## Qualitative decision theory: from Savage’s axioms to nonmonotonic reasoning (2002)

Venue: | Journal of the ACM |

Citations: | 32 - 0 self |

### BibTeX

@ARTICLE{Dubois02qualitativedecision,

author = {Didier Dubois and Hélène Fargier and Patrice Perny and Henri Prade and Lip Pole Ia},

title = {Qualitative decision theory: from Savage’s axioms to nonmonotonic reasoning},

journal = {Journal of the ACM},

year = {2002},

volume = {49},

pages = {455--495}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract: This paper investigates to what extent a purely symbolic approach to decision making under uncertainty is possible, in the scope of Artificial Intelligence. Contrary to classical approaches to decision theory, we try to rank acts without resorting to any numerical representation of utility nor uncertainty, and without using any scale on which both uncertainty and preference could be mapped. Our approach is a variant of Savage's where the setting is finite, and the strict preference on acts is a partial order. It is shown that although many axioms of Savage theory are preserved and despite the intuitive appeal of the ordinal method for constructing a preference over acts, the approach is inconsistent with a probabilistic representation of uncertainty. The latter leads to the kind of paradoxes encountered in the theory of voting. It is shown that the assumption of ordinal invariance enforces a qualitative decision procedure that presupposes a comparative possibility representation of uncertainty, originally due to Lewis, and usual in nonmonotonic reasoning. Our axiomatic investigation thus provides decision-theoretic foundations to preferential inference of Lehmann and colleagues. However, the obtained decision rules are sometimes either not very decisive or may lead to overconfident decisions, although their basic principles look sound. This paper points out some limitations of purely ordinal approaches to Savage-like decision making under uncertainty, in perfect analogy with similar difficulties in voting theory.

### Citations

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Citation Context ...pping (it has little intuitive appeal); three-valued logic semantics based on an abstraction of conditional probabilities (Dubois and Prade, 1994); infinitesimal probabilistic semantics (Adams, 1975; =-=Pearl, 1988-=-; Lehmann and Magidor, 1992); possibilistic semantics (Dubois and Prade, 1995a) whereby a set of conditional assertions is equivalent to a family of possibility orderings; finitistic probabilistic sem... |

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Citation Context ...eight bearing on W(f) is viewed as a degree of pessimism of the decision maker. Other decision rules have been proposed, especially some that generalize both EU(f) and W(f)�; see (Jaffray, 1989) and (=-=Schmeidler 1989-=-), among others. Dubois et al. (1995c, 2001) propose a non-numerical generalization of W(f) on an ordinal scale. However the expected utility criterion is by far the most commonly used one. Savage (19... |

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Citation Context ...ential consequence relation. Section 6 exhibits a model of the set of axioms, where uncertainty is represented by a single weak ordering of states, in the framework of comparative possibility theory (=-=Lewis, 1973-=-; Dubois and Prade, 1998). The relation on events induced by the likely dominance expresses relative likelihood and plays the role of a probability measure, but only its strict part is transitive. Thi... |

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Citation Context ... little intuitive appeal); three-valued logic semantics based on an abstraction of conditional probabilities (Dubois and Prade, 1994); infinitesimal probabilistic semantics (Adams, 1975; Pearl, 1988; =-=Lehmann and Magidor, 1992-=-); possibilistic semantics (Dubois and Prade, 1995a) whereby a set of conditional assertions is equivalent to a family of possibility orderings; finitistic probabilistic semantics by means of a very s... |

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Citation Context ...et S via a mapping (it has little intuitive appeal); three-valued logic semantics based on an abstraction of conditional probabilities (Dubois and Prade, 1994); infinitesimal probabilistic semantics (=-=Adams, 1975-=-; Pearl, 1988; Lehmann and Magidor, 1992); possibilistic semantics (Dubois and Prade, 1995a) whereby a set of conditional assertions is equivalent to a family of possibility orderings; finitistic prob... |

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Citation Context ...lyzed. Users of information systems are not necessarily capable of describing their state of uncertainty by means of a probability distribution, nor may they be willing to quantify their preferences (=-=Boutilier, 1994-=-). This is typical of electronic commerce, or recommender systems for instance. In many cases, it sounds more satisfactory to implement a choice method that is fast, and based on rough information abo... |

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Citation Context ... and uncertainty (i.e. more than separate ordinal scales for each of them). Condition OI also involves a condition of independence with respect to irrelevant alternatives (in the sense of Arrow 1969, =-=Sen 1986-=-). It says that the preference f ≥ g only depends on the relative positions of quantities f(s) and g(s) on the preference scale. This unnecessary part of the condition could be cancelled within the pr... |

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Citation Context ...xioms, ordinal invariance and preferential inference Property (N) and the orderly property of proposition 9 are closely related to one of the characteristic properties for uncertainty relations that (=-=Friedman and Halpern, 1996-=-) call "plausibility" relations, which are closely connected to non-monotonic consequence relations. Given a subjective likelihood relation > L , define an inference relation as follows: A Æ B if and ... |

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Citation Context ...y the utility function is specified via symbolic constraints (Lang, 1996; Bacchus and Grove, 1996 for instance). Others use sets of integers and the like to describe rough probabilities or utilities (=-=Tan and Pearl, 1994-=-). Lehmann (1996) injects some qualitative concepts of negligibility in the classical expected utility framework. However some approaches are genuinely qualitative in the sense that they do not involv... |

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Citation Context ...sual decision rule is based on the expected utility criterion�: EU(f)�= ∑ sŒS p(s)u(f(s))swhere by act f is preferred to act g if and only if EU(f) > EU(g). Another proposal is the maximin criterion (=-=Arrow and Hurwicz, 1972-=-). It applies when no information about the current state is available, and it ranks acts according to its worst consequence�: W(f) = min sŒS u(f(s)). This criterion, first proposed by Wald (1950), ha... |

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Citation Context ...greement on what "qualitative" means. Some authors assume incomplete knowledge about classical additive utility models, whereby the utility function is specified via symbolic constraints (Lang, 1996; =-=Bacchus and Grove, 1996-=- for instance). Others use sets of integers and the like to describe rough probabilities or utilities (Tan and Pearl, 1994). Lehmann (1996) injects some qualitative concepts of negligibility in the cl... |

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Citation Context ...uence relation. Section 6 exhibits a model of the set of axioms, where uncertainty is represented by a single weak ordering of states, in the framework of comparative possibility theory (Lewis, 1973; =-=Dubois and Prade, 1998-=-). The relation on events induced by the likely dominance expresses relative likelihood and plays the role of a probability measure, but only its strict part is transitive. This type of relation, refi... |

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Citation Context ...A concrete non-trivial example of a subjective likelihood relation that fits decision theory based on ordinal invariance can be provided in the setting of qualitative possibility theory (Lewis, 1973; =-=Dubois, 1986-=-; Dubois and Prade, 1998). 2 Except if there are no null events. Null events generally never appear in nonmonotonic reasoning nor belief revision theories. But supposing that there is no null event is... |

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Citation Context ...terpart, where the weight bearing on W(f) is viewed as a degree of pessimism of the decision maker. Other decision rules have been proposed, especially some that generalize both EU(f) and W(f)�; see (=-=Jaffray, 1989-=-) and (Schmeidler 1989), among others. Dubois et al. (1995c, 2001) propose a non-numerical generalization of W(f) on an ordinal scale. However the expected utility criterion is by far the most commonl... |

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Citation Context ...on ~L as usual: A ≥ L B if and only if not (B > L A); A ~L B iff A ≥ L B and B ≥ L A. The preference relation on the set of consequences X is supposed to be a weak order (a complete preordering, e.g. =-=Roubens and Vincke, 1985-=-). Namely, ≥ P is a reflexive and transitive relation, and completeness means x ≥ P y or y ≥ P x, "x, y Œ X. x ≥ P y means that consequence x is not worse than y. The induced strict preference relatio... |

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Citation Context ...partial ordering relation on an abstract set, projected to the set S via a mapping (it has little intuitive appeal); three-valued logic semantics based on an abstraction of conditional probabilities (=-=Dubois and Prade, 1994-=-); infinitesimal probabilistic semantics (Adams, 1975; Pearl, 1988; Lehmann and Magidor, 1992); possibilistic semantics (Dubois and Prade, 1995a) whereby a set of conditional assertions is equivalent ... |

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Citation Context ...is no real agreement on what "qualitative" means. Some authors assume incomplete knowledge about classical additive utility models, whereby the utility function is specified via symbolic constraints (=-=Lang, 1996-=-; Bacchus and Grove, 1996 for instance). Others use sets of integers and the like to describe rough probabilities or utilities (Tan and Pearl, 1994). Lehmann (1996) injects some qualitative concepts o... |

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Citation Context ...between the decision rules displayed above, that require Pareto-dominance of best acts over the most plausible states, and voting procedures found in social choice (Mas-Colell and Sonnenschein, 1972, =-=Weymark 1984-=-�; see also Sen,1986) by relaxing the transitivity of the social preference and assuming the transitivity of its strict part. They do find that decisions should be made unanimously by an oligarchy of ... |

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Citation Context ... g) and (f', g') will be called statewise orderequivalent if and only if "sŒS, f(s) ≥ P g(s) if and only if f'(s) ≥ P g'(s). This is denoted (f, g) ≡ (f', g'). The following Ordinal Invariance axiom (=-=Fargier and Perny, 1999-=-) fully embodies our requirement of a purely ordinal approach to DMU. OI: "f, f' g, g' Œ X S , if (f, g) ≡ (f', g') then ( f ≥ g ¤ f' ≥ g'). This axiom does express that what matters for preference be... |

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2 |
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(Show Context)
Citation Context ...ce relations on acts. This kind of impediment was already pointed out by Doyle and Wellman (1991) for preference-based default theories. This paper extends preliminary results that were presented in (=-=Dubois et al., 1997-=-). It is organized as follows. Section 2 recalls some classical decision rules and Savage's framework for DMU. Section 3 presents the likely dominance rule that can rank acts from a purely ordinal kno... |

1 | Prade (1997a) Nonmonotonic reasoning, conditional objects and possibility theory - Benferhat, Dubois, et al. |

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1 |
Prade (2001) A characterization of generalized concordance rules in multicriteria decision-making
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Citation Context ...lose to the framework of voting theories after Arrow (1951), Sen (1986) etc. The advantage is that it is faithful to the kind of elementary information users can directly provide. The other approach (=-=Dubois et al. 2001-=-) presupposes the existence of a totally ordered scale (typically a finite one) for grading both likelihood and utility. Decision rules generalizing maximin and maximax criteria can be defined on this... |

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1 |
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Citation Context ...omplex methods, because handling infinite state spaces. A qualitative, finitistic, description of the goals of the robot, and of its knowledge of the environment might lead to more tractable methods (=-=Sabbadin, 2000-=-, for instance). Besides, the expected utility criterion is often adopted because of its mathematical properties (it enables dynamic programming principles to be used). However it is not clear that th... |