Anonymous, non-manipulable, binary social choice
aa r X i v : . [ ec on . T H ] J u l Anonymous, non-manipulable, binary social choice
Achille Basile ∗ Dipartimento di Scienze Economiche e StatisticheUniversit`a Federico II, 80126 Napoli, ItalyE-mail: [email protected] RaoSchool of Business and EconomicsIndiana University Northwest, Gary, IN 46408E-mail: [email protected]. P. S. Bhaskara RaoDepartment of Computer Information SystemsIndiana University Northwest, Gary, IN 46408E-mail: [email protected] 6, 2020
Abstract
Let V be a finite society whose members express weak orderings (hence also indif-ference, possibly) about two alternatives. We show a simple representation formulathat is valid for all, and only, anonymous, non-manipulable, binary social choicefunctions on V . The number of such functions is 2 n +1 if V contains n agents. JEL Code: D71AMS Subject Classification: 91B14
Keywords: social choice functions, anonymity, strategy-proofness, committees, quota ma-jority, weak orderings. ∗ Corresponding author. epresentation of anonymous... Introduction
It is well known, after the celebrated Gibbard-Satterthwaite Theorem ([4], [8]), that the twoproperties of anonymity and non-manipulability of a social choice function, conflict everytime the collective choice is from among a set of at least three alternatives. Anonymityguarantees that all individuals of the collectivity are equally powerful in the social choicedetermination, whereas non-manipulability (or strategy-proofness) guarantees that tellingthe truth is strategically dominant for all individuals. The desirability of both propertiesresulted in a large literature that considers, with three or more alternatives, social choicefunctions over restricted domains or considers weaker properties to be satisfied by thecollective choice (a classical survey is [1]). At the same time, attention has been paid tothe case, quite common in many important practical situations, in which the society hasto decide by choosing between two alternatives a and b . In the latter, binary, case, if everyvoter is asked to declare a strict preference, one has that the anonymous, non-manipulablesocial choice functions are all, and only, the quota majority methods (see [7, Corollaryof page 63]), in the sense that they are the n + 2 functions µ k , with k = 0 , , , . . . , n + 1,defined as follows. For a profile P = ( P v ) v ∈ V of preferences , the corresponding socialchoice µ k ( P ) is a if the number of voters choosing a is at least k , otherwise the collectivechoice is b .This paper deals with the binary setting in which the voters are allowed to express indif-ference also. We shall describe in this setting all anonymous non-manipulable social choicefunctions by means of a representation formula that turns out to be quite simple to de-scribe and also is a direct extension of the quota majority rule (we call it extended quotamajority ). As a straightforward corollary, we shall show that there are 2 n +1 anonymous,non-manipulable, binary social choice functions when the society has n voters.The fact that, due to the transition from strict to weak orderings, the number of anonymous,non-manipulable, binary social choice functions depends exponentially on the number ofvoters, rather than linearly, suggests that obtaining a sound representation theorem forweak orderings is not an obvious task. To the best of our knowledge, this problem wasconsidered only recently in Lahiri and Pramanik ([5, Theorem 2]). In Section 4 we shallcompare the two representations. We point out that ours is not only glaringly a plainextension of the quota majority method (indeed a sequence of majority rules), but alsocan be made optimal in the sense of minimizing the set of the necessary parameters. Thenotion of extended quota majority method is quite intuitive. If one applies a quota majoritymethod µ k allowing for indifference, except for the cases in which the quota k ∈ { , n +1 } ,where we get a constant collective choice, there will always be profiles for which the methoddoes not provide a social choice. Then, the idea is to apply a further quota majority method µ k but only to profiles for which µ k has not given the value. Some more profiles will becovered, but also the application of the second quota may leave uncovered profiles. Then,one can apply a µ k , and so on. To ensure that this procedure stops, the sequence of quotas In this case every voter v is asked to declare either P v = a or P v = b . epresentation of anonymous... Results must involve at some point a quota k r ∈ { , n + 1 } . It turns out that this is the only way toproduce anonymous, non-manipulable binary social choice functions. We go beyond this.We show that the choice of the quotas can be done in an optimal way, in the sense thatthe dimension of the vector k = ( k , k , . . . , k r ) is minimum. We also characterize this byshowing that it happens correspondingly to an up-down course of the sequence like, forexample, the following · · · < k < k < k < k < k < . . . In [3] we have given a formula to represent all strategy-proof binary social choice functions,on an arbitrary set of voters (i.e. not necessarily finite, as we assume here), wherein votersare permitted to express indifference. We introduced a class of social choice functions, thatwe call ψ -type functions , and shown that these are all, and only, the binary social choicefunctions that cannot be manipulated. We have seen that some (not all such functions) canhave a simpler structure based on collections of committees (see [3, Remark 2.7, Proposition2.9]) potentially within the society. A significant example is the simple majority rule ([3,Proposition 4.3]). The results of this paper extend the exercise for the simple majority rule.We emphasize that they cannot be seen as straightforward consequences of the generalrepresentation theorem ([3, Theorem 4.2]) obtained in [3]. In particular this is true sinceour main result (Theorem 2.7 below) is also a uniqueness result of the representation ofa non-manipulable anonymous social choice function by means of some special extendedquota majority methods that we call proper extended quota majority methods .The paper proceeds as follows. Next Section presents our results. The third Sectioncontains all technical details of the needed proofs. Fourth Section concludes by comparingour representation formula with that proposed by Lahiri and Pramanik in [5]. Let V be a society of cardinality n . A profile P = ( P v ) v ∈ V consists of the declarations,denoted by P v , of agent v ’s preference between the alternatives a and b . The collectivity V will necessarily implement one of the two alternatives. Since we allow for indifference, thepossibilities for agent v are: to declare preference for a , or for b or to declare indifferencebetween a and b .A social choice function (scf, for short) φ is a mapping P φ ( P ), the value φ ( P ) being thealternative selected as the social outcome corresponding to the profile P . Since throughoutthe paper we only deal with binary (i.e. only the two alternatives a and b are considered)scfs, we shall use scf to mean binary scf.In order to ensure a fair consideration of the opinions of all agents, one may requireanonymity of scfs. Non-manipulability may be required to prevent strategical false dec-larations. The formal, well established definitions, are: Definition 2.1
A scf φ is: epresentation of anonymous... Results anonymous , if φ ( P ) = φ ( P ◦ σ ) = φ ( ( P σ ( v ) ) v ∈ V ) , for every profile P and for everypermutation σ of V . non-manipulable , if φ ( P v , P − v ) ≻ ∼ P v φ ( Q v , P − v ) , for every voter v , for every pro-file P , and for every weak ordering Q v . We introduce now extended quota majority methods, denoted by φ k , which can be de-scribed as follows. Let us call r -tuple , for 1 ≤ r ≤ n + 1, an ordered tuple k =( k , k , . . . , k r ) of distinct elements from the set { , , . . . , n + 1 } such that k r ∈ { , n + 1 } .For a profile P , let λ ( P ) be the smallest index λ for which either at least k λ voters prefer a , or at least n + 1 − k λ voters prefer b .Given these premises, we give the following definition. Definition 2.2
A scf is said to be an extended quota majority method if for some r -tuple k , we have that the scf is defined as follows φ k ( P ) def = (cid:26) a, if at least k λ ( P ) voters prefer a,b, if at least n + 1 − k λ ( P ) voters prefer b. Remark 2.3
Notice that:1. When P is a strict profile, then obviously the index λ ( P ) is zero, hence φ k ( P ) = µ k ( P ) , for every k , i.e. the extended method, restricted to strict profiles, gives backthe original quota majority method.2. It is immediate to recognize that, by Definition 2.2, k = n + 1 gives the collectivechoice being always b , irrespective of the profiles expressed by the collectivity. Anal-ogously, if k = 0 we get always a as the collective choice irrespective of the profilesexpressed by the collectivity. When ≤ k ≤ n , the range of the collective choice is { a, b } . ✷ Now our first representation theorem can be promptly stated.
Theorem 2.4
Extended quota majority methods φ k are anonymous and non-manipulable.Moreover, every anonymous non manipulable binary social choice function is an extendedquota majority method for some k . Definition 2.5
We say that the length of k is the smallest index λ for which k λ ∈ { , n +1 } . The possibility of representing anonymous, strategy-proof scfs as extended quota majorities,does not ensure uniqueness of the representation. In order to achieve representations thatare also unique, we introduce proper extended quota majorities.3 epresentation of anonymous... Results
Definition 2.6
We say that an extended majority method is proper if it satisfies one ofthe following up and down conditions.down-up: < k r − < ... < k < k < k < k < k < k < k < ... < n + 1 = k r k r < ... < k < k < k < k < k < k < k < ... < k r − < n + 1 up-down: n + 1 = k r > ... > k > k > k > k > k > k > k > ... > k r − > n + 1 > k r − > ... > k > k > k > k > k > k > k > ... > k r The following figure, illustrates, in a society V of 11 voters, relatively to the function φ k , the properness of the sequence k : the solid line involves a down-up sequence k =(4 , , , , , ,
12) of length 6; the dashed line an up-down sequence k = (4 , , , , ,
12) oflength 5.The next one, illustrates, in a society V of 11 voters, relatively to φ k , the properness of thesequence k : the solid line involves a down-up sequence k = (7 , , , , , , ,
0) of length7; the dashed line an up-down sequence k = (4 , , , ,
0) of length 4.4 epresentation of anonymous... Results
Our main results is the following.
Theorem 2.7
For every onto binary social choice function φ which is anonymous, andnon-manipulable there exists one and only one proper extended quota majority method φ k such that φ = φ k . Because of Theorems 2.4 and 2.7, the following is now obvious.
Corollary 2.8
The onto scfs that are anonymous, and strategy-proof are all, and only, theproper extended quota majority methods.
To determine the cardinality of the class of all anonymous, non-manipulable, binary socialchoice functions in a society with n agents, we can count the extended quota majority meth-ods. Indeed we count that there are 2 n anonymous, non-manipulable scfs corresponding tothe collective choice b for the unanimous indifference. Symmetrically, there are 2 n anony-mous, non-manipulable scfs corresponding to the collective choice a for the unanimousindifference.To see the first statement, let us take a subset J ⊆ { , . . . , n } . If J is empty, let us associate J with the constant scf b . Suppose J is nonempty and has cardinality r . In this case weassociate J with the scf φ ( k ,k ,...,k r − ,n +1) where the proper k presents: k r − = min J, k r − = max J \ { k r − } , k r − = min J \ { k r − , k r − } ,k r − = max J \ { k r − , k r − , k r − } , . . . , k = (cid:26) max J \ { k , . . . , k r − } , if k is a minmin J \ { k , . . . , k r − } , if k is a max . Theorems 2.4 and 2.7 guarantees that the correspondence defined above is a bijection.Hence by symmetry, we have proved the following.
Corollary 2.9
There are n +1 anonymous, non-manipulable, binary social choice func-tions if n voters choose between two alternatives, being allowed to express indifference. Remark 2.10
A few comments are in order.1. Definition 2.2 con be formally given with reference to an arbitrary sequence j =( j , j , ... ) . The only needed condition is that at least one of its values belongs to { , n + 1 } . When ≤ j ≤ n , if r is the smallest index for which j r ∈ { , n + 1 } ,the original sequence j and the truncated sequence ( j , j , ..., j r ) give rise to the samescf. In this case, j r = n + 1 corresponds to assign b to the profile where all agents areunanimously indifferent. An analogous comment applies when j r = 0 replaces n + 1 ,in that case a replaces b .2. Let φ h , h = ( h λ ) λ ∈{ ,..., } be defined with the help of an arbitrary sequence. If wehave h β ≥ h γ ≥ h α for indices α, β < γ , then φ h does not change if we remove h γ from the sequence h . The deletion of h γ is possible even if we have h β ≤ h γ ≤ h α (or h β = h γ ) for indices α, β < γ . It is sufficient to observe that the index of a profile P does not change if we remove h γ . Indeed it isobvious that the index λ ( P ) cannot be γ . epresentation of anonymous... Proofs
3. By applying to the truncated sequence ( j , j , ..., j r ) above the deletion of the repeatedindices, we recognize that the scf given by the formula of Definition 2.2 applied to anarbitrary j produces an extended quota majority method. ✷ Definition 2.11
We say that a representation of a scf φ as an extended quota majoritymethod is minimal if it has minimum length among all such representations of φ . We close this Section by observing that
Corollary 2.12
A representation is proper if and only if it is minimal.
The following example illustrates some of the concepts and it is relevant for Remark 3.13
Example 2.13
In this example, to modify the status quo (say, b ) to the new status (say, a ),the society V = { , . . . , } needs that at least two individuals wish doing that. However,if such individuals are no more than four, it is needed also that the voters in favor ofmaintaining the status quo are less than seven.The model for this situation is the scf φ defined as follows: φ ( P ) = a if either | D ( a, P ) | ≥ or ≤ | D ( a, P ) | < | D ( b, P ) | < . In all the othercases φ ( P ) = b .According to Remark 2.10, for such a φ possible defining sequences are (5 , , , , (5 , , , , (5 , , , the latter being the proper representation. ✷ The investigation of the scfs which are non-manipulable relies (see [6], [5], [3]) on the so-called committees and their duals . A committee is, by definition, a nonempty, closedunder superset, familiy of coalitions that can be formed in the society V . We are particularlyinterested in families G k = { E ⊆ V : | E | ≥ k } for k = 1 , . . . , n . The superset closed familydual to G k is G ◦ k = { E ⊆ V : | E | ≥ n + 1 − k } , i.e. G n +1 − k .Let V = {G k : k = 1 , . . . , n } be the set of all such committees on V that we refer to as committees of of cardinal type k . A synonym is superset closed family (SSCF,for short) of cardinal type k . It will be convenient to consider the power set of V andthe empty subset of the power set of V also as committees. They can be considered as ofcardinal type respectively zero ( G = 2 V ) and n + 1 ( G n +1 = Ø). They are also dual to eachother.We shall also consider, given a subsets I of V with cardinality ℓ, (0 ≤ ℓ < n ), SSCFs F on I c = V \ I of cardinal type. If we suppose that the type of F is k (necessarily we have We remind from [3], where the notion has been introduced, that the dual of a committee F is thecommittee F ◦ def = { E ⊆ V : V \ E / ∈ F} . epresentation of anonymous... Existence and uniqueness of proper representations ≤ k ≤ n − ℓ ), the corresponding dual (with respect to I c ) family is also of cardinal typeand has type m = n − ℓ − k + 1 . By comparing Definition 2.2 with [3, Remark 3.3], it is evident that:
Remark 3.1
An onto extended quota majority method φ k , k = ( k , k , ..., k r ) , is a scf ofthe form (see [3, Remark 3.3]) φ F , x , where x is either a or b according to the fact that thefirst k i / ∈ { , . . . , n } is either or n + 1 , and the collection F consists of the committees G k , . . . , G k i − . The above Remark and [3, Proposition 3.4] tell us that extended quota majorities are non-manipulable scfs. Anonymity being obvious, we have the first part of Theorem 2.4. We shallprove the second part, namely that every anonymous, non manipulable scf is an extendedquota majority method, in subsection 3.3. The fact that extended quota majority methodsadmit proper representations and that the proper representation is unique for onto scfs, isdiscussed in the subsection that follows.
Throughout the sequel of the paper we adopt the following notation: by D ( a, P ) , D ( b, P ) , and I ( P ) we denote the subsets of V consisting of voters that, respectively, choose a, b orare indifferent between the two alternatives. Proposition 3.2
Let φ j , j = ( j λ ) λ ∈{ ,...,r } , be an onto extended quota majority method.There is one and only one proper (sub)sequence k of j such that φ j = φ k . proof: We shall first show the existence of k . Without loss of generality we can assume that { j , j , . . . , j r − } ⊆ { , . . . , n } .We shall describe a procedure that, through the deletion of suitable indices j ’s, producesthe proper representation φ k of φ j . The initial element k of k is set to be j .We discuss the case j > k . The case j < k can be discussed in a similar way.Let us partition the sequence defining φ j as illustrated in the following figure j , ..., j i − j i , ..., j i − k ...j i , ..., j i − j i , ..., j i − The indices i , i , . . . are defined as follows: i = min { i > j i < j } ; i = min { i > i : j i > j } ; i = min { i > i : j i < j } ; . . . One can adopt the usual convention that the minimum of the empty set is ∞ and once i h = ∞ the sequence of indices i ’s stops. The top row contains values of (cardinalities) j ’sbigger than k , the bottom one smaller than k epresentation of anonymous... Existence and uniqueness of proper representations Let k be defined as the maximum of { j , ..., j i − } . It is a straightforward calculation toverify that φ j = φ k ,k ,j i ,...,j i − ,...,j r .Let k be defined as the minimum of { j i , ..., j i − } . It is a straightforward calculation toverify that φ j = φ k ,k ,k ,j i ,...,j i − ,...,j r .If one of the values in the set { j i , ..., j i − } is smaller than k , due to Remark 2.10, suchvalue can be deleted without modifying the scf φ j . Therefore, we can assume we are in asituation like this: k < k < k < { j i , ..., j i − } . Let k be defined as the maximum of { j i , ..., j i − } . It is a straightforward calculation toverify that φ j = φ k ,k ,k ,k ,j i ,...,j i − ,...,j r .If one of the values in the set { j i , ..., j i − } is bigger than k , due to Remark 2.10, suchvalue can be deleted without modifying the scf φ j . Therefore, we can assume we are in asituation like this: { j i , ..., j i − } < k < k < k < k . Let k be defined as the minimum of { j i , ..., j i − } . It is a straightforward calculation toverify that φ j = φ k ,k ,k ,k ,k ,j i ,...,,...,j r . We continue this way till we produce a proper k giving the same scf.We shall now see the uniqueness. For a scf φ giving value b for unanimous indiffer-ence, suppose that we have two proper representations φ k = φ k ,k ,...,k r − ,n +1 and φ h = φ h ,h ,...,h s − ,n +1 . We shall show first that k = h , then k = h , k = h , and so on.Afterwords we show that also the lengths coincide. We shall write n − k + 1 as k ◦ .To see k = h :Suppose, without loss of generality, that k < h . Let P be a profile with | D ( a, P ) | = k and | D ( b, P ) | = h ◦ . By using the k -representation, we get that the social choice is a . Byusing the h -representation, we get that the social choice is b and this is a contradiction.To see k = h :Suppose, without loss of generality, that k < h . We shall consider the following threecases: k < k < h , k < k < h , k < h < k , obtaining for everyone a contradictionto the assumption that φ k = φ h . When k < k < h , since k is proper, we have necessarily · · · < k < k < k < h . Let P be a profile with | D ( a, P ) | = k − | D ( b, P ) | = h ◦ . Along k the index of P is twoand the value of φ ( P ) is a . Along h the index of P is one and the value of φ ( P ) is b , acontradiction.For the other two cases k < k < h , and k < h < k , let us consider a profile P with | D ( a, P ) | = k and | D ( b, P ) | = k ◦ −
1. By using the k -representation, we get that thesocial choice is a . By using the representation φ = φ k ,h ,...,h s − ,n +1 , we get that the socialchoice is b and this is a contradiction. 8 epresentation of anonymous... Existence and uniqueness of proper representations To show that φ k ,h ,...,h s − ,n +1 ( P ) = b , in the case k < k < h , since k ◦ − ≥ h ◦ , clearly φ k ,h ,...,h s − ,n +1 ( P ) = b . In the case k < h < k , since the index of the profile P cannotbe zero, first notice that it is also not one. Indeed, not only | D ( a, P ) | < h , but also | D ( b, P ) | = n − k < n − h < h ◦ . Since h is proper, we have k < h < k < h , and theindex of P is two: | D ( b, P ) | = n − k > n − h ⇒ | D ( b, P ) | ≥ h ◦ .We can now suppose, without loss of generality, that we are in a situation like the following · · · < k < k < k < k < k < k < k < . . . · · · < h < h < k < k < h < h < h < . . . After having proved, for i ≤ min { r − , s − } , that k λ = h λ for λ < i , we prove bycontradiction that k i = h i . As in the previous steps, let us assume that k i < h i .We have φ k ,k ,...,k i − ,k i ,...,k r − ,n +1 = φ k ,k ,...,k i − ,h i ,...,h s − ,n +1 and one of the following two cases.Case 1: k i +1 < k i − < ... < k < k < k < ... < k i − < k i ∧ h i +1 < k i − < ... < k < k < k < ... < k i − < h i Let P be a profile with | D ( a, P ) | = k i − − | D ( b, P ) | = h ◦ i . By using the representation φ = φ k ,k ,...,k i − ,h i ,...,h s − ,n +1 , the index of the profile is i and the social choice is b .By using the representation φ = φ k ,k ,...,k r − ,n +1 , we get that the social choice is a and thisis a contradiction. To show that φ k ,k ,...,k r − ,n +1 ( P ) = a , notice that i < r − k r − and k r = n + 1 on the right of k against the properness of therepresentation. We see now that with respect to the representation φ = φ k ,k ,...,k r ,n +1 theprofile has index i + 1 and the social choice is a .Case 2: k i < k i − < ... < k < k < k < ... < k i − < k i +1 ∧ h i < k i − < ... < k < k < k < ... < k i − < h i +1 Let P be a profile with | D ( a, P ) | = k i and | D ( b, P ) | = k ◦ i − −
1. By using the represen-tation φ = φ k ,k ,...,k r − ,n +1 , we get that the social choice is a (index is i ). By using therepresentation φ = φ k ,k ,...,k i − ,h i ,...,h s − ,n +1 , we get that the social choice is b and this is acontradiction. To show that φ k ,k ,...,k i − ,h i ,...,h s − ,n +1 ( P ) = b , since the index of the profile P with respect to this representation cannot be less than i , first notice that it is also not i . Indeed, not only | D ( a, P ) | < h i , but also | D ( b, P ) | = n − k i − < n − h i < h ◦ i .9 epresentation of anonymous... Subsets of profiles If s − > i , along the sequence k , h , . . . , h s − , the index of P is i +1: | D ( b, P ) | = n − k i − >n − h i +1 ⇒ | D ( b, P ) | ≥ h ◦ i +1 . If s − i , then the index is s . In both cases the value ofthe social choice is b as announced.The indices r and s coincide.Suppose r > s , then necessarily we have a situation as the following k r − < ... < k s +1 < k s − < k s − < ... < k < k < k < ... < k s − < k s < ... < k r = n + 1 k s − < k s − < ... < k < k < k < ... < k s − < h s = n + 1If we take a profile P | D ( a, P ) | = k s − − | D ( b, P ) | = 0. By using the representation φ = φ k ,k ,...,k r − ,n +1 , we get that the social choice is a (index s + 1). By using the repre-sentation φ = φ k ,k ,...,k s − ,n +1 , we get that the social choice is b and this is a contradiction. ✷ proof of Corollary 2.12 :Let us take a representation φ k ,k ,...,k r − ,n +1 of φ whose length is minimum and supposethat it is not proper. Let us suppose that k > k , since a similar argument can be given ifthe reverse inequality holds true. Since the representation is not proper, we shall have forsome index i ≥ k i − < ... < k < k < k < ... < k i and k i +1 > k i − , or the condition k i < ... < k < k < k < ... < k i − and k i +1 < k i − . In the first case the deletion of the smallest between k i and k i +1 still produces a represen-tation of φ . In the second case the deletion of the largest between k i and k i +1 still producesa representation of φ . Hence φ k ,k ,...,k r − ,n +1 is not of minimum length. For the converse,namely, to show that every proper representation is minimal, if φ k ,k ,...,k r − ,n +1 is a properrepresentation of φ , by the uniqueness Theorem, it must be necessarily minimal. ✷ We shall introduce here some useful subsets of the set P of all profiles. Definition 3.3
Let the natural numbers ℓ, k, m be such that ≤ ℓ < n, ≤ k ≤ n − ℓ,m = n − ℓ − k + 1 . Let I be a subset of V with cardinality ℓ , and F a SSCF on I c ofcardinal type k (with respect to I c ). Minimal representation obviously exist and we have just seen they are proper. The fact that the numbers ℓ, k and m (possibly indexed) will be always such that 0 ≤ ℓ < n, ≤ k ≤ n − ℓ, m = n − ℓ − k + 1 , will be assumed throughout the sequel of the paper. epresentation of anonymous... Subsets of profiles
1. The sets P a ( ℓ, k ) and P b ( ℓ, k ) : P ∈ P a ( ℓ, k ) def ⇔ (cid:26) | D ( a, P ) | ≥ k, | D ( b, P ) | < m.P ∈ P b ( ℓ, k ) def ⇔ (cid:26) | D ( a, P ) | < k, | D ( b, P ) | ≥ m.
2. The sets P a ( I, F ) and P b ( I, F ) : P ∈ P a ( I, F ) def ⇔ (cid:26) | D ( a, P ) ∩ I c | ≥ k,D ( b, P ) ⊆ I c .P ∈ P b ( I, F ) def ⇔ (cid:26) D ( a, P ) ⊆ I c , | D ( b, P ) ∩ I c | ≥ m. In the particular case of I = Ø (i.e. ℓ = 0 ) we shorten P a (Ø , G k ) as P a ( G k ) and P b (Ø , G k ) as P b ( G k ) . P ( ℓ, k ) def = P a ( ℓ, k ) ∪ P b ( ℓ, k ) , and P ( G k ) def = P a ( G k ) ∪ P b ( G k ) . Trivially, the sets P a and P b are disjoint.The profiles belonging to P a ( ℓ, k ) have a stucture that can be also described as in the nextproposition. Indeed, if we take a profile P ∈ P a ( ℓ, k ), we can certainly fix a subset of D ( a, P ) with cardinality k . Let us call V such a set. Call V a subset of V \ V which is asuperset of D ( b, P ) with cardinality n − ℓ − k = m −
1. Finally, define I = V \ ( V ∪ V )(the cardinality of I is ℓ ). Hence, the profile P looks like in the following figure I a I ∼ V ∼ D ( b, P ) V a V Profile P top a { a, b } { a, b } b a aV I I c I ( P )where the sets I a , V a , I ∼ , V ∼ may be empty and D ( a, P ) = I a ∪ V a ∪ V . Similarly we cando for a profile P ∈ P b ( ℓ, k ). Therefore, the following proposition is proved. Proposition 3.4
The following equations hold true: P a ( ℓ, k ) = [ σ P a ( σ ( I ) , σ ( F )) , Hence: P ∈ P a ( G k ) means | D ( a, P ) | ≥ k , and P ∈ P b ( G k ) means | D ( b, P ) | ≥ n − k + 1. epresentation of anonymous... Subsets of profiles P b ( ℓ, k ) = [ σ P b ( σ ( I ) , σ ( F )) , where: σ runs over all permutations of V, I is a subset of V with cardinality ℓ , F is theSSCF on I c of cardinal type k (with respect to I c ). Since G ℓ + k ⊆ G k and dually G ◦ ℓ + k ⊇ G ◦ k , we also have the next proposition. Proposition 3.5 P ∈ P a ( ℓ, k ) ⇔ (cid:26) D ( a, P ) ∈ G k ,D ( b, P ) / ∈ G ◦ ℓ + k ⇔ P ∈ P a ( G k ) \ P b ( G ℓ + k ) ,P ∈ P b ( ℓ, k ) ⇔ (cid:26) D ( a, P ) / ∈ G k ,D ( b, P ) ∈ G ◦ ℓ + k ⇔ P ∈ P b ( G ℓ + k ) \ P a ( G k ) . We conclude by presenting the definition of a scf that will be useful for an intermediatedstep in the completion of the proof of Theorem 2.4. Given a sequence h ℓ , k i = ( ℓ λ , k λ ), for λ ∈ Λ = { , , . . . , | Λ |} , we say that a profile P has index λ ( P ) if λ ( P ) is the smallestindex λ for which P ∈ P λ def = P ( ℓ λ , k λ ). This notion replicates, mutatis mutandis, the onepreceding Definition 2.2 (see also [3, Definitions 3.1 and 3.2]). Definition 3.6
Given x ∈ { a, b } , and the sequence h ℓ , k i = ( ℓ λ , k λ ) , where λ ∈ Λ = { , , . . . , | Λ |} , we define the following scf: ψ h ℓ , k i , x ( P ) = a, if P ∈ P a ( ℓ λ ( P ) , k λ ( P ) ) b, if P ∈ P b ( ℓ λ ( P ) , k λ ( P ) ) x, if P / ∈ ( S λ ∈ Λ P λ ) . Remark 3.7
With reference to Definition 3.6, if we adopt, for λ = 0 , , , ..., | Λ | , thefollowing shorter notation: ψ λ def = ψ h ( ℓ ,...,ℓ λ ) , ( k ,...,k λ ) i ,x , we can notice that, obviously,on P we have ψ = ψ λ , ∀ λ ≥ . On P \ P we have ψ = ψ λ , ∀ λ ≥ . ... On P α \ ( S β<α P β ) we have ψ α = ψ λ , ∀ λ ≥ α ; On P \ ( S λ ∈ Λ P λ ) we have ψ λ ( · ) = x, ∀ λ ∈ Λ . For the definition of the SCF φ G , x employed in the following proposition we refer to [3,Remark 3.3]. Proposition 3.8
Assume for the sequences ( ℓ λ , k λ ) λ ∈ Λ ( with Λ = { , , . . . , | Λ |} ) that k is decreasing and ℓ + k is increasing.The scf ψ h ℓ , k i , x is identical to the scf φ G , x where the collection G is: G = G k , G ℓ + k , G k , G ℓ + k , G k , G ℓ + k , . . . , G k | Λ | , G ℓ | Λ | + k | Λ | , if x = a G ℓ + k , G k , G ℓ + k , G k , G ℓ + k , G k , . . . , G ℓ | Λ | + k | Λ | , G k | Λ | , if x = b being the committees constituting G , to be considered in the order from left to right describedin the above formula. epresentation of anonymous... Subsets of profiles Remark 3.9
Before giving the proof we observe that we are saying, on the base of Remark2.10, that ψ h ℓ , k i , x is the extended quota majority method defined:by the sequence ( k , ℓ + k , k , ℓ + k , k , ℓ + k , . . . , k | Λ | , ℓ | Λ | + k | Λ | , , if x = a ,by the sequence ( ℓ + k , k , ℓ + k , k , ℓ + k , k , . . . , ℓ | Λ | + k | Λ | , k | Λ | , n + 1) if x = b . ✷ proof: Let us shorten the notation for the two functions, by using simply ψ and φ . Also,according to our notation, m (possibly indexed, as it is now) is n − ℓ − k + 1.We show that for every profile P , we have φ ( P ) = ψ ( P ) . Notice that P a ( G ℓ + k ) ⊆ P a ( G k ); and P b ( G k ) ⊆ P b ( G ℓ + k ) . To prove that the two scfs coincide on P we distinguish two cases. Let us consider firstthe case that P ∈ S λ ∈ Λ P λ . Suppose that λ is the index of the profile P . We have either P ∈ P a ( ℓ λ , k λ ) or P ∈ P b ( ℓ λ , k λ ), respectively giving ψ ( P ) = a or ψ ( P ) = b . We showthat correspondingly φ attains the same value on P .Case P ∈ P a ( ℓ λ , k λ ).Since by Proposition 3.5, P a ( ℓ λ , k λ ) = P a ( G k λ ) \ P b ( G ℓ λ + k λ ) ⊆ [ P a ( G ℓ λ + k λ ) ∪ P a ( G k λ )] \ P b ( G ℓ λ + k λ ), if we prove that P does not belong to P ( G ℓ β + k β ) ∪ P ( G k β ) whenever β < λ , weshall have that φ ( P ) = a . In other words we have to exclude that P belongs to one of thefollowing four sets: P a ( G ℓ β + k β ) ⊆ P a ( G k β ), P b ( G k β ) ⊆ P b ( G ℓ β + k β ).We show that P / ∈ P b ( G ℓ β + k β ).Suppose the contrary, then | D ( b, P ) | ≥ m β . On the other hand since P / ∈ P b ( G ℓ λ + k λ ), wehave | D ( b, P ) | < m λ against the assumption that m is decreasing.Now we can show that P / ∈ P a ( G k β ). Indeed, if not by Proposition 3.5 we have P ∈ P a ( ℓ β + k β ), against the definition of index.Case P ∈ P b ( ℓ λ , k λ ).By Proposition 3.5 P ∈ [ P b ( G ℓ λ + k λ ) ∪ P ( G k λ )] \ P a ( G k λ ). An argument similar to theprevious one applies. If the index β is less than λ , the profile P cannot belong to P a ( G k β )otherwise the monotonicity of k is violated. Now, the profile cannot be in P b ( G ℓ β + k β ),otherwise it belongs to P b ( ℓ β + k β ), against the definition of index.It remains to consider the case that P / ∈ S λ ∈ Λ P λ .In this case the value of ψ ( P ) is x , and we show that even φ ( P ) is x . We have to investigatethe case that P ∈ S λ ∈ Λ [ P ( G k λ ) ∪ P ( G ℓ λ + k λ )], since in the opposite case the assertion comesfrom the definition of φ . So, let us suppose that P ∈ S λ ∈ Λ [ P ( G k λ ) ∪ P ( G ℓ λ + k λ )] and let α be the first (moving from left to right in the definition of G ) index such that P ∈ P ( G α ).We are done if we show that x = a ⇒ P / ∈ P b ( G α ) . Indeed, if this is not the case, P ∈ P b ( G α ) . But α is either some k λ or an ℓ λ + k λ and inboth cases we can write P ∈ P b ( G ℓ λ + k λ ) . Since
P / ∈ S λ ∈ Λ P λ , by Proposition 3.5 we musthave P ∈ P a ( G k λ ) . Hence α = k λ and P ∈ P a ( G α ) ∩ P b ( G α ) , a contradiction.13 epresentation of anonymous... Anonymous, non manipulable is extended quota majority Similarly, we can show that x = b ⇒ P ∈ P b ( G α ) , and we are done. ✷ First we see that committees of cardinal type arise when anonymous scfs are considered.Compare indeed next proposition with [3, Proposition 4.1]. We recall that the committeesof cardinal type on V are V = {G k : k = 1 , . . . , n } , G k = { E ⊆ V : | E | ≥ k } . We alsoconsider the power set of V and the empty subset of the power set of V as committees ofcardinal type respectively zero ( G = 2 V ) and n + 1 ( G n +1 = Ø). Proposition 3.10
Let φ be a non-manipulable scf. If φ is anonymous, the (unique) com-mittee F such that for every profile P one has D ( a, P ) ∈ F ⇒ φ ( P ) = a, and D ( b, P ) ∈ F ◦ ⇒ φ ( P ) = b, is of cardinal type. proof: If we denote by ̺ ( F ) the least cardinality of the coalitions belonging to F , we have toprove that | F | ≥ ̺ ( F ) ⇒ F ∈ F . Let D ∈ F be such that | D | = ̺ ( F ) . Let E ⊆ F be such that | D | = | E | . Then take a permutation σ of V such that the σ -imageof E is D .Take a strict profile S that ranks a as top on D and b as top on E c . We hence have φ ( S ) = a .Let Q be the profile S ◦ σ , i.e. Q v = S σ ( v ) . Because of anonymity and since Q is strict, wehave that D ( a, Q ) ∈ F .Clearly v ∈ D ( a, Q ) is same as σ ( v ) ∈ D ( a, S ) = D , which is the same as v ∈ E . So E isin F and therefore the superset F too is in F . ✷ In the above proposition
F ∈ V when φ is onto. F ∈ {G , G n +1 } for the constant scfs a and b respectively.The following is the crucial step to obtain the results presented in Section 2. Theorem 3.11
Let φ be a scf which is onto, anonymous, and non-manipulable. Say x isthe collective choice corresponding to the unanimous indifference.Correspondingly, we canfind a sequence h ℓ , k i = ( ℓ λ , k λ ) λ ∈ Λ such that: epresentation of anonymous... Anonymous, non manipulable is extended quota majority
1. the sequence ℓ is strictly increasing and ℓ = 0 ,
2. in case x = b (respectively, x = a ), the sequence k is strictly decreasing (respectively,decreasing) and the sequence ℓ + k is increasing (respectively, strictly increasing).3. φ = ψ h ℓ , k i , x . proof: For the proof, we shall go along steps that echo those for proving [3, Theorem 4.2].However, a deeper argument is needed to achieve the result.We remind that once the sequences ℓ and k are given, the sequence m is also fixed for thedual committees. We also remind the notation introduced in Remark 3.7.Due to Proposition 3.10, let F be the SSCF on V of cardinal type, say k , such that | D ( a, P ) | ≥ k ⇒ φ ( P ) = a, and | D ( b, P ) | ≥ n − k + 1 ⇒ φ ( P ) = b. Set: I = Ø, ℓ = | I | , m = n − ℓ − k + 1, P = P ( ℓ , k ).Clearly on P we have that φ = ψ h ℓ ,k i ,x = ψ , hence, if the set of profiles { P / ∈ P : φ ( P ) = x } is empty, the theorem is proved, being h ℓ , k i the desired sequence. So, letus suppose it is nonempty and let P be one of its members with I ( P ) =: I of smallestcardinality, that we denote by ℓ . By definition 1 ≤ ℓ < n .Applying Proposition 3.10 to the scf defined for the society I c as follows: P I c −→ φ ([ ∼ I , P I c ]) , we determine a SSCF F on I c of cardinal type (with respect to I c ), say k , such that,with m = n − ℓ − k + 1,(+) | D ( a, P ) ∩ I c | ≥ k ⇒ φ ([ ∼ I , P I c ]) = a, and | D ( b, P ) ∩ I c | ≥ m ⇒ φ ([ ∼ I , P I c ]) = b. Set P = P ( ℓ , k ).Notice that if for a profile P we have I ( P ) = I , necessarily it is true that P ∈ P .Since P ∈ P \ P , we must have | D ( a, P ) | < k and | D ( b, P ) | < m .To fix ideas, let us assume, throughout the sequel, that x = b . The argument for the case x = a is the same, except for reversing the role of the the sequences k and m .Since φ ( P ) = a we have that necessarily | D ( a, P ) | ≥ k and therefore k < k . Let Q be a profile identical to P on I , identical to P on a subset of D ( a, P ) withcardinality k , and reporting b as the top choice of the remaining m − epresentation of anonymous... Anonymous, non manipulable is extended quota majority For the new profile Q we have φ ( Q ) = a , and also Q ∈ P \ P . The latter implies that m ≤ m . CLAIM: on P ∪ P the scfs φ and the scf ψ h ℓ , k i ,x (= ψ ), where ℓ = ( ℓ , ℓ ) and k = ( k , k ),coincide.Indeed, let us consider a profile P ∈ P \ P . We have to show that P ∈ P a ( ℓ , k ) ⇒ φ ( P ) = a and P ∈ P b ( ℓ , k ) ⇒ φ ( P ) = b. We show the first implication only. Applying Proposition 3.4 we can find a permutation σ of V such that P ∈ P a ( σ ( I ) , σ ( F )), namely such that the profile ( P ◦ σ ) belongs to P a ( I , F ). By Definition 3.3 2., (+) and strategy-proofness, we have that φ ( P ◦ σ ) = a .By anonymity we get φ ( P ) = a .Now it is clear that if the set of profiles { P / ∈ ( P ∪ P ) : φ ( P ) = x } is empty, the theoremis proved, being ℓ = ( ℓ , ℓ ) and k = ( k , k ) the desired sequences.If the set { P / ∈ ( P ∪ P ) : φ ( P ) = x } is nonempty, we shall apply the Lemma 3.12 below,which is really the induction step.We apply repeatedly Lemma 3.12 till we stop, that is,when { P / ∈ ( P ∪ · · · ∪ P r ) : φ ( P ) = x } is empty reaching the desired representation. ✷ Lemma 3.12
Let φ be a scf which is onto, anonymous, non-manipulable, and assigns x ∈ { a, b } to the profile in which all agents are indifferent. Let the sequences ℓ λ , k λ and m λ (for λ ∈ { , , . . . , r − } ) be such that ≤ ℓ λ < n, ≤ k λ ≤ n − ℓ λ , m λ = n − ℓ λ − k λ + 1 . Assume further that ℓ = 0 , ℓ λ is strictly increasing, andA1 for every ≤ λ ≤ r − , there is a profile P λ ∈ P λ \ ( P ∪· · ·∪ P λ − ) with: φ ( P λ ) = x ; I λ := I ( P λ ) ; ℓ λ = | I λ | = min {| I ( P ) | : P / ∈ ( P ∪ · · · ∪ P λ − ) , φ ( P ) = x } ,A2 in case x = b (respectively, x = a ), k λ is strictly decreasing, m λ is decreasing (respec-tively, k λ is decreasing, m λ is strictly decreasing)A3 φ = ψ r − on P ∪ · · · ∪ P r − A4 { P / ∈ ( P ∪ · · · ∪ P r − ) : φ ( P ) = x } is nonempty.Then we can find ℓ r > ℓ r − , k r < k r − , m r ≤ m r − (in case x = b , otherwise for x = a we shall have k r ≤ k r − , m r < m r − ), and a profile P r ∈ P r \ ( P ∪ · · · ∪ P r − ) with φ ( P r ) = x , such that | I ( P r ) | = ℓ r = min {| I ( P ) | : P / ∈ ( P ∪ · · · ∪ P r − ) , φ ( P ) = x } , and φ = ψ r on P ∪ · · · ∪ P r . If we assume that Q ∈ P , then ψ ( Q ) = a . Hence necessarily Q ∈ P a ( ℓ , k ). This leads to k ≥ k wich is false.Having seen that Q / ∈ P , also Q / ∈ P b ( ℓ , k ), namely we have | D ( b, Q ) | < m hence m − ≤ m − epresentation of anonymous... Anonymous, non manipulable is extended quota majority proof: Notice that whenever P is a profile with I ( P ) = I λ , we necessarily have P ∈ P λ .Now, using A4 , let P r be a profile in { P / ∈ ( P ∪ · · · ∪ P r − ) : φ ( P ) = x } with I ( P r ) =: I r of smallest cardinality, that we denote by ℓ r . By definition ℓ r − ≤ ℓ r < n .CLAIM 1: ℓ r − < ℓ r . Suppose the contrary, i.e. that ℓ r − = ℓ r . Let σ be a permutation of V that maps I r − onto I r . Since for the profile P r ◦ σ we have I ( P r ◦ σ ) = I r − , certainly we also have that P r ◦ σ ∈ P ( ℓ r − , k r − ). Hence P r ∈ P ( ℓ r − , k r − ), against the definition of P r . ✷ Applying Proposition 3.10 to the scf defined for the society I cr as follows: P I cr −→ φ ([ ∼ I r , P I cr ]) , we determine a SSCF F r on I cr of cardinal type k r (with respect to I cr ) such that, with m r = n − ℓ r − k r + 1,(+) | D ( a, P ) ∩ I cr | ≥ k r ⇒ φ ([ ∼ I r , P I cr ]) = a, and | D ( b, P ) ∩ I cr | ≥ m r ⇒ φ ([ ∼ I r , P I cr ]) = b. Set P r = P ( ℓ r , k r ).Notice that whenever we have I ( P ) = I r , necessarily it is true that P ∈ P r . Hence P r ∈ P r \ ( P ∪ · · · ∪ P r − ).Claims 2 and 3 below, refers to x = b . For the case x = a just reverse the role of k ’s and m ’s first proving m r < m r − , then k r ≤ k r − .CLAIM 2: k r < k r − .Let us suppose on the contrary that k r ≥ k r − . Since φ ( P r ) = a , we have necessarily that | D ( a, P r ) | ≥ k r . Hence we have | D ( a, P r ) | ≥ k r − . Showing also that | D ( b, P r ) | < m r − ,we have P r ∈ P r − which is a contradiction. For our purpose it is enough to use CLAIM1. Indeed we can write: | D ( b, P r ) | ≤ n − ℓ r − k r < n − ℓ r − − k r − < m r − . ✷ CLAIM 3: m r ≤ m r − .Again by contradiction assume that m r > m r − .Let Q r be a profile identical to P r on I r , identical to P r on a subset of D ( a, P r ) withcardinality exactly k r , and reporting b as the top choice of the remaining m r − Q r we have φ ( Q r ) = a .Now, since m r − ≥ m r − , we have | D ( b, Q r ) | = m r − ≥ m r − . But also, | D ( a, Q r ) | = k r If in Theorem 3.11 we had that the sequence ℓ + k is strictly increasing, wewould have obtained the proper representation directly by means of Proposition 3.8. Butin general, it is possible that ℓ + k is not strictly increasing. A relevant example is the scf φ of Example 2.13.According to Proposition 3.8, from the procedure of the proof of Theorem 3.11, we get forthe mentioned φ the sequence (5 , , , , , , , , , where ℓ + k is constantly 5. We shallshow this below. Computing ℓ and k : ℓ = 0 and k = 5 are easily obtained by the restriction to strictprofiles. It is obvious that { P / ∈ P : φ ( P ) = b } is nonempty and consists of the profiles P for which ≤ | D ( a, P ) | < | D ( b, P ) | < . Therefore ℓ = 1 .Considering the restriction φ ( ∼ , P { ,..., } ) on the society { , . . . , } , we get that k = 4 ,and consequently m = 7 .Hence, { P / ∈ P ∪ P : φ ( P ) = b } consists of the profiles P for which ≤ | D ( a, P ) | < | D ( b, P ) | < . Therefore ℓ = 2 .Considering the restriction φ ( ∼ { , } P { ,..., } ) on the society { , . . . , } , we get that k = 3 ,and m = 7 .Hence, { P / ∈ P ∪ P ∪ P : φ ( P ) = b } consists of the profiles P for which ≤ | D ( a, P ) | < | D ( b, P ) | < . We conclude by determining ℓ = 3 , k = 2 , m = 7 . ✷ The proper representation of φ is given by (5,2,12). In [5], Lahiri and Pramanik show that the anonymous, onto, non-manipulable scfs are (alland only) quota rules either with indifference default a , denoted by f r, x a , or with indifference epresentation of anonymous... Comparison default b , denoted by f r, y b .In the above notation Lahiri and Pramanik assume that r ∈ { , , . . . , n } , x , y are r -dimensional vectors such that • x i ≤ x i +1 ≤ x i + 1, for all i ∈ { , . . . , r − }• x ∈ { } × { , } × · · · × { , . . . r }• y i ≤ y i +1 ≤ y i + 1, for all i ∈ { , . . . , r − }• y ∈ { ( n − r ) + 1 } × { ( n − r ) + 1 , ( n − r ) + 2 } × · · · × { ( n − r ) + 1 , ( n − r ) + 2 , . . . n } .The quota rules can be described as in the table below (compare with [5, Definitions 10and 11]) Cardinality of I ( P ) Collective choice f r, x a ( P ) Collective choice f r, y b ( P ) r , or more a br − a, if | D ( a, P ) | ≥ x , otherwise b b, if | D ( b, P ) | ≥ y ′ , otherwise ar − a, if | D ( a, P ) | ≥ x , otherwise b b, if | D ( a, P ) | ≥ y ′ , otherwise ar − a, if | D ( a, P ) | ≥ x , otherwise b b, if | D ( a, P ) | ≥ y ′ , otherwise a ... ... ... r − i a, if | D ( a, P ) | ≥ x i , otherwise b b, if | D ( a, P ) | ≥ y ′ i , otherwise a ... ... ...1 a, if | D ( a, P ) | ≥ x r − , otherwise b b, if | D ( a, P ) | ≥ y ′ r − , otherwise a a, if | D ( a, P ) | ≥ x r , otherwise b b, if | D ( a, P ) | ≥ y ′ r , otherwise a where, for 1 ≤ i ≤ r , we have set, for symmetry, y ′ i = ( n − r + 1) − y i + i ∈ { i, i − , . . . , } . Notice that, similarly to x , for y ′ we have • y ′ ∈ { } × { , } × · · · × { , . . . r } (and y ′ i ≤ y ′ i +1 ≤ y ′ i + 1, for all i ∈ { , . . . , r − } ).Also observe that x r corresponds to k in our representation.The difference between Lahiri and Pramanik representation and ours is evident. Moreoverthe final example shows that our representation theorem is simpler, involving a smallernumber of parameters.Given a scf φ , anonymous and non-manipulable, after determining the indifference quota r that indisputably gives rise to the default collective choice, Lahiri and Pramanik haveto discuss all cases r > | ( I ( P ) | ≥ 0, by means of the further parameters x [ r −| I ( P ) | ] in casedefault is a ( y ′ [ r −| I ( P ) | ] , for default b ).Using an approach different from that of Lahiri and Pramanik [5], we have provided analgorithm that produces a unique, up and down, sequence of majority quotas, that appliedin the given order gives back the scf φ . 19 epresentation of anonymous... References Example 2.13 continued. For the scf φ of this example, we know that our properrepresentation formula gives φ = φ (5 , , . Let us determine the representation of φ as φ = f r, y b , according to Lahiri and Pramanik [5]. We need to determine r and the vector y .It is clear that r = 10. Hence we have that y has dimension 10, and, for i ∈ { , . . . , } ,we have y i = i + 2 − y ′ i where( y ′ , y ′ , . . . , y ′ ) = (1 , , , , , , 7) and ( y ′ , y ′ , y ′ ) = (7 , , . Finally y = (2 , , , , , , , , , . Notice that other choices for ( y ′ , y ′ , y ′ ) are possible, so that the Lahiri and Pramanikrepresentation is not unique, contrary to ours. References [1] Barber`a, S., Strategy-Proof Social Choice, in K. J. Arrow, A. K. Sen and K. Suzumura (Eds.)Handbook of Social Choice and Welfare. Volume 2. Netherlands: North-Holland, chapter 25,731-831, 2011. ISSN: 1574-0110.[2] Barber`a, S., Berga, D. and Moreno, B., Group strategy-proof social choice functions with bi-nary ranges and arbitrary domains: characterization results, Int. J. Game Theory, 41(2012),791-808.[3] Basile, A., Rao, S., and Bhaskara Rao, K.P.S., Binary strategy-proof social choice functionswith indifference, Economic Theory, (2020), https://doi.org/10.1007/s00199-020-01273-1.[4] Gibbard, A., Manipulation of voting schemes: a general result, Econometrica, 41(1973),587-601.[5] Lahiri, A. and Pramanik, A., On Strategy-proof Social Choice between Two Alternatives,Soc Choice Welf (2019). https://doi.org/10.1007/s00355-019-01220-7[6] Larsson, B. and Svensson, L.G., Strategy-proof voting on the full preference domain, Math.Soc. Sci., 52(2006), 272-287.[7] Moulin, H., The Strategy of Social Choice, North-Holland Publishing Company, 1983, Am-sterdam.[8] Satterthwaite, M.A., Strategy-proofness and Arrow’s conditions: existence and correspon-dence theorems for voting procedures and social welfare functions, J. Econ. Theory, 10(1975),187-217.[1] Barber`a, S., Strategy-Proof Social Choice, in K. J. Arrow, A. K. Sen and K. Suzumura (Eds.)Handbook of Social Choice and Welfare. Volume 2. Netherlands: North-Holland, chapter 25,731-831, 2011. ISSN: 1574-0110.[2] Barber`a, S., Berga, D. and Moreno, B., Group strategy-proof social choice functions with bi-nary ranges and arbitrary domains: characterization results, Int. J. Game Theory, 41(2012),791-808.[3] Basile, A., Rao, S., and Bhaskara Rao, K.P.S., Binary strategy-proof social choice functionswith indifference, Economic Theory, (2020), https://doi.org/10.1007/s00199-020-01273-1.[4] Gibbard, A., Manipulation of voting schemes: a general result, Econometrica, 41(1973),587-601.[5] Lahiri, A. and Pramanik, A., On Strategy-proof Social Choice between Two Alternatives,Soc Choice Welf (2019). https://doi.org/10.1007/s00355-019-01220-7[6] Larsson, B. and Svensson, L.G., Strategy-proof voting on the full preference domain, Math.Soc. Sci., 52(2006), 272-287.[7] Moulin, H., The Strategy of Social Choice, North-Holland Publishing Company, 1983, Am-sterdam.[8] Satterthwaite, M.A., Strategy-proofness and Arrow’s conditions: existence and correspon-dence theorems for voting procedures and social welfare functions, J. Econ. Theory, 10(1975),187-217.