1,721,002 research outputs found
Objective rationality and uncertainty averse preferences
Full text linkAs in Gilboa, Maccheroni, Marinacci, and Schmeidler \cite{GMMS}, we consider a decision maker characterized by two binary relations: and . The first binary relation is a Bewley preference. It\ models the rankings for which the decision maker is sure. The second binary relation is an uncertainty averse preference, as defined by Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio \cite{CMMM}. It models the rankings that the decision maker expresses if he has to make a choice. We assume that is a completion of .\ We identify axioms under which the set of probabilities and the utility index representing are the same as those representing . In this way, we show that Bewley preferences and uncertainty averse preferences, two different approaches in modelling decision making under Knightian uncertainty, are complementary. As a by-product, we extend the main result of Gilboa, Maccheroni, Marinacci, and Schmeidler \cite{GMMS}, who restrict their attention to maxmin expected utility completions
Ambiguity aversion and wealth effects
Full text linkWe study how changes in wealth affect ambiguity attitudes. We define a decision maker as decreasing (resp., increasing) absolute ambiguity averse if he becomes less (resp., more) ambiguity averse as he becomes richer. Our definition is behavioral. We provide different characterizations of these attitudes for a large class of preferences: monotone and continuous preferences which satisfy risk independence. We then specialize our results for different subclasses of preferences. Inter alia, our characterizations provide alternative ways to test experimentally the validity of some of the models of choice under uncertainty
Orthogonal decompositions in Hilbert A-modules
No full textPre-Hilbert A-modules are a natural generalization of inner product spaces in which the scalars are allowed to be from an arbitrary algebra. In this perspective, submodules are the generalization of vector subspaces. The notion of orthogonality generalizes in an obvious way too. We provide necessary and sufficient topological conditions for a submodule to be orthogonally complemented. Then, we present four applications of our results. The most important ones are Doob's and Kunita–Watanabe's decompositions for conditionally square-integrable processes. They are obtained as orthogonal decomposition results carried out in an opportune pre-Hilbert A-module. Second, we show that a version of Stricker's Lemma can be also derived as a corollary of our results. Finally, we provide a version of the Koopman–von-Neumann decomposition theorem for a specific pre-Hilbert module which is useful in Ergodic Theory
A characterization of probabilities with full support and the Laplace method
Full text linkWe show that a probability measure on a metric space X has full support if and only if the set of all probability measures that are absolutely continuous with respect to it is dense in the set of all probability measures on X. We illustrate the result through a general version of Laplace method, which in turn leads to a general stochastic convergence result to global maxima
Abstract integration of set-valued functions
No full textWe develop an abstract notion of integration for Effros measurable correspondences whose values are weakly compact subsets of a separable Banach space. This notion is built on a basic monotonicity hypothesis and the simple requirements that the integral assigns at most one value to any single-valued correspondence and evaluates the constant functions in the obvious way; linearity of the integral is not required. These hypotheses alone guarantee that the abstract integral is relatively weakly compact-valued, and its closed convex hull decomposes into the abstract integrals of the measurable selections from that correspondence. We use this decomposition theorem to prove a Fatou-type lemma and a monotone convergence theorem, and to derive necessary and sufficient conditions for the linearity and parametric continuity of the abstract integral. In turn, we apply our main results to obtain simple characterizations of some classical set-valued integrals, and derive (possibly nonadditive) aggregation methods for correspondences. All in all, we find that abstract integration theory yields many results about particular integrals for set-valued maps in a unified manner, often with minimal recourse to measure-theoretic arguments
Equilibria of nonatomic anonymous games
Full text linkWe define a new notion of equilibrium for nonatomic anonymous games, termed epsilon-estimated equilibrium, and prove its existence for any positive s. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk-Nash. This augmented scope is our main motivation. Our approach also resolves some conceptual problems present in Schmeidler (1973) pointed out by Shapley
Caution and reference effects
Full text linkWe introduce Cautious Utility, a new model based on the idea that individuals are
unsure of trade-offs between goods and apply caution. The model yields an endowment effect, even when gains and losses are treated symmetrically. Moreover, it implies either loss aversion or loss neutrality for risk, but in a way unrelated to the endowment effect, and it captures the certainty effect, providing a novel unified explanation of all three phenomena. Cautious Utility can help organize empirical evidence, including some that directly contradicts leading alternatives
Cautious expected utility and the certainty effect
No full textMany violations of the Independence axiom of Expected Utility can be traced to subjects' attraction to risk-free prospects. The key axiom in this paper, Negative Certainty Independence (Dillenberger, 2010), formalizes this tendency. Our main result is a utility representation of all preferences over monetary lotteries that satisfy Negative Certainty Independence together with basic rationality postulates. Such preferences can be represented as if the agent were unsure of how to evaluate a given lottery p; instead, she has in mind a set of possible utility functions over outcomes and displays a cautious behavior: she computes the certainty equivalent of p with respect to each possible function in the set and picks the smallest one. The set of utilities is unique in a well-defined sense. We show that our representation can also be derived from a "cautious" completion of an incomplete preference relation
The structure of variational preferences
No full textMaccheroni, Marinacci, and Rustichini (2006), in an Anscombe–Aumann framework, axiomatically characterize preferences that are represented by the variational utility functional V. In this paper, for a given variational preference, we study the class C of functions c that represent V. Inter alia, we show that this set is fully characterized by a minimal and a maximal element. The minimal element, also identified by Maccheroni, Marinacci, and Rustichini (2006), fully characterizes the decision maker’s attitude toward uncertainty, while the novel maximal element characterizes the uncertainty perceived by the decision maker
Ambiguity and robust statistics
No full textStarting with the seminal paper of Gilboa and Schmeidler (1989) an analogy between the maxmin approach of decision theory under Ambiguity and the minimax approach of Robust Statistics e.g., Blum and Rosenblatt (1967) has been hinted at. The present paper formally clarifies this relation by showing the conditions under which the two approaches are actually equivalent
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