Jean-Yves Jaffray

Some Characterizations of Lower Probabilities and Other Monotone Capacities through the Use of Mobius Inversion

Monotone capacities (on finite sets) of finite or infinite order (lower probabilities) are characterized by properties of their Mobius inverses. A necessary property of probabilities dominating a given capacity is demonstrated through the use of Gale’s theorem for the transshipment problem. This property is shown to be also sufficient if and only if the capacity is monotone of infinite order. A characterization of dominating probabilities specific to capacities of order 2 is also proved.

On the extension of additive utilities to infinite sets

Abstract Independence condition C is known as necessary and sufficient for the existence of an additive utility on a finite subset X of a Cartesian product. A stronger necessary condition, H, interpreted as both an independence and Archimedean condition, is derived. It is shown to be sufficient when X is countable by constructing an additive utility as the limit of a sequence of additive utilities on finite subsets of X. When X is not countable, but is a Cartesian product, another necessary condition, the existence of A, a countable perfectly (order-) dense subset of X, is added to H; an additive utility is constructed by extension to X of an additive utility on a countable set linked to A. An application to a no-solvability case is given.

An experimental study of updating ambiguous beliefs

‘Ambiguous beliefs’ are beliefs which are inconsistent with a unique, additive prior. The problem of their update in face of new information has been dealt with in the theoretical literature, and received several contradictory answers. In particular, the ‘maximum likelihood update’ and the ‘full Bayesian update’ have been axiomatized. This experimental study attempts to test the descriptive validity of these two theories by using the Ellsberg experiment framework.

Coherent bets under partially resolving uncertainty and belief functions

By considering situations of partially resolving uncertainty, a one-to-one correspondence between belief functions and coherent betting systems à la de Finetti is shown to exist.

Application of Linear Utility Theory to Belief Functions

Dempster-Shafer theory (Dempster[ 3 ], Sharer[ 9 ], [10] ) represents knowledge about events through the use of a generally non-additive set-function, termed a lower probability function (Dempster) or a belief function (Shafer). As shown in [3 ], there exists a wide class of uncertainty situations in which the objective information available naturally takes this form.[ 9 ]emphasizes the usefulness Of belief functions in the describing of subjective judgments. Dempster-Shafer theory is of little interest for decision analysts in the absence of a complementary decision model. This paper proposes that the mode] resulting from the application of von Neumann-Morgenstern linear utility theory to belief functions be adopted in such cases. The terminology and notations of Dempster-Shafer theory as they are presented in this paper are roughly those found in[ 9 ]. Proofs of the main properties of belief functions can be found in[ 3 ], [9 ], or in Chateauneuf and Jaffray[ 2 ]. For linear utility theory we have followed Fishburn [5 ].

Some experimental findings on decision making under risk and their implications

Abstract Expected utility (EU) theory is the standard model of decision analysis. Recent experimental studies have however revealed a major problem related to the construction of the utility function which characterizes preferences in the model: The function constructed depends on the assessment method used. These findings are reported and the psychological factors which appear to be responsible for these inconsistencies—the certainty effect and a decision framing effect (reference point shifting)—are described. The applicability of EU theory as a prescriptive model is discussed.

Dynamic decision making without expected utility: An operational approach

Non-expected utility theories, such as rank dependent utility (RDU) theory, have been proposed as alternative models to EU theory in decision making under risk. These models do not share the separability property of expected utility theory. This implies that, in a decision tree, if the reduction of compound lotteries assumption is made (so that preferences at each decision node reduce to RDU preferences among lotteries) and that preferences at different decision nodes are identical (same utility function and same weighting function), then the preferences are not dynamically consistent; in particular, the sophisticated strategy, i.e., the strategy generated by a standard rolling back of the decision tree, is likely to be dominated w.r.t. stochastic dominance. Dynamic consistency of choices remains feasible, and the decision maker can avoid dominated choices, by adopting a non-consequentialist behavior, with his choices in a subtree possibly depending on what happens in the rest of the tree. We propose a procedure which: (i) although adopting a non-consequentialist behavior, involves a form of rolling back of the decision tree; (ii) selects a non-dominated strategy that realizes a compromise between the decision maker’s discordant goals at the different decision nodes. Relative to the computations involved in the standard expected utility evaluation of a decision problem, the main computational increase is due to the identification of non-dominated strategies by linear programming. A simulation, using the rank dependent utility criterion, confirms the computational tractability of the model.

Approximations of rational criteria under complete ignorance and the independence axiom

Rational decision making under complete ignorance, a limit case of uncertainty, is defined. Through a concept of approximation, a meaning is given to a criterion almost possessing a property. Rational criteria depend almost on the sole bounds of the outcome range of each decision, they are almost continuous, they can almost possess a transitive indifference relation; however, under Savages Independence Axiom, this last property restricts possible criteria to those which can be approximated, at least partly, by either the Maximin or the Maximax criterion.

Decision making in a case of mixed uncertainty: A normative model

Abstract We consider a case of uncertainty which is frequently met in various fields, e.g., in parametric statistics: Events {θ}, θ ∈ ∵, are members of family E on which the decision maker possesses no information at all; however, conditionally on the realization of {θ}, he is able to affix probabilities to all members of another family of events, F . We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of E ; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of F ; (3) has (unconditional) preferences which are consistent with his conditional ones. These assumptions are shown to be sufficient to ensure an approximate representation of the decision makers preference by a real-valued function W which has the form W(f) = v[ Inf θ∈∵ E θ (u∘f), Sup θ∈∵ E θ (u∘f)] , where u and v, respectively, characterize the decision makers attitudes toward risk and toward complete ignorance.

Preponderence of the Certainty Effect Over Probability Distortion in Decision Making Under Risk

The appeal of expected utility (EU) theory — the standard model of decision analysis — as a normative model is not necessatily lessened by its poor quality as a descriptive model. As a matter of fact, observations of systematic violations of EU in Allais [1], Kahneman and Tversky [7], Mac Crimmon and Larsson [15], Kunreuther [10], Schoemaker [17] and elsewhere have had no apparent effect on its popularity.