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From decision theory to decision aiding methodology (my very personal version of this history and some related reflections)
, 2003
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Hidden uncertainty in the logical representation of desires
 In Proceedings of Eighteenth International Joint Conference on Artificial Intelligence (IJCAI’03
, 2003
"... In this paper we introduce and study a logic of desires. The semantics of our logic is defined by means of two ordering relations representing preference and normality as in Boutilier’s logic QDT. However, the desires are interpreted in a different way: “in context A, I desire B ” is interpreted as ..."
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In this paper we introduce and study a logic of desires. The semantics of our logic is defined by means of two ordering relations representing preference and normality as in Boutilier’s logic QDT. However, the desires are interpreted in a different way: “in context A, I desire B ” is interpreted as “the best among the most normal A ∧ B worlds are preferred to the most normal A ∧ ¬B worlds”. We study the formal properties of these desires, illustrate their expressive power on several classes of examples and position them with respect to previous work in qualitative decision theory. 1
Great Expectations. Part II: Generalized Expected Utility as a Universal Decision Rule
 In Proc. of the 18th International Joint Conference on Artificial Intelligence (IJCAI03
"... Abstract Many different rules for decision making have been introduced in the literature. We showthat a notion of generalized expected utility proposed in [Chu and Halpern 2003] is a universal ..."
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Cited by 14 (3 self)
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Abstract Many different rules for decision making have been introduced in the literature. We showthat a notion of generalized expected utility proposed in [Chu and Halpern 2003] is a universal
Computer science and decision theory
 Annals of Operations Research
"... This paper reviews applications in computer science that decision theorists have addressed for years, discusses the requirements posed by these applications that place great strain on decision theory/social science methods, and explores applications in the social and decision sciences of newer decis ..."
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This paper reviews applications in computer science that decision theorists have addressed for years, discusses the requirements posed by these applications that place great strain on decision theory/social science methods, and explores applications in the social and decision sciences of newer decisiontheoretic methods developed with computer science applications in mind. The paper deals with the relation between computer science and decisiontheoretic methods of consensus, with the relation between computer science and game theory and decisions, and with “algorithmic decision theory.” 1
Bridging the Gap between Discrete Sugeno and Choquet
"... This paper deals with decision making under uncertainty when the worth of acts is evaluated by means of Sugeno integral. One limitation of this approach is the coarse ranking of acts it produces. In order to refine this ordering, a mapping from the common qualitative utility and uncertainty scale to ..."
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This paper deals with decision making under uncertainty when the worth of acts is evaluated by means of Sugeno integral. One limitation of this approach is the coarse ranking of acts it produces. In order to refine this ordering, a mapping from the common qualitative utility and uncertainty scale to the reals is proposed, whereby Sugeno integral is changed into a Choquet integral. This work relies on a previous similar attempt at refining possibilistic preference functionals of the maxmin and minmax type into a socalled bigstepped expected utility, encoding a very refined qualitative lexicographic ordering of acts. 1
Divergent Mathematical Treatments in Utility Theory
"... Abstract In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standa ..."
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Abstract In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection of settheoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper. Model Theory and Scientific Models In a 1960 paper, Patrick Suppes claimed that: [...] in the exact statement of the theory or in the exact analysis of data the notion of model in the sense of logicians provides the appropriate intellectual tool for making the analysis both precise and clear. (Suppes 1960: 295) This claim was defended against the background thesis that the meaning of the concept of model is the same in mathematics and the empirical sciences (Suppes 1960: 289). Suppes' view of models is too restrictive in two distinct ways: one of these has become clear through the recent literature on modelling, whereas the other has been neglected and provides the main motivation for the discussion presented in Arguments about the model as a theory, or a method, or a distortion of reality, all focus on the model as a scientific object and how it functions in science. Without denying this dimension of the model at all, this paper wants to broaden the perspective by claiming that the model is also a social and political device. The model will be understood as a practice connecting data, index numbers, national accounts, equations, institutes, trained personnel, laws, and policymaking. (van den Bogaard 1999: 283) It is clear that, if one is ready to accept the qualification of models for methods, distortions of reality and social devices, Suppes' more stringent semantic qualification must appear to impose very narrow, perhaps unrealistic, constraints on the study of modelling practices. It does not follow that Suppes' appeal to notions and techniques from mathematical logic should be deemed irrelevant to the study of modelling practices in general. In this paper, I seek to defend the opposite point of view by applying a modeltheoretic approach to the study of mathematical modelling within utility theory. While doing so, I depart from Suppes' original aims, which presuppose, in my opinion, too strict a delimitation of the ways in which modeltheoretic considerations may support philosophical investigations of scientific models. The quotation opening this section spells out the delimitation in question by restricting the mobilisation of settheoretical semantics to the purposes of formulating scientific theories (typically as classes of models defined by a settheoretical predicate, an approach whose abstract development has been presented in Da Costa and Chuaqui 1988) or carrying out an exact analysis of data (e.g. by the embedding of a data structure into a representing structure, a strategy adopted in Da 123 (for continua) to infinity. 1 Niederée's results have shown, among other things, how one can identify measuring numbers with sets of experimental data, as well as the equivalence between certain mathematical assumptions (e.g. the Archimedean property) and properties of experimental procedures. These results shed light on important features of scientific models (especially measurement models) by modeltheoretic means, without pursuing any of the tasks recommended by Suppes, i.e., theory formulation or data analysis. The main objective of this paper is to extend along a further direction the same modeltheoretic style of investigation, which recognises the value of Suppes' original proposal but transcends its limited scope. The modeltheoretic machinery I shall rely upon comes from Nonstandard Analysis: it is briefly surveyed in Sect. 2 (more details are found in the ''Appendix''). I shall apply Nonstandard Analysis to two models from utility theory in order to construct an alternative mathematical treatment for the economic setups they are supposed to describe. This will allow me to show that the fragments of economic theory based on these models are crucially sensitive to a choice of mathematical treatment, more precisely, a selection of settheoretic parameters. What this suggests is that economic theory is, at an abstract level, significantly sensitive to the choice of mathematical resources employed in its articulation. The existence of distinct choices leads to bifurcations in the kind of result one may hope to obtain. In particular, if one wishes to uphold certain normative constraints or introduce certain formal approaches, it is sometimes mandatory to drop traditional mathematical environments based on the real numbers. These remarks will be illustrated in full detail in Sects. 47, after a brief semitechnical preliminary. Classical and Nonstandard Analysis A vast amount of work in mathematical social science (especially economics) relies on the availability of the objects of classical analysis in the semantic metatheory. For example, in consumer theory utilities are real numbers, bundles of goods are realvalued vectors and their totality is canonically a subset of some Euclidean space. In many interesting cases there is no particular empirical motivation to select certain specific analytical objects in modelbuilding, either because they (e.g. the metric structure on a set of alternatives ranked by a preference relation) support abstract models without having any empirical interpretation or because, even when they represent some nonmathematical content, they enter a model also as carriers of properties that have no particular connection with this content (e.g. the topological separability of the real numbers representing utilities) and yet influence what can be established about the given model. Because of this, it is of interest to consider what happens if one replaces certain canonically employed analytical objects with alternative objects. In this paper, I consider the objects of Nonstandard Analysis, 123 which share a number of properties with classical objects but are, at the same time, significantly different. I shall focus on their application to two mathematical models from utility theory. In each case, I study the consequences of using certain extensions of classical numerical sets within a Nonstandard universe as codomains of functions that are canonically selected to be realvalued. While a standard mathematisation based on realvalued functions gives rise to negative results, a Nonstandard mathematisation replaces them by positive results (which may hold under stronger conditions than were sufficient to deduce the negative results by standard means). This divergence highlights the essential relativity (i.e., with respect to a selection of mathematical resources) of negative results in economic theory, since the remarks that hold for the utility models discussed in detail admit of a general reformulation. Such a reformulation will be presented in Sect. 7, after a full discussion of the main examples has taken place, in Sects. 46. It is appropriate to note at this point, by way of a concluding remark, that applications of Nonstandard Analysis are not new to the field mathematical economics (see for instance, Skala 1975; Fishburn and Lavalle 1991; Lehmann 2001). However, all those of which I am aware adopt a local point of view, i.e., they construct an ultraproduct of some real structure suitable to specific modelling purposes. Moreover, they are not concerned with showing how canonical and nonstandard resources affect in divergent ways the results of modelling. My approach, on the contrary, is global in the sense that it relies on a Nonstandard universe in which several results involving nonstandard models are simultaneously obtained (this point will be clarified in Sect. 3). Moreover, it is primarily concerned with showing how canonical and nonstandard resources affect the results of modelling. A Formal Preliminary The Nonstandard universe I shall make use of in the next sections can be constructed from a collection S 0 of Urelemente (informally speaking, nonsets) that contains a copy of the set of real numbers R. One can use S 0 to generate a hierarchy of settheoretical objects by means of the following inductive condition: where PðS n Þ is the powerset of S n . The union U of all the S i (with i a natural number) is a very rich object. It clearly contains the real numbers and all of their subsets, but it also contains the Cartesian product R n , for any natural number n, and, as a consequence, every finitary relation on the real numbers, all functions of one or several real variables and so on. The object U can then be described by means of a firstorder language with identity L which contains, apart from connectives and quantifiers, the symbol 2 for settheoretical membership and a name for every entity Footnote 2 continued refinements, D. Rizza 123 in U (including all relations, functions, sets of relations or functions etc.). 3 Thus, in particular, there are Lnames for N, the set of natural numbers, or S, the set of all sequences of real numbers, both of which will be considered later. One may then use the compactness theorem of firstorder logic to obtain an enlargement U 0 of the structure U ¼ hU; 2i (how this can be done is outlined in the ''Appendix''). What matters for present purposes is that an arbitrary enlargement will contain extended numerical sets Ã N and Ã R that are richer than their respective counterparts N; R, in the sense that the latter numerical sets can be embedded in their starred extensions and these include additional elements. The only additional numerical elements that will be extensively relied upon in what follows are the infinitely large numbers in Ã N, i.e., the elements in this set that are greater than any n 2 N (where 'greater than' is a binary relation on Ã N that agrees with the usual ordering on the standard numbers). The language L introduced above will also play an important role, since it allows one to state sentences that may contain names for N or R as parameters: if these sentences are true in the standard universe U, then they are true in its enlargement, with respect to Ã N and Ã R. In Sect. 6, this will make it possible to 'extend' properties of arbitrary, standard numbers to infinitely large numbers in particular. This preliminary is sufficient to introduce the models alluded to in the previous section. Ranking Information States My first example is taken from choice theory. A typical aim in this setting is to show that the choicebehaviour of an agent can be represented by a utility function. Formally, one introduces a space A of alternatives over which the agent is supposed to express or reveal her preferences by way of binary comparisons. Thus, a preference is understood to be a binary relation P on A, in particular a complete preorder, i.e., a relation that is transitive and complete. The aim is then to show the existence of a function u: A ! R, from A into the set of real numbers R such that, for any x; y 2 A: xPy iff uðxÞ uðyÞ: It is not unusual to encounter in the literature models where P is defined on an uncountably large set. A fairly recent collection of models based on this setting, the simplest of which I am going to discuss, is found in Divergent Mathematical Treatments in Utility Theory 123 main aim is to retain them in order to show that, once they have been deployed, the highly idealised character of the resulting models makes them amenable to distinct mathematical treatments that, in turn, give rise to vastly different results. In brief, substantive appeals to mathematical idealisation give rise to divergent modelling trajectories. Although I shall focus on the most basic model studied by Dubra end Echenique, it is worth noting that my remarks about it also apply, essentially unchanged, to more sophisticated variants presented in the same paper. The model in question involves an uncountably large set X of possible 'states of nature' endowed with the family of its partitions. Each partition divides the whole X into disjoint subsets and a partition P is finer than a partition Q if every element of P is included in some element of Q (if strictly included, then P is strictly finer than Q). This formal model is meant to describe in mathematical terms the informational states of a decisionmaker: one interprets an arbitrary partition of X as resulting from an equivalence relation of informational indifference. For an agent in possession of Q, any two possible states of nature within an element of Q carry the same amount of information and are therefore indistinguishable from this point of view. It is also assumed that any decisionmaker would find the transition from Q to a strict refinement P desirable, since it leads from a less informative to a more informative state. With this description of the model in place, taking PðXÞ to be the set of all partitions of X, Dubra and Echenique consider the complete preorders on PðXÞ that rank any partition strictly below any strict refinement (a condition they call monotonicity). According to them, any rational decisionmaker must rank PðXÞ on one such preorder. It can however be shown that none of them is representable by a realvalued utility function. Dubra and Echenique draw the following conclusion: Our result is important because it shows that utility theory is not likely to be a useful tool in the analysis of the value of information. This finding should be contrasted with the existing literature on the value of information, where utility representations are used. The use of a utility implies that preferences are not monotone (Dubra and Echenique 2001: 1). The point of the above quotation is that the existence of a utility representation is incompatible with the assumption that an agent should prefer more information to less. This remark is certainly correct, but it hides the following dilemma: is the problem inherent in utility theory as a formal approach to the study of idealised rankings or is it an effect of restricting attention to realvalued utility functions and, thus, of certain properties of R? This dilemma refines the formulation of the problem highlighted by Dubra and Echenique because it does not presuppose that the codomain of a utility function should be R. No particular feature of the space of informational states suggests that such a codomain should be selected. It is therefore meaningful to look for alternative numerical codomains, on which utility functions may exist. In other words, it is reasonable to conjecture that a lack of fit exists not between utility functions and spaces of information states, but between these spaces and the ordered reals.
Great Expectations. Part II: Generalized Expected Utility as a Universal Decision Rule*
"... Many different rules for decision making have been introduced in the literature. We show that a notion of generalized expected utility proposed in a companion paper [Chu and Halpern, 2003] is a universal decision rule, in the sense that it can represent essentially all other decision rules. 1 ..."
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Many different rules for decision making have been introduced in the literature. We show that a notion of generalized expected utility proposed in a companion paper [Chu and Halpern, 2003] is a universal decision rule, in the sense that it can represent essentially all other decision rules. 1