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Identifying two piecewise linear additive value functions from anonymous preference information
Vincent Auriau, Khaled Belahcene, Emmanuel Malherbe, Vincent Mousseau, Marc Pirlot
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
The paper addresses the problem of eliciting two distinct additive value functions (UTA models) from anonymous preference information provided by two decision-makers. The authors propose an elicitation procedure that identifies these piecewise linear marginal value functions using a finite sequence of indifference queries. The method relies on two types of geometric constraints: single-rectangle queries to determine slope pairs and neighboring-rectangle queries to couple and assign slopes to specific models, ensuring identifiability without knowing which answer belongs to which decision-maker.
Entities (9)
Relation Signals (7)
Additive Value Function → hascomponent → Marginal Value Function
confidence 95% · the value of x is defined by u(x)= ∑ i u i (x i ), where u i is the marginal value function on criterion i.
Marginal Value Function → istype → Piecewise Linear Function
confidence 92% · we consider that these marginals u i are strictly increasing and piecewise linear
Elicitation Procedure → uses → Indifference Query
confidence 90% · It is possible to elicit two piecewise linear additive preference models based on anonymized preference information using a finite number of indifference queries.
Neighboring Rectangles Constraint → assigns → Slope
confidence 88% · neighboring rectangles queries provide coupling constraints allowing to assign the slopes to the two value functions.
Single Rectangle Constraint → determines → Slope Pair
confidence 88% · Single rectangle queries make it possible to elicit the anonymized pair of slopes of marginal value functions on a specific interval
Commuter → haspreferencemodel → Additive Value Function
confidence 85% · Commuter will use the car in a urban context... The marginal value functions for both DMs (Traveler & Commuter) are provided
Traveler → haspreferencemodel → Additive Value Function
confidence 85% · Traveler will drive long distances... The marginal value functions for both DMs (Traveler & Commuter) are provided
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Abstract
Abstract:Eliciting a preference model involves asking a person, named decision-maker, a series of questions. We assume that these preferences can be represented by an additive value function. In this work, we query simultaneously two decision-makers in the aim to elicit their respective value functions. For each query we receive two answers, without noise, but without knowing which answer corresponds to which this http URL propose an elicitation procedure that identifies the two preference models when the marginal value functions are piecewise linear with known breaking points.
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- Source: https://arxiv.org/abs/2602.20638v1
- Canonical: https://arxiv.org/abs/2602.20638v1
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Identifying two piecewise linear additive value functions from anonymous preference information Vincent Auriau a, b, * , Khaled Belahcène a , Emmanuel Malherbe b , Vincent Mousseau a and Marc Pirlot c a MICS, CentraleSupélec, Université Paris Saclay, France b Artefact Research Center, France c MATHRO, Université de Mons, Belgique ORCID (Vincent Auriau): https://orcid.org/0009-0002-9640-2639, ORCID (Khaled Belahcène): https://orcid.org/0000-0003-4502-9539, ORCID (Emmanuel Malherbe): https://orcid.org/0009-0006-0898-6873, ORCID (Vincent Mousseau): https://orcid.org/0000-0001-8574-3337, ORCID (Marc Pirlot): https://orcid.org/0000-0002-3689-0944 Abstract. Eliciting a preference model involves asking a person, named decision-maker, a series of questions. We assume that these pref- erences can be represented by an additive value function. In this work, we query simultaneously two decision-makers in the aim to elicit their respective value functions. For each query we receive two answers, without noise, but without knowing which answer corre- sponds to which decision-maker. We propose an elicitation procedure that identifies the two preference models when the marginal value functions are piecewise linear with known breaking points. 1 Introduction The additive value model is standard to specify preferences over a set of alternatives evaluated on several criteria, see [11]. More precisely, given an alternativexdefined by its evaluation onncriteriax= (x 1 ,...,x n ), wherex i is the evaluation ofxon criterioni, the value ofxis defined byu(x)= ∑ i u i (x i ), whereu i is the marginal value function on criterioni. In this paper, we consider that these marginals u i are strictly increasing and piecewise linear, with an a priori defined linear decomposition. In the following, we will call such preference model a UTA model. To elicit such a UTA model, it is usual to define standard queries corresponding to matching questions [16] involving two alternativesxandywhose evaluations differ on two criteriaiand jonly. Three evaluations out ofx i ,x j ,y i ,andy j are given, and the respondent provides the unfixed value in such a way that indifference holds betweenxandy(which is notedx∼y). It implies thatu i (x i )+ u j (x j )=u i (y i )+u j (y j ). The set of points(x i ,x j )in the(i,j) plane for whichu i (x i )+u j (x j )is a constant forms an indifference curve in the(i,j)plane. Using matching questions, one can identify points belonging to each indifference curve [3], which will prove useful in the sequel. We consider a framework in which we have to elicit simultane- ously two UTA models. In this context, the answer to a matching query consists of two evaluationsaanda ′ for the unfixed compo- nent. These responses are provided anonymously, without the ability to know to which model each answer corresponds. In this paper, our ∗ Corresponding Author. Email: vincent.auriau@artefact.com purpose is to study this problem from an identifiability perspective. In other words, we investigate whether it is always possible to define a finite sequence of queries whose answers enable the identification of the two piecewise linear additive value functions models. The iden- tifiability problem can also be formulated as a game in which a first player specifies the queries, and a second player provides (thruthfull) answers. The question becomes: is there a finite winning strategy for the first player? In another variant of the game (not considered in this paper), the second player can answer with an (adversarial) noise. Although the problem is purely formulated as a formal one, it is worth noticing that it corresponds to real-world problems. For in- stance, a supermarket is willing to adapt the list of products to its customer base. The products selected should align with the client’s preferences. Yet, not all customers necessarily have the same pref- erences, and one can consider a market segmentation in which each segment represents a group of clients with homogeneous preferences. Example 1.In this paper, we will use, as a running example, the choice among electric cars that differ on two criteria: autonomy (from 100km to 600km) and price (from 10keto 50ke). Answers to preference queries are obtained through an online system that guar- antees anonymity. Two decision-makers express their preferences: Commuterwill use the car in a urban context for relatively short distances, whileTravelerwill drive long distances.Commuterand Travelercan use respectively 30keand 40keliquid savings; they need to get a loan for car that are more expensive than their savings. Such type of problem has already been tackled from a preference learning perspective [2], and the identifiability result proposed here is of high importance to justify the search for efficient learning algo- rithms that infer multiple UTA models from preference statements. Our aim is to provide a systematic elicitation procedure to identify two additive piecewise linear value functions from a finite sequence of matching queries. In this introduction, we provide an overall pre- sentation of this procedure; the following sections aim at providing a detailed and justified description of this elicitation procedure. Hence, the main result of the paper can be formulated as follows. Theorem 1.It is possible to elicit two piecewise linear additive pref- erence models based on anonymized preference information using a finite number of indifference queries. arXiv:2602.20638v1 [cs.AI] 24 Feb 2026 In our setting, each matching query involves alternatives that vary on two criteriaiandjonly, with three values fixed by the Eliciter and the responses are provided by the different Decision-Makers (DMs) in order to obtain an indifference statement. The value space of cri- teria(i,j)defines a plane where the queries can be fully represented since everything else is equal. In particular, given that marginal value functions are piecewise linear, the indifference curves are also repre- sented by (different) piecewise linear functions. In this criteria plane, it is convenient to define two types of queries with a specific geome- try that we will use as “building blocks” of our elicitation procedure: ●Single Rectangle constraint:The query values as well as the an- swers are within a single rectangle, defined by the a priori defined linear segments on both criterion scales. ●Neighboring Rectangles constraint:The query values are selected such that it implies two neighboring rectangles defined by the lin- ear segments on the criterion scales. Single rectangle queries make it possible to elicit the anonymized pair of slopes of marginal value functions on a specific interval, while neighboring rectangles queries provide coupling constraints allowing to assign the slopes to the two value functions. Using these building blocks, the procedure can be summarized as follows: ●Initialization: the procedure starts with the two first criteria and postulates w.l.o.g. that the slope on criterion 1 are equal to one on the first segment. An Single-Rectangle query identifies the slope of each model for the first segment on criterion 2. ●Iteration: given an elicited segmentℓon criterion1, a succession of queries allows to identify the slopes of both models on segment ℓ+1; the same applies to criterion 2. ●Finally, in order to obtain all marginal value functions, one can apply the same principle for all pairs of criteria (1,i),i≠1. The paper is organized as follows. After a review of related works in Section 2, Section 3 provides general notation. Section 4 presents the two query types which constitute “building blocks” for the iden- tification procedure. Section 5 presents results justifying the iterative nature of the general procedure presented in Section 6. 2 Background and Related Work The additive value (or utility) function (AVF) model is the most widely used and studied preference model. It may represent pref- erence relations on sets of objectsx=(x 1 ,...,x n )described byn attributes or criteria. If attributeitakes values in a setX i (criterion scale), let us consider a preference relation≿on the Cartesian product X=Π n i=1 X i . Under well understood conditions on the preference≿ [11, 10, 16], there exist real valued functionsu i ∶X i →Rsuch that, for allx,y∈X, we have x≿yiff n ∑ i=1 u i (x i )≥ n ∑ i=1 u i (y i ).(1) The marginal value (or utility) functionsu i are unique up to positive affine transformations. The main property required of≿for admitting such a representation is an independence condition allowing force- teris paribusreasoning. Specifically, one may compare objects with- out specifying the values of the attributes on which they agree. Such characterization results allow one to devise query strategies for elicit- ing the marginal value functions, hence describing the preferences on the whole Cartesian product by a numerical representation. Among the query strategies, the method of indifference judgments builds a “standard sequence” of points on a criterion scale, which are equally spaced (in terms of value). This is done by using a fixed interval on the scale of another criterion as a unit measure. Queries are designed as follows. Letq i ,p i be two distinct values on the scaleX i of cri- terioni, andq j a value on the scaleX j of criterionj. The query is: “What is the valuea j ∈X j such that the value difference betweenq i andp i onX i is exactly compensated by the value difference between q j anda j onX j (all other things being equal, i.e., ceteris paribus)?” The answera j is a first point in a standard sequence. The next point is the answer to the modified query obtained by substitutingq j bya j in the initial query, and so on. The pairs(q i ,q j ),(p i ,a j )belong to an indifference curve in the domainX i ×X j (hence the name “indif- ference judgments”). Jacquet-Lagrèze and Siskos [8] developed a method, called UTA, to learn marginal value functions of an AVF that is compatible with known preferences expressed on a set of pairs of objects. In order to exploit linear programming they assume that the marginal value functions are piecewise linear. Let the pointsx i,ℓ ,ℓ=0,1,...,Lpar- tition the scaleX i of criterioniintoLsubintervals[x i,ℓ ,x i,ℓ+1 ], and the marginal value function be piecewise linear with respect to this decomposition. The latter is known as soon as the valuesu i (x i,ℓ ) at the intervals endpoints are determined. Alternatively, it suffices to determine the slopes of the linear segments of the marginal value functionγ i,ℓ =[u i (x i,ℓ )−u i (x i,ℓ−1 )]/(x i,ℓ −x i,ℓ−1 ). This method has been subsequently widely developed and applied [9]. In gen- eral, there are several models (actually, an infinite number of models) compatible with the set of preference examples. Some papers [e.g., 7] deal with formulating robust conclusions by considering the set of all models compatible with the examples. In this work, we consider the case in which two DMs reply to elicitation queries each using her own AVF model, with piecewise linear marginals. The answers to the queries are not tagged by the name of the DM. The question we tackle is whether it is possible to identify each DM’s model without ambiguity, by asking queries of the type “indifference judgments”. Similar work about model identifiability problems have been ad- dressed in multicriteria decision analysis in the case of the Choquet integral [12], the hierarchical Choquet integral [5] and more [4, 14]. This question of model identifiability can be approached from differ- ent angles. At its core is a parametric family of functions(f ω ) ω∈Ω whereΩis the latent parameter set, and eachf ω is a function map- ping queries inQto answers inA. Abstract identifiability can be cast as the existence of a queryq∈Qsuch that two distinct parameters ω≠ω ′ ∈Ωyield distinct outcomes, i.e.f ω (q)≠f ω ′ (q). More opera- tional definitions exist. For instance, various notions of combinatorial dimension have been put forward so as to upper bound the number of queries needed to distinguish concepts [1]. In turn, these bounds can be used to obtain statistical guarantees w.r.t. the estimation of those parameters from data, playing a key role in probably approxi- mately correct machine learning (see e.g. [15, 13]). The problem we address here is not standard w.r.t. those approaches, because neither the queries nor the latent parameters are. Indifference queries do not have a binary outcome, as expected from concept learning theory. More importantly, our latent parameter space encompasses both the preference models of the DMs but also the identity of the DM asso- ciated to each answer (and thus grows with the number of queries). A more fruitful analogy comes from adversarial game-playing: con- sider two players, the Eliciter and the Adversary, and a fixed param- eter space, such thatωcompletely specifies two UTA models. The Eliciter can either query the Adversary, or reveal two UTA models. The Adversary answers queries as he wishes. He immediately loses if there are no parameterωcompatible with all answers, and wins if the Eliciter reveals a parameterω E and he can provide an alternate ω ′ ≠ωwhich is compatible with all answers. A procedure based on mixed-integer linear programming is described in [2] and allows to decide whether a set of preference statements can be partitioned into two DMs represented by a UTA model, and can serve as a judge in this game. In effect, this allows the Adversary to cheat and modify the model behind the scene, as long as this behavior cannot be de- tected, and forces the Eliciter to reduce the set of all possible models to a singleton. Such approach can be found in [6]. Here, we show that the Eliciter has indeed a winning strategy. 3 Problem setting We consider a setN=1,...,nof criteria. A criterioni∈Nis defined on its scale, notedX i ⊂R. For each criterioni, we are given a positive integerL i andL i +1valuesx i,0 <x i,1 <⋅<x i,L i ∈X i . The value function of a DMκin the form of a UTA model can be defined by its values at the edges of the intervals, (u κ i (x i,ℓ )) ℓ≤L i ,i∈N . An equivalent formulation uses the slopes of this value functionγ κ i,ℓ = u κ i (x i,ℓ )−u κ i (x i,ℓ−1 ) x i,ℓ −x i,ℓ−1 for1≤ℓ≤L i . We consider the functions to be strictly increasing, meaning thatγ κ i,ℓ >0. Example 2.(Ex. 1 cont.) The marginal value functions for both DMs (Traveler & Commuter) are provided in Figure 1(a) for the price criterion, and in Figure 1(b) for the autonomy criterion. These value functions induce interpretable preferences: for instance, an increase in autonomy from 100km to 200km is valued 2 (10, resp.) forTraveler (forCommuter, resp.). Indeed, 200km autonomy is a poor value for Travelerwho drives long distances, while it is already reasonably good forCommuterwho drives shorter distances. Given the value functionu κ of DMκ, one can defineIndifference curveswhich are the sets of pointsx∈ ∏ i∈N Xin the Cartesian product of criteria having the same value, i.e., such thatu κ (x)= z(z∈R). All pairs of objects on an Indifference Curve are said to be indifferent, meaning they belong to the symmetric part∼of the≿ relation defined in (1). Example 3.(Ex. 2 cont.) Indifference curves induced by the value functions ofTravelerandCommuterare depicted in Figure 1(c). Ob- viously, for a given DM, indifference curves do not cross. On the con- trary the indifference curves of the different DMs do cross, meaning their preferences are different. One can note that the criteria spaceX i ×X j can be seen as a grid where each rectangle element is defined by the breaking points on each criterion: ( x i,ℓ i −1 ,x j,ℓ j −1 ) ; ( x i,ℓ i ,x j,ℓ j ) . In this specific rect- angle denotedR ℓ i ,ℓ j (corresponding to theℓ th i interval of criterioni and the theℓ th j interval of criterionj), the indifference represented by the value functions is a straight line, neither horizontal nor vertical, whose slope isγ κ i,ℓ i /γ κ j,ℓ j >0. We consider two DMs, notedαandβ, that can be queried at will. We define a queryQas questioning the DMs concerning alterna- tives whose evaluations vary on two criteria only. More specifically, we formulateQas the specification of an ordered pair of criteria, (i,j)∈N 2 and a triplet of values on these criteria:(q i ,q j ,p i ), with (q i ,p i )∈X 2 i andq j ∈X j . This query triggers the collection of an- swersA=a κ j ,a κ ′ j ⊂X j from the two DMs. We arbitrarily decide to noteκandκ ′ such thata κ j ≤a κ ′ j . We also sometimes omit to spec- ify the selected criteria, when it is already clear enough. The elements ofAare such that either[(q i ,q j )∼ α (p i ,a κ j ),(q i ,q j )∼ β (p i ,a κ ′ j )] or[(q i ,q j )∼ β (p i ,a κ j ),(q i ,q j )∼ α (p i ,a κ ′ j )], with∼ α (resp.∼ β ) denoting indifference for DMα(resp.β). In other words, the an- swers are non-identifiable, meaning that we don’t known which DM they come from.Some queries cannot be answered by one, or both DMs, when the difference of value on the first criterion is too large to be compensated on the second one. In this case, we are provided with the answer “None”. Example 4.(Ex. 3 cont.) a queryQ=(200km, 30ke)∼(400km, ?) can be interpreted as: given a car with a 30keprice having 200km autonomy, how much more would you be willing to pay to obtain 400km autonomy? the two answers are (34ke, 43ke) without know- ing which DM gave each answer. Eliciting the value function of both DMs involves a sequence of queries, and for each of these queries, there is a hidden bit of infor- mation. Hence, the problem of attributing each answer to the correct DM compounds at each step–giving a combinatorial aspect, poten- tially explosive, to this problem. A summary of the notation can be found in Table 1. Table 1: Notations summary DescriptionNotation Criteria setN=1,...,ni,j∈N Criterion scaleX i =[x i,0 ,x i,L ] Criterion interval[x i,ℓ ,x i,ℓ+1 ]ℓ∈0,...,L Decision makersα,β Value functionsu κ (⋅)κ∈α,β Marginal value functions u κ i (⋅)κ∈α,β,i∈N Value function slopes γ κ i,ℓ κ∈α,β,i∈N, 1≤ℓ≤L QueriesQ=(q i ,q j )∼(p i ,?)q i ∈X i ,p j ∈X j Answersa κ i ,a κ ′ i i∈N,a κ i ∈X i 4 Play Patterns Our perfect Eliciter relies on two basic play patterns or sequences of moves (i.e. indifference queries) yielding preference information that allows it to reach specific short-term goals. Given two criteriai≠j∈ N, these goals can be specified geometrically in the planeX i ×X j : obtaining indifference statements between points located either in the same rectangle (Section 4.1) or adjacent rectangles (Section 4.2). 4.1 Obtaining Single-Rectangle Preference Information We are first interested in collecting preference information in the form of anonymized indifference statements between points lying in the same rectangleR ℓ i +1,ℓ j +1 . Play Pattern 1 ensures this informa- tion can be obtained in at most two queries. The play pattern opens with the following query from the bottom right corner to the left side:(i∶x i,ℓ i +1 ,j∶x j,ℓ j )∼(i∶x i,ℓ i ,j∶?), yielding two answersa 1 <a 2 ∈X i ∪None. There are three cases: ●Ifx j,ℓ j ≤a 1 ≤a 2 ≤x j,ℓ j +1 (Case 1, illustrated by Fig. 2). Both indifference curves lie below the diagonal and hit the left side of the rectangle. The specified goal is reached in a single query. ●Else, ifa 1 ≤x j,ℓ j (Case 2, illustrated by Fig. 3). Exactly one DM has an indifference curve (in blue) steeper than the diagonal and that hits the top side of the rectangle. Querying on the bottom side Price (in k$) u i ( x i ) 5040302010 0 10 14 28 35 Traveler Value Function Commuter Value Function (a)Marginal value of Price Autonomy (in km) u i ( x i ) 100200400600 0 2 10 17 20 30 Traveler Value Function Commuter Value Function (b)Marginal value of Autonomy Autonomy (in km) Price ( ink $) 100200400600 50 40 30 10 Traveler Indifference Curve Commuter Indifference Curve (c)Indifference curves for Price vs Autonomy Figure 1: Preferences of the DMs from the example, serving as ground truth for the elicitation. Marginal values for Price (left), Autonomy (middle), and resulting indifference curves (right). from the point given by the other DM (in yellow), makes sure to obtain two valuesb 1 <b 2 ∈X j such thatb 2 =x j,ℓ j +1 , yielding two points on the bottom side of the rectangle. ●Finally, Case 3 covers all situations where both indifference curves are steeper than the diagonal and hit the top side of the rectangle, as illustrated by Fig. 4. Querying from the top left corner ensures getting both answers on the bottom side of the rectangle. x j, j = q j a j x j, j + 1 a j A B C DM DM x i, 0 x j, 0 x i, i = p i x i, i + 1 = q i Figure 2: A successful Single Rectangle query in the spaceX i ×X j , corresponding to case 1. Colored lines represent DMs’ indifference curves. x j, j = q j a j x j, j + 1 a j x i, 0 x j, 0 x i, i = p i x i, i + 1 = q i x j, j = p j x j, j + 1 = q j A BC DM DM Query Answers x i, 0 x j, 0 x i, i = q i x i, i + 1 = a i a i Figure 3: Outcome and second query for case 2 of the Single Rectangle Constraint. Exploiting Single-Rectangle Preference InformationSuppose that we have three points in[x i,ℓ i ,x i,ℓ i +1 ]×[x j,ℓ j ,x j,ℓ j +1 ] A(A i ,A j ),B(B i ,B j )≠A,C(C i ,C j )≠Asuch that one DM – sayκ∈α,β– is indifferent betweenAandBand the other –say κ ′ ≠κ∈α,β– is indifferent betweenAandC. x j, j = q j a j x j, j + 1 a j x i, 0 x j, 0 x i, i = p i x i, i + 1 = q i x j, j = p j x j, j + 1 = q j A BC DM DM x i, 0 x j, 0 x i, i = q i x i, i + 1 a i a i Figure 4: Outcome and second query for case 3 of the Single Rectan- gle Constraint. Play Pattern 1:to get a Single Rectangle Constraint Input:two criteriai≠j, two interval labelsℓ i ∈J0,L i −1K, ℓ j ∈J0,L j −1K Output:three pointsA,B≠A,C≠Ain[x i,ℓ i ,x i,ℓ i +1 ]× [x j,ℓ j ,x j,ℓ j +1 ]s.t. one DM is indifferent betweenA andBand the other is indifferent betweenAandC a 1 ,a 2 ←Query(i∶x i,ℓ i +1 ,j∶x j,ℓ j )∼(i∶x i,ℓ i ,j∶?); ifa 2 ≤x j,ℓ j +1 then/ * Case 1 return(x i,ℓ i +1 ,x j,ℓ j ),(x i,ℓ i ,a 1 ),(x i,ℓ i ,a 2 ); ifa 1 ≤x j,ℓ j +1 then/ * Case 2 b 1 ,b 2 ←Query(i∶x i,ℓ i ,j∶a 1 )∼(j∶x j,ℓ j ,i∶?); return(x i,ℓ i ,a 1 ),(b 1 ,x j,ℓ j ),(b 2 ,x j,ℓ j ); else/ * Case 3 b 1 ,b 2 ←Query(i∶x i,ℓ i ,j∶x j,ℓ j +1 )∼(j∶x j,ℓ j ,i∶?); return(x i,ℓ i ,x j,ℓ j +1 ),(b 1 ,x j,ℓ j ),(b 2 ,x j,ℓ j ); (A i ,A j )∼ κ (B i ,B j ) (A i ,A j )∼ κ ′ (C i ,C j ) ⇔ u κ ((A i ,A j ))=u κ ((B i ,B j )) u κ ′ ((A i ,A j ))=u κ ′ ((C i ,C j )) The additive value function is such thatu κ ((A i ,A j ))=u κ i (x i,ℓ i )+ γ κ i,ℓ i +1 ⋅(A i −x i,ℓ i )+u κ j (x j,ℓ j )+γ κ j,ℓ j +1 ⋅(A j −x j,ℓ j ), with a similar form foru κ ′ . More details are shared in Appendix A. It leads to: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ κ j,ℓ j =Λ κ ⋅γ κ i,ℓ i γ κ ′ j,ℓ j =Λ κ ′ ⋅γ κ ′ i,ℓ i with: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Λ κ = B i −A i A j −B j Λ κ ′ = C i −A i A j −C j IfB=Cthenγ α i /γ α j =γ β i /γ β j . Otherwise, letk∈0,1such that k=0iffαanswersBandβanswersC. We have: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α j,ℓ j =(k⋅Λ κ +(1−k)⋅Λ κ ′ )⋅γ α i,ℓ i γ β j,ℓ j =((1−k)⋅Λ κ +k⋅Λ κ ′ )⋅γ β i,ℓ i (2) 4.2 Obtaining Neighboring-Rectangles Preference Information We are also interested in collecting preference information in the form of anonymized indifference statementsA∼ κ B,A∼ κ ′ C between points lying in adjacent rectangles, i.e.A∈R ℓ i ,ℓ j −1 and B,C∈R ℓ i ,ℓ j +1 . Play Pattern 2 ensures this information can be ob- tained with a finite number of queries. The play pattern is based on the iterated uses of queries of the form(i∶x i,ℓ i +δ i ,j∶ 1 2 (x j,ℓ j −1 +x j,ℓ j )+ λ 2 δ i )∼(i∶x i,ℓ i ,j∶?). The valuesλandδ i are initialized so that the first query point lies at the center of rectangleR ℓ i ,ℓ j . These values are then adjusted ac- cording to the answers until both answers are in the sought rectangle and the pattern terminates, as illustrated by Figure 5. Case 1 corre- sponds to the situation where both answers are abovex j,ℓ j but at least one is not belowx j,ℓ j +1 . In such a case,δ i is divided by two and the query point moves left (see Figure 6)- along a line passing through the point(x i,ℓ i ,x j,ℓ j )whose slope is controlled byλ. The variableλis initialized so that this line coincides with the diagonal of the lower rectangle. If, at the first query, the lowest answer is be- lowx j,ℓ j we obtain single-rectangle preference information for one of the DMs and can compute the actual value of the slope of its indif- ference curve, which is stored inλ(Case 2, illustrated by Figure 7). In effect, the querying point moves towards the upper-left vertex of the lower rectangle along a more gentle slope, ensuring Case 2 never occurs again. Exploiting Neighboring-Rectangles Preference Information. Suppose that we haveA(A i ,A j )in[x i,ℓ i ,x i,ℓ i +1 ]×[x j,ℓ j −1 ,x j,ℓ j ] andB(B i ,B j )≠A,C(C i ,C j )≠Ain[x i,ℓ i ,x i,ℓ i +1 ]× [x j,ℓ j ,x j,ℓ j +1 ]such that one DM is indifferent betweenAandB and the other, is indifferent betweenAandC. At this point, we at- tributeBto the DMκandCtoκ ′ , but we do not know whether (κ,κ ′ )is(α,β)or(β,α). (A i ,A j )∼ κ (B i ,B j ) (A i ,A j )∼ κ ′ (C i ,C j ) Contrary to the Single Rectangle, we have:u κ j (B j )−u κ j (A j )= γ κ j,ℓ j +1 ⋅(B j −x j,ℓ j )+γ κ j,ℓ j ⋅(x j,ℓ j −A j ), leading to: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ κ j,ℓ j +1 =γ κ i,ℓ i ⋅Θ κ +γ κ k,ℓ j ⋅Φ κ γ κ ′ j,ℓ j +1 =γ κ ′ i,ℓ i ⋅Θ ′ κ +γ κ ′ j,ℓ j ⋅Φ κ ′ Play Pattern 2:to get a Neighboring Rectangles Constraint Input:two criteriai≠j, two interval labelsℓ i ∈J1,L i K, ℓ j ∈J1,L j −1K Output:three pointsA,B,CwithA∈[x i,ℓ i ,x i,ℓ i +1 ]× [x j,ℓ j −1 ,x j,ℓ j ]andB,C∈[x i,ℓ i ,x i,ℓ i +1 ]× [x j,ℓ j ,x j,ℓ j +1 ]such that one DM is indifferent betweenAandBand the other is indifferent betweenAandC δ i ← 1 2 ⋅(x i,ℓ i+1 −x i,ℓ i ); λ←(x j,ℓ j −x j,ℓ j −1 )/(2δ i ); a 1 ,a 2 ←Query(x i,ℓ i +δ i ,x j,ℓ j −1 −λ⋅δ i )∼(x i,ℓ i ,?); whilenot(a 1 >x j,ℓ j anda 2 ≤x j,ℓ j +1 )do ifa 1 >x j,ℓ j thenδ i ←δ i /2;/ * Case 1 elseλ←(x j,ℓ j −a 1 )/(2δ i );/ * Case 2 a 1 ,a 2 ←Query(x i,ℓ i +δ i ,x j,ℓ j −λ⋅δ i )∼(x i,ℓ i ,?); end return(x i,ℓ i +δ i ,x j,ℓ j −λ⋅δ i ),(x i,ℓ i ,a 1 ),(x i,ℓ i ,a 2 ); with: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Θ κ = A i −B i B j −x j,ℓ j ; Θ κ ′ = A i −C i C j −x j,ℓ j Φ κ = A j −x j,ℓ j B j −x j,ℓ j ; Φ κ ′ = A j −x j,ℓ j C j −x j,ℓ j Letk∈0,1such thatk=0iffαanswersBandβanswersC: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α j,ℓ j +1 =γ α i,ℓ i (kΘ κ +(1−k)Θ κ ′ )+γ α j,ℓ j (kΦ κ +(1−k)Φ κ ′ ) γ β j,ℓ j +1 =γ β i,ℓ i ((1−k)Θ κ +kΘ κ ′ )+γ β j,ℓ j ((1−k)Φ κ +kΦ κ ′ ) (3) x j, j 1 x j, j x j, j + 1 a j a j q j A B C / i i DM DM x i, 0 x j, 0 x i, i = p i x i, i + 1 q i Figure 5: Example of a Query with a neighboring rectangles con- straint. Colored lines represent the indifference curve of each DM. 5 Propagation Patterns At this point, each low-level pattern introduced in Section 4 yields preference information about the slopes of the indifference curves. They are exploited by introducing a binary variablekrepresenting two possible worlds per query, according to which DM provide each answer. Thus, carelessly chainingnof those patterns opens2 n pos- sible worlds, as illustrated by Figure 8. To avoid this combinatorial explosion, we devise two higher-level sequences of moves that aim x j, j 1 x j, j x j, j + 1 a j a j q j x i, 0 x j, 0 x i, i = p i x i, i + 1 q i x j, j 1 x j, j x j, j + 1 a j a j q j q ′ j DM DM x i, 0 x j, 0 x i, i = p i x i, i + 1 q i q ′ i Figure 6: Outcome and second query for case 1 of Play Pattern 2. x j, j 1 x j, j x j, j + 1 a j a j q j x i, 0 x j, 0 x i, i = p i x i, i + 1 q i x j, j 1 x j, j x j, j + 1 a j a j q j q ′ j DM DM x i, 0 x j, 0 x i, i = p i x i, i + 1 q i Figure 7: Outcome and second query for case 2 of Play Pattern 2. and succeed at correctly assigning the disambiguation variablesk, effectively pruning the worlds corresponding to an incorrect assign- ment. These sequences differ in their requirements: in Section 5.1, we assume we know the slopes of the marginal value functions of each DM on two intervals ofX i and we obtain the slopes and their assignment to each DM on an interval of the other criterion; in Sec- tion 5.2 we assume we know the slopes of the marginal value func- tions of each DM on an interval ofX i and an interval ofX j , and we obtain the slopes and their assignment to each DM on a neighboring interval. 5.1 Identification with two single rectangle constraints Play Pattern 3:Identification with two Single-Rectangle Constraints Input:A criterionjand an interval labelsℓ j ∈J1,L j K; a second criterioni≠j, and two interval labels ℓ i ≠ℓ ′ i ∈J1,L i Ksuch thatγ α i,ℓ i ,γ β i,ℓ i ,γ α i,ℓ ′ i andγ β i,ℓ ′ i are known. Output:the identified values ofγ α j,ℓ j ,γ β j,ℓ j A,B,C←Play Pattern 1((i,j),(ℓ i ,ℓ j )); A ′ ,B ′ ,C ′ ←Play Pattern 1((i,j),(ℓ ′ i ,ℓ j )); System1a←Equation 2 andA,B,C; System1b←Equation 2 andA ′ ,B ′ ,C ′ ; γ α j,ℓ j ,γ β j,ℓ j ←Solve(1a)=(1b), see Appendix B; returnγ α j,ℓ j ,γ β j,ℓ j ; Our play patterns have been defined to be able to identify the slopes of the value functions of both DMs on a specific intervalℓ j of a criterionj. The idea is to use two intervalsℓ i ≠ℓ ′ i on a crite- rionifor which the value functions are known and withγ α i,ℓ i /γ α i,ℓ ′ i ≠ γ β i,ℓ i /γ β i,ℓ ′ i . Using Play Pattern 1, on[x i,ℓ i ,x i,ℓ i +1 ]× [ x j,ℓ j ,x j,ℓ j +1 ] , we obtain a system, from Equation 2 forγ α,j,ℓ j andγ β,j,ℓ j . Doing the same for[x i,ℓ ′ i ,x i,ℓ ′ i +1 ]× [ x j,ℓ j ,x j,ℓ j ] we get a second system with different expression for these two slopes. Equating these expres- sions, we obtain two equations with two variablesk,k ′ : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (kΛ κ +(1−k)Λ κ ′ )γ α i,ℓ i = ( k ′ Λ ′ κ +(1−k ′ )Λ ′ κ ′ ) γ α i,ℓ ′ i ((1−k)Λ κ +kΛ κ ′ )γ β i,ℓ i = ( (1−k ′ )Λ ′ κ +k ′ Λ ′ κ ′ ) γ β i,ℓ ′ i (4) This system is non-singular. Its resolution yields the values of vari- ableskandk ′ and the respective slopes of the marginal value func- tions of each DM. Details can be found in Appendix B and the de- scribed procedure is summarized in Algorithm 3. This result means that once one marginal value function has been identified on at least two distinct intervals, it is possible to generalize the identifiability of any interval on another criterion. 5.2 Identification with Single Rectangle and a Neighboring Rectangles queries Play Pattern 4:Identification with a Single-Rectangle and a Neighboring-Rectangles Constraints Input:A criterionjand an interval labelsℓ j ∈J2,L j Kfor whichγ α j,ℓ j −1 andγ β j,ℓ j −1 are known; a second criterioni≠jand an interval labelℓ i ∈J1,L i Ks.t. γ α i,ℓ i andγ β i,ℓ i are known. Output:the identified values ofγ α j,ℓ j ,γ β j,ℓ j A,B,C←Play Pattern 1((i,j),(ℓ i ,ℓ j )); A ′ ,B ′ ,C ′ ←Play Pattern 2((i,j),(ℓ i ,ℓ j −1)); System1a←Equation 2 andA,B,C; System1b←Equation 3 andA ′ ,B ′ ,C ′ ; γ α j,ℓ j ,γ β j,ℓ j ←Solve(1a)=(1b), see Appendix C; returnγ α j,ℓ j ,γ β j,ℓ j ; Our second identification pattern, described in Play Pattern 4, is relatively similar to the first one. The difference lies in the use of a neighboring interval of the criterion interval to be elicited instead of a second interval on the other criterion. Therefore, we will make use of one - instead of two - Single Rectangle query and one Neigh- boring Rectangles query. Considering an interval labeledℓ j −1on a criterionjthat has been elicited, we choose an additional inter- valℓ i on a different criterionithat has also been elicited and for whichγ α i,ℓ i /γ α j,ℓ j −1 ≠γ β i,ℓ i /γ β j,ℓ j −1 . We use the Play Pattern 1 on [ x j,ℓ j ,x j,ℓ j +1 ] ×[x i,ℓ i ,x i,ℓ i +1 ]to obtain the triplet(A,B,C)and Play Pattern 2 on [ x j,ℓ j −1 ,x j,ℓ j +1 ] ×[x i,ℓ i ,x i,ℓ i +1 ]to obtain the triplet(A ′ ,B ′ ,C ′ ). It leads to two systems from Equations 2, 3. These systems express different equations forγ α j,ℓ j andγ β j,ℓ j and therefore can be reduced to : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (kΛ κ +(1−k)Λ κ ′ )γ α i,ℓ i =γ α i,ℓ i ( k ′ Θ ′ κ +(1−k ′ )Θ ′ κ ′ ) +γ α j,ℓ j −1 ( k ′ Φ ′ κ +(1−k ′ )Φ ′ κ ′ ) ((1−k)Λ κ +kΛ κ ′ )γ β i,ℓ i =γ β i,ℓ i ( k ′ Θ ′ κ +(1−k ′ )Θ ′ κ ′ ) +γ β j,ℓ j −1 ( k ′ Φ ′ κ +(1−k ′ )Φ ′ κ ′ ) (5) This system with two equations and two unknown variablesk,k ′ can be solved. With these values, we can finally obtain the values ofγ α j,ℓ j andγ β j,ℓ j using Equation 2 for example. U U x j, 0 x j, 1 p i q i u . , j (x j, 0 ) u , j (x j, 1 ) u , j (x j, 1 ) Elicited, Elicited, U U x j, 0 p j x j, 1 a , j a , j x j, 2 u . , j (x j, 0 ) u , j (x j, 0 ) u , j (x j, 0 ) Value Function, Value Function, Other potential slopes after 1st Query Figure 8: Possible value functions after Play Pattern 1. The colored lines represent the marginal value functions of each DM. 6 General Elicitation Procedure We propose a general elicitation procedure that can be used in order to identify a set of two DMs in our described setup. The procedure can be found in the Algorithm 5 and is described in the following sec- tions. A Python implementation and a few examples will be shared in a GitHub repository shared upon acceptance. Algorithm 5:General Strategy for the Elicitation Initialize criterioni: Setγ α i,ℓ i =γ β i,ℓ i =1for all DMs on[x i,ℓ i −1 ,x i,ℓ i ] Initialize criterionj: Play Pattern 1 on[x i,ℓ i −1 ,x i,ℓ i ]× [ x j,ℓ j ,x j,ℓ j +1 ] ; Attribute slopesγ α j,ℓ j ,γ β j,ℓ j on [ x j,ℓ j −1 ,x j,ℓ j ] ; Fully Elicit Criterioni,j: forℓ ′ j ∈Jℓ j ,L j K Play Pattern 4 on[x j,ℓ ′ j ,x j,ℓ ′ j +1 ]×[x i,ℓ i −1 ,x i,ℓ i ]; Solve Eq. 5 and deduct slopesγ α j,ℓ ′ j ,γ β j,ℓ ′ j ; forℓ ′ i ∈Jℓ i ,L i K Play Pattern 4 on[x i,ℓ ′ i ,x i,ℓ ′ i +1 ]× [ x j,ℓ j −1 ,x j,ℓ j ] ; Solve Eq. 5 and deduct slopesγ α i,ℓ ′ i ,γ β i,ℓ ′ i ; forcriterionj ′ ∈N,j≠i,j Initialize criterionj ′ : Play Pattern 3 on[x j ′ ,0 ,x j ′ ,1 ]×[x i,ℓ i ,x i,ℓ i +1 ]; Solve Eq. 4 and deduct slopesγ α j ′ ,1 ,γ β j ′ ,1 ; Fully Elicit Criterionj ′ : forℓ j ′ ∈J1,L j ′ K Play Pattern 4 on [x j ′ ,ℓ j ′ ,x j ′ ,ℓ ′ j +1 ]×[x i,ℓ i ,x i,ℓ i +1 ]; Solve Eq. 5 and deduct slopesγ α j ′ ,ℓ ′ j ,γ β j ′ ,ℓ ′ j ; 6.1 Initialization We begin by choosing two criteriai≠jand a rectangleR ℓ i ,ℓ j ⊂ X i ×X j and following Play Pattern 1 so as to obtain Single Rectan- gle preference information describing twointersectingindifference curves (i.e. distinct pointsBandC). If this is not possible, the DMs have the exact same preferences and the identification task is over. Otherwise, we set: γ α i,ℓ i =1;γ α j,ℓ j = B i −A i A j −B j ;γ β i,ℓ i =1;γ β j,ℓ j = C i −A i A j −C j Indeed, without loss of generality, we can arbitrarily assign the an- swerBto the DMα(resp.Ctoβ) and normalize the UTA model of α(resp.β) so that it has unit slope on[x i,ℓ i −1 ,x i,ℓ i ] 1 . In order to find a suitable rectangle, we traverse the intervals inX i andX j in increasing order. Thus, when one is found, all rectangles with lower indices have unanimous DMs and can be identified at this point. 6.2 Full elicitation of criteriaiandj It is now possible to use the Play Pattern 4 to elicit interval by inter- val the criteriaiandj, in increasing order. Let us consider that the criterionihas been elicited up to theℓ-th interval, we formulate: ●a Single Rectangle Query on [ x j,ℓ j −1 ,x j,ℓ j ] ×[x i,ℓ+1 ,x i,ℓ+2 ] ●aNeighboringRectanglesQueryon [ x j,ℓ j −1 ,x j,ℓ j ] × [x i,ℓ ,x i,ℓ+2 ] Following Equation 5, we are able to obtain the slopes valuesγ α i,ℓ+1 andγ β i,ℓ+1 . Using this procedure iteratively for2≤ℓ≤L i , it is pos- sible to fully elicit the criterioni. The same procedure can be applied for criterionj. The only change needed is the use of the interval [ x j,ℓ j −1 ,x j,ℓ j ] instead of [x i,ℓ i −1 ,x i,ℓ i ]for the different queries. The iterative steps for two criteria are illustrated in Figure 9, assuming initialization occurs in the rectangle[x i,0 ,x i,1 ]×[x j,0 ,x j,1 ]. Observe it is necessary and sufficient to exploit preference information concerning exactly one rectangle per column and per row (i.e.L i +L j −1rectangles) to fully identifyu α i ,u α j ,u β i ,u β j . 6.3 Elicitation of the remaining criteria The remaining criteria can finally be elicited independently. For a criterionj ′ ∉i,j, an initialization is done in order to get the slopes on the first interval, following the Play Pattern 3: ●a Single Rectangle Query on[x i,ℓ i −1 ,x i,ℓ i ]×[x j ′ ,0 ,x j ′ ,1 ] ●a Single Rectangle Query on[x i,ℓ i −1 ,x i,ℓ i ]×[x j ′ ,0 ,x j ′ ,1 ] Following Equation 4, we obtain the slopes valuesγ α i,1 andγ β i,1 . At this point, it is possible to use the procedure for criterioniorj described in Section 6.2 to fully elicit the criterionj ′ . x 1, 0 x 1, 1 x 1, 2 x 1, 3 x 2, 0 x 2, 1 x 2, 2 x 2, 3 Criterion 1 Criterion 2 Elicited Indifference Slope, DM Elicited Indifference Slope, DM Elicitation Order 123 4 5 Figure 9: General Procedure sequencing for the elicitation of two dif- ferent DMs on two criteria. 1 This normalization is licit but not standard in MAVT. The usual conven- tion of having values ranging from 0 to 1 can be easily restored after the complete identification of the two AVF. Conclusion In this paper, we study the identifiability of two additive piecewise linear value functions using matching queries. Such queries provide two answers (those of the two additive models), but without speci- fying to which model each answer corresponds. We provide a sys- tematic elicitation procedure that makes it possible to obtain the two additive piecewise linear models from a finite sequence of matching queries, hence proving identifiability. This work leaves open questions that should be studied in fur- ther research. First, our work is limited to additive value functions for which the marginals are piecewise linear; it would be interest- ing to study this identifiability problem when marginals can be any monotonically increasing function. A second extension relates to the fact that our work accounts for two DMs only; extending Single- and Neighboring-Rectangle queries to more than two DMs seems straightforward, but defining an elicitation procedure to identifyn additive models (n>2) would be of great interest. Finally, the posi- tive results concerning the identifiability of such model could help to analyze the behavior of statistical learning algorithms. References [1] D. Angluin. 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A Details for Equation 2 (A i ,A j )∼ κ (B i ,B j ) (A i ,A j )∼ κ ′ (C i ,C j ) ⇔ u κ ((A i ,A j ))=u κ ((B i ,B j )) u κ ′ ((A i ,A j ))=u κ ′ ((C i ,C j )) SinceA i ,B i ,C i ∈[x i,ℓ i ,x i,ℓ i +1 ]andA j ,B j ,C j ∈ [ x i,ℓ j ,x j,ℓ j +1 ] : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u κ i (x i,ℓ i )+γ κ i,ℓ i +1 ⋅(A i −x i,ℓ i )+u κ j (x j,ℓ j )+γ κ j,ℓ j +1 ⋅(A j −x j,ℓ j )=u κ i (x i,ℓ i )+γ κ i,ℓ i +1 ⋅(B i −x i,ℓ i )+u κ j (x j,ℓ j )+γ κ j,ℓ j +1 ⋅(B j −x j,ℓ j ) u κ ′ i (x i,ℓ i )+γ κ ′ i,ℓ i +1 ⋅(A i −x i,ℓ i )+u κ ′ j(x j,ℓ j )+γ κ ′ j,ℓ j +1 ⋅(A j −x j,ℓ j )=u κ ′ i (x i,ℓ i )+γ κ ′ i,ℓ i +1 ⋅(C i −x i,ℓ i )+u κ ′ j(x j,ℓ j )+γ κ ′ j,ℓ j +1 ⋅(C j −x j,ℓ j ) ⇔ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ κ i,ℓ+1 ⋅(A i −B i )+γ κ j,ℓ+1 ⋅(A j −B j )=0 γ κ ′ i,ℓ i +1 ⋅(A i −C i )+γ κ ′ j,ℓ j +1 ⋅(A j −C j )=0 ⇔Equation2 B Condition of unicity of the system resulting from Play Pattern 3 Using Play Pattern 1, on[x i,ℓ i ,x i,ℓ i +1 ]× [ x j,ℓ j ,x j,ℓ j +1 ] , we obtain a system, from Equation 2: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α j,ℓ j =(k⋅Λ κ +(1−k)⋅Λ κ ′ )⋅γ α i,ℓ i γ β j,ℓ j =((1−k)⋅Λ κ +k⋅Λ κ ′ )⋅γ β i,ℓ i (6) Following the same procedure with[x i,ℓ ′ i ,x i,ℓ ′ i +1 ]× [ x j,ℓ j ,x j,ℓ j ] we get a second system with different expression for our two slopes: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α i,ℓ i = ( k ′ ⋅Λ ′ κ ′ +(1−k ′ )⋅Λ ′ κ ′ ) ⋅γ α j,ℓ j γ β i,ℓ i = ( (1−k ′ )⋅Λ ′ κ +k ′ ⋅Λ ′ κ ′ ) ⋅γ β j,ℓ j (7) Equating the two systems, we obtain two equations with two variablesk,k ′ , which we can solve: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (kΛ κ +(1−k)Λ κ ′ )γ α i,ℓ i = ( k ′ Λ ′ κ +(1−k ′ )Λ ′ κ ′ ) γ α i,ℓ ′ i ((1−k)Λ κ +kΛ κ ′ )γ β i,ℓ i = ( (1−k ′ )Λ ′ κ +k ′ Λ ′ κ ′ ) γ β i,ℓ ′ i (8) ⇔ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [k(Λ κ −Λ κ ′ )+Λ κ ′ ]γ α i,ℓ i = [ k ( Λ ′ κ −Λ ′ κ ′ ) +Λ ′ κ ′ ] γ α i,ℓ ′ i [k(Λ κ ′ −Λ κ )+Λ κ ]γ β i,ℓ j = [ k ( Λ ′ κ ′ −Λ ′ κ ) +Λ ′ κ ] γ β (9) If the equations are linearly correlated, it means that the coefficients ink,k ′ are correlated between the equations. Differently said, it would mean that∃z∈Rsuch that: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (Λ κ −Λ κ ′ )γ α i,ℓ i =z⋅(Λ κ ′ −Λ κ )γ β i,ℓ i (Λ ′ κ ′ −Λ ′ κ )γ β i,ℓ ′ i =z⋅(Λ ′ κ −Λ ′ κ ′ )γ α i,ℓ ′ i ⇔ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ γ α i,ℓ i =−z⋅γ β i,ℓ i γ β i,ℓ ′ i =−z⋅γ α i,ℓ ′ i The second system holds if theΛvalues are different attandt+1. Which is the case if the answers of the two DMs are different. It results that the solution of the system can be found and is unique unless: ●the answers from the querytort ′ are the same for both DMs ●the slopes ratios of each DM are the same on the two intervals on criterioni, γ α i,ℓ i γ α i,ℓ ′ i = γ β i,ℓ i γ β i,ℓ ′ i C Condition of unicity of the system resulting from Play Pattern 4 Using Play Pattern 1, on[x i,ℓ i ,x i,ℓ i +1 ]× [ x j,ℓ j ,x j,ℓ j +1 ] , we obtain a system, from Equation 2: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α j,ℓ j +1 =(k⋅Λ κ +(1−k)⋅Λ κ ′ )⋅γ α i,ℓ i γ β j,ℓ j +1 =((1−k)⋅Λ κ +k⋅Λ κ ′ )⋅γ β i,ℓ i (10) Following Play Pattern 2 with[x i,ℓ i ,x i,ℓ i +1 ]× [ x j,ℓ j −1 ,x j,ℓ j +1 ] we get a second system from Equation 3 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α j,ℓ j +1 =γ α i,ℓ i ( k ′ Θ κ +(1−k ′ )Θ κ ′ ) +γ α j,ℓ j ( k ′ Φ κ +(1−k ′ )Φ κ ′ ) γ β j,ℓ j +1 =γ β i,ℓ i ( (1−k ′ )Θ κ +k ′ Θ κ ′ ) +γ β j,ℓ j ( (1−k ′ )Φ κ +k ′ Φ κ ′ ) (11) Equating the two systems, and following Appendix B, we find that if the equations are linearly correlated, it means that the coefficients ink,k ′ are correlated between the equations. Differently said, it would mean that∃z∈Rsuch that: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (Λ κ −Λ κ ′ )γ α i,ℓ i =z⋅(Λ κ ′ −Λ κ )γ β i,ℓ i (Θ ′ κ −Θ ′ κ ′ )γ α i,ℓ i +(Φ ′ κ −Φ ′ κ ′ )γ α j,ℓ j =z⋅[(Θ ′ κ ′ −Θ ′ κ )γ β i,ℓ i +(Φ ′ κ ′ −Φ ′ κ )γ β j,ℓ j ] ⇔ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ γ α i,ℓ i =−z⋅γ β i,ℓ i γ α j,ℓ j =z⋅γ β j,ℓ j Similarly to B, we find that the solution of the system is unique unless: ●the answers from the querytort ′ are the same for both DMs ●the slopes ratios of each DM are the same on the two intervals on criterionj, γ α i,ℓ i γ β i,ℓ i = γ α j,ℓ j γ β j,ℓ j