Fuzzy Group Identification Problems
aa r X i v : . [ ec on . T H ] D ec Fuzzy Group Identification Problems
Federico Fioravanti Fernando Tohm´eINMABB, Universidad Nacional del SurCONICETAv. Alem 1253, (8000) Bah´ıa Blanca, Argentinae-mail: [email protected], [email protected]
Abstract
We present a fuzzy version of the Group Identification Problem(“Who is a J?”) introduced by Kasher and Rubinstein (1997). Weconsider a class N = { , . . . , n } of agents, each one with an opinionabout the membership to a group J of the members of the society,consisting in a function p i : N → [0 , J .We consider the problem of aggregating those functions, satisfyingdifferent sets of axioms and characterizating different aggregators.While some results are analogous to those of the originally crisp model,the fuzzy version is able to overcome some of the main impossibilityresults of Kasher and Rubinstein. People usually classifies other people, objects or entities in groups. Some-times these classifications are obvious, as in the assignment of countries tothe continent to which they belong. But in many cases, opinions may not beeven clear cut and thus it becomes hard to reach a consensus. For instance,if a group of people wants to identify which of them should be considered“tall”, finding out whether a member is such may be far from evident. Ofcourse, a consensus could be reached on that people that are 2.00 metersheight or more are “tall”. But it is not clear whether someone who’s heightis 1.75 meters can be considered “tall”.This kind of classification problems, prone to vagueness and imprecision,gave the impetus for the introduction of fuzzy sets. Zadeh (1965) defined afuzzy subset U of a set A as a membership function f : A → [0 , ( a ) indicates the degree of membership of a in U . This allows to represent,in particular, preferences as degrees. In turn, there are several approachesto the aggregation of fuzzy preferences. Just to mention older contributions,(Dutta et al., 1987) deal with exact choices under vague preferences while(Dutta, 1986) investigates the structure of fuzzy aggregation rules determin-ing fuzzy social orderings.In this paper we introduce a fuzzy approach to the group identification prob-lem formalized by Kasher and Rubinstein (K-R) in “Who is a J?”(1997).They consider a finite society that has to determine which of its subsets ofmembers consists of exactly those individuals that can be deemed to be J s.By a slight abuse of language, this subset is denoted J . Each different setof axioms postulated to yield a solution to this problem characterizes a classof aggregation functions, called Collective Identity Functions (CIFs) . The“Liberal” one labels as a J any individual that deems himself to be a J ; the“Dictatorial” CIF is such that a single individual decides who is a J . Finally,the “Oligarchic” CIF determines that somebody is a J if all the members ofa given group agree on that.Since K-R, the group identification problem in a crisp setting has been ap-proached in many ways. Sung and Dimitrov (2005), refine the characteri-zation of the liberal CIF, Miller (2008) study the problem of defining morethan two groups, Saporiti and Cho (2017) analyze the incentives of voters inan incomplete preferences setting, etc.In our model, every agent, instead of labeling any of the individuals in N = { , . . . , n } (the society) as belonging or not to J , assigns a value inthe [0 ,
1] interval to each agent, representing the degree in which that agentis believed to belong to J . The axioms in K-R have counterparts in our frame-work. On the other hand, we also consider another axiom drawn from theliterature, namely the Extreme Liberalism axiom (Fioravanti-Tohm´e, 2018).We present fuzzy versions of those crisp axioms. We deal first with the ax-ioms defining the “Liberal” aggregator, showing that the fuzzy versions ofthe results in K-R remain valid. But when we turn to the axioms definingthe Dictatorial aggregator, our results differ from those in the crisp setting.More specifically, K-R prove an impossibility result when the domain andthe range of the aggregator are restricted, indicating that the DictatorialCIF is the only one satisfying those conditions. But there does not exist aclear “translation” of those restrictions into our framework, allowing differentinterpretations. In some of these we obtain more aggregators verifying theaxioms, while in others there does not exist any of them.Fuzzy settings of this problem have been studied several times in the lastyears. So, for instance, Cho and Park (2018) present a model of groupidentification for more than two groups, allowing fractional opinions, while2allester and Garc´ıa-Lapresta (2008) deal with fuzzy opinions in a sequentialmodel. An interesting paper related to ours, is Alcantud and Andr´es Calle(A-AC, 2017), which presents a deep analysis of the aggregation problemof fuzzy opinions yielding fuzzy subsets. This contribution introduces theFuzzy Collective Identity Functions (FCIFs), an expression that we adoptin this paper, defining the liberal and dictatorial aggregators, among others,presenting ways of circumventing the classical impossibility result from K-R.The main difference between this and our work is that we find a characteriza-tion of the liberal FCIF, and determine different domain and range conditionsaccording to which the impossibility result can obtain or be eluded.The plan of the paper is as follows. Section 2 is devoted to present the modeland the set of axioms considered here. In Section 3 we deal with “Liberal” ag-gregators while in Section 4 we focus on the Dictatorial aggregator. Finally,Section 5 concludes. Let N = { , . . . n } be a set of agents that have to define who of them belongsto the group of J s (which by a slight abuse of language is denoted J ). Theopinion of agent i is characterized by a function p i : N → [0 ,
1] where p i ( j )indicates the assessment of agent i of the degree of membership of j to J .Agent i has thus a vector of opinions P i = { p i (1) , . . . , p i ( n ) } . A profile ofopinions P is a n × n matrix P = { P , . . . , P n } . With P we denote the setof all the profiles of opinions.A fuzzy subset J of N is characterized by a membership (characteristic) func-tion f J : N → [0 , J . Let F = { f J : N → [0 , } be the set of possible membership functions ofa fuzzy subset J . We denote by FJ the Fuzzy Collective Identity Function (FCIF) such that FJ : P → F .A FCIF takes a profile of opinions and returns a membership function forthe fuzzy subset J . More precisely, the membership function of the set J associated to the profile P ∈ P is denoted f PJ . We do not impose furtherrestrictions on this membership function. It can be different for every i , forexample, f PJ (1) = p (1) while f PJ (2) = p (2)2 .In what follows we will present in an axiomatic way the properties that asocial planner would like to see implemented by a “fair” aggregation pro-cess, several of them introduced already by K-R in their seminal work. Thefollowing two axioms state that, if the opinion about an agent changes, sayby increasing (decreasing) his degree of membership, then the aggregatedopinion should reach at least (at most) the previous degree.3 Fuzzy Monotonicity (FMON): let P ∈ P be such that f PJ ( i ) = a and let P ′ be a profile such that P ′ k,i > P k,i for some k with P ′ h,j = P h,j for all ( h, j ) = ( k, i ), then f P ′ J ( i ) ≥ a .If P ′′ is a profile such that P ′′ k,i < P k,i for some k with P ′′ h,j = P h,j forall ( h, j ) = ( k, i ), then f P ′′ J ( i ) ≤ a . • Fuzzy Strong Monotonicity (FSMON): let P ∈ P be such that f PJ ( i ) = a and let P ′ be a profile such that P ′ k,i > P k,i for some k with P ′ h,j = P h,j for all ( h, j ) = ( k, i ),then f P ′ J ( i ) > a .If P ′′ is a profile such that P ′′ k,i < P k,i for some k with P ′′ h,j = P h,j forall ( h, j ) = ( k, i ), then f P ′′ J ( i ) < a .It is easy to see that if a FCIF verifies FSMON then it satisfies FMON.The next axiom states that the aggregate opinion about an agent is boundedby the upper and lower bounds of all the individual opinions about thatagent. That is: • Fuzzy Consensus (FC): if p i ( j ) ≥ a j and p i ( j ) ≤ b j for some a j , b j ∈ R , a j , b j > i ∈ N , then a j ≤ f PJ ( j ) ≤ b j . The following axiom states that if two agents are evaluated in a similar way,the FCIF must classify them also similarly: • Symmetry (SYM): agents j and k are symmetric if: – p i ( j ) = p i ( k ) for all i ∈ N − { j, k } – p j ( i ) = p k ( i ) for all i ∈ N − { j, k } – p j ( k ) = p k ( j ) – p j ( j ) = p k ( k )Then, if two agents j and k are symmetric it follows that f PJ ( j ) = f PJ ( k ).The following property is our first true “fuzzy” axiom. It distinguishes be-tween being more approved than disapproved: an agent that gets a mem-bership degree of more than 0 . J s. Alcantud and Andr´es Calle (2017) call this property
Unanimity . We could have taken any number in [0 ,
1] as threshold, instead of 0 .
5. The main resultsin this paper do not depend on this particular choice. Fuzzy Symmetry (FSYM): agents j and k are fuzzy-symmetric if: – p i ( j ) , p i ( k ) ≥ . p i ( j ) , p i ( k ) < . i ∈ N − { j, k } – p j ( i ) , p k ( i ) ≥ . p j ( i ) , p k ( i ) < . i ∈ N − { j, k } – p j ( k ) , p k ( j ) ≥ . p j ( k ) , p k ( j ) < . – p j ( j ) , p k ( k ) ≥ . p j ( j ) , p k ( k ) < . j and k are fuzzy-symmetric then f PJ ( j ) , f PJ ( k ) ≥ . f PJ ( j ) , f PJ ( k ) < . f PJ ( j ) = f PJ ( k ) for all j, k ∈ N , FSYM does notimpose such a strong condition. The following examples may be useful tounderstand the difference. Example 1.
Consider N = { , } . • The FCIF such that f J ( i ) = p ( i )+ p ( i )2 satisfies SYM but not FSYM.To see this, consider profile P = ( { . , . } , { . , . } ) . We have that f PJ (1) = 0 . < . and f PJ (2) = 0 . > . . • The FCIF such that f J (1) = 0 . p (1)+0 . p (1) and f J (2) = 0 . p (2) + 0 . p (2) verifies FSYM but not SYM. The next axiom states that the aggregate opinion about an agent shouldonly take into account the individual opinions about her.The other is its “fuzzy” version. • Independence (I): let P and P ′ be two profiles such that given anagent j ∈ N , p i ( j ) = p ′ i ( j ) for all i ∈ N . Then f PJ ( j ) = f P ′ J ( j ). • Fuzzy Independence (FI): let P and P ′ be two profiles such thatgiven an agent j ∈ N and for every i ∈ N we have p i ( j ) ≥ . p ′ i ( j ) ≥ . p i ( j ) < . p ′ i ( j ) < . f PJ ( j ) ≥ . f P ′ J ( j ) ≥ . f J ( i ) = f ( p ( i ) , . . . , p n ( i )), FI allows every agent to be affected by theopinion about the other agents.The next two examples provide a more clear view of how this axioms affectthe FCIF. Example 2.
Consider N = { , } . The FCIF such that f J ( i ) = p ( i )+ p ( i )2 verifies I but not FI as it can beseen in profiles P = ( { . , . } , { . , . } ) and P ′ = ( { . , . } , { . , . } ) .We have that f PJ (1) = 0 . < . and f P ′ J (1) = 0 . > . . • The FCIF such that f J (1) = 0 . p (1)+0 . p (2) and f J (2) = 0 . p (1)+0 . p (2) verifies FI but not I. The following axiom states that the opinion that an agent has abouthimself should be considered important to determine if she is considered a J . • Liberalism (L): if p i ( i ) = 1 for some i ∈ N , then f PJ ( k ) = 1 for some k ∈ N .If p i ( i ) = 0 for some i ∈ N , then f PJ ( k ) = 0 for some k ∈ N .A fuzzy version of this axiom is: • Fuzzy Liberalism (FL): if p i ( i ) ≥ . i ∈ N , then f PJ ( k ) ≥ . k ∈ N .If p i ( i ) < . i ∈ N , then f PJ ( k ) < . k ∈ N .The following example shows the difference between these two axioms: Example 3.
Consider N = { , . . . , n } . • The FCIF such that f J ( i ) = 1 if p i ( i ) = 1 and f J ( i ) = 0 otherwise,verifies L but not FL. • The FCIF such that f J ( i ) = 0 . if p i ( i ) ≥ . and f J ( i ) = 0 . if p i ( i ) < . , verifies FL but not L. Other relevant properties concern the capacity any agent has of beingdeterminant on the characterization as J of another one (like a parent aboutthe religious affiliation of his kids): • Extreme Liberalism (EL):(i) If p i ( j ) = 1 for some i, j ∈ N , then f PJ ( k ) = 1 for some k ∈ N .(ii) If p i ( j ) = 0 for some i, j ∈ N , then f PJ ( k ) = 0 for some k ∈ N .A fuzzy version is: • Fuzzy Extreme Liberalism (FEL):(i) If p i ( j ) ≥ . i, j ∈ N , then f PJ ( k ) ≥ . k ∈ N .(ii) If p i ( j ) < . i, j ∈ N , then f PJ ( k ) < . k ∈ N .6ome final examples illustrate the difference between these last two ax-ioms. Example 4.
Consider N = { , . . . , n } . • A FCIF that satisfies EL but not FEL: f J such that f J ( i ) = 1 for all i ∈ N − { } and f J (1) = 0 if there exist a j, k ∈ N such that p j ( k ) = 0 or f J (1) = 1 otherwise. • A FCIF that verifies FEL but not EL: consider N = { , } and f J suchthat f J (1) = 0 . and f J (2) = 0 . for every P ∈ P . We define the Strong Liberal FCIF as: L ( P , . . . , P N )( i ) = L J ( i ) = p i ( i )for all i ∈ N .It is straightforward to see that this FCIF is analogous to the Strong LiberalCIF that K-R introduced in their work. It verifies FMON, FC, FI and FL.Moreover, it is the only FCIF that verifies this set of axioms: Theorem 1.
The only FCIF that verifies FMON, FC, FI and FL is theStrong Liberal FCIF.Proof.
It is clear that the Liberal FCIF verifies these 4 axioms. Supposethere exists another FCIF satisfying them. Let P be a profile such that p i ( i ) ≥ . f PJ ( i ) < .
5. Using FMON several times we can create aprofile P ′ identical to P except that p ′ j ( i ) < . j = i , such that f P ′ J ( i ) < .
5. Consider a profile P ′′ such that p ′′ i ( i ) ≥ . p ′′ j ( k ) < . j, k ∈ N (except when j = k = i ). By FC we have that f P ′′ J ( k ) < . k = i . So the set of agents such that f P ′′ J ( k ) ≥ . {∅ , { i }} .Because of FL we have f P ′′ J ( i ) ≥ .
5. But then we have a contradiction withFI, because agent i is treated similarly (in the sense of FI) in profiles P ′ and P ′′ but f P ′ J ( i ) < . f P ′′ J ( i ) ≥ . Corollary 1.
The only FCIF that verifies FMON, FC, I and L is the StrongLiberal FCIF.
From the uniqueness of the Strong Liberal FCIF we obtain the followingresult: 7 orollary 2.
If a FCIF verifies FMON, FC, FI and FL then it verifies SYMand FSYM.
From the fact that the Liberal FCIF does not verify FSMON and thatFSMON implies FMON we get:
Corollary 3.
There is no FCIF that verifies FSMON, FC, FI and FL.
The following two FCIFs are the fuzzy counterparts of the Unanimity andInclusive CIF: • The
Unanimity
FCIF is defined as: U ( P , . . . , P N )( i ) = U J ( i ) = min j p j ( i )for all i ∈ N . • The
Inclusive
FCIF is defined as:
Inc ( P , . . . , P N )( i ) = Inc J ( i ) = max j p j ( i )for all i ∈ N . As in the crisp case, under extreme concepts of liberalism like EL or FEL,we obtain the same uniqueness results:
Theorem 2.
Inc is the only FCIF that verifies FMON, FC, FI and EL (i)or FEL (i).U is the only FCIF that verifies FMON, FC, FI and EL(ii) or FEL(ii).Proof.
A similar construction as the used for proving Theorem 1 yields theproof of the two statements.We can derive the following impossibility result from Theorem 2:
Corollary 4.
There is no FCIF that verifies FMON, FC, FI and FEL orEL. Alcantud and Andr´es Calle (2017) define this aggregator as the
Conjunctive
FCIF. Alcantud and Andr´es Calle (2017) call this aggregator the
Benevolent
FCIF. Dictatorship
Kasher and Rubinstein use, in a section of their paper, a slightly modifiedversion of the CIFs. They assume that there is a consensus in the societythat there exists someone who is a J and someone who is not a J .Then, it follows that the only CIF (with the alternative domain and rangeconditions established for this case) that verifies Consensus and Independenceis the Dictatorial one.Here we define the Dictatorial FCIF with the agent j as a dictator as: D ( P , . . . , P N )( i ) = D J ( i ) = p j ( i )for all i ∈ N .There are many ways to interpret these restrictions in our framework. Onepossibility is that in a profile P i , there exists at least one j such that p i ( j ) = 1and one k such that p i ( k ) = 0. We call this set of profiles P ∗ .An alternative set of profiles is P ∗∗ , in which for every agent i there existsat least one k and one j such that p i ( k ) ≥ . p i ( j ) ≤ . p i = and p i = . We call this set P ∗∗∗ .With respect to the membership functions, we can consider the case in whichthere exist at least one j and one k such that f J ( j ) = 1 and f J ( k ) = 0. Theclass of such function is denoted F ∗ .We define F ∗∗ as the set of membership functions such that for every profile P there exists at least one k and one j such that f PJ ( k ) ≥ . f PJ ( k ) ≤ . f J = and f J = . Wecall this set F ∗∗∗ .It is easy to verify that: P ∗ ⊂ P ∗∗ ⊂ P ∗∗∗ and F ∗ ⊂ F ∗∗ ⊂ F ∗∗∗ A social planner may require different properties to be satisfied by the do-mains and ranges of membership functions. Depending on those specifica-tions, there are various possibilities:
Theorem 3.
Consider FCIFs that satisfy axioms FC and FI.1. Dictatorial is not the only FCIF such that
F J : P ∗∗∗ → F ∗∗∗ .2. Dictatorial is the only FCIF such that F J : P ∗ → F ∗∗ or F J : P ∗∗ → F ∗∗ , and there is no FCIF such that F J : P ∗∗∗ → F ∗∗ . . Dictatorial is the only FCIF such that F J : P ∗ → F ∗ , and there is noFCIF such that F J : P ∗∗ → F ∗ or F J : P ∗∗∗ → F ∗ .Proof.
1. We represent with | P i | > the number of p j ( i )s that are largerthan 0 . | P i | < the number of those less than 0 . P ∈ P .Now we consider the following FCIF: f PJ ( i ) = min j p j ( i )+ max j p j ( i )2 if | P i | > = n . max j p j ( i )2 if | P i | > > | P i | < p j ( i ) if p j ( i ) = p k ( i ) f or all j, k ∈ N . if | P i | < = | P i | > . min j p j ( i )2 if | P i | < > | P i | >min j p j ( i )+ max j p j ( i )2 if | P i | < = n This FCIF verifies FC and FI and is not the Dictatorial FCIF.2. We denote with | f PJ | > the number of f PJ ( i )s larger or equal to 0 . | f PJ | < are those less than 0 . f ∈ F .We say that a coalition L ⊆ N is fuzzy semidecisive for agent i , if thefollowing conditions are satisfied for every profile P ∈ P :[for all j ∈ L, p j ( i ) ≥ . and for all j / ∈ L, p j ( i ) < . ⇒ f Pj ( i ) ≥ . j ∈ L, p j ( i ) < . and for all j / ∈ L, p j ( i ) > . ⇒ f Pj ( i ) < . . (2)A coalition L ⊆ N is called fuzzy semidecisive if it is fuzzy semidecisivefor every agent i in N .Analogously, we say that L ⊆ N is fuzzy decisive over agent i if thefollowing conditions are satisfied for every profile P ∈ P :[for all j ∈ L, p j ( i ) ≥ . ⇒ f Pj ( i ) ≥ . j ∈ L, p j ( i ) < . ⇒ f Pj ( i ) < . .
10n the same way, L ⊆ N is said fuzzy decisive if it is fuzzy decisive forevery agent i in N .We will first prove the existence of a semidecisive coalition for an agent i , and then show that is semidecisive for all i ∈ N . Without loss ofgenerality we start assuming that N = 3 (the result extends easily toall other cases).Consider, also w.l.g. the profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By hypothesis, | f P J | > = 3 and | f P J | < = 3.Suppose | f P J | > = 2, and that f P J (1) , f P J (2) ≥ .
5. Now consider theprofile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By FI, f P J (1) , f P J (2) ≥ . f P J (3) ≥ .
5, a contradiction.Then | f P J | > = 1.Suppose now that f P J (2) ≥ .
5. By FI,for all P ∈ P , [ p (2) ≥ . , p (2) < . , p (2) < . ⇒ f PJ (2) ≥ . . (3)It follows that (1) is verified by L = { } for i = 2.Consider now profile P = ( { . , . , . } , { . , . , . } , { . , . , . } )and suppose by contradiction that f P J (2) ≥ . f P J (3) < .
5. Moreover, f P J (1) < . f P J (1) ≥ .
5, the profile P = ( { . , . , . } , { . , . , . } , { . , . , . } )would lead to a contradiction, because by FI, f P J (1) , f P J (2) ≥ .
5; andby FC, f P J (3) > . f P J (2) > . P ∈ P , [ p (2) < . , p (2) > . , p (2) > . ⇒ f PJ (2) > . . Consider the following profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) .
11y FC f P J (2) < . f P J (3) ≥ .
5, then the profile P = ( { . , . , . } , { . , . , . } , { . , . , . } )yields | f P J | > = 3, a contradiction.Then, the only possibility is that f P J (1) ≥ . P ∈ P , [ p (1) < . , p (1) ≥ . , p (1) ≥ . ⇒ f PJ (1) ≥ . . (4)Finally, consider P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By (3) and (4) we have that f P J (1) , f P J (2) ≥ . f P J (3) ≥ .
5, a contradiction.Thus (2) is verified for agent 1, that is fuzzy semidecisive for agent 2.Now, we can prove that if there exists a fuzzy decisive coalition L ⊆ N for some i ∈ N , then L is fuzzy semidecisive. Without loss of generality,we can suppose that L = { } is fuzzy decisive for 2.Let P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By FC f P J (2) < . f P J (1) ≥ .
5, then by FI, FC and the fact that agent1 is fuzzy decisive over 2, we have that | f P J | > = 3 with P = ( { . , . , . } , { . , . , . } , { . , . , . } ) , a contradiction.Then f P J (3) ≥ . P ∈ P , [ p (3) ≥ . , p (3) < . , p (3) < . ⇒ f PJ (3) ≥ . . Consider the profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . If f P J (3) ≥ .
5, then by FI, FC and the fact that agent 1 is fuzzydecisive over 2, we have that | f P J | > = 3 with P = ( { . , . , . } , { . , . , . } , { . , . , . } ) , a contradiction.Thus f P J (3) < . P ∈ P , [ p (3) < . , p (3) ≥ . , p (3) ≥ . ⇒ f PJ (3) < . . L = { } is fuzzy semidecisive.Now we show that the intersection of two fuzzy semidecisive coalitionsis fuzzy semidecisive. First we prove that L ∩ L ′ = ∅ . Suppose thatthis is not the case.Without loss of generality, let L = { } y L ′ = { , } .Consider the profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . Then f PJ (1) , f PJ (2) ≥ . | f P ′ J | > = 3 with P ′ = ( { . , . , . } , { . , . , . } , { . , . , . } ) , a contradiction.Then L ∩ L ′ = ∅ .Second we prove that L ∩ L ′ is fuzzy semidecisive. Without loss ofgenerality, let L = { , } and L ′ = { , } .Consider another profile P = ( { . , . , . } , { . , . , . } , { . , . , . } )and suppose that, f PJ (1) < . f PJ (2) ≥ .
5, then by FI, FC and the fact that L ′ is fuzzy semidecisive,we have that | f P ′ J | > = 3 with P ′ = ( { . , . , . } , { . , . , . } , { . , . , . } ) , a contradiction.Alternatively, if f PJ (3) ≥ .
5, then consider the following profile P ′′ = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By FI, f P ′′ J (3) ≥ . L and L ′ are fuzzy semidecisive, f P ′′ J (1) , f P ′′ J (2) ≥ . | f P ′′ J | > = 3, a contradiction.Thus, f PJ (1) ≥ .
5, and FI imply thatfor all P ∈ P , [ p (1) ≥ . , p (1) < . , p (1) < . ⇒ f PJ (1) ≥ . . (5)13ow consider another profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . If f PJ (1) ≥ .
5, then it follows from FI and the fact that L and L ′ arefuzzy semidecisive, that | f P ′ J | > = 3 if P ′ = ( { . , . , . } , { . , . , . } , { . , . , . } ) , a contradiction.Thus f PJ (1) < . P ∈ P , [ p (1) < . , p (1) ≥ . , p (1) ≥ . ⇒ f PJ (1) < . . (6)Then we have that L = { } is fuzzy semidecisive.We prove that given a coalition L ⊆ N , either L is fuzzy semidecisiveor N \ L is fuzzy semidecisive.By FC, N is fuzzy semidecisive over N . Without loss of general-ity, fix L = { , } and suppose by contradiction that L is not fuzzysemidecisive. Then it must exist a profile P and an individual i ∈ N ,such that p ( i ) ≥ . , p ( i ) ≥ . , p ( i ) < . f PJ ( i ) < .
5; or p ( i ) < . , p ( i ) < . , p ( i ) ≥ . f PJ ( i ) ≥ . i = 1. By FI,for all P ∈ P , [ p (1) < . , p (1) < . , p (1) ≥ . ⇒ f PJ (1) ≥ . . (7)We want to prove that N \ L = { } is fuzzy semidecisive for agent 1.To do this, we have to show that the following is the case:for all P ∈ P , [ p (1) ≥ . , p (1) ≥ . , p (1) < . ⇒ f PJ (1) < . . (8)Consider the profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . If f P J (1) ≥ .
5, the wanted result follows from FI. If not, i.e. if f P J (1) < . f P J (3) < . f P J (2) ≥ .
5, then we have that | f P J | > = 3 if P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . Then it must be that case that f P J (1) ≥ .
5, and by FI,for all P ∈ P , [ p (1) ≥ . , p (1) ≥ . , p (1) < . ⇒ f PJ (1) ≥ . . (9)14onsider profile P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By FC f P J (1) < . f P J (2) ≥ .
5, then by FC and FI we have that | f P J | > = 3 if P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . Thus, it must be f P J (3) ≥ .
5; and by FIfor all P ∈ P , [ p (3) ≥ . , p (3) ≥ . , p (3) < . ⇒ f PJ (3) ≥ . . (10)But then, by FI, (7) y (10), we have that | f P J | > = 3 if P = ( { . , . , . } , { . , . , . } , { . , . , . } ) , a contradiction.Then we have that N \ L = { } is fuzzy semidecisive for agent 1. Fi-nally we obtain that L = { } is fuzzy semidecisive for N .Now we prove that if a coalition L is fuzzy semidecisive, with | L | > L are also fuzzy semidecisive.Let L ⊆ L ′ ⊆ N , with L fuzzy semidecisive. If L ′ is not fuzzy semide-cisive, then N \ L ′ is fuzzy semidecisive.But then L ⊆ L ′ , ( N \ L ′ ) ∩ L = ∅ , contradicting our previous proof.Thus L ′ must be fuzzy semidecisive over N .Now consider h ∈ L . If L \ { h } is fuzzy semidecisive, the result isproved. If not, we have that N \ ( L \ { h } ) is fuzzy semidecisive.Then we have that N \ ( L \ { h } ) ∩ L = { h } .The next step is to prove that there always exists an agent h ∈ N suchthat { h } is fuzzy semidecisive.By FC, N is fuzzy semidecisive, thus there exists L ′ ⊆ N such that N \ L ′ is fuzzy semidecisive. Then there must exist L ′′ such that( N \ L ′ ) \ L ′′ is fuzzy semidecisive.Because N is finite, by iterating this process we find an h ∈ N that isfuzzy semidecisive over N .Finally we prove that if a coalition L ⊆ N is fuzzy semidecisive, thenit is fuzzy decisive. For that, consider a fuzzy semidecisive coalition L ⊆ N .Then there exists h ∈ L that is fuzzy semidecisive over N . Withoutloss of generality, suppose that h = 1. Suppose that { } is not fuzzydecisive for agent 2.Then it must exist a profile P such that15a) p (2) ≥ . f PJ (2) < . p (2) < . f PJ (2) ≥ . { } is fuzzy semidecisive for agent 2, there has to exist a j = 1 such that p j (2) > . k ∈ N \ { , j } such that p k (2) < .
5. Otherwise, by FC we get f PJ (2) ≥ .
5. Without loss of generality,consider the case where P = ( { . , . , . } , { . , . , . } , { . , . , . } ) . By FC, f PJ (1) < .
5. Then f PJ (3) ≥ . P ∈ P , [ p (3) < . , p (3) ≥ . , p (3) ≥ . ⇒ f PJ (3) ≥ . . (11)Consider the following profile P ′ = ( { . , . , . } , { . , . , . } , { . , . , . } ) . If f P ′ J (3) < .
5, then by FIfor all P ∈ P , [ p (3) ≥ . , p (3) < . , p (3) ≥ . ⇒ f PJ (3) < . . (12)Because { } is fuzzy semidecisive over 3, by (11) and (12), we get thatit is fuzzy semidecisive over N .But, { } ∩ { } = ∅ , a contradiction.Then f P ′ J (3) ≥ . P ∈ P , [ p (3) ≥ . , p (3) < . , p (3) ≥ . ⇒ f PJ (3) ≥ . . Because { } is fuzzy semidecisive over 3, we also know thatfor all P ∈ P , [ p (3) ≥ . , p (3) < . , p (3) < . ⇒ f PJ (3) ≥ . . Then if for all P ∈ P ,[ p (3) ≥ . , p (3) < . , p (3) < . ⇒ f PJ (3) ≥ . , we would get the desired result and { } would be fuzzy decisive foragent 3. 16therwise, we can repeat the previous argument and show that { } isfuzzy semidecisive over N , which would lead us to a contradiction since { } ∩ { } = ∅ .Because 1 ∈ L , we have that L is fuzzy semidecisive for agent 3.Finally, a similar argument shows that L is fuzzy decisive for all i ∈ N .So we have that whenever a FCIF verifies FC and FI, there must existan agent fuzzy decisive over N .The impossibility of a FCIF is proved by set inclusion, since the Dic-tatorial FCIF does not verify the hypothesis in P ∗∗∗ → F ∗∗ .3. The proof is similar to (2).We see in the following example that the aggregator proposed in part 1of Theorem 3 is not the Dictatorial FCIF. Example 5.
If we consider the following profile P = ( { . , . , . } , { . , . , } , { , , . } ) ,we obtain f PJ (1) = 0 . , f PJ (2) = 0 . and f PJ (3) = 0 . , which is differentto the opinion of any agent. When we use the crisp version of the Independence axiom, we find a sim-ilar result using an analogous proof.
Theorem 4.
Consider FCIFs that verify the axioms FC and I.1. Dictatorial is not the only FCIF such that
F J : P ∗∗∗ → F ∗∗∗ .2. Dictatorial is the only FCIF such that F J : P ∗ → F ∗∗ or F J : P ∗∗ → F ∗∗ ,and there is no FCIF such that F J : P ∗∗∗ → F ∗∗ .3. Dictatorial is the only FCIF such that F J : P ∗ → F ∗ , and there is noFCIF such that F J : P ∗∗ → F ∗ or F J : P ∗∗∗ → F ∗ .Proof. We only prove (1) since the proofs of (2) and (3) are similar to theproofs of (2) and (3) of Theorem (3).1. We call a FCIF
Democratic if f J ( i ) = p ( i ) + . . . + p n ( i ) n for all i ∈ N .This FCIF verifies FC and I. 17he fuzzy structure provides the opportunity to find non dictatorial rules.If a FCIF verifies I, f J ( i ) will only depend on the opinions p j ( i ).A function f : R k → R that verifies min { a , . . . , a k } ≤ f ( a , . . . , a k ) ≤ max { a , . . . , a k } is called a k - dimensional mean (Hajja, 2013). A FCIF thatverifies FC is then, a n -dimensional mean.The mean can yield different levels of “democracy”, in the sense that theDictatorial FCIF is a mean while the Democratic FCIF is also a mean.A natural question is whether every n -dimensional mean yields a fuzzy ag-gregation function.The following example shows that the answer is negative. Example 6.
The Inclusive and Unanimous FCIFs are n -dimensional means.Consider the Unanimous FCIF and the profile P = { (0 , . , . , (1 , , . , (0 . , . , } . We obtain U ( i ) = 0 for all i ∈ N , a membership function that does notbelong to F ∗∗∗ . In this work we analyze, from a fuzzy point of view, the Group IdentificationProblem. The opinions of the agents are no longer crisp statements about themembership or not to a group. Instead of that, their opinions are expressedin terms of degrees of membership to the class of J s.We presented the axioms that have been already analyzed in the literatureand gave fuzzy versions of them. In the case of ‘Liberal’ aggregators, theuniqueness and impossibility results from K-R still remain.The axioms FL and FEL (L and EL) are very restrictive, and do not allowmore rules other than the Strong Liberal FCIF.When we deal with the ‘Dictatorial’ aggregator, the results are richer thanin the crisp version, due to the different interpretations that we may give tothe domain and range conditions postulated in K-R.Depending on the goals of the social planner we can have several, just oneor no rules satisfying the desired properties.The binary nature of determining if someone has more or less than 0 . eferences [1] Alcantud, J. C., de Andr´es Calle, R.: “The problem of collectiveidentity in a fuzzy environment”, Fuzzy Sets and Systems , 315:57–75, 2017.[2] Ballester, M.A., Garca-Lapresta, J.L.: “A Model of Elitist Quali-fication”,
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