Sharing profits in formal fuzzy contexts

Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems 466 (2023) 108452 www.elsevier.com/locate/fss Sharing profits in formal fuzzy contexts A. Jiménez-Losada ∗, M. Ordóñez Escuela Técnica Superior de Ingenieros, Matemática Aplicada II, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Received 10 June 2022; received in revised form 15 November 2022; accepted 9 December 2022 Available online 15 December 2022 Abstract Cooperative game theory is concerned with situations where a group of agents coordinate their actions to get a common benefit. An allocation rule for these situations is a way to share the common benefit among the agents. The search for a fair allocation rule may depend on the information one has about these agents. A formal context represents information about certain attributes of a set of objects in a table, and they have been used in the literature to describe information about the agents in a game. More recently, formal contexts are extended to the fuzzy setting. Now in this paper we establish a methodology to share the profits of a group of agents that have some information about them collected in a fuzzy formal context when those benefits depend on a set of attributes. ©2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Cooperative games; Fuzzy concept lattices; Shapley value; Fuzzy sets 1. Introduction The theory of cooperative games analyzes situations in which a finite set of agents must share a common benefit or cost among themselves. For such a goal, the values that different subsets of agents (coalitions) could obtain in similar situations are used as data. So, a cooperative game of transferable utility for a set of agents is a function on the family of coalitions called the characteristic function. However, if some additional information about the agents and their relationships is known, then the sharing formulae are modified as the possibilities of cooperation between them change. The most widely used sharing rule is the Shapley value [24], introduced in 1953. Since then, numerous modifications of the rule have been proposed by introducing different kinds of information about the players: communication situations [22], apriori unions [26], conference systems [23], convex geometries [6], permission structures [8]or restricted formation of coalitions [20]. The definition of a sharing rule must be based on the verification of certain properties that make it reasonable. These properties are usually linked to the conditions imposed by the given information. Later, cooperative games have been extended to the fuzzy setting. Aubin uses fuzzy coalitions *Corresponding author. E-mail address: [email protected] (A. Jiménez-Losada). https://doi.org/10.1016/j.fss.2022.12.008 0165-0114/©2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 in [2], Mares works with fuzzy payoffs in [21] and Borkotokey mixes both tools in [7]. Also, the Shapley value is analyzed for cooperative games with additional fuzzy information about the agents. So, fuzzy communication situations [18], proximity relations among players [11], cohesion indices [12], fuzzy permission structures [13]or fuzzy authorization structures [14]have been studied. In all of them, fuzzy definitions of existing crisp structures were used. A formal context represents, by means of a table (a binary relation), the relationships between a set of objects and a set of attributes applicable to those objects. A concept [15,16]1in a context is a pair formed by a subset of objects and a subset of attributes in such a way that all attributes fulfilling all those objects and vice versa are agglutinated. Therefore, concepts represent those groups of objects and attributes that are well defined in the context. The family of concepts in a context has the structure of a lattice. Fuzzy contexts were introduced in 2002 [3]using a fuzzy binary relation between objects and attributes. There are numerous definitions of the fuzzy formal concept in the literature, in [5] several of them are compared. Some are defined as a pair of fuzzy sets of objects and attributes, others are mixed in the sense that they use a fuzzy set and a crisp set, and finally there are others that are crisp concepts but in the fuzzy setting. In this paper, we deal with a notion of concept fully fuzzy. Formal contexts and fuzzy formal contexts have been applied in several fields to analyzed data: software engineering, social network analysis, web services, security analysis, text mining, information retrieval, linguistics, ontology engineering, risk analysis or KDD process. Singh et al. [25]did a compilation of references before 2015 about applications of the concept analysis in both, crisp and fuzzy settings. More recently, Alwersh and Kovács [1] cited numerous applications of the fuzzy formal contexts in the above fields. Game theory and formal concept analysis were related in two different ways. Ignatov and Kwuida in [17]used cooperative games to measure the importance of objects and attributes in the construction of a concept, namely they used game theory to study formal contexts. Faigle et al. in [10] introduced concept games. They proposed cooperative games whose characteristic function measures the benefits obtained by the concepts in a formal context. So, agents are the objects and attributes are those aspects that can diminish the profit obtained by these agents. The concept is then a stable coalition in the context with respect to the attributes. A Shapley value-kind solution was defined in [10] for this type of games and also for other values such as the Banzhaf value in [19]. Now, in this paper, our aim is to analyzed games in a fuzzy formal context following [10]. Particularly we look for an allocation rule similar to the Shapley value but for a particular family of concept games in a fuzzy formal context. For this goal, the paper is organized as follows. Section 2presents an allocation problem to be solved that motivates the theory developed in the following sections. This example is used throughout the paper to illustrate definitions and results. Section 3contains several preliminaries about cooperative games and fuzzy formal concepts. In Section 4we define a Shapley value for the proposed situations, and in Section 5we study several properties of our allocation rule. Also, there is a conclusion section at the end. 2. Motivation The types of problem to be solved in this paper are similar to the following. Assume a set of agents cooperating to sell a software product that performs as many applications as possible at the best level for a given environment. The set of agents involved and the desired applications are therefore known as a given, as well as the profits obtained according to the applications that can be safely offered. But in reality, it is possible to sell intermediate products with certain levels of risk of failure in some of the applications, by lowering the price appropriately. Moreover, what we consider known regarding the relationship between agents and applications is the level of risk with which each of them is able to generate each application. We consider three companies N={1, 2, 3}collaborating for the development of a certain software product. This software is intended to provide three different applications M={a, b, c}to users. The level of quality achieved in these three utilities will determine the final price of the product. The following table Ishows for each company its ability to produce the product in terms of the quality it is capable of developing in each application through its error rate. So, for example, the company 2is able to obtain a product without errors in terms of utility a, but is unable to make utility cwork. 1There exists an older paper about formal concept analysis in German. 2 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Table 2.1 Table Iof risks of failure. abc 10.10.60.8 20 0.31 30.50.10.8 The next function w, in Table 2.2, is the price of the product according to the services it has (assuming maximum quality). We assume that this price decreases if the quality of the applications is not maximal in proportion to their error levels. Also, we consider that if firms do not obtain useful software, then they sell their research to another external interested agent. Table 2.2 Prices of the product depending on the applications not covered. A M ab ac bc a b c ∅ w(A) 10 20 30 40 60 100 120 250 So, for instance w(ab) is the price of a product contains only application c, w(∅)is the best price because all the applications work and w(M) is the profit for selling the research work (the product does not work). Obviously wis decreasing in the family of subsets of M(according to the containment relationship) and non-negative. We seek to distribute among agents the benefit that is obtained with the best product that they can build, adapting the price to the risks of failure of each application, according to the capacity that each agent has been contributing to the final project. Hence, the problem data are (N, M, I, w). The next sections present a methodology to carry out this distribution of benefits. The function wis seen as a cooperative game over Mand the table Ias a fuzzy formal context over (N, M). The fuzzy concepts of the context serve as a transmitter element of the information on the benefits given by attributes to obtain information on the involvement of each agent in these benefits. 3. Preliminaries 3.1. Lattices and formal contexts Let (L, ≤)be a (finite) partially ordered set with La finite set and x, y∈L. The supremum of x, yexists if there is only one element of L, x∨y, such that x, y≤x∨yand any other z∈Lwith x, y≤zsatisfies x∨y≤z. The infimum is the only element x∧ysuch that x∧y≤x, yand if z∈Lverifies z≤x, ythen z≤x∧y. A lattice is a partially ordered set (L, ≤)in which each pair of elements has the supremum and infimum. The top of the lattice is an element top ∈Lwith x≤top for all x∈L, and the bottom is another element bottom ∈Lwith bottom ≤x for all x∈L. We will use x<yto say x≤yand x= y. Element ycovers xin the lattice, yx, if x<yand there is no z∈Lwith x<z<y. The elements coatoms (atoms) in Lare those elements x∈Lthat satisfy top x (xbottom). A (maximal) chain in the lattice is an ordered set C={x0, x1, ..., xp}such that x0=bottom, xp=top and xk−1xkfor all k=1, ..., p. The family of chains of the lattice (L, ≤)is denoted by CH(L) and its cardinality ch(L) =|CH(L)|. A formal context is a triple C=(N, M, I)where: Nis a finite set whose elements are named objects, Mis another finite set whose elements are called attributes, and Iis a binary relation between Nand M. Therefore, Iis a {0, 1}- matrix such that I(i, a) =1if object iis related to attribute aand I(i, a) =0 otherwise. The derivation operators in the formal context Care defined for each S⊆Nand A ⊆Mas S C={a∈M:I(i,a)=1∀i∈S}and A C={i∈N:I(i,a)=1∀a∈A}.(1) A concept in Cis a pair (S, A) with S⊆Nand A ⊆Msuch that S C=Aand A C=S. If (S, A) is a concept, then Sis called its extent and Aits intent. The family of concepts of Cis denoted by LC. If (S, A), (T , B) ∈LCthen (S, A) (T , B) if S⊆T(or equivalently B⊆A). The pair (LC, )is a lattice with supremum and infimum 3 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 (S, A) (T , B) =(A ∩B) C,A∩Band (S, A) (T , B) =S∩T,(S∩T)  C.(2) The bottom of this lattice is (M C, M) and the top is (N, N C). 3.2. Cooperative games and concept games Consider Na finite set of agents (usually called players) in a cooperation situation, that is, a situation where they cooperate to get a common profit. We denote by RNthe |N|-dimensional real space using the labels of Nin the axes. A payoff vector for Nis an element x∈RNsuch that xiis understood as the individual outcome for player i∈N from the common profit. The main goal in a cooperation situation is to find a way to divide the common profit among the agents. We represent by 2Nthe power set of N, this is the family of all subsets (called coalitions) of N. The Boolean algebra of Nis the lattice (2N, ⊆)formed with the elements of the power set ordered by the inclusion relation. It is known that ch(2N) =|N|!. A cooperative game (with transferable utility) is a pair (N, v) where Nis the set of players and vis a mapping v:2N→Rwith v(∅) =0 called characteristic function. Number v(S) represents the worth of coalition Sfor the cooperation of its players. Cooperative games represent cooperation situations adding information about the different coalitions. A value for cooperative games φis an application that obtains a payoff vector φ(N, v) for each game (N, v). A well-known value for cooperative games is the Shapley value [24], given (N, v) and i∈N the outcome of iis Shi(N, v) =1 |N|!  C∈CH(2N)v(Si C)−v(Si C\{i})(3) where Si Cis the first coalition in Cfrom the bottom ∅containing i. If STin 2Nthen v(S) −v(T ) is called the contribution of Sfrom T, this contribution is marginal because it is assignable to only one player, S\T={i}. Therefore, the Shapley value of iis the average of the marginal contributions obtained for this player iin each chain in 2N. Given a nonempty coalition T⊆N, unanimity game uTis defined as uT(S) =1if T⊆Sand uT(S) =0 otherwise. The Shapley value of the unanimity game uTfor each player iis Shi(N, uT) =1 |T|if i∈Tand Shi(N, uT) =0 otherwise. Faigle et al. in [10] study games defined in formal contexts. A concept game is a pair (C, v) where C=(N, M, I)is a formal context and v:LC→Rwith v(M C, M) =0if M C=∅. In the context Cthe set Nis the set of players and M is the set of handicaps that affect the worth of the collaborations. If I(i, a) =1, then the player isatisfies the handicap a. The authors consider that the only well-defined coalitions are the extents of the different concepts; moreover, the worth can depend on the players and also on the handicaps. Hence, the characteristic function is restricted to the concept lattice. For each (S, A) ∈LC, number v(S, A) is understood as the profit obtained by Sif the attributes (handicaps) of Aoccur and v(N, N C)is considered as the common profit. So, concept games represent cooperative situations that add information from a formal context and a characteristic function. A value for concept games φ is a mapping that obtains a payoff vector for each concept game, that is, for every (C, v) we get φ(C, v) ∈RNif C=(N, M, I). The extent Shapley value is a version of the classic Shapley value for concept games defined in [10].2 For any concept game (C, v) with C=(N, M, I) the payoff of a player i∈Nis Shex i(C,v)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ v(M C,M) |M C|if i∈M C 1 ch(C) C∈CH(C) 1 |Si C\Ti C|vSi C,(Si C) C−vTi C,(Ti C) C otherwise (4) where Si Cis the extent of the first concept in Cfrom the bottom (M C, M) containing iand Ti Cthe extent of the concept covered by Si Cin C. Thus, the extent Shapley value is similar to the Shapley value; it is an average of payoff vectors, one for each chain in the concept lattice. The extent Shapley value shares v(N, N C)among players. 2Although the construction is feasible with all the concept games, they consider only formal contexts C=(N, M, I) such that there are not superfluous attribute, i.e. N C=∅. 4 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 3.3. Fuzzy sets and fuzzy formal contexts We will use the triangular norm, its associated conorm and the implication of Gödel [9] over the interval [0, 1], this is for all t1, t2∈[0, 1]respectively: t1∧t2=min(t1, t2), t1∨t2=max(t1, t2)and t1→t2=1ift1≤t2 t2if t1>t 2. Let Lbe a finite set. A fuzzy set τof Lis defined by its membership function μτ:L →[0, 1]. Without loss of generality, we will identify τwith μτ, so τ(x) =μτ(x) for all x∈L. The family of fuzzy sets of Lis denoted by [0, 1]L. Classic sets are also fuzzy sets (called crisps), each S⊆Lis identified with S∈[0, 1]Lwith S(x) =1if x∈S and S(x) =0 otherwise. We will use 0to represent ∅. The support of τ∈[0, 1]Lis the set supp(τ) ={x∈L :τ(x) > 0}, and the image of τis supposed to be an ordered set, im(τ) ={t∈[0, 1] :∃ x∈L with τ(x) =t} ={t1<t 2<···< tp}. If t∈[0, 1], then the t-cut of τis the set [τ]t={x∈L :τ(x) ≥t}. The minimum and the maximum of τare, respectively, the numbers ∧ τ=∧{τ(x) :x∈supp(τ)}(∧0 =0) and ∨τ=∨{τ(x) :x∈L}. Next, we define several operations and relations for fuzzy sets. If τ, σ∈[0, 1]Lthen τ≤σif τ(x) ≤σ(x) for all x∈L. The intersection is (τ ∩σ)(x) =τ(x) ∧σ(x) and the union (τ ∪σ )(x) =τ(x) ∨σ(x). The fuzzy sets τ, σare complementary if τ(x) +σ(x) ≤1for each x∈L. We can define the addition and the subtraction for particular cases. If τ, σare complementary, then (τ +σ )(x) =τ(x) +σ(x) is well defined; and if σ≤τ, then (τ −σ)(x) =τ(x) −σ(x) is also well done. If t∈[0, 1], then (tτ)(x) =tτ(x). Finally, if τ∈[0, 1]Land L ⊆ˆ Lthen we will use τ0∈[0, 1]ˆ Las the fuzzy set given by τ0(x) =τ(x) if x∈L 0ifx∈ˆ L\L. Let v:2L→Rbe a function over the power set of L. The (non-monotonic) Choquet integral [28]of a fuzzy set τ∈[0, 1]Lwith regard to vis defined as c τdv= p  k=1 (tk−tk−1)v[τ]tk−v(∅),(5) where im(τ) ∪{0, 1} =(t0<···<t p). Next, we show several known properties of the Choquet integral. Let τ, σ∈ [0, 1]Land v, ware functions over 2L: (P1) If t1, t2∈R+then c τd(t 1v1+t2v2) =t1c τdv+t2c τdw, (P2) if t∈[0, 1]then c tτ dv =tc τdv, (P3) if S⊆Nthen c Sdv=v(S) −v(∅), and (P4) if L ⊆¯ Land vis a function over 2¯ Lthen c τ0dv =c τdv|2L. Generally c (τ +σ) dv = c τdv+c σdv. A fuzzy formal context is a triple C=(N, M, I); where Nis a finite set of objects, Mis a set of attributes and Iis a fuzzy binary relation between Nand M. So, Iis now a [0, 1]-matrix, where I(i, a) indicates the degree of fulfilment of the attribute aby the object i. For each i∈Nwe represent by Ii∈[0, 1]Mthe i-row of Iand for each a∈Mwe use Ia∈[0, 1]Nas the a-column of I. Obviously, a formal context is a fuzzy context with a {0, 1}-matrix, and we will call it crisp. Therefore, Cis a fuzzy context non-crisp if there exists (i, a) ∈N×Mwith I(i, a) ∈(0, 1). The derivation operators are defined in the fuzzy setting following [3]. If σ∈[0, 1]Nand α∈[0, 1]Mthen σ C∈[0, 1]M and α C∈[0, 1]Nwith σ C(a) = i∈N [σ(i)→I(i,a)]and α C(i) = a∈M [α(a) →I(i,a)],(6) with i∈Nand a∈M. A fuzzy concept3for the fuzzy context Cis a pair (σ, α) ∈[0, 1]N×[0, 1]Mwith σ C=α and α C=σ. Let LCbe the set of fuzzy concepts for the fuzzy context C, which is a finite set. If (σ, α) ∈LCthen σ C=(σ C) C=σand α C=(α C) C=α. As in the crisp case, if we define (σ, α) (τ, β) when σ≤τ(or equivalent α≥β) then (LC, )is a lattice with the following supremum and infimum 3There are several definitions of concepts in the fuzzy framework, Belohlavek in [5] comments on all the different notions. We use one of the most general definitions in the sense that both elements, extent and intent, are fuzzy. 5 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 (σ, α) (τ, β) =(α ∩β) C,α∩βand (σ, α) (τ, β) =σ∩τ,(σ ∩τ) C.(7) Also, as in the crisp case, the bottom of this lattice is (M C, M) and the top (N, N C). We use the algorithms in [4]to calculate fuzzy concepts. 4. Intent fuzzy concept games: the extent Shapley value We define the model showed in Section 2using the notions given in preliminaries. Following Section 2, observe that C=(N, M, I)with Ithe risks of failure given in Table 2.1 for the agents in Nand the application in Mis a fuzzy formal context. Definition 4.1. An intent fuzzy concept game is a pair (C, w) where C=(N, M, I) is a fuzzy context and w:2M→ R+is a function satisfying: w(A) ≤w(B) if A ⊇Band w(M) =0if M C=0. The family of this kind of games is denoted by . In the crisp case, when Cis a formal context, we associate to (C, w) ∈a concept game (see Subsection 3.2) given by vw(S, A) =w(A) for any concept (S, A) ∈LC. We get a solution for (C, w) applying formula (4)to the associated concept game, Shex i(C,w)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ w(M) |M C|,if i∈M C 1 ch(C) C∈CH(C) 1 |Si C\Ti C|w(Si C) C−w(T i C) C,otherwise. (8) The payoff vector Shex(C, w) is an allocation of value w(N C). The allocation is done as the average in the concept lattice of a payoff vector for each chain. In each step of a chain, we have two concepts (S, A) (T , B) obtaining the contribution w(A) −w(B). This contribution is not marginal for only one player and is allocated among the involved players S\T, in an egalitarian way. In this paper we deal with games defined in the environment of a fuzzy context, as in Section 2, following this idea, the intent fuzzy concept games. Hence, we study games where the potential value that players can obtain depends only on the requirements or handicaps (the attributes) that limit its attainment but considering the relation between players and attributes fuzzy. We consider4wproportional in the sense that if A ⊆Mis satisfied at level t∈[0, 1]then the expected worth of tA ∈[0, 1]Mis tw(A). Hence, the Choquet integral is a good tool for managing fuzzy information. Therefore, the value by wof α∈[0, 1]Mis defined by w(α) =w(∅)+c αdw= p  k=1 (tk−tk−1)w [α]tk,(9) with im(α) ∪{0, 1} =(t0<···<t p). Since property (P3) we have constructed an extension of the initial w. The top of the fuzzy concept lattice LCis (N, N C), so the profit to allocate, top profit, is w(N C). The bottom profit is the worth of the bottom (M C, M), w(M). We denote them by w(C)=w(N C)and w⊥(C)=w(M). (10) Example 4.1. Following the example in Section 2, the top profit using (9) and (10)is determined as w(C)=w(0,0.1,0.8)=0.1·40 +0.7·120 +0.2·250 =138. This number is the price to allocate among the players because it corresponds to the best product that they can obtain: a product with risk rate of 0.1for application band 0.8for c. Also we can calculate the price of the bottom, the worst option: to do a useless job. This price following (10)is w⊥(C) =10. 4This is a usual consideration in the fuzzy setting for games, see [2,27]. 6 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Our goal is to obtain a payoff vector for the players of (C, w) ∈following the philosophy of the classic Shapley value. A value for intent fuzzy concept games is a mapping φover that obtains a payoff vector φ(C, w) ∈RNfor each (C, w) ∈with C=(N, M, I). Following formula (8), we go through the chains of the concept lattice distributing the value obtained by the increment of intent. Example 4.2. Fig. 4.1 shows the fuzzy concept lattice LCof the context in Table 2.1. Fig. 4.1. Fuzzy concept lattice of context C. The fuzzy concepts in this example identify the possible products that can transfer information to the levels of participation of the players. Hence, the extents are fuzzy coalitions explaining the capacity with which each player engages in the collaboration. For example, it is possible to make a product with α=(1, 0.3, 1) ∈[0, 1]M, namely a software with only the application band with an error risk level of 0.3. But this fuzzy set is not the intent of a fuzzy concept in LC(see Fig. 4.1) and this fact means that the participation levels of the firms given in α C=(0.1, 0, 0.5)can get a better product, α C=(1, 0.1, 1). But now we need to define how to allocate this contribution among the players. The extent of each fuzzy concept represents the participation levels of the players required to get that intent, considering the marginalities of the players of the increment of extent. We define the notion of contribution in each step of a chain in the fuzzy concept lattice as in formula (8). If (C, w) ∈and (σ, α) (τ, β) in LCthen the contribution of the first fuzzy concept to the second one is w(α) −w(β). As in the crisp case, this contribution is not always assignable to only one player, but neither to a crisp set of them. We need to explain how to allocate this contribution among the involved players taking into account their participation levels, the marginality. Suppose a unit of profit such that a particular set of players Sis necessary to get it, the outcome of the classic Shapley value (3)for the unanimity game uS(see Subsection 3.2) gives us the share for each player of this unit. This is the motivation of the next definition. Definition 4.2. Let Nbe a set of players. The Shapley power function of a player i∈Nis the function hi:2N→R defined as hi(S) =Shi(uS). 7 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 We use the Shapley power functions of the players to introduce a concept of marginality in an intent fuzzy concept game. Definition 4.3. Let (C, w) ∈with C=(N, M, I). If (σ, α) (τ, β) in LCthen the contribution of the first fuzzy concept to the second is w(α) −w(β). The marginality of player i∈Nfor that contribution is given by maσ τ(i, C)=1 ∨(σ −τ) c (σ −τ)dhi. Particularly, the marginality of each player iin the bottom (M C, M) is defined by maM C(i, C)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 ∨M Cc M Cdhiif M C= 0 1 |N|if M C=0. We compare these numbers with regard to the crisp case. Suppose (C, w) with Ccrisp. If we consider two concepts (S, A) (T , B) in LCas we said the contribution is w(B) −w(A). In formula (8)this contribution is allocated among all players in S\Tin the egalitarian way. Now, following Definition 4.2, the contribution by (P3) is the same if we see the context as a particular case of fuzzy one. Moreover, we have S−T=S\Tand then the marginality for each player iis maS T(i, C)=1 ∨(S \T)c (S \T)dh i=hi(S \T) If i∈S\Tthen hi(S \T) =1 |S\T|. So, the player igets in this step 1 |S\T|[w(A) −w(B)] as in formula (8). If i/∈S\Tthen hi(S \T) =0, and hence igets nothing. Therefore, the marginality of a player for a contribution w(α) −w(β) generated at step (σ, α) (τ, β) means his individual capacity to intervene in the generation of that contribution. This capacity may be different for the players involved, those in supp(σ −τ). The next proposition shows that the marginalities in each step always sum one. Proposition 4.1. Let C=(N, M, I)be a fuzzy context and (σ, α) (τ, β) in LC. The marginalities of that step satisfy  i∈N maσ τ(i, C)=1. The same happened with the marginalities in the bottom. Proof. We do the sum helped by property (P1),  i∈N maσ τ(i, C)=1 ∨(σ −τ)  i∈Nc (σ −τ)dhi=1 ∨(σ −τ) c (σ −τ)d i∈N hi For each non-empty coalition S⊆Nwe have  i∈N hi(S) = i∈S 1 |S|=1 and  i∈N hi(∅)=0. Let im(σ −τ) =(t1<···<t p)be the image of σ−τwhere tp=∨(σ −τ), and t0=0. Using (5)we get 8 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452  i∈N maσ τ(i, C)=1 tp p  k=1 (tk−tk−1)=1. The proof for the marginalities in the bottom is the same.  We define now the extent Shapley value for intent fuzzy concept games. If Cis a fuzzy context and C∈CH(LC) then we take C=(σ0,σ 1, ..., σp), (11) the ordered set from the bottom to the top of the intents in the chain. So σ0=M Cand σp=N. Definition 4.4. Let (C, w) ∈be an intent fuzzy concept game with C=(N, M, I). The extent fuzzy Shapley value of a player i∈Nis SHex i(C,w)=maM C(i, C)w(M) +1 ch(LC) C∈CH(LC) C=(σ0,σ 1, ..., σp) p  k=1 maσk σk−1(i, C)[w(σk) C−w(σk−1) C]. The first term of the formula allocates the worth of the bottom in the fuzzy concept lattice, and later, for each chain, we allocate the contributions given by the intents of the steps considering the marginalities given by the extents. If we take a crisp context Cthen SHex(C, w) =Shex(C, w) following formula (8). We show the computation of the fuzzy extend Shapley value in an example. Example 4.3. The extent Shapley value obtains an allocation of w(C)following the chains of the fuzzy concept lattice as the classic Shapley value. We have ch(LC) =13. The Shapley power functions for the firms are in Table 4.1. Table 4.1 Shapley power functions. S∅12312 13 23 N h1(S) 01001/21/20 1/3 h2(S) 00101/20 1/21/3 h3(S) 00010 1/21/21/3 The bottom profit is allocated using M C=(0.1, 0, 0.1)as w⊥(C)maM C(i, C)=10 1 2,0,1 2=(5,0,5). Consider for example the chain, given by its concepts ordered from the bottom, C=(C1,C 2,C 5,C 8,C 13,C 15,C 16). In the first step of C, (C1, C2), we obtain by (9)the contribution w(0.1,1,1)−w(M) =27. As (0.6, 0, 0.1) −(0.1, 0, 0.1) =(0.5, 0, 0)then the marginalities of the players using Table 4.1 are ma(0.6,0,0.1) (0.1,0,0.1)=(1,0,0), namely player 1 gets the whole contribution, players get (27, 0, 0). In the last step (C15, C16)we have the contribution 9 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Table 5.2 Fuzzy context C2. de 40.10.5 50.70.4 Table 5.3 contains the price of the second product. Worth w2(M2)again means the sale of research on applications if the product does not work. Table 5.3 Prices of the new product, w2. AM 2de ∅ w2(A) 10 80 100 200 In Fig. 5.2 we see the fuzzy concept lattice LC2. Fig. 5.2. Fuzzy concept lattice of context C2. The top profit is w 2(C)=w2(∅)+c (0.1,0.4)dw 2=151. The solution of the extent Shapley value for this new intent fuzzy concept game is SHex(C2,w 2)=(98.25,52.75) Suppose now that the first three firms decide to invite the two new ones to add their applications to the product to improve it being the price the sum of the independent prices. They are interested in this fact because they can increase the number of sales. Obviously the allocation of the price of one new product with all the five applications must be the same as in the independent problems. Concatenation expresses this idea in only one structure. Table 5.4 is the concatenation of the fuzzy contexts. 16 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Table 5.4 Concatenation C1⊕C2. abcde 10.10.60.81 1 20 0.31 1 1 30.50.10.81 1 40 0 0 0.10.5 50 0 0 0.70.4 Following Fig. 5.1, as (N1) C1= 0 and (M2) C2= 0the Fig. 5.3 shows the fuzzy concept lattice of the concatenation. Observe that Uis isomorphic to LC2and Dis isomorphic to LC1. Fig. 5.3. Fuzzy concept lattice of C1⊕C2. The list of fuzzy concepts of concatenation is in Table 5.5. In the case of the function w12, let us look at some examples of its evaluation. If A ={a, b, d, e}, then w12(A) = w1({a, b}) =20 because A M2. If A ={d}, then w12 =w1(∅) +w2(d) =330. Finally, although this option does not appear in the calculation of the value, if A ={a, e}then w12(A) =w1(a) +w2(e) =160. The concatenation property satisfied by SHex says that SHex(C1⊕C2,w 12)=(76.9,15.8,45.3,98.25,52.75). Finally, we will see that under certain conditions it is feasible to determine the solution by decomposition of the fuzzy concept lattice into groups of chains. Let C=(N, M, I) be a fuzzy context with N C=0. For each a∈M, we take the fuzzy set in N, σC a(i) =1,if I(i,a)= 0 0,if I(i,a)=0(13) 17 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Table 5.5 Fuzzy concepts of the concatenation C1⊕C2. Concept Extent Intent Concept Extent Intent C1(0.1,0,0.1,0,0)(1,1,1,1,1)C12 (1,0,1,0,0)(0.1,0.1,0.8,1,1) C2(0.6,0,0.1,0,0)(0.1,1,1,1,1)C13 (0.8,1,0.1,0,0)(0,0.3,1,1,1) C3(0.1,0,0.5,0,0)(1,0.1,1,1,1)C14 (1,1,0.1,0,0)(0,0.3,0.8,1,1) C4(0.8,0,0.1,0,0)(0.1,0.6,1,1,1)C15 (0.8,1,0.8,0,0)(0,0.1,1,1,1) C5(0.6,0.3,0.1,0,0)(0,1,1,1,1)C16 (1,1,1,0,0)(0,0.1,0.8,1,1) C6(0.1,0,0.8,0,0)(0.5,0.1,1,1,1)C17 (1,1,1,0.1,0.4)(0,0,0,1,1) C7(1,0,0.1,0,0)(0.1,0.6,0.8,1,1)C18 (1,1,1,0.1,0.7)(0,0,0,1,0.4) C8(0.8,0.3,0.1,0,0)(0,0.6,1,1,1)C19 (1,1,1,0.5,0.4)(0,0,0,0.1,1) C9(0.8,0,0.8,0,0)(0.1,0.1,1)C20 (1,1,1,0.1,1)(0,0,0,0.7,0.4) C10 (0.1,0,1,0,0)(0.5,0.1,0.8,1,1)C21 (1,1,1,1,0.4)(0,0,0,0.1,0.5) C11 (1,0.3,0.1,0,0)(0,0.6,0.8,1,1)C22 (1,1,1,1,1)(0,0,0,0.1,0.4) Observe that σC a= Nbecause as N C=0 then there exists iwith I(i, a) =0. We say that a∼bwith a, b∈Mif σC a=σC band let MCthe set of equivalence classes in Mby ∼. If A ∈MCthen σC A=σC afor any a∈A. We endow MCwith the following partial order: for all A, B∈MCwe have B<Aif σC B<σC A. Definition 5.3. Let C=(N, M, I) be a fuzzy context with N C=0. The set of separators of Cis Sep (C)={A∈MC:Amaximal in (MC,<)}. Observe that σC A=0if I(i, a) =0for all i∈Nand any a∈A, so or M/∈Sep (C)or Sep (C) ={M}and we have the trivial case I=0. The following results will explain the idea of the separator notion in a context. Lemma 5.6. Let C=(N, M, I) be a fuzzy context with N C=0. The set of coatoms of LCis (σC A,αC A):A∈Sep (C), being αC A(b) ={i∈N:I(i,a)=0}I(i, b) for all b∈Mand any a∈A. Proof. Let (σ C A, αC A)with A ∈MC. First, we prove the claim (σ C A, αC A) ∈LC. Let a∈A. For all b∈M, (σC A) C(b) = i∈N σC a(i) →I(i,b) =⎛ ⎝ {i∈N:I(i,a)=0} 1→I(i,b) ⎞ ⎠∧⎛ ⎝ {i∈N:I(i,a)=0} 0→I(i,b) ⎞ ⎠ = {i∈N:I(i,a)=0} I(i,b)=αC A(b). We define the fuzzy set ¯αa∈[0, 1]Mwith ¯αC a(b) ={i∈N:I(i,a)=0}I(i,a), if b=a 0,otherwise. It holds for any i∈N (¯αC a) C(i) = b∈M ¯αa(b) →I(i,b)=⎛ ⎝ {i∈N:I(i,a)=0} I(i,a) ⎞ ⎠→I(i,a)=σC A(i). Therefore, αC A=(¯αC a) Cand the claim is true. 18 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Now, we consider A ∈Sep (C). Suppose there exists another fuzzy concept (σ, α) (σC A, αC A)with (σ, α) = (N, 0). As α= 0 there is b∈Mwith ¯αC b≤α. So, (¯αC b) C≥σ>σ A. But we have already proved that (¯αC b) C=σC Bfor some B∈MC, so Ais not maximal in (MC, <). Each separator defines a new fuzzy context such that its fuzzy concept lattice is a certain group of chains in the old fuzzy concept lattice. Definition 5.4. Let C=(N, M, I) be a fuzzy context with N C=0. For each A ∈Sep (C)the context restricted by A is CA=(N, M, IA)with IA(i, b) =I(i,b), if I(i,a)= 0 0,if I(i,a)=0 for all i∈N, b∈Mand any a∈A. Lemma 5.7. Let C=(N, M, I) be a fuzzy context with N C=0. It holds LCA={(σ, α) ∈LC:α≥αC A}. Proof. If I=0 then Sep(C) ={M}and obviously CM=C. Suppose then I= 0 and then σC A= 0for all A ∈M(C). First, we see that given b∈M, for all i∈Nand any a∈A σCA b(i) =σC b(i), if I(i,a)= 0 0,if I(i,a)=0.(14) If i∈Nverifies I(i, a) = 0, then we get IA(i, b) =I(i, b). So, in that case, σCA b(i) =σC b(i). If I(i, a) =0, then IA(i, b) =0 and we get σCA b(i) =0. Hence we have from (14) that σCA A=σC Aand σCA B≤σC Bfor every B∈M\{A}. Following Lemma 5.6, for all b∈Mand any a∈A αCA A(b) = {i∈N:IA(i,a)=0} IA(i, b) = {i∈N:I(i,a)=0} I(i,b)=αC A(b) because IA(i, a) = 0if and only if I(i, a) = 0, and, moreover IA(i, b) =I(i, b) when I(i, a) = 0. We obtain the result that the only coatom of LCAis (σC A, αC A). Let σ∈[0, 1]Nbe a fuzzy set of players with σ≤σC A. As σ≤σC Athen σ(i) =0if I(i, a) =0for a∈A. We calculate the derivation in both contexts, Cand CA. Let b∈Mand a∈M, σ C(b) = i∈N σ(i)→I(i,b)= {i∈N:I(i,a)=0} σ(i)→I(i,b), σ CA(b) = i∈N σ(i)→IA(i, b) = {i∈N:I(i,a)=0} σ(i)→IA(i, b) = {i∈N:I(i,a)=0} σ(i)→I(i,b) because IA(i, b) =I(i, b) when I(i, a) = 0. Let α∈[0, 1]Mbe a fuzzy set of attributes with α≥αC A. As α≥αC Athen α(b) ≥ {i∈N:I(i,a)=0} I(i,b) for all b∈Mand any a∈A. In particular, if a∈Mthen α(a) >0. We calculate the derivation in both contexts, C and CA. Let i∈Nand a∈M. If I(i, a) = 0 then IA(i, b) =I(i, b) and, obviously, α CA(i) =α C(i). If I(i, a) =0, and then IA(i, a) =0, we obtain α(a) →0 =0. Hence α C(i) = b∈M α(b) →I(i,b)=0= b∈M α(b) →IA(i, b) =α CA(i).  19 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Theorem 5.8. Let (C, w) ∈with C=(N, M, I) and N C=0. The extent fuzzy Shapley value satisfies ch(C)SHex(C,w)= A∈Sep (C) ch(CA)SH ex i(CA,w). Proof. Let i∈N. As M CA=M Cthen maM CA(i, CA) =maM C(i, C)for all A ∈Sep (C). From the above proposition we have CH(C)=$ A∈Sep (C) CH(CA), CH(CA)∩CH(CB)=∅∀A,B ∈Sep (C). Let C∈CH(CA)with C={σ0, σ1, ..., σp}and A ∈Sep (C). Also, from Lemma 5.7 we get for all k=1, ..., pthat maσk σk−1(i, C)=maσk σk−1(i, CA). Finally, (σk) CA=(σk) Cimplies ch(C)SHex i(C,w)= A∈Sep (C) ch(CA)SHex i(CA,w).  We explain the property of decomposition seen in the above theorem. Example 5.2. The fuzzy context of Section 2, see Table 2.1, does not satisfy the condition N C=0. Following the literature of the Example, that condition means that all the applications of the products can be reached by the firms without risk of error. We change the matrix looking for the condition. Now consider the fuzzy context of Table 5.6. Table 5.6 Fuzzy context Cin Example 5.2. abc 10.10.60 2001 30.50.10.8 Now N C=0. Fig. 5.4 shows the fuzzy concept lattice of this new context. We determine the set of separators in C. As σa=(1, 0, 1), σb=(1, 0, 1)and σc=(0, 1, 1)then a≡band following Definition 5.3 Sep (C)= {{a,b},{c}}. We can test Lemma 5.6 seeing that the fuzzy concepts (σC {a,b},αC {a,b})=((1,0,1), (0.1,0.1,0)) (σC c,αC c)=((0,1,1), (0,0,0.8)) are the coatoms of LCin Fig. 5.4. Now, in Table 5.7 we construct the fuzzy contexts C{a,b}and Cc. Table 5.7 Fuzzy contexts C{a,b}and Cc. C{a,b}abc 10.10.60 2 000 3 0.5 0.1 0.8 Ccabc 10 0 0 20 0 1 3 0.5 0.1 0.8 Fig. 5.5 shows the fuzzy concept lattices of these two new fuzzy contexts. 20 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 Fig. 5.4. Fuzzy concept lattice of context Cin Example 4.2. Fig. 5.5. Fuzzy concept lattices LC{a,b}and LCc. Theorem 5.8 allows for the decomposition of the game by chains if N C=0, moreover, the decomposition is easy to find by looking at the coatoms of the fuzzy concept lattice. But it is not true that we can always guarantee a proper decomposition if we have only one coatom, we have only one separator, and the result says nothing. If N C= 0 then it is hard to identify the fuzzy contexts that can be properly decomposed and also to find the decomposition. For example, consider the fuzzy context C2in Table 5.2 using two players {4, 5}and two attributes {d, e}with the fuzzy concept lattice in Fig. 5.2. We will see that it is not possible to divide the lattice in chains using fuzzy formal contexts 21 A. Jiménez-Losada and M. Ordóñez Fuzzy Sets and Systems 466 (2023) 108452 although it has two coatoms. The concept lattice LC2only has two maximal chains, so we can only separate the lattice one by one. If we take the lattice formed by the chain L ={C1, C2, C4, C6}in Fig. 5.2 then we will test that there is no fuzzy context Cwith table Isuch that LC=L. The context Cmust contain in I4the number 0.1, and, moreover, ∧I4=0.1. The raw I5must contain the numbers 0.4 and 0.7. Column Idnumbers 0.1 and 0.7, and finally column Ie number 0.4. So Ihas the following structure, d e 4 0.1p 5 0.7 0.4 with p≥0.4 Therefore, (p, 0.4) ∈[0, 1]{4,5}verifies (p, 0.4) C=(0.1, 1) C=(p, 0.4), but then ((p, 0.4), (0.1, 1)) ∈LCand this is not true. The above fact shows an important difference between the analysis of concepts in the fuzzy setting and this analysis in the crisp case. In the crisp setting [10]it is clear that any chain (any closure system really) represents the concept lattice of a formal context, but now we have proved that in the fuzzy case there exist chains which are not the fuzzy concept lattice of a fuzzy context. 6. Conclusions In this paper we have introduced an allocation rule for a family of profit-sharing problems with fuzzy data. In these problems (see Section 2) we have a set of agents, a fuzzy formal context that relates them to certain properties, and a profit function that depends on these properties. The proposed allocation rule follows the Shapley value philosophy. In Section 5, we have shown that the solution is well constructed in the sense that it meets reasonable conditions (Theorems 5.1, 5.2 and 5.3), as well as having good operational properties on data expansion or decomposition (Theorems 5.5 and 5.8). We have chosen the Choquet integral as a tool for aggregating information, but it is an open problem to define similar solutions using other integrals. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability No data was used for the research described in the article. 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