSBEREICH 504 Rationalitätskonzepte, Entscheidungsverhalten und ökonomische Modellierung No. 08-32 Case-Based Expected Utility: references over Actions and Data Jürgen Eichberger and Ani

1 Hypothesis Testing. H 0 : population parameter = hypothesized value:

1 Hypothesis Testing In Statistics, a hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. Data

Nash Was a First to Axiomatize Expected Utility

Nash Was a First to Axiomatize Expected Utility Han Bleichrodt a, Chen Li a, Ivan Moscati b, & Peter P. Wakker a,c a: Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam, the Netherlands;

ANTICIPATING REGRET: WHY FEWER OPTIONS MAY BE BETTER. TODD SARVER Northwestern University, Evanston, IL 60208, U.S.A.

http://www.econometricsociety.org/ Econometrica, Vol. 76, No. 2 (March, 2008), 263 305 ANTICIPATING REGRET: WHY FEWER OPTIONS MAY BE BETTER TODD SARVER Northwestern niversity, Evanston, IL 60208,.S.A.

Subjective Beliefs and Ex-Ante Trade Luca Rigotti Duke University Fuqua School of Business Tomasz Strzalecki Northwestern University Department of Economics This version: January 8, 2008 Chris Shannon

DECISION MAKING UNDER UNCERTAINTY:

DECISION MAKING UNDER UNCERTAINTY: Models and Choices Charles A. Holloway Stanford University TECHNISCHE HOCHSCHULE DARMSTADT Fachbereich 1 Gesamtbibliothek Betrtebswirtscrtaftslehre tnventar-nr. :...2>2&,...S'.?S7.

Cowles Foundation for Research in Economics at Yale University

Cowles Foundation for Research in Economics at Yale University Cowles Foundation Discussion Paper No. 896R THE LIMITS OF PRICE DISCRIMINATION Dirk Bergemann, Benjamin Brooks, and Stephen Morris May 03

Choice 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

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

Applied Mathematics and Computation

Applied Mathematics Computation 219 (2013) 7699 7710 Contents lists available at SciVerse ScienceDirect Applied Mathematics Computation journal homepage: www.elsevier.com/locate/amc One-switch utility

Inter-temporal Poverty Measures: The Impact of A uence

Inter-temporal Poverty Measures: The Impact of A uence Laurence Roope University of Manchester 24th July 2010 (University of Manchester) 24th July 2010 1 / 25 Agenda Introduction Our new class of inter-temporal