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Academic Commons Search Resultsen-usAnalogical reasoning in decision processes
http://academiccommons.columbia.edu/catalog/ac:114940
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:450Mon, 28 Mar 2011 00:00:00 +0000We give a definition of reasoning by analogy, which is tailored to a setting of decision making under uncertainty. We present a model of decision making which is based on such a definition, and show that it is compatible with a large class of preferences.Economic theoryma734EconomicsWorking papersStates, models and information: A reconsideration of Ellsberg's paradox
http://academiccommons.columbia.edu/catalog/ac:114913
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:449Fri, 25 Mar 2011 00:00:00 +0000We show that a general process of decision making involves uncertainty about two different sets: the domain of the acts and another set, which we call the set of models for the decision maker. We study the effect of different information structures on the set of models, and prove the existence of a dichotomy: either the decision maker's ranking of the acts obeys Subjective Expected Utility theory or there are many events to which probabilities cannot be assigned. We use this result to formalize the idea of Knightian Uncertainty. The relevance of information structures associated to Knightian Uncertainty is shown by means of examples, one of which is a version of Ellsberg's experiments. Our findings show that a decision maker faces, generally speaking, uncertainties of two different types - "uncertainty about which state obtains" and "uncertainty about how the world works" and that Savage's theory considers only uncertainty of the first type. Finally, in situations of Knightian Uncertainty, we identify the class of events to which probabilities can be assigned, and study the relation with the class of unambiguous events in the sense of [13] and [25].Economic theoryma734EconomicsWorking papersStates, 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 papersCores of non-atomic market games
http://academiccommons.columbia.edu/catalog/ac:115124
Amarante, Massimiliano; Maccheroni, Fabio; Marinacci, M.; Montrucchio, L.http://hdl.handle.net/10022/AC:P:458Fri, 25 Mar 2011 00:00:00 +0000We study the cores of non-atomic market games, a class of transferable utility cooperative games introduced by Aumann and Shapley [2], and, more in general, of those games that admit a na-continuous and concave extension to the set of ideal coalitions, studied by Einy, Moreno, and Shitovitz [12]. We show that the core of such games is norm compact and we provide some representation results. We also give a Multiple Priors interpretation of some of our results.We study the cores of non-atomic market games, a class of transferable utility cooperative.Economic theoryma734EconomicsWorking papersAmbiguous events
http://academiccommons.columbia.edu/catalog/ac:116095
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:499Thu, 24 Mar 2011 00:00:00 +0000We focus on a class of Multiple Prior Models. Those characterized by nonatomic countably additive priors. Preferences generating such representations have been recently axiomatized in [17]. We argue that this is the proper setting for comparing the notions of unambiguous event given by Epstein and Zhang in [7] and by Ghirardato, Maccheroni and Marinacci in [10]. The two definitions are known to be nonequivalent. Our main result is that an event T is unambiguous in the sense of Epstein and Zhang if and only if either (i) it is unambiguous in the sense of [10]; or (ii) conditional on T, the decision maker is an expected utility maximizer. We also provide an easy operational criterion for establishing whether or not an event is unambiguous in the sense of Epstein and Zhang.Economic theoryma734EconomicsWorking papersOn the uniqueness of convex-ranged probabilities
http://academiccommons.columbia.edu/catalog/ac:116218
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:504Thu, 24 Mar 2011 00:00:00 +0000We provide an alternative proof of a theorem of Marinacci [2] regarding the equality of two convex-ranged measures. Specifically, we show that, if P and Q are two nonatomic, countably additive probabilities on a measurable space (S, Σ), the condition [∃A∗ ∈ Σ with 0 < P(A∗) < 1 such that P(A∗) = P(B)=⇒ Q(A∗) = Q(B) whenever B∈Σ] is equivalent to the condition [∀A,B ∈ Σ P(A) > P(B)=⇒ Q(A) ≥ Q(B)]. Moreover, either one is equivalent to P = Q.Economic theoryma734EconomicsWorking papersAmbiguity, measurability and multiple priors
http://academiccommons.columbia.edu/catalog/ac:116241
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:505Thu, 24 Mar 2011 00:00:00 +0000The paper provides a notion of measurability which is suited for a class of Multiple Prior Models. Those characterized by nonatomic countably additive priors. Preferences generating such representations have been recently axiomatized in [12]. A notable feature of our definition of measurability is that an event is measurable if and only if it is unambiguous in the sense of Ghirardato, Maccheroni and Marinacci [8]. In addition, the paper contains a thorough description of the basic properties of the family of measurable/unambiguous sets, of the measure defined on those and of the dependence of the class of measurable sets on the set of priors. The latter is obtained by means of an application of Lyapunov's convexity theorem.Economic theoryma734EconomicsWorking papersEquivalence of public mixed-strategies and private behavior-strategies in games with public monitoring
http://academiccommons.columbia.edu/catalog/ac:116333
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:509Thu, 24 Mar 2011 00:00:00 +0000In repeated games with public monitoring, the consideration of behavior strategies makes relevant the distinction between public and private strategies. Recently, Kandori and Obara [5] and Mailath, Matthews and Sekiguchi [7] have provided examples of games with equilibrium payoffs in private strategies which lie outside the set of Public Perfect Equilibrium payoffs. The present paper focuses on another distinction, that between mixed and behavior strategies. It is shown that, as far as with mixed strategies one is concerned, the restriction to public strategies is not a restriction at all. Our result provides a general explanation for the findings of Kandori and Obara [5] and Mailath, Matthews and Sekiguchi [7] as well as a general method for constructing examples of that sort.Economic theoryma734EconomicsWorking papersEquivalence of public mixed-strategies and private behavior strategies in games with public monitoring
http://academiccommons.columbia.edu/catalog/ac:113060
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:370Wed, 23 Mar 2011 00:00:00 +0000In repeated games with public monitoring, the consideration of behavior strategies makes relevant the distinction between public and private strategies. Recently, Kandori and Obara [6] and Mailath, Matthews and Sekiguchi [8] have provided examples of games with equilibrium payoffs in private strategies which lie outside the set of Public Perfect Equilibrium payoffs. The present paper focuses on another distinction, that between mixed and behavior strategies. It is shown that, as far as with mixed strategies one is concerned, the restriction to public strategies is not a restriction at all.Economic theoryma734EconomicsWorking papersRecursive structure and equilibria in games with private monitoring
http://academiccommons.columbia.edu/catalog/ac:113083
Amarante, Massimilianohttp://hdl.handle.net/10022/AC:P:371Wed, 23 Mar 2011 00:00:00 +0000In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming.Economic theoryma734EconomicsWorking papers