Optimal Risk Sharing with Optimistic and Pessimistic Decision Makers
|
|
- Patrick Bradley
- 6 years ago
- Views:
Transcription
1 Optimal Risk Sharing with Optimistic and Pessimistic Decision Makers A. Araujo 1,2, J.M. Bonnisseau 3, A. Chateauneuf 3 and R. Novinski 1 1 Instituto Nacional de Matematica Pura e Aplicada 2 Fundacao Getulio Vargas 3 Paris School of Economics, Université Paris 1 Panthéon-Sorbonne Centre d Economie de la Sorbonne, UMR CNRS 8174 Economic Theory Conference University of Kansas December 9-10, 2011 Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 1
2 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 2
3 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 3
4 Introduction Main motivation: we aim at proving that under mild conditions individually Pareto rational optima will exist even in presence of non-convex preferences. We essentially consider preferences on l + hence accommodating the behavior of DM in front of a countable flow x of payoffs, or else choosing among financial assets x whose outcomes depend on the realization of a countable set of states of the world. Therefore, our conditions for the existence of P.O. can be interpreted as myopia in the first context and as pessimism or reasonable optimism in the second context. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 4
5 Introduction (continued) Replacing the uncertain context by a risky one, we then derive some necessary properties of risk sharing. For strong risk averters these conditions are exactly the usual ones of pairwise comonotonicity, for strong risk lovers exclusively corner conditions or pairwise comonotonicity can appear. A crucial result for our work is the proof that the closed unit ball B(0, 1) for the normed dual l of the normed space l 1 is compact in the Mackey topology τ(l, l 1 ). Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 5
6 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 6
7 Theorem Let B(S) be the normed space of bounded real mappings on S endowed with the sup-norm. Then: Corollary 1) B(S) is the norm dual of the space rca(s) of all regular and bounded Borel measures for S endowed with the discrete topology. 2) The closed unit ball of B(S) is compact for the Mackey topology τ(b(s), rca(s)). The closed unit ball B(0, 1) for the normed dual l of the normed space l 1 is compact in the Mackey topology τ(l, l 1 ). Proof. Taking S = N, clearly l = B(S). Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 7
8 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 8
9 Existence of individually rational Pareto efficient allocations under Mackey upper-semicontinuous preferences Assumption (1) For every i = 1,..., m, i is a transitive reflexive preorder on l +. (ii) For every i and every y i l +, {x i l + x i i y i } is τ-closed where τ is the Mackey topology τ(l, l 1 ). Theorem Under Assumptions (1) and (2), individually rational Pareto efficient allocations exist. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 9
10 Examples of Mackey upper-semicontinuous preferences Definition Let (S, A) be a measurable space and B (A) be the space of bounded A-measurable mappings from S to R. v is a capacity on A if v( ) = 0, v(s) = 1, A, B A, A B v(a) v(b). x B (A), the Choquet integral of x with respect to v, denoted xdv is defined by 0 (v(x t) 1)dt v(x t)dt. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 10
11 >From Araujo et al. General Equilibrium, Wariness and efficient bubbles", JET forthcoming, we know that preferences on l represented by a Choquet integral is Mackey usc if and only if is strongly myopic where: Definition A preference relation on l, is strongly myopic if (x, y) l l x y, z l +, implies that there exists n large enough such that x y + z En where z En (m) = 0 if m < n and z(m) if m n. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 11
12 Pessimistic Mackey upper-semicontinuous preferences Definition A Choquet DM will be said pessimistic if v is convex, i.e., A, B A, v(a B) + v(a B) v(a) + v(b). Recall that if v is convex, then xdv = min{e P (x) P additive probability measure v} Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 12
13 Proposition For a pessimistic decision maker, the following assertions are equivalent: (1) is Mackey upper-semicontinuous; (2) v is outer continuous, i.e., A n, A A = P(N), A n A v(a n ) v(a). Example I(x) = inf N {x(n)} or equally I(x) = xdv where v(a) = 0 A N, v(n) = 1 Remark that I is not Mackey lower semicontinuous. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 13
14 Optimistic Mackey upper-semicontinuous preferences Definition A Choquet DM will be said optimistic if v is concave, i.e., A, B A, v(a B) + v(a B) v(a) + v(b). Recall that if v is concave, xdv = max{e P (x) P additive probability measure v} or else x, y B (A), x y, α ]0, 1[ αx + (1 α)y y. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 14
15 Proposition For an optimistic decision maker the following assertions are equivalent: (1) is Mackey upper-semicontinuous; (2) is Mackey lower-semicontinuous; (3) v is outer continuous at, i.e., A n P(N), A n v(a n ) 0. Remark. I(x) = sup N {x(n)}, i.e., I(x) = xdv with v(a) = 1 A, v( ) = 0 is not Mackey upper-semicontinuous. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 15
16 Example where no existence of nontrivial individually rational Pareto optima if such a too optimistic" DM is present Consumption space X = l +, p s = 1 2. s DM1 U 1 (x) = s N p sx s, initial endowments ω 1 = (2 1 s+1 ) s N DM2 U 2 (x) = sup s N x s, ω 2 = ω 1. The unique Pareto optimal allocation is ( x 1 = ω 1 + ω 2, x 2 = 0). No individually rational Pareto optimal allocation. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 16
17 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 17
18 for risk lovers and risk averse decision makers Interpret l as the set of all bounded A measurable mappings x from N to R where A = P(N) and assume that a σ-additive probability P is given on (N, P(N)). So any x can now be interpreted as a random variable. Definition x is less risky than y for the second order stochastic dominance denoted x SSD y if t P(x u)du t P(y u)du for all t R. x is strictly less risky than y denoted x SSD y if one of the previous inequality is strict. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 18
19 Definition Let i be the preference relation of a DM. i is a strict (strong) risk averter if x SSD y x i y and x SSD y x i y. i is a strict (strong) risk lover if x SSD y x i y and x SSD y x i y. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 19
20 Examples of Mackey upper-semicontinuous strict risk lover of strict risk averter Consider a Choquet decision maker where v = f P, i.e., v is a distortion of a probability P, i.e., f : [0, 1] [0, 1], f (0) = 0, f (1) = 1, f strictly increasing. If f is strictly concave and continuous, then x l I(x) = xdv f with v f = f P defines a Mackey upper-semicontinuous strict risk lover. If f is strictly convex and continuous, then x l I(x) = xdv f with v f = f P defines a Mackey upper-semicontinuous strict risk averter. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 20
21 The main result Theorem Consider m strict risk averters i = 1,..., m with Mackey upper-semicontinuous preferences i and n strict risk lovers j = 1,..., n with Mackey upper-semicontinuous preferences j with initial endowments ω i l ++, ω j l ++. Then individual rational Pareto efficient allocations (in (l + ) m+n ) exist. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 21
22 Theorem continued Theorem For such PO (x i, i = 1,..., m; y j, j = 1,..., n) we have: 1) The allocations of risk averters are pairwise comonotonic, i.e., (x i1 (s) x i1 (t))(x i2 (s) x i2 (t)) 0 (s, t) N 2, (i 1, i 2 ) {1,..., m} 2. 2) For two risk lovers j 1, j 2 : (y j1 (s) y j1 (t))(y j2 (s) y j2 (t)) < 0 (y j1 (s) y j1 (t))(y j2 (s) y j2 (t)) = 0 (i.e. in at least one state, one of these two risk lovers gets 0). 3) If all allocations of risk lovers are strictly positive, then these allocations are pairwise comonotonic. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 22
Could Nash equilibria exist if the payoff functions are not quasi-concave?
Could Nash equilibria exist if the payoff functions are not quasi-concave? (Very preliminary version) Bich philippe Abstract In a recent but well known paper (see [11]), Reny has proved the existence of
More informationThe Existence of Equilibrium in. Infinite-Dimensional Spaces: Some Examples. John H. Boyd III
The Existence of Equilibrium in Infinite-Dimensional Spaces: Some Examples John H. Boyd III April 1989; Updated June 1995 Abstract. This paper presents some examples that clarify certain topological and
More informationDocuments de Travail du Centre d Economie de la Sorbonne
Documents de Travail du Centre d Economie de la Sorbonne The no-trade interval of Dow and Werlang : some clarifications Alain CHATEAUNEUF, Caroline VENTURA 2008.65 Maison des Sciences Économiques, 106-112
More informationMultidimensional inequalities and generalized quantile functions
Multidimensional inequalities and generalized quantile functions Sinem Bas Philippe Bich Alain Chateauneuf Abstract In this paper, we extend the generalized Yaari dual theory for multidimensional distributions,
More informationChoice under Uncertainty
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) Group 2 Dr. S. Farshad Fatemi Chapter 6: Choice under Uncertainty
More informationNotes on Recursive Utility. Consider the setting of consumption in infinite time under uncertainty as in
Notes on Recursive Utility Consider the setting of consumption in infinite time under uncertainty as in Section 1 (or Chapter 29, LeRoy & Werner, 2nd Ed.) Let u st be the continuation utility at s t. That
More informationRegularity of competitive equilibria in a production economy with externalities
Regularity of competitive equilibria in a production economy with externalities Elena del Mercato Vincenzo Platino Paris School of Economics - Université Paris 1 Panthéon Sorbonne QED-Jamboree Copenhagen,
More informationAbout partial probabilistic information
About partial probabilistic information Alain Chateauneuf, Caroline Ventura To cite this version: Alain Chateauneuf, Caroline Ventura. About partial probabilistic information. Documents de travail du Centre
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationExistence of Equilibria with a Tight Marginal Pricing Rule
Existence of Equilibria with a Tight Marginal Pricing Rule J.-M. Bonnisseau and B. Cornet December 1, 2009 Abstract This paper deals with the existence of marginal pricing equilibria when it is defined
More informationChapter 1 - Preference and choice
http://selod.ensae.net/m1 Paris School of Economics (selod@ens.fr) September 27, 2007 Notations Consider an individual (agent) facing a choice set X. Definition (Choice set, "Consumption set") X is a set
More informationRandom sets. Distributions, capacities and their applications. Ilya Molchanov. University of Bern, Switzerland
Random sets Distributions, capacities and their applications Ilya Molchanov University of Bern, Switzerland Molchanov Random sets - Lecture 1. Winter School Sandbjerg, Jan 2007 1 E = R d ) Definitions
More informationRisk Measures in non-dominated Models
Purpose: Study Risk Measures taking into account the model uncertainty in mathematical finance. Plan 1 Non-dominated Models Model Uncertainty Fundamental topological Properties 2 Risk Measures on L p (c)
More informationDual utilities under dependence uncertainty
Dual utilities under dependence uncertainty Ruodu Wang Zuo Quan Xu Xun Yu Zhou 5th July 218 Abstract Finding the worst-case value of a preference over a set of plausible models is a well-established approach
More informationZero sum games Proving the vn theorem. Zero sum games. Roberto Lucchetti. Politecnico di Milano
Politecnico di Milano General form Definition A two player zero sum game in strategic form is the triplet (X, Y, f : X Y R) f (x, y) is what Pl1 gets from Pl2, when they play x, y respectively. Thus g
More informationChoquet Integral and Its Applications: A Survey
Choquet Integral and Its Applications: A Survey Zengwu Wang and Jia-an Yan Academy of Mathematics and Systems Science, CAS, China August 2006 Introduction Capacity and Choquet integral, introduced by Choquet
More informationPatryk Pagacz. Equilibrium theory in infinite dimensional Arrow-Debreu s model. Uniwersytet Jagielloński
Patryk Pagacz Uniwersytet Jagielloński Equilibrium theory in infinite dimensional Arrow-Debreu s model Praca semestralna nr 1 (semestr zimowy 2010/11) Opiekun pracy: Marek Kosiek Uniwersytet Jagielloński
More informationJournal of Mathematical Economics. Coherent risk measures in general economic models and price bubbles
Journal of Mathematical Economics ( ) Contents lists available at SciVerse ScienceDirect Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco Coherent risk measures in general
More informationLectures for the Course on Foundations of Mathematical Finance
Definitions and properties of Lectures for the Course on Foundations of Mathematical Finance First Part: Convex Marco Frittelli Milano University The Fields Institute, Toronto, April 2010 Definitions and
More informationFinancial asset bubble with heterogeneous agents and endogenous borrowing constraints
Financial asset bubble with heterogeneous agents and endogenous borrowing constraints Cuong Le Van IPAG Business School, CNRS, Paris School of Economics Ngoc-Sang Pham EPEE, University of Evry Yiannis
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationRobust Comparative Statics of Non-monotone Shocks in Large Aggregative Games
Robust Comparative Statics of Non-monotone Shocks in Large Aggregative Games Carmen Camacho Takashi Kamihigashi Çağrı Sağlam June 9, 2015 Abstract A policy change that involves redistribution of income
More informationIntertemporal Risk Aversion, Stationarity, and Discounting
Traeger, CES ifo 10 p. 1 Intertemporal Risk Aversion, Stationarity, and Discounting Christian Traeger Department of Agricultural & Resource Economics, UC Berkeley Introduce a more general preference representation
More informationMultivariate comonotonicity, stochastic orders and risk measures
Multivariate comonotonicity, stochastic orders and risk measures Alfred Galichon (Ecole polytechnique) Brussels, May 25, 2012 Based on collaborations with: A. Charpentier (Rennes) G. Carlier (Dauphine)
More informationNoncooperative Games, Couplings Constraints, and Partial Effi ciency
Noncooperative Games, Couplings Constraints, and Partial Effi ciency Sjur Didrik Flåm University of Bergen, Norway Background Customary Nash equilibrium has no coupling constraints. Here: coupling constraints
More informationA SHORT INTRODUCTION TO BANACH LATTICES AND
CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,
More informationInverse Stochastic Dominance Constraints Duality and Methods
Duality and Methods Darinka Dentcheva 1 Andrzej Ruszczyński 2 1 Stevens Institute of Technology Hoboken, New Jersey, USA 2 Rutgers University Piscataway, New Jersey, USA Research supported by NSF awards
More informationAnti-comonotone random variables and Anti-monotone risk aversion
Anti-comonotone random variables and Anti-monotone risk aversion Moez Abouda Elyéss Farhoud Abstract This paper focuses on the study of decision making under risk. We, first, recall some model-free definitions
More informationOn Kusuoka Representation of Law Invariant Risk Measures
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of
More informationPareto Optimal Allocations for Law Invariant Robust Utilities
Pareto Optimal Allocations for Law Invariant Robust Utilities on L 1 Claudia Ravanelli Swiss Finance Institute Gregor Svindland University of Munich and EPFL November 2012 Abstract We prove the existence
More informationElimination of Arbitrage States in Asymmetric Information Models
Elimination of Arbitrage States in Asymmetric Information Models Bernard CORNET, Lionel DE BOISDEFFRE Abstract In a financial economy with asymmetric information and incomplete markets, we study how agents,
More informationAmbiguity Aversion and Trade
Ambiguity Aversion and Trade Luciano I. de Castro Alain Chateauneuf First Version: March 2008 This version: April 2011 forthcoming, Economic Theory Abstract What is the effect of ambiguity aversion on
More informationDocuments de Travail du Centre d Economie de la Sorbonne
Documents de Travail du Centre d Economie de la Sorbonne Intertemporal equilibrium with production: bubbles and efficiency Stefano BOSI, Cuong LE VAN, Ngoc-Sang PHAM 2014.43 Maison des Sciences Économiques,
More informationWhen does aggregation reduce risk aversion?
When does aggregation reduce risk aversion? Christopher P. Chambers and Federico Echenique April 22, 2009 Abstract We study the problem of risk sharing within a household facing subjective uncertainty.
More informationMeasurable Ambiguity. with Wolfgang Pesendorfer. August 2009
Measurable Ambiguity with Wolfgang Pesendorfer August 2009 A Few Definitions A Lottery is a (cumulative) probability distribution over monetary prizes. It is a probabilistic description of the DMs uncertain
More information(c) For each α R \ {0}, the mapping x αx is a homeomorphism of X.
A short account of topological vector spaces Normed spaces, and especially Banach spaces, are basic ambient spaces in Infinite- Dimensional Analysis. However, there are situations in which it is necessary
More informationNear-Potential Games: Geometry and Dynamics
Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo January 29, 2012 Abstract Potential games are a special class of games for which many adaptive user dynamics
More informationAliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide
aliprantis.tex May 10, 2011 Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide Notes from [AB2]. 1 Odds and Ends 2 Topology 2.1 Topological spaces Example. (2.2) A semimetric = triangle
More informationAsymptotics of minimax stochastic programs
Asymptotics of minimax stochastic programs Alexander Shapiro Abstract. We discuss in this paper asymptotics of the sample average approximation (SAA) of the optimal value of a minimax stochastic programming
More informationOn a Class of Multidimensional Optimal Transportation Problems
Journal of Convex Analysis Volume 10 (2003), No. 2, 517 529 On a Class of Multidimensional Optimal Transportation Problems G. Carlier Université Bordeaux 1, MAB, UMR CNRS 5466, France and Université Bordeaux
More informationReward-Risk Portfolio Selection and Stochastic Dominance
Institute for Empirical Research in Economics University of Zurich Working Paper Series ISSN 1424-0459 Published in: Journal of Banking and Finance 29 (4), pp. 895-926 Working Paper No. 121 Reward-Risk
More informationConditional and Dynamic Preferences
Conditional and Dynamic Preferences How can we Understand Risk in a Dynamic Setting? Samuel Drapeau Joint work with Hans Föllmer Humboldt University Berlin Jena - March 17th 2009 Samuel Drapeau Joint work
More informationA Note on Robust Representations of Law-Invariant Quasiconvex Functions
A Note on Robust Representations of Law-Invariant Quasiconvex Functions Samuel Drapeau Michael Kupper Ranja Reda October 6, 21 We give robust representations of law-invariant monotone quasiconvex functions.
More informationConvex Analysis and Economic Theory AY Elementary properties of convex functions
Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory AY 2018 2019 Topic 6: Convex functions I 6.1 Elementary properties of convex functions We may occasionally
More informationLecture 1. Stochastic Optimization: Introduction. January 8, 2018
Lecture 1 Stochastic Optimization: Introduction January 8, 2018 Optimization Concerned with mininmization/maximization of mathematical functions Often subject to constraints Euler (1707-1783): Nothing
More informationNear-Potential Games: Geometry and Dynamics
Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo September 6, 2011 Abstract Potential games are a special class of games for which many adaptive user dynamics
More informationFunctional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32
Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak
More informationIntegral Jensen inequality
Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a
More informationof the set A. Note that the cross-section A ω :={t R + : (t,ω) A} is empty when ω/ π A. It would be impossible to have (ψ(ω), ω) A for such an ω.
AN-1 Analytic sets For a discrete-time process {X n } adapted to a filtration {F n : n N}, the prime example of a stopping time is τ = inf{n N : X n B}, the first time the process enters some Borel set
More informationDecision-making with belief functions
Decision-making with belief functions Thierry Denœux Université de Technologie de Compiègne, France HEUDIASYC (UMR CNRS 7253) https://www.hds.utc.fr/ tdenoeux Fourth School on Belief Functions and their
More informationValid Inequalities and Restrictions for Stochastic Programming Problems with First Order Stochastic Dominance Constraints
Valid Inequalities and Restrictions for Stochastic Programming Problems with First Order Stochastic Dominance Constraints Nilay Noyan Andrzej Ruszczyński March 21, 2006 Abstract Stochastic dominance relations
More informationExhaustible Resources and Economic Growth
Exhaustible Resources and Economic Growth Cuong Le Van +, Katheline Schubert + and Tu Anh Nguyen ++ + Université Paris 1 Panthéon-Sorbonne, CNRS, Paris School of Economics ++ Université Paris 1 Panthéon-Sorbonne,
More informationEquilibrium Theory under Ambiguity
Equilibrium Theory under Ambiguity Wei He and Nicholas C. Yannelis July 3, 2015 Abstract We extend the classical results on the Walras-core existence and equivalence to an ambiguous asymmetric information
More informationSOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS
APPLICATIONES MATHEMATICAE 22,3 (1994), pp. 419 426 S. G. BARTELS and D. PALLASCHKE (Karlsruhe) SOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS Abstract. Two properties concerning the space
More informationDecomposability and time consistency of risk averse multistage programs
Decomposability and time consistency of risk averse multistage programs arxiv:1806.01497v1 [math.oc] 5 Jun 2018 A. Shapiro School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta,
More informationEconomics 201B Second Half. Lecture 12-4/22/10. Core is the most commonly used. The core is the set of all allocations such that no coalition (set of
Economics 201B Second Half Lecture 12-4/22/10 Justifying (or Undermining) the Price-Taking Assumption Many formulations: Core, Ostroy s No Surplus Condition, Bargaining Set, Shapley-Shubik Market Games
More informationPortfolio optimization with stochastic dominance constraints
Charles University in Prague Faculty of Mathematics and Physics Portfolio optimization with stochastic dominance constraints December 16, 2014 Contents Motivation 1 Motivation 2 3 4 5 Contents Motivation
More informationMathematics II, course
Mathematics II, course 2013-2014 Juan Pablo Rincón Zapatero October 24, 2013 Summary: The course has four parts that we describe below. (I) Topology in Rn is a brief review of the main concepts and properties
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationAggregate Risk. MFM Practitioner Module: Quantitative Risk Management. John Dodson. February 6, Aggregate Risk. John Dodson.
MFM Practitioner Module: Quantitative Risk Management February 6, 2019 As we discussed last semester, the general goal of risk measurement is to come up with a single metric that can be used to make financial
More informationWORKING PAPER NO. 443
WORKING PAPER NO. 443 Cones with Semi-interior Points and Equilibrium Achille Basile, Maria Gabriella Graziano Maria Papadaki and Ioannis A. Polyrakis May 2016 University of Naples Federico II University
More informationGeneralized quantiles as risk measures
Generalized quantiles as risk measures F. Bellini 1, B. Klar 2, A. Müller 3, E. Rosazza Gianin 1 1 Dipartimento di Statistica e Metodi Quantitativi, Università di Milano Bicocca 2 Institut für Stochastik,
More informationMarket Equilibrium and the Core
Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange
More informationDecision principles derived from risk measures
Decision principles derived from risk measures Marc Goovaerts Marc.Goovaerts@econ.kuleuven.ac.be Katholieke Universiteit Leuven Decision principles derived from risk measures - Marc Goovaerts p. 1/17 Back
More informationIntroduction to Functional Analysis
Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture
More informationAllais, Ellsberg, and Preferences for Hedging
Allais, Ellsberg, and Preferences for Hedging Mark Dean and Pietro Ortoleva Abstract Two of the most well-known regularities observed in preferences under risk and uncertainty are ambiguity aversion and
More informationColumbia University. Department of Economics Discussion Paper Series. The Knob of the Discord. Massimiliano Amarante Fabio Maccheroni
Columbia University Department of Economics Discussion Paper Series The Knob of the Discord Massimiliano Amarante Fabio Maccheroni Discussion Paper No.: 0405-14 Department of Economics Columbia University
More informationLecture Notes October 18, Reading assignment for this lecture: Syllabus, section I.
Lecture Notes October 18, 2012 Reading assignment for this lecture: Syllabus, section I. Economic General Equilibrium Partial and General Economic Equilibrium PARTIAL EQUILIBRIUM S k (p o ) = D k k (po
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationIn the Name of God. Sharif University of Technology. Microeconomics 1. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 1 44715 (1396-97 1 st term) - Group 1 Dr. S. Farshad Fatemi Chapter 10: Competitive Markets
More informationStrategic Uncertainty and Equilibrium Selection in Discontinuous Games.
Strategic Uncertainty and Equilibrium Selection in Discontinuous Games. Philippe Bich October 21, 2016 Abstract We introduce the new concept of prudent equilibrium to model strategic uncertainty, and prove
More informationSocial Welfare Functions for Sustainable Development
Social Welfare Functions for Sustainable Development Thai Ha-Huy, Cuong Le Van September 9, 2015 Abstract Keywords: criterion. anonymity; sustainable development, welfare function, Rawls JEL Classification:
More informationBayesian Persuasion Online Appendix
Bayesian Persuasion Online Appendix Emir Kamenica and Matthew Gentzkow University of Chicago June 2010 1 Persuasion mechanisms In this paper we study a particular game where Sender chooses a signal π whose
More informationMULTIPLE-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA IN OLG MODELS WITH AMBIGUITY
MULTIPLE-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA IN OLG MODELS WITH AMBIGUITY by Kiyohiko G. Nishimura Faculty of Economics The University of Tokyo and Hiroyuki Ozaki Faculty of Economics Tohoku University
More informationTheoretical Foundation of Uncertain Dominance
Theoretical Foundation of Uncertain Dominance Yang Zuo, Xiaoyu Ji 2 Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 84, China 2 School of Business, Renmin
More informationRobust Comparative Statics for Non-monotone Shocks in Large Aggregative Games
Robust Comparative Statics for Non-monotone Shocks in Large Aggregative Games Carmen Camacho Takashi Kamihigashi Çağrı Sağlam February 1, 2016 Abstract A policy change that involves a redistribution of
More informationRepresentation theorem for AVaR under a submodular capacity
3 214 5 ( ) Journal of East China Normal University (Natural Science) No. 3 May 214 Article ID: 1-5641(214)3-23-7 Representation theorem for AVaR under a submodular capacity TIAN De-jian, JIANG Long, JI
More informationRISK AND RELIABILITY IN OPTIMIZATION UNDER UNCERTAINTY
RISK AND RELIABILITY IN OPTIMIZATION UNDER UNCERTAINTY Terry Rockafellar University of Washington, Seattle AMSI Optimise Melbourne, Australia 18 Jun 2018 Decisions in the Face of Uncertain Outcomes = especially
More informationMULTI-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA: INDETERMINACY, COMPLEXITY AND SUSTAINED DEFLATION
JSPS Grants-in-Aid for Scientific Research (S) Understanding Persistent Deflation in Japan Working Paper Series No. 058 December 2014 MULTI-BELIEF RATIONAL-EXPECTATIONS EQUILIBRIA: INDETERMINACY, COMPLEXITY
More informationCore equivalence and welfare properties without divisible goods
Core equivalence and welfare properties without divisible goods Michael Florig Jorge Rivera Cayupi First version November 2001, this version May 2005 Abstract We study an economy where all goods entering
More informationLower semicontinuous and Convex Functions
Lower semicontinuous and Convex Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2017 Outline Lower Semicontinuous Functions
More informationCertainty Equivalent Representation of Binary Gambles That Are Decomposed into Risky and Sure Parts
Review of Economics & Finance Submitted on 28/Nov./2011 Article ID: 1923-7529-2012-02-65-11 Yutaka Matsushita Certainty Equivalent Representation of Binary Gambles That Are Decomposed into Risky and Sure
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationThe Ohio State University Department of Economics. Homework Set Questions and Answers
The Ohio State University Department of Economics Econ. 805 Winter 00 Prof. James Peck Homework Set Questions and Answers. Consider the following pure exchange economy with two consumers and two goods.
More informationDifferentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries
Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries Econ 2100 Fall 2017 Lecture 19, November 7 Outline 1 Welfare Theorems in the differentiable case. 2 Aggregate excess
More informationB. Appendix B. Topological vector spaces
B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function
More informationMATH 426, TOPOLOGY. p 1.
MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p
More informationPersuading a Pessimist
Persuading a Pessimist Afshin Nikzad PRELIMINARY DRAFT Abstract While in practice most persuasion mechanisms have a simple structure, optimal signals in the Bayesian persuasion framework may not be so.
More informationStrictly monotonic preferences on continuum of goods commodity spaces
Strictly monotonic preferences on continuum of goods commodity spaces Carlos Hervés-Beloso RGEA. Universidad de Vigo. Paulo K. Monteiro Fundação Getúlio Vargas. VI Encuentro de Análisis Funcional y Aplicaciones
More informationTWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES. S.S. Dragomir and J.J.
RGMIA Research Report Collection, Vol. 2, No. 1, 1999 http://sci.vu.edu.au/ rgmia TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES S.S. Dragomir and
More informationThe newsvendor problem with convex risk
UNIVERSIDAD CARLOS III DE MADRID WORKING PAPERS Working Paper Business Economic Series WP. 16-06. December, 12 nd, 2016. ISSN 1989-8843 Instituto para el Desarrollo Empresarial Universidad Carlos III de
More informationA NICE PROOF OF FARKAS LEMMA
A NICE PROOF OF FARKAS LEMMA DANIEL VICTOR TAUSK Abstract. The goal of this short note is to present a nice proof of Farkas Lemma which states that if C is the convex cone spanned by a finite set and if
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationRandom Process Lecture 1. Fundamentals of Probability
Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/43 Outline 2/43 1 Syllabus
More informationNotes on Distributions
Notes on Distributions Functional Analysis 1 Locally Convex Spaces Definition 1. A vector space (over R or C) is said to be a topological vector space (TVS) if it is a Hausdorff topological space and the
More informationJanuary 29, Introduction to optimization and complexity. Outline. Introduction. Problem formulation. Convexity reminder. Optimality Conditions
Olga Galinina olga.galinina@tut.fi ELT-53656 Network Analysis Dimensioning II Department of Electronics Communications Engineering Tampere University of Technology, Tampere, Finl January 29, 2014 1 2 3
More informationUSING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION
USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye towards defining weak solutions to elliptic PDE. Using Lax-Milgram
More informationSecond Welfare Theorem
Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part
More informationChoice under uncertainty with the best and worst in mind: Neo-additive capacities
Journal of Economic Theory 37 (2007) 538 567 www.elsevier.com/locate/jet Choice under uncertainty with the best and worst in mind: Neo-additive capacities Alain Chateauneuf a, Jürgen Eichberger b,, Simon
More information