Maxmin Expected Utility with a Non-Unique Prior

Applied Mathematical Finance, 2009

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.

Social Science Research Network, 2000

Annals of Statistics, 1995

This essay considers decision-theoretic foundations for robust Bayesian statistics. We modify the approach of Ramsey, de Finetti, Savage and Anscombe and Aumann in giving axioms for a theory of robust preferences. We establish that preferences which satisfy axioms for robust preferences can be represented by a set of expected utilities. In the presence of two axioms relating to state-independent utility, robust preferences are represented by a set of probability/utility pairs, where the utilities are almost state-independent (in a sense which we make precise). Our goal is to focus on preference alone and to extract whatever probability and/or utility information is contained in the preference relation when that is merely a partial order. This is in contrast with the usual approach to Bayesian robustness that begins with a class of "priors" or "likelihoods," and a single loss function, in order to derive preferences from these probabilityfutility assumptions.

2018

We contrast three decisions rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: -Maximin, Maximality, and -admissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that carries greater expected utility for each probability in a (closed) convex set. If the convex set is a singleton, then each rule agrees with maximizing expected utility. We show that, even when the option set is convex, this pairwise comparison between acts may fail to identify those acts which are Bayes for some probability in a convex set that is not closed. This limitation affects two of the decision rules but not -admissibility, which is not a pairwise decision rule. -admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes.

We prove the existence of an expected utility function for preferences over probabilistic prospects satisfying Strict Monotonicity, Indifference, the Common Ration Property, Substitution and Reducibility of Extreme Prospects. The example in Rubinstein (1988) that is inconsistent with the existence of a von Neumann-Morgenstern for preferences over probabilistic prospects, violates the Common Ratio Property. Subsequently, we prove the existence of expected utility functions with piecewise linear Bernoulli utility functions for preferences that are piece-wise linear. For this case a weaker version of the Indifference Assumption that is used in the earlier existence theorems is sufficient. We also state analogous results for probabilistic lotteries. We do not require any compound prospects or mixture spaces to prove any of our results. In the second last section of this paper, we “argue” that the observations related to Allais paradox, do not constitute a violation of expected utility maximization by individuals, but is a likely manifestation of individuals assigning (experiment or menu-dependent?) subjective probabilities to events which disagree with their objective probabilities.

Building a model of individual preferences is a key for rational decision-making under uncertainty. Since preference can hardly be studied completely, its approximation from partially known preference is very important. The present paper provides a framework for building such approximation for regular preferences on abstract partially ordered set, and then applies the results to preferences on the set of distributions, thus establishing a link to decision-making.

The Review of Economic Studies, 1991

Economic Theory, 2009

This paper provides a new axiomatization for expected utility from a frequentist perspective. Given a set of outcomes, we consider preference relations on the set of infinite sequences of the outcomes with well-defined relative frequencies (wdf sequences), instead of the set of probability distributions. Each wdf sequence represents a probability distribution. We propose a system of axioms analogous to those for expected utility theory, and prove a representation theorem that uses the long-term average criterion with a uniquely determined instantaneous utility function up to linear transformation.

2006

In this paper we extend Savage's theory of decision-making under uncertainty from a classical environment into a non-classical one. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility.

2003

Abstract Intelligent agents often need to assess user utility functions in order to make decisions on their behalf, or predict their behavior. When uncertainty exists over the precise nature of this utility function, one can model this uncertainty using a distribution over utility functions. This view lies at the core of games with incomplete information and, more recently, several proposals for incremental preference elicitation.

Theory and decision, 1988

Metroeconomica, 1978

In this paper we would like to do two things: firstly, to show that the proof of the von-Neumann-Moorgenstern linear utility theorem can be simplified by noting the isomorphism of the problem with Rn secondly, we would like to argue that there is hidden within the von-Neumann-Morgenstern assumptions an unacceptable condition on commodity preferences.

Kybernetika, 2015

A generalized notion of lottery is considered, where the uncertainty is expressed by a belief function. Given a partial preference relation on an arbitrary set of generalized lotteries all on the same finite totally ordered set of prizes, conditions for the representability, either by a linear utility or a Choquet expected utility are provided. Both the cases of a finite and an infinite set of generalized lotteries are investigated.

Economic Theory, 1993

We provide necessary and sufficient conditions for weak (semi)continuity of the expected utility. Such conditions are also given for the weak compactness of the domain of the expected utility. Our results have useful applications in cooperative solution concepts in economies and games with differential information, in noncooperative games with differential information and in principal-agent problems. I Introduction Recent work on cooperative solution concepts in economies and games with differential information (e.g. Yannelis [25,1, Krasa-Yannelis 1-16-1, Allen [2,3,1, Koutsougeras-Yannelis [17], Page [22]) has necessitated the consideration of conditions that guarantee the (semi)continuity of an agent's expected utility.1 Specifically, in this paper (g2, ~, P) is a probability space, representing the states of the world and their governing distribution, (V, I1" 11) a separable Banach space of commodities, and X :,(2 ~ 2 v a set-valued function, prescribing for each state ~o of the world the set X(~o) of possible consumptions. We define the set L~a~ of feasible state contingent consumption plans to consist of all Bochner integrable a.e. selections of X, that is, the set of all x~&~ such that x(~o)~X(og) a.e. in .(2. As usual, 5e~, stands for the (prequotient) set of all Bochner-integrable V-valued functions on (s ~, P); the ~ 1-seminorm on this space is defined by LIx tll:= LIx(,.

Lecture Notes in Computer Science, 2007

The aim of the paper is to extend the Savage like axiomatization of possibilistic preference functionals in qualitative decision theory to conditional acts, so as to make a step towards the dynamic decision setting. To this end, the de Finetti style approach to conditional possibility recently advocated by Coletti and Vantaggi is exploited, extending to conditional acts the basic axioms pertaining to conditional events.