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Academic Commons Search Resultsen-usStates, models and unitary equivalence I: Representation theorems and analogical reasoning
http://academiccommons.columbia.edu/catalog/ac:115407
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:470Fri, 25 Mar 2011 00:00:00 +0000I show that virtually any model of decision making under uncertainty is associated to a certain structure. This contains three fundamental ingredients: (1) The domain of the acts; (2) Another set, which is called the set of models for the decision maker; and (3) The decision maker's information about the set of models (an algebra of subsets of the set of models). A consequence of this finding is that that the decision maker's choices can be viewed as the outcome of a two-stage process. First, the set of acts is mapped into a system of hypothetical bets on the set of models. Then, the latter are ranked by the decision maker. I show that this procedure can be thought of as describing a general form of analogical reasoning. I also observe that the appearance of two different sets implies that the decision maker is uncertain about two different objects and that he may receive information about any of them. In particular, information about the set of models affects the decision maker's ranking of the available alternatives. In the sequel to this paper, I show that certain natural information structures lead to an inherent inability of assigning probabilities on the domain of the acts. In a formal sense, their properties describe the idea of Knightian Uncertainty.Economic theoryma734EconomicsWorking papersAmbiguous events and Maxmin Expected Utility
http://academiccommons.columbia.edu/catalog/ac:115430
Amarante, Massimiliano; Filiz, Emelhttp://hdl.handle.net/10022/AC:P:471Fri, 25 Mar 2011 00:00:00 +0000We study the properties associated to various definitions of ambiguity ([8], [9], [18] and [23]) in the context of Maximin Expected Utility (MEU). We show that each definition of unambiguous events produces certain restrictions on the set of priors, and completely characterize each definition in terms of the properties it imposes on the MEU functional. We apply our results to two open problems. First, in the context of MEU, we show the existence of a fundamental incompatibility between the axiom of "Small unambiguous event continuity" ([8]) and the notions of unambiguous event due to Zhang [23] and Epstein-Zhang [8]. Second, we show that, in the context of MEU, the classes of unambiguous events according to either Zhang [23] or Epstein-Zhang [8] are always λ-systems. Finally, we reconsider the various definitions in light of our findings, and identify some new objects (Z-filters and EZ-filters) corresponding to properties which, while neglected in the current literature, seem relevant to us.Economic theoryma734, ef2011EconomicsWorking papersThe knob of the discord
http://academiccommons.columbia.edu/catalog/ac:115381
Amarante, Massimiliano; Maccheroni, Fabiohttp://hdl.handle.net/10022/AC:P:469Fri, 25 Mar 2011 00:00:00 +0000For (S, Σ) a measurable space, let C1 and C2 and be convex, weak* closed sets of probability measures on Σ. We show that if C1 ∪ C2 satisfies the Lyapunov property, then there exists a set A ∈ Σ such that minμ1 ∈ C1 μ1(A) > maxμ2 ∈ C2 (A). We give applications to Maxmin Expected Utility and to the core of a lower probability.Economic theoryma734EconomicsWorking papers