2. One can also consider the corresponding symbolic base as the union of these local symbolic bases. Then the possibility distribution would be computed via vector comparisons. To find the same ordering as π-pref net, we should use symmetric Pareto to compare vectors of configurations. This **representation** has no advantage compared to π-pref nets since we cannot apply inference rules on it. There were several attempts to represent CP-nets orderings using **a** symbolic **possibilistic** logic base. The construction of the local logic bases is exactly as presented here. However, to each node they associate only one symbolic weight. Therefore, to each node is associated at most one logic formula. Besides, they do not consider any indifference between variable values. See [Dubois et al., 2015] for **a** bibliography and **a** discussion. It was also observed that an exact logical **representation** of CP-nets was not possible when nodes in the network have several children even though good approximations can be considered. This is because additional constraints in that **framework** compare individual symbolic weights, not product thereof. In addition, they show that Symmetric Pareto and Leximin orderings respectively lower and upper bound the CP-net ordering. These results can be exploited with **a** π-pref-net since it represent its **graphical** counterpart [Dubois et al., 2013b, Dubois et al., 2013a].

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This method is inspired from CP-nets, **a** well-known **framework** for representing pref- erences in AI [6]. It is **a** **graphical** **representation** that exploits conditional preferential independence in structuring the preferences provided by **a** user. These preferences take the form u : x i > ¬x i , i.e., x is preferred to ¬x in context u, (u can be tautological). CP-nets are underlain by **a** ceteris paribus principle that amounts to giving priority to preferences attached to parent nodes over preferences attached to children nodes in the CP-net structure. Besides, it has been noticed that **a** CP-net ordering can be approxi- mated by **a** **possibilistic** logic **representation** with symbolic weights [19,8]. The priority in favor of father nodes carries over to the **possibilistic** setting in the following way. For each pair of formulas of the form (¬u ∨ x i , α i ) and (¬u ∨ ¬x i ∨ x j , α j ), x i plays the role of the father of x j in **a** CP-net. Indeed, the first formula expresses **a** prefer- ence in favor of having x i true (in context u), while in the second formula the context is refined from u to u ∧ x i , which establishes **a** particular type of links between the two formulas where the second formula is in some sense **a** descendant of the first one. Then, the following constraint between the corresponding weights is applied α i > α j , in **a** CP-net spirit. These constraints between symbolic weights can be obtained system- atically by Algorithm 2, which computes the partial order between symbolic weights from **a** **possibilistic** logic base. Applying this procedure allows us to add constraints among symbolic weights and to get **a** more refined ranking of items, as we notice in the following example.

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Now, several key notions in game theory, such as pure strategy Nash equilibria or dominated strategies, need only ordinal preferences. **Graphical** languages for ordinal **preference** **representation** have been used in some places for representing and analyzing games, such as Section 5 of [3]: there, **a** game is described as **a** set of agents, **a** set of (binary) variables, **a** control assignment function assigning each variable to an agent, and finally, **a** compact description of the agents’ preferences **under** the form of **a** collec- tion of CP-nets. The structure of the agent’s preferences can sometimes guarantee the existence or the unicity of **a** pure Nash equilibrium. For instance, assume we have two agents {1,2} and two variables **A** and B, with domains {**a**, ¯**a**} and {b, ¯b}, such that 1 controls **A** and 2 controls B. Preferential dependencies are as follows:

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1 Introduction
Since the direct assessment of **a** **preference** relation between elements of Carte- sian products is usually not feasible, current work in **preference** modeling aims at proposing compact **preference** models achieving **a** good compromise between elic- itation easiness and computational efficiency. Conditional **preference** networks (CP-nets) [4] are **a** popular example of such setting. However, in spite of their appealing **graphical** nature, CP-nets may induce debatable priorities between de- cision variables and lack **a** logical counterpart. Symbolic **possibilistic** logic bases stand as another approach to represent preferences [9]. This setting overcomes the above mentioned CP-nets limitations. Moreover, it leaves complete freedom for stating relative priorities between variables. But, it is not **a** **graphical** model. This paper explores the **representation** of preferences by **possibilistic** net- works, outlined in [1] and establishes formal results about them. This approach preserves **a** **possibilistic** logic **representation**, while offering **a** **graphical** compact format convenient for elicitation.

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i.e., below the lowest one where houses are preferred to them and in the highest one where there is an incomparable item.
4.2 Constraints between Weights in CP-Theories Style
CP-theories as introduced in [11], are **a** generalization of CP-nets. Also based on **a** **graphical** **representation**, CP-theories offer **a** more expressive language where prefer- ence priority can be made explicit between the **preference** constraints. Thus, such con- straints have the same form as in CP-nets u : x > ¬x [W ]; in addition we have the set of variables (attributes) W for which it is known that the **preference** associated to x does not depend on any value assignment of an attribute in W (i.e., the **preference** attached to the concerned attribute holds irrespective of values of attributes in W ). It has been suggested that **possibilistic** logic is able to approximate this **representation** by adding more priority constraints over the symbolic weights [20]. Formally, **a** possibilis- tic **preference** constraint of the form u : x > ¬x [W ], with an irrespective requirement w. r. t. variables in W is encoded by **a** **possibilistic** **preference** statement (¬u ∨ x, α i ),

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The paper is organized as follows. First, **a** short background on **possibilistic** logic, on CP-nets and its encoding with **possibilistic** logic formulas having sym- bolic weights is provided in Sections 2 and 3. Then in Section 4 we discuss the different partial orders that can be used for comparing the vectors of symbolic weights which reflect the violation of preferences and are associated with each interpretation. Used as such, each of the considered orders are successful for re- trieving the CP-net ordering on specific **graphical** structures and fail on others, as shown in Section 5. Section 6 identifies on which particular structures the ex- isting **possibilistic** **representation** is exact, and shows more generally how lower and upper representations can be obtained. Section 7 briefly discusses the related work and exhibits **a** final example that points out the difficulty of capturing the CP-net ordering exactly in **a** logical way.

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k > 1). On the other hand, as we have seen in Sect. 3 , representing utility functions in the
k -additive form rather than the bundle form can be significantly more succinct, particularly in cases where **a** **representation** with **a** small value for k is possible.
We have also explored connections to well-known combinatorial optimisation problems, which has allowed us to establish complexity results for the problem of finding **a** socially optimal allocation with respect to different representations of utility functions (Sect. 5 ). In this context, we have also briefly discussed the relation of our negotiation **framework** to combinatorial auctions for different kinds of bidding languages. While our negotiation **framework** is clearly not an auction (it is, for instance, not concerned with the aspect of agreeing on the price for **a** set of items), the abstract “centralised” problem of finding **a** socially optimal allocation (which is not itself **a** problem faced by the agents participat- ing in **a** negotiation process) directly corresponds to the winner determination problem in combinatorial auctions. **Under** this view, the languages used to represent utility functions correspond to bidding languages for such auctions. As regards our complexity results, it is important to stress that the high complexity of the distributed negotiation **framework** does not, at least not necessarily, mean that it cannot be usefully applied in practice. This view is supported by the fact that, in recent years, several algorithms for winner determination in combinatorial auctions (**a** problem of comparable complexity to the problems arising here) have been proposed and applied successfully (Rothkopf et al. 1998 ; Fujishima et al. 1999 ; Sandholm 2002 ).

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4 Conditional **Preference** **Possibilistic** networks
Marginal networks are inspired from Bayesian networks. Similarly, one may use possi- bilistic networks [2], **a** possibility theory counterpart to Bayes nets, for modeling pref- erences rather than uncertainty (understanding the possibility degrees as satisfaction levels). Possibility theory relies on the idea of **a** possibility distribution π, which is **a** mapping from **a** universe of discourse Ω to the unit interval [0, 1], or to any bounded totally ordered scale. Two forms of conditioning, respectively based on minimum and product, make sense in possibility theory, leading to two types of chain rules. We may then compute satisfaction values for configurations, taking advantage of Markov prop- erty, and obtain **a** total order between configurations in both cases. In the absence of available quantitative values, one may think of keeping the possibility degrees unspec- ified (which also preserves the ability of representing partial orders). This led us to propose **a** new **graphical** **preference** model based on **possibilistic** networks [3, 4], called π-Pref nets. In **a** π-Pref net, for each variable **A** i ∈ V , for each instantiation u i of

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1 Introduction
Since the direct assessment of **a** **preference** relation between elements of Carte- sian products is usually not feasible, current work in **preference** modeling aims at proposing compact **preference** models achieving **a** good compromise between elic- itation easiness and computational efficiency. Conditional **preference** networks (CP-nets) [4] are **a** popular example of such setting. However, in spite of their appealing **graphical** nature, CP-nets may induce debatable priorities between de- cision variables and lack **a** logical counterpart. Symbolic **possibilistic** logic bases stand as another approach to represent preferences [9]. This setting overcomes the above mentioned CP-nets limitations. Moreover, it leaves complete freedom for stating relative priorities between variables. But, it is not **a** **graphical** model. This paper explores the **representation** of preferences by **possibilistic** net- works, outlined in [1] and establishes formal results about them. This approach preserves **a** **possibilistic** logic **representation**, while offering **a** **graphical** compact format convenient for elicitation.

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While there are several different formats for formally specifying **a** game (most notably extended form and normal form, which coincide as far as static games are concerned), utility functions are usually represented explicitly, by listing the values for each combination of strategies (an exception being the recent **framework** of **graphical** games, which we discuss later in the paper). However, the number of utility values which must be specified, that is, the number of possible combinations of strategies, is exponential in the number of players, which renders such an explicit way of representing the preferences of the players unreasonable when the number of players is not very small. This becomes even more problematic when the set of strategies available to an agent consists in assigning **a** value from **a** finite domain to each of **a** given set of variables (which is the case in many real-world domains). In this case, representing utility functions explicitly leads to **a** description whose size is exponential both in the number of agents (n × 2 n values for n agents each with two available strategies) and in the number of variables controlled by the agents (2 × 2 p × 2 p values for two agents each controlling p Boolean variables). Thus, in all these cases, specifying players’ preferences explicitly is clearly unreasonable, both because it would need exponential space, and because studying these games (for instance by computing solution concepts such as pure-strategy Nash equilibria) would require accessing all of these utility values at least once and hence would take time exponential in the numbers of agents and variables in all cases.

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In this paper, we address two major issues. The first one concerns the problem of aggregation of mono-criterion **preference** relations **under** uncertainty. In fact, the aggregation of mono-criterion preferences for collective decision making is **a** very traditional problem, also known as voting problem. Studies on different electoral voting systems can be found in [9]. The majority of voting methods require voters to make binary comparisons, declaring that one alternative is preferred to another one, without taking into account uncertainty in the **preference** relationship. Belief function theory is **a** mathe- matical **framework** for representing and modeling uncertainty [10]. In [11], [12], the authors proposed **a** belief-function- based model for **preference** fusion, allowing the expression of uncertainty over the lattice order (i.e. **preference** struc- ture). However, this modeling approach does not constitute an optimal **representation** of preferences in the presence of uncertain and voluminous information. Based on [11], the model of uncertainty proposed by Masson, et al. in [13] allows the expression of uncertainty on binary relations (i.e. **preference** relations). Precisely, this approach proposes to model **preference** uncertainty over binary relations between pairs of alternatives, and to infer partial orders. The **preference** relations considered in this method are : “strict **preference**” and “incomparability”. The“indifference” relation has not been addressed. Furthermore, this approach defines only one mass function on each alternative pair, which is not expressive enough. In fact, this does not allow to express uncertainty over “strict **preference**” and “incomparability” at the same time.

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Thanks to some resemblances between those models many transformations can be considered and are depicted by dashed lines in Figure 7. UCP-nets are **a** restriction of GAI-nets and **a** generalization of CP-nets. Indeed, **a** UCP-net structure can be trans- formed into **a** junction tree such that for each clique we sum up the local utilities of the variables belonging to it, just leading to **a** GAI net. However, due to the acyclic restriction of UCP-nets and the necessary, commitment with Ceteris Paribus, not any GAl-net can be represented by **a** UCP-net. Besides, when handled symbolically, π-Pref nets and marginal utility nets lead to the same orderings. Indeed comparing configu- rations is nothing but comparing vectors of weights. Therefore, product and addition make no difference on symbolic weights. Transformation from π-Pref nets to GAI-nets might also be considered since, as for Bayesian nets, **possibilistic** nets can be trans- lated into junction trees. However, an important difference between these two settings lie in the meaning of values. Both utilities and possibility degrees express levels of sat- isfaction, but the latter are bounded. In GAI-nets, what really matters is the difference between utilities. Thus, representing the same information in π-Pref nets is not possible; one may only try to induce the same qualitative order between the configurations. The opposite transformation is not obvious. In fact, it requires **a** two level transformation. First, translating utilities to possibility degrees. Second, moving from **a** junction tree to **a** **possibilistic** network. This procedure was never studied in the literature.

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Abstract. Among several **graphical** models for preferences, CP-nets are
often used for learning and **representation** purposes. They rely on **a** sim- ple **preference** independence property known as the ceteris paribus inde- pendence. Our paper uses **a** recent symbolic **graphical** model, based on **possibilistic** networks, that induces **a** **preference** ordering on configura- tions consistent with the ordering induced by CP-nets. Ceteris paribus preferences in the latter can be retrieved by adding suitable constraints between products of symbolic weights. This connection between possi- bilistic networks and CP-nets allows for an extension of the expressive power of the latter while maintaining its qualitative nature. Elicitation complexity is thus kept stable, while the complexity of dominance and optimization queries is cut down.

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When targeting 3D objects, **a** crucial model property is viewpoint invariance, i.e. the ability to encode object information independently of the viewpoint. Although viewpoint invariance has been achieved in 2D [31], an increasingly popular means of achieving viewpoint invariance is to represent object geometry directly in 3D. Very interesting work [11], [12], [32] takes the ap- proach of representing an object with **a** large set of local affine-invariant descriptors and the relative 3D spatial re- lationships between the corresponding surface patches. Other methods organize local appearance descriptors on **a** 3D shape model, obtained by CAD [16] or 3D homog- raphy [14]. To detect an object in an image, these meth- ods start with an appearance-based matching of image descriptors with model descriptors. In **a** second phase, global optimization is used to evaluate the geometrical consistency of the appearance-based matches using the geometrical information contained in the model, and compute **a** 2D bounding box [14] or **a** 3D pose [16]. Rothganger et al. [11] do not speak of 3D pose in their results, although it seems obvious that **a** 3D pose is implicitly computed during detection.

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The absence of **a** distinction between “beginners” and “experts” in our analysis seems problematic as well. This is clearly the case in the ATC example: we know from previous observation that ATC controllers do not scan the strips the way we described the process above. Instead, they rely heavily on their knowledge of the sector, recurrent problems and recurrent aircraft to detect conflicts. Again, our description aimed at eliciting what the visualization enables for **a** reader that only uses information extracted from the **representation**. However, during normal operations, ATC controllers regularly do what they call **a** “tour of the radar image” or **a** “tour of the strip board”, in order to check “everything”. In this case, they are supposed to heavily scan both representations and may exhibit some of the theorized behavior. Furthermore, we observed that ATC controllers make more errors when training on **a** new sector, at least partly because of **representation** flaws. These flaws are compensated for by expertise, which is somewhat related to knowledge in the head and memory (in some cases, an ATC controller is considered as expert on **a** sector only after 2 years of training). However, in high-load situations, with lots of aircraft, or with particular problematic conditions such as unexpected storms, the **representation** becomes more important and controllers seem more likely to exhibit the theorized behavior.

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As is well-known, **a** choice function C is rationalizable by **a** binary relation R on X m if
and only if, for each feasible set of prospects S in the domain Σ, C selects the R-greatest elements in S. However, in the present context of choice **under** uncertainty, we might want to impose more than just the standard rationalizability property in order to think of **a** choice function as representing **a** plausible method of selecting from sets of available prospects. Clearly, there are many theories of choice **under** uncertainty such as those pioneered and discussed by von Neumann and Morgenstern (1944), Milnor (1954), Savage (1954), Fishburn (1970), Arrow and Hurwicz (1972), G¨ ardenfors (1976), Kim and Roush (1980), Barber` **a**, Barrett and Pattanaik (1984), Barber` **a** and Pattanaik (1984), Kannai and Peleg (1984), Barber` **a**, Bossert and Pattanaik (2004), to name but **a** few.

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

To conclude, the question of an exact **representation** of any CP-net by **a** partially ordered set of propositional formulae remains open, but this note suggests that the discrepancies between the two **representation** settings look more important than expected (see [ 4 ] for additional discrepancies between CP-net and **possibilistic** logic, pertaining to the transitivity of priorities between father nodes and children nodes in CP-nets). All that can be expected is **a** formal proof that for general acyclic CP-net structures the ordering ≻ N can only be

the regression. In addition, noise in the experimental data can ob- scure the behavior of the corrected phase angle at high frequencies. At high frequencies, Z r → R e,est . Thus, the argument to the inverse tangent in Eq. 4 will have **a** sign controlled by noise in the denomi- nator, and the phase calculated from Eq. 4 will have values scattered about ± adj 共⬁兲. This scatter will of course be evident as well in the ohmic-resistance-corrected magnitude plots. Nevertheless, Bode plots corrected for ohmic resistance are useful as **a** pedagogical tool and for determining whether CPE behavior is evident in the data. Real and imaginary components.— The difficulty with using the ohmic-resistance-corrected Bode plots presented in the previous sec- tion is that an accurate estimate is needed for the electrolyte resis- tance and that, at high frequencies, the difference Z r − R e,est is de- termined by stochastic noise. These difficulties can be obviated by plotting the real and imaginary components of the impedance 共see, for example, Betova et al. 17 兲. The real part of the impedance, shown in Fig. 4a, provides the same information as is available from the modulus plots presented in Fig. 2b. The high-frequency asymptote reveals the ohmic electrolyte resistance, and the low-frequency asymptote reveals the sum of the polarization impedance and the electrolyte resistance. The imaginary part of the impedance, pre- sented in Fig. 4b, has the significant advantage that the characteristic

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