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104
Ambiguity Made Precise: A Comparative Foundation
, 2000
"... The theory of subjective expected utility (SEU) has been recently extended to allow ambiguity to matter for choice. We propose a notion of absolute ambiguity aversion by build#wJ on a notion of comparative ambiguity aversion. We characterize it for a preference mo d#w which encompasses some of the m ..."
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Cited by 141 (23 self)
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The theory of subjective expected utility (SEU) has been recently extended to allow ambiguity to matter for choice. We propose a notion of absolute ambiguity aversion by build#wJ on a notion of comparative ambiguity aversion. We characterize it for a preference mo d#w which encompasses some of the most popular mod#wk in the literature. We nextbuild on these id#se to provid# ad efinition of unambiguous actand event, and showthe characterization of the latter. As an illustration, we consider the classical Ellsberg 3color urn problemand find that the notions developed in the paper provide intuitive answers.
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 100 (14 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
Densities of Idempotent Measures and Large Deviations
 AMS, AND INRIA REPORT N
, 1995
"... Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation i ..."
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Cited by 43 (10 self)
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Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle as defined by Varadhan.
On Independence for NonAdditive Measures, with a Fubini Theorem
, 1997
"... An important technical question arising in economic and financial applications of decision models with nonadditive beliefs is how to define stochastic independence. In fact the straightforward generalization of independence does not in general yield a unique product. I discuss the problem of indepe ..."
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Cited by 31 (0 self)
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An important technical question arising in economic and financial applications of decision models with nonadditive beliefs is how to define stochastic independence. In fact the straightforward generalization of independence does not in general yield a unique product. I discuss the problem of independence, with specific focus on the validity of the Fubini theorem. The latter holds in general only for a special class of functions. It also requires a stronger notion of independent product. This is unique when the product must be a belief function. Finally I discuss an application to the issue of randomization in decision making.
Potential estimates for a class of fully nonlinear elliptic equations
 Duke Math. J
"... We study the pointwise properties of ksubharmonic functions, that is, the viscosity subsolutions to the fully nonlinear elliptic equations Fk[u] = 0, where Fk[u] is the elementary symmetric function of order k, 1 ≤ k ≤ n, of the eigenvalues of [ D 2 u], F1[u] = �u, Fn[u] = det D 2 u. Thus 1subh ..."
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Cited by 29 (0 self)
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We study the pointwise properties of ksubharmonic functions, that is, the viscosity subsolutions to the fully nonlinear elliptic equations Fk[u] = 0, where Fk[u] is the elementary symmetric function of order k, 1 ≤ k ≤ n, of the eigenvalues of [ D 2 u], F1[u] = �u, Fn[u] = det D 2 u. Thus 1subharmonic functions are subharmonic in the classical sense; nsubharmonic functions are convex. We use a special capacity to investigate the typical questions of potential theory: local behaviour, removability of singularities, and polar, negligible, and thin sets, and we obtain estimates for the capacity in terms of the Hausdorff measure. We also prove the Wiener test for the regularity of a boundary point for the Dirichlet problem for the fully nonlinear equation Fk[u] = 0. The crucial tool in the proofs of these results is the Radon measure Fk[u] introduced recently by N. Trudinger and X.J. Wang for any ksubharmonic u. We use ideas from the potential theories both for the complex MongeAmpère and for the pLaplace equations.
DIRICHLET DUALITY AND THE NONLINEAR DIRICHLET PROBLEM ON RIEMANNIAN MANIFOLDS
"... In this paper we study the Dirichlet problem for fully nonlinear secondorder equations on a riemannian manifold. As in our previous paper [HL4] we define equations via closed subsets of the 2jet bundle where each equation has a natural dual equation. Basic existence and uniqueness theorems are esta ..."
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Cited by 24 (13 self)
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In this paper we study the Dirichlet problem for fully nonlinear secondorder equations on a riemannian manifold. As in our previous paper [HL4] we define equations via closed subsets of the 2jet bundle where each equation has a natural dual equation. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex MongeAmpère equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain Ω ⊂ ⊂ X, we give geometric conditions which imply that the Dirichlet problem on Ω is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local
Normed groups: dichotomy and duality
"... The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a ri ..."
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Cited by 21 (14 self)
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The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a rightinvariant metric carrying ‡ows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closedgraph theorem, to the Birkho¤Kakutani metrization theorem and further back still to Banach’s Théorie des opérations linéaires. We collect together known salient features and develop their theory including Steinhaus theory uni…ed by the Category Embedding Theorem [BOst11], the associated themes of subadditivity and convexity, and a topological duality inherent to topological
On the measurement of inequality under uncertainty
 Journal of Economic Theory
, 1997
"... To take into account both ex ante and ex post inequality considerations, one has to deal with inequality and uncertainty simultaneously. Under certainty, much of the literature has focused on ‘‘comonotonically linear’ ’ indices: functionals that are linear on cones of income profiles that agree on t ..."
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Cited by 21 (0 self)
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To take into account both ex ante and ex post inequality considerations, one has to deal with inequality and uncertainty simultaneously. Under certainty, much of the literature has focused on ‘‘comonotonically linear’ ’ indices: functionals that are linear on cones of income profiles that agree on the social ranking of the individuals. This family generalizes both the Gini index and the egalitarian index (minimal income). However, it does not include functionals such as the average of expectedGini and Giniofexpectation. In contrast, the family of minofmeans functionals is rich enough for this purpose. Journal of Economic Literature
Capacities and probabilistic beliefs: A precarious coexistence
 Mathematical Social Sciences
"... materials circulated to invite discussion and critical comment. These papers may be freely circulated but to protect their tentative character they ar'e nor to be quoted without the permission of the author. ..."
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Cited by 21 (1 self)
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materials circulated to invite discussion and critical comment. These papers may be freely circulated but to protect their tentative character they ar'e nor to be quoted without the permission of the author.
A Formula for Incorporating Weights into Scoring Rules
 Theoretical Computer Science
, 1998
"... A "scoring rule" is an assignment of a value to every tuple (of varying sizes). This paper is concerned with the issue of how to modify a scoring rule to apply to the case where weights are assigned to the importance of each argument. We give an explicit formula for incorporating weights t ..."
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Cited by 20 (1 self)
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A "scoring rule" is an assignment of a value to every tuple (of varying sizes). This paper is concerned with the issue of how to modify a scoring rule to apply to the case where weights are assigned to the importance of each argument. We give an explicit formula for incorporating weights that can be applied no matter what the underlying scoring rule is. The formula is surprisingly simple, in that it involves far fewer terms than one might have guessed. It has three further desirable properties. The first desirable property is that when all of the weights are equal, then the result is obtained by simply using the underlying scoring rule. Intuitively, this says that when all of the weights are equal, then this is the same as considering the unweighted case. The second desirable property is that if a particular argument has zero weight, then that argument can be dropped without affecting the value of the result. The third desirable property is that the value of the result is a continuous ...