On the multi-utility representation of preference relations (original) (raw)
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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.
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Annals of Operations Research, 2008
In the field of Artificial Intelligence many models for decision making under uncertainty have been proposed that deviate from the traditional models used in Decision Theory, i.e. the Subjective Expected Utility (SEU) model and its many variants. These models aim at obtaining simple decision rules that can be implemented by efficient algorithms while based on inputs that are less rich than what is required in traditional models. One of these models, called the likely dominance (LD) model, consists in declaring that an act is preferred to another as soon as the set of states on which the first act gives a better outcome than the second act is judged more likely than the set of states on which the second act is preferable. The LD model is at much variance with the SEU model. Indeed, it has a definite ordinal flavor and it may lead to preference relations between acts that are not transitive. This paper proposes a general model for decision making under uncertainty tolerating intransitive and/or incomplete preferences that will contain both the SEU and the LD models as particular cases. Within the framework of this general model, we propose a characterization of the preference relations that can be obtained with the LD model. This characterization shows that the main distinctive feature of such relations lies in the very poor relation comparing preference differences that they induce on the set of outcomes.
2001
A qualitative counterpart to Von Neumann and Morgenstern's Expected Utility Theory of decision under uncertainty was recently proposed by Dubois and Prade. In this model, belief states are represented by normalised possibility distributions over an ordinal scale of plausibility, and the utility (or preference) of consequences of decisions are also measured in an ordinal scale. In this paper we extend the original Dubois and Prade's decision model to cope with partially inconsistent descriptions of belief states, represented by non-normalised possibility distributions. Subnormal possibility distributions frequently arise when adopting the possibilistic model for casebased decision problems. We consider two qualitative utility functions, formally similar to the original ones up to modifying factors coping with the inconsistency degree of belief states. We provide axiomatic characterizations of the preference orderings induced by these utility functions.
Incomplete preference relations: An upper bound condition
In decision making, consistency in fuzzy preference relations is associated with the study of transitivity property. While using additive consistency property to complete incomplete preference relations, the preference values found may lie outside the interval [0, 1] or the resultant relation may itself be inconsistent. This paper proposes a method that avoids inconsistency and completes an incomplete preference relation using an upper bound condition. Additionally, the paper extends the upper bound condition for multiplicative reciprocal preference relations. The proposed methods ensure that if (n − 1) preference values are provided by an expert, such that they satisfy the upper bound condition, then the preference relation is completed such that the estimated values lie inside the unit interval [0, 1] in the case of preference relations and [1/9, 9] in the case of multiplicative preference relation. Moreover, the resultant preference relation obtained using the proposed method is transitive.
CANONICAL REPRESENTATION OF PREFERENCES OVER A SET OF PROBABILITY DISTRIBUTIONS
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.
Lexicographic Combinations of Preference Relations in the Context of Possibilistic Decision Theory
2006
In Possibilistic Decision Theory (PDT), decisions are ranked by a pes- simistic or by an optimistic qualitative criteria. The preference relations induced by these criteria have been axiomatized by corresponding sets of rationality postulates, botha la von Neumann and Morgenstern anda la Savage. In this paper we first address a particular issue regarding the ax- iomatic systems of PDTa la von Neumann and Morgenstern. Namely, we show how to adapt the axiomatic systems for the pessimistic and optimistic criteria when some finiteness assumptions in the original model are dropped. Second, we show that a recent axiomatic approach by Giang and Shenoy using binary utilities can be captured by preference relations defined as lexi- cographic refinements of the above two criteria. We also provide an axiomatic characterization of these lexicographic refinements.