New Characterizations of Strategy-Proofness under Single-Peakedness
Andrew Jennings, Rida Laraki, Clemens Puppe, Estelle Varloot
aa r X i v : . [ c s . G T ] F e b New Characterizations of Strategy-Proofnessunder Single-Peakedness
Andrew B Jennings , Rida Laraki , , Clemens Puppe , and Estelle Varloot Arizona State University University of Liverpool Universit´e Paris Dauphine-PSL Karlsruhe Institute of Technology
Abstract.
We provide novel simple representations of strategy-proofvoting rules when voters have uni-dimensional single-peaked preferences(as well as multi-dimensional separable preferences). The analysis recov-ers, links and unifies existing results in the literature such as Moulin’sclassic characterization in terms of phantom voters and Barber`a, Gul andStacchetti’s in terms of winning coalitions (“generalized median voterschemes”). First, we compare the computational properties of the var-ious representations and show that the grading curve representation issuperior in terms of computational complexity. Moreover, the new ap-proach allows us to obtain new characterizations when strategy-proofnessis combined with other desirable properties such as anonymity, respon-siveness, ordinality, participation, consistency, or proportionality. In theanonymous case, two methods are single out: the –well know– ordinalmedian and the –most recent– linear median.
In mechanism design, strategy-proofness (SP) is a desirable property. It impliesthat whatever agents’ beliefs are about others’ behavior, their best strategyis to sincerely submit their privately-known types, even when their beliefs arewrong or mutually inconsistent. Consequently, strategy-proofness guarantees tothe designer that she has implemented the intended choice function, i.e. that thefinal decision is indeed linked in the intended way to the agents’ true types. Ofcourse, depending on the context there are other desirable properties that onewould like to satisfy, such as unanimity, voter sovereignty, efficiency, anonymity,neutrality, consistency, participation (e.g. the absence of the no-show paradox),or fairness properties such as proportionality.When side payments are possible and utilities are quasi-linear, anonymousand efficient strategy-proof mechanisms can be designed (the well-known Vickrey-Clarke-Groves mechanisms). By contrast, in contexts of ‘pure’ social choice (‘vot-ing’), the Gibbard-Satterthwaite [16,7] Theorem shows that only dictatorial rulescan be sovereign (‘onto’) and strategy-proof on an unrestricted domain of pref-erences. In particular, no onto voting rule can be anonymous and strategy-proofwithout restrictions on individual preferences.o overcome the impossibility, several domain restrictions have been investi-gated. One of the most popular is one-dimensional single-peakedness. Under thisdomain restriction, the path-breaking paper by Moulin [10] showed that thereis a large class of onto, anonymous and strategy-proof rules. All of them canbe derived by simply adding some fixed ballots (called ‘phantom’ votes) to theagents’ ballots and electing the median alternative of the total. Moulin’s paperinspired a large literature that obtained related characterizations for other par-ticular domains or proved impossibility results (see, among many others, Borderand Jordan [4] and Barber`a, Gul and Stacchetti [3], Nehring and Puppe [14]and, recently, Freeman, Pennock, Peters and Wortman-Vaughan [6]).
Our Contribution
In contrast to Moulin’s elegant and simple phantom voter characterization inthe anonymous case, the more general characterizations in terms of winningcoalitions (called ‘generalized median voter schemes’ in [3], and ‘voting by issues’in [14]), as well as Moulin’s own ‘inf-sup’ characterization in the appendix ofhis classic paper are complex. One of the basic objectives of our paper is toprovide simpler alternative representations in the one-dimensional single-peakedsetting that apply also to the non-anonymous case. Non-anonymous methods areimportant in practice because some voters may represent more people or someexperts (because of their past performances) may be more qualified than others.In the present paper, we provide two new (and equivalent) alternative char-acterizations of strategy-proof voting rules in the non-anonymous case. Our firstrepresentation is a natural extension of Moulin’s idea: the selected outcome isthe median of voters’ peaks after the addition of new peaks computed from suit-able phantom functions . The second representation has a compact functionalform which we refer to as grading curve in the anonymous case. We comparethe computational properties of the various representations and show that thegrading curve representation is superior in terms of computational complexity.The new representations have additional merits. They can be used to con-nect the phantom characterization with the ‘voting by issues’ one, and to providenew characterizations of and insights into special cases. For instance, we showthat the case of uniformly distributed phantom voters (cf. Caragiannis, Procac-cia, and Shah [5]) corresponds to a linear grading curve which in turn is fullycharacterized by a particular “compromising” axiom: proportionality.
Further Related Literature
A link between Moulin’s [10] inf-sup characterization and his phantom votercharacterization in the anonymous case has been provided by Weymark [18] whoshowed how to derive the phantom voter representation from the inf-sup one.In the present paper we show how both results can be derived from our general Curiously, Moulin provided completely different proofs for his two characterizations. . Section7 concludes. Appendix contains additional results (e.g. a multi-dimensional sep-arable preferences extension) and some missing proofs. The voting problem we are considering can be described by the following el-ements. First, an ordered set of alternatives Λ . For example candidates on aleft-right spectrum, a set of grades or a set of locations on the line. Second afinite set of voters N = { , . . . , n } . A typical element of r ∈ Λ N is called a vot-ing profile. Finally, a voting rule φ is a function that associates to each votingprofile in Λ N an element in Λ .This is interpreted as follows: each voter i ∈ N has a single peaked preferenceover the linearly ordered set Λ (see the definition 1 of ”single peaked” below).He submits his peak (or a strategically chosen ballot) r i ∈ Λ to the designer whothen implements φ ( r , .., r n ) = φ ( r ) and returns the chosen alternative.Without loss of generality we will assume Λ ⊆ R and use the notations m := inf Λ , M := sup Λ and ˜ Λ = Λ ∪ { m, M } . In Moulin’s [10] paper, Λ = R , m = −∞ and M = + ∞ . Barber`a, Gul and Stacchetti [3] assume Λ is finite. Definition 1 (Single peaked preference).
The preference order of voter i over the alternatives Λ is single peaked if there is a unique alternative x ∈ Λ such that for any y, z ∈ Λ if z is between x and y then voter i prefers x to y and To our knowledge, consistency and participation are rarely been studied in the non-anonymous case. to z . The alternative x is the peak of the preference order. It is voter i favoritealternative.Remark 1. In terms of utility function this means that the utility is increasingfrom m to the peak and then decreasing from the peak to M .We wish the voting rule to satisfy some desirable properties. The main focusof this paper is strategy proofness (SP). Sections 5 and 6 will explore variouscombinations with other axioms. Axiom 1 (Strategy-Proofness: SP) . A voting rule ϕ is strategy-proof if for everyvoting profile r and voter i ∈ N , if s differs from r only in dimension i then: ϕ ( s ) ≥ ϕ ( r ) ≥ r i or ϕ ( s ) ≤ ϕ ( r ) ≤ r i . Here are some important consequences of strategy proofness.
Lemma 1 (Group Strategy-Proofness).
If a voting rule is strategy proofthen no coalition can manipulate to obtain a better outcome for all its members.Proof.
Denote by r the honest voting profile and let S be a coalition of voterswhose strategic manipulation leads to the profile s . If for all i ∈ S , r i ≥ ϕ ( r )then by SP, going from r to s one dimension at a time, imply that ϕ ( s ) ≤ ϕ ( r ):no voter in S benefits from the manipulation. Similarly, if for all i ∈ S , r i ≤ ϕ ( r )then ϕ ( s ) ≥ ϕ ( r ). Finally, if at least two voters in S have their peaks atdifferent sides of ϕ ( r ) and if ϕ ( s ) = ϕ ( r ), then one of the two voters has autility decrease. Consequently, no group of voters S can change the outcome inthe benefice of all its members. Definition 2.
A voting rule ϕ : Λ N → Λ is responsive if for all voters i , andfor all r and s that only differ in dimension i , if r i < s i then ϕ ( r ) ≤ ϕ ( s ) . Responsiveness is sometimes called weak monotonicity.
Lemma 2 (Responsiveness).
If a voting rule ϕ : Λ N → Λ is strategy proofthen it is responsive.Proof. Let ϕ be a voting rule that is SP. If r and s only differ in i with r i < s i . – If r i < ϕ ( r ) then by strategy-proofness ϕ ( r ) ≤ ϕ ( s ). – If s i > ϕ ( s ) then by strategy-proofness ϕ ( r ) ≤ ϕ ( s ). – Else ϕ ( r ) ≤ r i < s i ≤ ϕ ( s ) 4 emma 3 (Continuity). If a voting rule ϕ : Λ N → Λ is strategy proof then itis Lipschitz continuous.Proof. Suppose that there is r and s that differ only in dimension i such that r i < s i and | ϕ ( s ) − ϕ ( r ) | > s i − r i .If s i < ϕ ( s ) (resp. r i > ϕ ( r )) then by strategy-proofness and responsiveness ϕ ( s ) = ϕ ( r ). This contradicts | ϕ ( s ) − ϕ ( r ) | > s i − r i . Therefore s i ≥ ϕ ( s )(resp. r i ≤ ϕ ( r )). It follows that r i ≤ ϕ ( r ) ≤ ϕ ( s ) ≤ s i . This contradicts | ϕ ( s ) − ϕ ( r ) | > s i − r i .Therefore we have shown that | ϕ ( s ) − ϕ ( r ) | ≤ s i − r i .Now let us use this property to show that ϕ is Lipschitz continuous.Now for any r and s such that | r j − s j | ≤ ǫn for all i . Then : | ϕ ( r ) − ϕ ( s ) | ≤ X i | ϕ ( r , . . . , r i , s i +1 , . . . , s n ) − ϕ ( r , . . . , r i − , s i , . . . , s n ) |≤ X i | r i − s i |≤ ǫ Lemma 4 (Continuous Extension).
If a voting rule ϕ : Λ N → Λ is strategyproof then it has a continuous extension in ˜ Λ N → ˜ Λ .Proof. We prove the result by induction with respect to the set of players.Suppose that for a given i we have shown that we can extend ϕ : Λ N → Λ to˜ Λ i − × Λ n − i +1 → ˜ Λ . We will now show that we can extend ˜ Λ i − × Λ n − i +1 → ˜ Λ to ˜ Λ i × Λ n − i → ˜ Λ .In the following r and s have fixed values in all dimensions except for i andonly differ in dimension i and r i > m . We seek to extend ϕ to s i = m (the prooffor s i = M is symmetrical). – If r i < ϕ ( r ) for a r i small enough then by strategy-proofness:lim s i → m ϕ ( s ) = ϕ ( r )Therefore we can extend by continuity ϕ ( s ) to s i = m . – If there exist x ∈ Λ such that r i = ϕ ( r ) for all r i < x then by continuity ϕ ( s ) = m when s = m . – If ϕ ( r ) = m then ϕ ( s ) = m when s = m .By induction we therefore obtain a continuous extension of ϕ .It is therefore natural to ask ourselves what are the SP voting rule in ˜ Λ N → ˜ Λ ,that are not continuous extensions of voting rules in Λ N → Λ .5 emma 5. A SP voting rule in ˜ Λ N → ˜ Λ is not an extension of a voting rule in Λ N → Λ iff it is identically constant whose value is not in Λ .Proof. ⇒ : Suppose that ϕ : ˜ Λ N → ˜ Λ is not an extension of a function from Λ N → Λ . Therefore there is a voting profile r ∈ Λ N such that ϕ ( r ) Λ . Let ϕ ( r ) = m (resp. M ). By strategy-proofness for all s ∈ ˜ Λ N , ϕ ( s ) = m (resp. M ). Therefore ϕ is a constant (equals to m or to M ) and its value is not in Λ . ⇐ : Immediate.A such from now on we will consider without loss of generality that ˜ Λ = Λ . In this section we will introduce the concept of phantom functions and establishan intuitive characterization of strategy-proof voting rule. We show later that itwill imply all the other representations.Let Γ := { m, M } N be the set of voting profiles where voters have extremepositions. For X ∈ Γ , we denote m ( X ) (resp. M ( X )) the set of voters i ∈ N such that X i = m (resp. X i = M ). Definition 3 (Phantom function).
A function α : Γ → Λ is a phantomfunction if α is weakly increasing ( X ≤ Y = ⇒ α ( X ) ≤ α ( Y ) ). We will use theshorthand α X := α ( X ) . Intuitively, the phantom functions provide the outcome of the election whenall voters vote at the extremes. It is immediate that there is a unique phantomfunction α for each strategy proof voting rule ϕ : ∀ X ∈ Γ, α ( X ) := ϕ ( X ) . (1)The phantom function is necessarily weakly increasing because strategy-proofvoting rules are responsive. Conversely, the next lemma proves that each phan-tom function is associated with a unique strategy-proof voting rule. Theorem 1 (Phantom Identification).
The voting function ϕ is strategy-proof iff there exists a phantom function α : Γ → Λ such that: ∀ r ; ϕ ( r ) := α X if ∃ X s.t. M ( X ) ⊆ { j | α X ≤ r j } ∧ m ( X ) ⊆ { j | α X ≥ r j } r i if ∃ X, Y s.t. α X ≤ r i ≤ α Y and M ( X ) = { j | r i < r j } ∧ m ( Y ) = { j | r i > r j } (2) Proof.
See the appendix A where it is proved in particular that ϕ is well defined(e.g. its value is independent on the choice of X or Y ).6he intuition behind this identification is simple. By strategy-proofness,when a voter’s ballot is smaller than the final outcome then, it can be replacedby the minimal ballot m without changing the final outcome. Symmetrically, ifit is larger than the final outcome, it can be replaced by M without changingthe outcome. As such, if the outcome is not one of the ballots, then it must be α X where M ( X ) is the set of voters whose ballots are greater than the outcome.On the other hand, if the outcome is one of the ballots then by responsiveness itis in between an α X and α Y where M ( X ) is the set of voters whose ballots arehigher than the outcome and m ( Y ) is the set of voters whose ballots are smallerthan the outcome. Remark 2.
The characterization of ϕ from α in lemma 1 can be used to find ϕ with an algorithm of complexity O ( n + 2 n f ( n )) where f ( n ) is the complexityof α for a given X (Algorithm 1 in the appendix). Proof.
See the appendix A.
In his appendix, Moulin [10] proved the following inf-sup characterization ofstrategy-proof voting rules without anonymity, or voter sovereignty.
Theorem 2 (Moulin Inf-Sup Characterization).
A voting rule ϕ is strategy-proof iff for each subset S ⊆ N (including the empty set), there is a value β S ∈ Λ such that: ∀ r , ϕ ( r ) = sup S ⊆ N min ( β S , inf i ∈ S { r i } ) . Remark 3.
Moulin observed in the proof that without loss, S → β S can be takenweakly increasing.Moulin’s result can easily be derived from phantom identification theorem 1. Corollary 1 (Inf-Sup with Phantom functions).
A function ϕ is strategy-proof iff there exists a phantom function α (the same as the one in lemma 1)that verifies: ∀ r , ϕ ( r ) = sup X ∈ Γ min ( α X , inf i ∈ M ( X ) { r i } ) . (3) .Proof. Let ϕ be a strategy-proof voting rule defined by a phantom function α .Let µ : Λ N → Λ be defined as ∀ r , µ ( r ) = sup X ∈ Γ min ( α X , inf i ∈ M ( X ) { r i } ) . . Since ϕ is the unique strategy-proof voting rule defined by α we only needto prove that ϕ = µ . 7 If µ ( r ) = α X then for any Y ∈ Γ, min ( α Y , inf j ∈ M ( Y ) { r j } ) ≤ α X in particu-lar for X we have { j | r j > α X } ⊆ M ( X ) ⊆ { r j ≥ α X } . Therefore ϕ ( r ) = α X – If µ ( r ) = r i then for any X ∈ Γ, min ( α X , inf j ∈ M ( X ) { r j } ) ≤ r i . Thereforefor X such that M ( X ) = { j | r i < r j } we have α X ≤ r i .For Y such that M ( Y ) = { j | r i ≤ r j } . Suppose that α Y < r i then since α isincreasing for all Z ∈ Γ either there is a j ∈ Z such that r j < r i or α Z < r i .This contradicts µ ( r ) = r i . Therefore α Y ≥ r i . As such ϕ ( r ) = r i . Therefore ϕ = µ . The most popular of Moulin’s representation (Theorem 3) assumes anonymity.
Axiom 2 (Anonymity) . A voting rule φ is anonymous if for any permutation σ and for all voting profile r : φ ( r σ (1) , . . . , r σ ( n ) ) = φ ( r , . . . , r n ) . Anonymity states that all voters must be treated equally. This is natural inseveral voting situations, but sometimes it is not a suitable model. For exampleshareholders should be weighted according to their rights and experts must beweighted in accordance with the accuracy of their predictions.
Theorem 3 (Moulin’s Phantom-Characterization: Anonymous Case).
A function φ is strategy-proof and anonymous iff there is a set of n + 1 phantomvoters α i chosen arbitrarily in Λ such that : ∀ r ; φ ( r ) = med ( r , . . . , r n , α , . . . , α n +1 ) . where med denotes the median operator. The two characterizations of Moulin stated above look quite different. Theirproofs and the statements are separated in his article. In order to link the twowe need to be able to choose n + 1 phantoms voters among the 2 N outputs ofour phantom function. In order to do this we will use the phantom functions andthe following θ function. Definition 4 (The θ function ). θ : R N × R → Γθ : r , x → X = θ ( r , x ) Such that ∀ i ; X i = m ⇔ r i < x. It is well defined as we have an odd number of input values. Observe that the definition of θ is asymmetric because it sends values strictly be-low the cut-off to m and greater than or equal to M . That’s why our followingcharacterizations “look” asymmetric. They are not. heorem 4 (Phantom Characterization: General Case). A function ϕ isstrategy-proof iff there exists a phantom function α (the same as in theorem 1)that satisfies: ∀ r ; ϕ ( r ) := med ( r , . . . , r n , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) ) . (4)As shown in section 5, anonymity implies that the phantom function α ( X )depends only on { i | X i = m } which, combined with Theorem 4 allows deducingdirectly Moulin’s characterization in the anonymous case. Proof.
This theorem is a direct consequence of the phantom identification the-orem 1.Let ϕ be a strategy-proof voting rule defined by a phantom function α . Let µ : Λ N → Λ be defined as ∀ r , µ ( r ) := med ( r , . . . , r n , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) ) . Since ϕ is the unique strategy-proof voting rule defined by α we only needto prove that ϕ = µ .If ϕ ( r ) = α X then there are M ( X ) elements of { r j } (resp. of { α θ ( r ,r j ) } )that are greater (resp. lesser) or equal to α X and m ( X ) elements of { r j } (resp.of { α θ ( r ,r j ) } ) that are lesser (resp. greater) to α X .Similarly, if ϕ ( r ) = r i , then there are M ( θ ( r , r i )) elements of { r j } (resp. of { α θ ( r ,r i ) } ) that are greater (resp. lesser) or equal to r i and m ( { α θ ( r ,r i + ǫ ) } )elements of { r j } (resp. of { α θ ( r ,r j ) } ) that are lesser (resp. greater) to α X .Therefore ϕ = µ . Remark 4.
This novel characterization can be used to compute ϕ ( r ) with thecomplexity O ( nlog ( n )+ nf ( n )) (Algorithm 2 in the appendix). This is better thanthe complexity of the phantom lemma characterization ( O ( n + nf ( n ))), which isconsiderably better than the 2 n f ( n ) complexity of the inf-sup characterization. In an unpublished PhD thesis [9] Jennings showed that Moulin’s characterisationin the anonymous case is equivalent to the following characterization.
Theorem 5 (Functional Characterization: Anonymous Case).
A votingrule φ : [0 , R ] n → [0 , R ] is strategy-proof and anonymous iff there is an increasingfunction g : [0 , R ] → [0 , R ] with g ( y ) > for y > such that: ∀ r , φ ( r ) := sup (cid:26) y | { r i ≥ y } n ≥ g ( y ) (cid:27) . Weymark [18] also showed how to derive the phantom voter representation from theinf-sup characterization. g functions are called the grading curves. This characterization in theanonymous case can easily be derived (see section 5) from the following newrepresentation, also a direct consequence of the phantom identification 1. Theorem 6 (Functional Characterization: General Case).
A voting rule ϕ is strategy-proof iff there exists a phantom function α (the same as the one inthe phantom identification 1) such that: ∀ r ; ϕ ( r ) := sup (cid:8) y ∈ Λ | α θ ( r ,y ) ≥ y (cid:9) . Proof.
Let ϕ be a strategy-proof voting rule defined by a phantom function α .And let µ be defined as ∀ r ; µ ( r ) := sup (cid:8) y ∈ Λ | α θ ( r ,y ) ≥ y (cid:9) . Let us show that ϕ = µ .First notice that y → α θ ( r ,y ) is decreasing. – Case ϕ ( r ) = α X : Let x = α X , we therefore have x = α θ ( r ,x ) therefore µ ( r ) = α X . – Case ϕ ( r ) = r i : Then for any ǫ > α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) therefore α θ ( r ,r i + ǫ ) < r i + ǫ and r i ≤ α θ ( r ,r i ) . As such µ ( r ) = r i .Therefore µ = ϕ . Remark 5.
This compact characterization provides the Algorithm 3 that returns ϕ ( r ) with a complexity O ( log ( n )( n + f ( n ))). This improves upon the phantomcharacterization. See the appendix. We can also ask how the problem would be described as a vote by issue. A voteby issue consists of a property space ( Ω, H ) where Ω is a set of alternatives and H is a set of subsets of Ω called issues. Each voter i provides a ballot r i ∈ Ω .Each issue H ∈ H is then resolved separably as a binary election. If r i ∈ H thenwe say that voter i votes for issue H or that his ballot verifies issue H . And if r i H then we say that he voted against H . If an issue is elected then the resultof the vote will be one of the elements of the issue. As such, the result of theelection is the intersection of elements of H .A coalition is a subset of voters. A coalition W ⊆ N is said to be winningfor H ∈ H , if when all voters in W voted for H , H is elected. Given an issue H ,strategy-proofness also implies that if W is winning for H than any super set of W is also winning for H . Let W H be the set of winning coalitions for H . Theorem 7 (Barber`a, Gul and Stacchetti [3]).
Let Λ be finite. A votingrule ϕ : Λ N → Λ is strategy-proof and voter-sovereign iff it is voting by issuessatisfying, for all G, H, G ⊆ H ⇒ W G ⊆ W H . Λ finite, assuming voter-sovereignty. We provenext that it holds in general. Moreover, we explicitly define the winning coalitionsin terms of the phantom function. Theorem 8 (A New Voting by Issue Characterization).
A voting rule ϕ is strategy-proof iff it is a vote by issues on the property space ( Λ, H ) where H = S a ∈ Λ { x ∈ Λ : x ≤ a } ∪ { x ∈ Λ : x ≥ a } where for every a ∈ Λ : – M ( X ) is a winning coalition for H = { y ∈ Λ : y ≥ a } if and only if α X ≥ a ; – m ( X ) is a winning coalition for H = { y ∈ Λ : y ≤ a } if and only if α X ≤ a .The phantom function α is the one associated to ϕ (as in phantom identificationtheorem 1.)Proof. Let ϕ be a strategy-proof voting rule defined by a phantom function α .Let µ be the vote by issue given in the theorem.Let us show that ϕ = µ . – Case ϕ ( r ) = α X : For a = α X we have M ( X ) = { r j ≥ a } is a winningcoalition for { y ≥ a } and m ( X ) = { r j ≤ a } is a winning coalition for { y ≤ a } . Therefore µ ( r ) = a . – Case ϕ ( r ) = r i : Let XY ∈ Γ be the same voting profiles as found in thephantom identification 1. For a = r i we have α Y ≥ a therefore M ( Y ) = { r j ≥ a } is a winning coalition for { y ≥ a } and α X ≤ a therefore m ( X ) = { r j ≤ a } is a winning coalition for { y ≤ a } . Therefore µ ( r ) = a .We have shown that ϕ = µ. The voting by issue representation looks less practical than the other formu-lations but it has the great power of being extendable to non-separable multidi-mensional not necessarily separable single peaked domains (e.g. median spaces),see Nehring and Puppe [13,14].
The following table recalls the different characterizations given. For the sameinput and the same phantom function α they all return the same output. Thevalue f ( n ) is the time complexity of α when the input is of size n .11 haracterization Formula ComplexityPhantom Lemma ϕ ( r ) := α X if ∀ j, M ( X ) = { j | α X ≤ r j } ∧ m ( X ) = { j | α X ≥ r j } r i if α X ≤ r i ≤ α Y where M ( X ) = { j | r i < r j } ∧ m ( Y ) = { j | r i > r j } n f ( n )Moulin’s Inf-Sup ϕ ( r ) = sup X ∈ Γ min ( α X , inf i ∈ M ( X ) { r i } ) 2 n f ( n )Phantomcharacterization ϕ ( r ) := med ( r , . . . , r n , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) ) n ( log ( n ) + f ( n ))Functionalcharacterization ϕ ( r ) := sup { y | α θ ( r ,y ) ≥ y } log ( n )( n + f ( n ))Voting by issues ϕ ( r ) := a : { r i ≤ a } ∈ { M ( X ) | α X ≥ a } ∧ { r i ≥ a } ∈ { m ( X ) | α X ≤ a } n f ( n ) The phantom function and the different representations will be very helpful inunderstanding the effects of imposing more axioms on the voting rule. This isthe subject of this section.
It makes little sense in some applications to vote for an alternative that is nevera possible output. As such we may wish for the voting rule to be voter-sovereign.
Axiom 3 (Voter sovereignty) . A voting rule φ is voter sovereign if for all x ∈ Λ there is a preference profile r such that: φ ( r ) = x . Voter sovereignty is, mathematically speaking, surjectivity. It states that anyalternative that the voters can vote for must be a possible outcome of the mech-anism.
Theorem 9 (Characterizing voter sovereignty).
A strategy-proof votingrule ϕ : Λ N → Λ is voter-sovereign iff its phantom function α satisfies α m = m and α M = M . In that case, ϕ is unanimous ( ϕ ( a, . . . , a ) = a for all a ∈ Λ ).Proof. ⇒ If α m > m (resp. α M < M ) then by our characterizations (for exampleTheorem 4), there is no way to obtain m (resp. M ) as the output of φ . Thiscontradicts voter sovereignty. ⇐ Suppose that α m = m and α M = M . Then ϕ ( a, . . . , a ) = a for all a ∈ Λ (use for example the characterization in Theorem 4 to deduce it).12 efinition 5 (Pareto optimality). A voting rule is called Pareto optimal ifno different alternative could lead to improved satisfaction for some voter withoutloss for any other voter.
Corollary 2 (Pareto optimal).
If a voting rule ϕ : Λ N → Λ is voter-sovereignand strategy-proof then it is efficient (the selected alternative is Pareto optimal).Proof. If ϕ is strategy-proof and voter-sovereign then min { r j } = r k ≤ ϕ ( r ) ≤ r l = max { r j } (use the characterization of Theorem 4 to deduce it).As such for any voting profile s if ϕ ( s ) < ϕ ( r ) (resp. ϕ ( s ) > ϕ ( r )) thenvoter l (resp. k ) has a worse outcome in s then in r according to peak r l (resp. r k ). It follows that no voting profile can obtain a result that is strictly better forat least one voter without being worse for another. A voting rule ϕ : Λ N → Λ is strictlyresponsive if for any r and s such that for all j , r j < s j we have ϕ ( r ) < ϕ ( s ) . Strict responsiveness is sometimes called strict monotonicity.
Definition 7 (Ordinals).
A voting rule ϕ : Λ N → Λ is ordinal if for all strictlyresponsive and surjective functions π : Λ → Λ we have: ϕ ( π ( r ) , . . . , π ( r n )) = π ◦ ϕ ( r ) . Remark 6.
Observe that ordinality says nothing when Λ is finite as, in this case,the identity is the only strictly responsive and surjective function from Λ toitself. Theorem 10 (Characterizing ordinals and strict responsiveness).
Fora strategy-proof voting rule ϕ : Λ N → Λ the following are equivalent:1. The phantom function α verifies α ( Γ ) = { m, M } ϕ is strictly responsive.And when Λ is a rich , the two are equivalent to (3) ϕ is ordinal and notconstant.Proof. (2) → (1) : Suppose that ϕ is strategy-proof and strictly-responsive. Ifthere exists X ∈ Γ such that α X
6∈ { m, M } , then for r and s such that if j ∈ M ( X ) then r j = α X , s j = M and if j ∈ m ( X ) then r j = m, s j = α X . Wehave ϕ ( s ) = ϕ ( r ) which contradicts strict responsiveness. Therefore α ( Γ ) ⊆{ m, M } . If α ( Γ ) = 1 then the function is a constant and is therefore notstrictly responsive. Consequently, α ( Γ ) = { m, M } . Λ is called rich if for any α < β in Λ there exists a γ ∈ Λ such that α < γ < β . → (2) : Suppose that ϕ is strategy-proof and α ( Γ ) = { m, M } . For any ϕ ( r ) and ϕ ( s ) be such that for each i , r i < s i . It follows that there is i and j such that r i = ϕ ( r ) ≤ ϕ ( s ) = s j . Let t be defined as for all k if r k < r i then t k = r k else t k = s k . We have ϕ ( t ) ∈ { t k } therefore by responsiveness and since r i
6∈ { t k } we have r i < ϕ ( t ) ≤ s j (3) → (1) : Suppose that ϕ is strategy-proof and ordinal. If there exists X ∈ Γ such that α X
6∈ { m, M } , let r be such that there are two alternatives a < b < α X such that if i ∈ M ( X ) then r i = b else r i = a . Let p i be bijectiveand π ( a ) < α X < π ( b ). Then: α X = ϕ ( π ( r ) , . . . , π ( r n ))= π ◦ ϕ ( r )= π ( b ) > α X We have reached a contradiction, therefore α ( Γ ) = { m, M } . (1) → (3) : Suppose that ϕ is strategy-proof and α ( Γ ) = { m, M } . For anystrictly responsive and bijective π and for any voting profile r : ϕ ( π ( r ) , . . . , π ( r n )) = med ( π ( r ) , . . . , π ( r n ) , α m , α θ ( π ( r ) ,π ( r )) , . . . , α θ ( π ( r ) ,π ( r n )) )= med ( π ( r ) , . . . , π ( r n ) , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) )= med ( π ( r ) , . . . , π ( r n ) , π ( α m ) , π ( α θ ( r ,r ) ) , . . . , π ( α θ ( r ,r n ) ))= π ◦ med ( r , . . . , r n , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) )= π ◦ ϕ ( r )Therefore ϕ is ordinal.In the anonymous case, the ordinal / strictly monotonic strategy-proof votingrules are the order statistics (see [2], chapter 10 and chapter 11). A voter i is said to be dummy (or invisible)if for all r and for all s that only differs from r in dimension i we have, ϕ ( r ) = ϕ ( s ) . Theorem 11 (Characterizing dummy voters).
A voting rule ϕ is strategy-proof and voter i is dummy iff for all X and Y that only differ in i , α X = α Y Proof. ⇒ : Suppose that voter i is dummy then for r ∈ Γ and s ∈ Γ that onlydiffer in i : α X = ϕ ( X ) = ϕ ( Y ) = α Y . : Suppose that for all X and Y that only differ in dimension i , we have α X = α Y :Let r differ from s only in dimension i . For all y , θ ( r , y ) and θ ( s , y ) differat most in dimension i therefore: α θ ( r ,y ) = α θ ( s ,y ) .It follows that: ϕ ( r ) = sup (cid:8) y ∈ Λ | α θ ( r ,y ) ≥ y (cid:9) = sup (cid:8) y ∈ Λ | α θ ( s ,y ) ≥ y (cid:9) = ϕ ( s ) . As seen above anonymity states that all voters must be treated equally.
Theorem 12 (Characterizing Anonymity).
A strategy-proof voting rule ϕ : Λ N → Λ is anonymous iff its phantom function α is anonymous.Proof. ⇒ : It is immediate that if ϕ is anonymous then so is α. ⇐ : By the Median representation we have : ϕ ( r ) = med ( r , . . . , r n , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) ) . For any permutation σ : ϕ ( r σ (1) ) , . . . , r σ ( n ) ) = med ( r σ (1) , . . . , r σ ( n ) , α m , α θ ( r ,r σ (1) ) , . . . , α θ ( r ,r σ ( n ) ) )= med ( r , . . . , r n , α m , α θ ( r ,r ) , . . . , α θ ( r ,r n ) )= ϕ ( r ) . Corollary 3 (Grading curves dependent on electorate).
A strategy-proofvoting rule ϕ : Λ N → Λ is anonymous iff there exists an increasing function g n : [0 , → Λ such that: ∀ X ∈ Γ ; α ( X ) = g n (cid:18) M ( X ) n (cid:19) . Where n is the cardinality of N .Proof. ⇒ : Simply use an increasing g n such that for all k ∈ [ | , n | ] α i = g n ( in ). ⇐ : Suppose ∀ X, α X = g n (cid:16) M ( X ) n (cid:17) . It follows that if Card ( X ) = M ( X ′ )then α X = α X ′ .To get rid of the dependency of the grading curve g n on the number of voters n , we need to impose electorate consistency (Smith [17], Young [19]). This is theobjective of the next section. 15 Variable Electorate: Consistency and Participation
We wish to consider situations where casting a vote or not is a choice. As suchwe need to make a distinction between the electorate E (that is potentiallyinfinite) and the set of voters N ⊆ E (which will be assumed finite). We definea voter as someone who casts a ballot and an elector someone who can cast aballot. Similarly a ballot is cast by a voter while a vote is the response of anelector. A vote that is not a ballot is represented by the symbol ∅ (interpreted asabstention). As such we represent the set of votes as an element of Λ ∗ = { r ∈ ( Λ ∪ ∅ ) E : r i = ∅ < + ∞} .Since we seek for strategy-proof methods, it may also be of interest to ensurethat no elector can benefit from not casting a ballot (no no-show paradox orparticipation [12]). Another desirable property is consistency. It states that whenwe obtain the same outcome for the voting profiles of two disjoint group of voters(with fixed ballots) then that outcome is also the outcome of the union of theirvoting profiles. Definition 9 (Voting function).
A voting function ϕ ∗ : Λ ∗ → Λ is a functionsuch that for any finite set of electors N there is a voting rule ϕ N : Λ N → Λ such that ϕ N is the restriction of ϕ ∗ to the set of voters N . Intuitively once our set of voters is fixed then we are considering a votingrule and we can use our previous results. Furthermore in the general case thevoting rules are independent even if they use very similar (but not identical) setof voters. As such we start by determining what our set of voters is and thenwe use the corresponding voting rule. This is coherent with the non-anonymoussetting where the result of the election strongly depends on who casts a ballot.
Definition 10 (Redefining Properties).
A voting function ϕ ∗ verifies one ofthe previously mentioned properties (strategy-proofness, continuity, voter sovereignty,invisible voter, strict responsiveness, anonymity) if for all sets of voters N therestriction of ϕ ∗ to N verifies the property. As with voting rules, we would like to define a phantom function that com-pletely characterizes the strategy-proof voting functions. We will then be able tocharacterize participation (e.g. no no-show paradox) and consistency by usingthe phantom functions.Let Γ ∗ := { m, M, ∅} E be the set of voting profiles where voters have extremepositions. (Recall that the set of voters is always finite). As before we wish for aphantom α : Γ ∗ → Λ such that: ∀ X ∈ Γ ∗ , α X =: ϕ ∗ ( X )The bijection over each set of voters gives us the bijection between a strategy-proof voting function and its associated phantom function. The definition ofa phantom function therefore corresponds to all functions that are phantomfunctions for each voting rule corresponding to a restriction of the voting functionto a fixed set of voters. 16 efinition 11 (Phantom function). A function α : Γ ∗ → ˜ Λ is a phantomfunction if α verifies for any X and Y that differ only in dimension i with X i = m and Y i = M we have α X ≤ α Y . Definition 12 (The θ function). θ :( R ∪ {∅} ) E × R → { m, M, ∅} E θ : r , x → X = θ ( r , x ) Such that ∀ i ; X i = m ⇔ r i < x and ∀ i ; X i = M ⇔ r i ≥ x . It follows that X i = ∅ means that the elector i did not vote ( r i = ∅ ). The intuition behind the θ function remains the same. We divide the votersinto two groups, those whose ballot is greater or equal to x and those whoseballot is lower than x . Remark 7 (Properties are transmitted to variable electorate).
All the previousresults remain true with the new definitions. In particular the characterizationsof strategy-proof voting functions are the same as those of strategy-proof votingrules.
A voting function ϕ ∗ : Λ ∗ → Λ is said toverify participation if for all r where r i = ∅ and s that only differs from r indimension i with s i = ∅ we have: ϕ ∗ ( s ) ≥ ϕ ∗ ( r ) ≥ r i or ϕ ∗ ( s ) ≤ ϕ ∗ ( r ) ≤ r i . Participation is a natural extension of strategy-proofness. ”Strategy-proofness+ participation” is equivalent to no matter what the elector does, no strategygives a strictly better outcome than an honest ballot.
Theorem 13 (Characterizing participation).
A strategy-proof voting rule ϕ ∗ : Λ ∗ → Λ verifies participation iff with the order m < ∅ < M , α is weaklyincreasing.Remark 8. Strategy-proofness is obtained by requiring a phantom function tobe increasing. Similarly ”strategy-proofness + participation” is obtained by im-posing that the phantom function is increasing for the order m < ∅ < M . Proof. ⇒ : Suppose m < ∅ < M . We will prove by reductio ad absurdum . Assumethat α is not increasing. Therefore there is an elector i , such that for X and Y that only differ in i we have X i < Y i and α ( X ) > α ( Y ). By definition of aphantom function either X i = ∅ or Y i = ∅ .17uppose X i = ∅ (therefore Y i = M ), then for r = Y we have that by remov-ing his vote voter i contradicts the participation property (e.g. participation).A similar proof works for Y i = ∅ . ⇐ : Suppose that α is weakly increasing. Let ϕ ∗ ( r ) = a where elector i didnot cast a ballot. Let s be the voting profile that is identical to r except that s i is a ballot.We will use the functional characterisation: ∀ r ; ϕ ∗ ( r ) := sup (cid:8) y ∈ Λ | α θ ( r ,y ) ≥ y (cid:9) . – If s i ≤ a then α θ ( r ,s i ) ≥ α θ ( r ,a ) ≥ s i therefore ϕ ( s ) ≥ s i . We also havethat α θ ( r ,a ) ≥ α θ ( s ,a ) therefore ϕ ∗ ( r ) ≥ ϕ ∗ ( s ). – If s i ≥ a then α θ ( r ,s i ) ≤ α θ ( r ,a ) ≤ s i therefore ϕ ( s ) ≥ s i . We also havethat α θ ( r ,a ) ≤ α θ ( s ,a ) therefore ϕ ∗ ( r ) ≤ ϕ ∗ ( s ). Corollary 4 (Participation winning coalitions).
In a vote by issue, a strategy-proof voting function verifies participation iff when a electorate i decides to be-come a voter with ballot x then for any property H containing x , if W H was awinning coalition of H for the initial set of voters then W H ∪ { i } is a winningcoalition for the new set of voters.Proof. ⇒ : Suppose that α is weakly increasing for the order m < ∅ < M . Let V be a fixed set of voters such that i V . For H = { y ≥ a } , W ⊆ V is a winningcoalition iff X ∈ Γ ∗ such M ( X ) = W verifies α X ≥ a . Let Y that differs from X only in dimension i with Y i = M , then α Y ≥ a . Therefore M ( Y ) is a winningcoalition.A similar proof works for H = { y ≤ a } . ⇐ : Suppose that when a electorate i decides to become a voter with ballot x then for any property H containing x , if W H was a winning coalition of H forthe initial set of voters then W H ∪ { i } is a winning coalition for the new set ofvoters.Let us take x = α X where i is not a voter for X . Then M ( X ) ∪ { i } (resp. m ( x )) is a winning coalition for { y ≥ x } (resp. { y ≤ x } ) therefore α Y ≥ α X (resp. α Y ≤ α X ) where Y differs from X only in dimension i and Y i = M (resp. Y i = m ). In order to define consistency, we introduce the function merge : Λ ∗ × Λ ∗ → Λ ∗ that takes the ballots of two disjoint set of voters and returns the union of theballots. Definition 14 (Consistency).
A voting function ϕ ∗ is said to be consistent if for any two disjoint set of voters R and S , when r represents the ballots of R and s represents the ballots of S , we have: ∗ ( r ) = ϕ ∗ ( s ) ⇒ ϕ ∗ ( r ) = ϕ ∗ ( merge ( r , s )) . Theorem 14 (Characterizing consistency).
A strategy-proof voting func-tion ϕ ∗ : Λ ∗ → Λ verifies consistency iff for all X, Y ∈ Γ ∗ with disjoint set ofvoters and α X ≤ α Y we have: α X ≤ α merge ( X,Y ) ≤ α Y . Proof. ⇒ : By reductio ad absurdum . Let us suppose α merge ( X,Y ) < α X ≤ α Y .Let us define s as: – Y j = m ⇒ s j = m ; – Y j = M ⇒ s j = α X .Therefore ϕ ∗ ( s ) = α X and ϕ ∗ ( X + s ) = α merge ( X,Y ) . This contradicts consis-tency.A similar proof shows that we cannot have α X ≤ α Y < α merge ( X,Y ) . ⇐ : Suppose that for all X, Y ∈ Γ ∗ that correspond to two disjoint set ofvoters such that α X ≤ α Y we have α X ≤ α merge ( X,Y ) ≤ α Y . If ϕ ∗ ( r ) = ϕ ∗ ( s ) = a then: – If a = r i = s k : then for all ǫ > α θ ( r + s,r i + ǫ ) ≤ max ( α θ ( r ,r i + ǫ ) , α θ ( s ,r i + ǫ ) ) ≤ r i and r i ≤ min ( α θ ( r ,r i ) , α θ ( s ,r i ) ) ≤ α θ ( r + s,r i ) . Therefore ϕ ( r + s ) = a . Consistency is verified. – If a = r i = α X where X and s have the same voters: α θ ( r ,r i + ǫ ) ≤ α θ ( r + X,r i + ǫ ) ≤ α X = r i and r i = α X ≤ α θ ( r + X,r i ) ≤ α θ ( r ,r i ) . Therefore ϕ ( r + s ) = a . Consistency is verified. – If a = α X = α Y where the voters of X are the voters of r and the voters of Y are the voters of s : α merge ( X,Y ) = α X = α Y . Therefore ϕ ( r + s ) = a . Consistency is verified. Corollary 5 (Consistent voting coalitions).
In a vote by issue, a strategy-proof voting function verifies consistency iff for any a ∈ Λ if X and Y ∈ Γ ∗ have disjoint set of voters: When M ( X ) and M ( Y ) are winning coalitions for H = { y ≥ a } then M ( X ) ∪ M ( Y ) is a winning coalition for H . – When m ( X ) and m ( Y ) are winning coalitions for H = { y ≤ a } then m ( X ) ∪ m ( Y ) is a winning coalition for H .Proof. ⇒ : Suppose that our strategy-proof voting verifies α X ≤ α mege ( X,Y ) ≤ α Y for all X and Y that correspond to disjoint set of voters.A simple inequality consideration for any a gives the result. ⇐ : Suppose α X ≤ α Y . M ( X ) and M ( Y ) are winning coalitions for { y ≥ α X } therefore α merge ( X,Y ) ≥ α X . Conversely m ( X ) and m ( Y ) are winning coalitionsfor { y ≤ α X } therefore α merge ( X,Y ) ≤ α Y . Corollary 6 (Voter sovereignty and consistency imply participation).
A strategy-proof voting function ϕ ∗ : Λ ∗ → Λ that verifies voter sovereignty andconsistency verifies participation.Proof. Let X ∈ Γ ∗ and Y ∈ Γ ∗ be the voting profile where only i is a voterand where he votes respectively m and M . By voter sovereignty, α X = m and α Y = M . By consistency, for any Z ∈ Γ ∗ where i is not a voter: α X ≤ α merge ( X,Z ) ≤ α Z ≤ α merge ( Y,Z ) ≤ α Y In this section we wish to show that consistency results in removing the depen-dency in n of the grading curves. Theorem 15 (Grading curve characterization).
A strategy-proof anony-mous voting function ϕ ∗ = ( ϕ n ) : Λ ∗ → Λ is consistent iff there is an increasingfunction g : [0 , → Λ (electorate size independent) and a constant x ∈ Λ suchthat the phantom function α : Γ ∗ is defined as: α X := g (cid:18) M ( X ) N (cid:19) if N 6 = 0 x if N = 0 Furthermore the voting function verifies participation iff x ∈ g ([0 , .Proof. ⇒ : Let us first show the existence of x and g : [0 , → Λ such thatforall X ∈ Γ ∗ the equation holds. x = α ∅ therefore x exists. For any other X if q = M ( X ) m ( X )+ M ( X ) we define g ( q ) = α X . Observe that this is well defined asby consistency and anonymity, we can duplicate and merge the electorate andso we must have α X = α Y = g ( q ) whenever M ( Y ) m ( Y )+ M ( Y ) = q .Now let us show that g is increasing. For any X and Y such that M ( X ) m ( X )+ M ( X ) ≤ M ( Y ) m ( Y )+ M ( Y ) by consistency we can duplicate X and Y into X ′ and Y ′ that20ave the same number of voters. As such α X ≤ α Y . It follows that g is increasingover the set of rationals. Since the values over irrational does not mater we cancomplete the definition with an increasing g without loss. ⇐ : For any X ∈ Γ ∗ we have α X = α merge ( X, ∅ ) .For any X and Y with at least one voter each by barycentric weights: M ( X ) m ( X ) + M ( X ) ≤ M ( merge ( X, Y )) m ( merge ( X, Y )) + M ( merge ( X, Y )) ≤ M ( Y ) m ( Y ) + M ( Y )Therefore since g is increasing. α X ≤ α merge ( X,Y ) ≤ α Y . We have shownconsistency.Finally we wish to show that if x ∈ g ([0 , n > ≤ k < n we have: kn + 1 < kn < k + 1 n + 1Therefore since g is increasing α is increasing except maybe in α ∅ . Thereforethe function verifies participation iff x ∈ g ([0 , Definition 15 (Homogeneity).
A voting function is homogeneous if for anytwo voting profiles where the proportion of ballots of each possible type is thesame they share the same outcome.
Corollary 7 (Characterizing homogeneity).
A strategy-proof voting func-tion is consistent and anonymous iff it is homogeneous.Proof. ⇒ : Immediate due to the fact that α X only depends on the fraction M ( X ) m ( X )+ M ( X ) . ⇐ : Suppose that we have an homogenous strategy-proof voting function. Bydefinition this implies anonymity. For any X and Y , we can duplicate in orderto have X ′ , Y ′ and merge ( X, Y ) ′ with the same number of voters in each and α Z = α Z ′ for Z ∈ { X, Y, merge ( X, Y ) } . By barycentric considerations: M ( X ′ ) ≤ M ( merge ( X, Y ) ′ ) ≤ M ( Y ′ )Therefore by definition of a phantom function α X ≤ α merge ( X,Y ) ≤ α Y . Definition 16 (Continuity with respect to new members). [17,19]. Avoting function ϕ ∗ is said to be continuous with respect to new members if: ∀ r , s , lim n → + ∞ ϕ ∗ ( merge ( n z }| { r , . . . , r , s )) = ϕ ( r ) Corollary 8 (Characterizing continuity with respect to new members).
A strategy proof, homogeneous voting function ϕ ∗ : Λ ∗ → Λ is continuous withrespect to new members iff its grading curve g is continuous. roof. To simply notation we replace merge by +. ⇒ : We have ∀ r , s , lim n → + ∞ ϕ ( n r + s ) = ϕ ( r ). Therefore for any X, Y ∈ Γ ∗ with at least one voter each:lim n → + ∞ g (cid:18) n M ( X ) + M ( Y ) n ( m ( X ) + M ( X )) + m ( Y ) + M ( Y ) (cid:19) = g (cid:18) M ( X ) m ( X ) + M ( X ) (cid:19) Therefore since g is increasing g is continuous in all rational numbers. There-fore by monotonicity and density of the rationals within the real numbers wehave that g is continuous. ⇐ : Let g be continuous. Then for all X, Y :lim n → + ∞ g (cid:18) n M ( X ) + M ( Y ) n ( m ( X ) + M ( X )) + m ( Y ) + M ( Y ) (cid:19) = g (cid:18) M ( X ) m ( X ) + M ( X ) (cid:19) Therefore by continuity of ϕ . ∀ r , s , lim n → + ∞ ϕ ( n r + s ) = ϕ ( r ) . In this subsection we suppose that Λ = [0 , Definition 17 (Linear median).
The strategy-proof voting function ϕ ∗ : [0 , ∗ → [0 , defined for any n = N and X ∈ { , } N , by α ( X ) = P i X i n is called thelinear median. It corresponds to the grading curve g ( x ) = x . Corollary 9.
The linear median satisfies anonymity, continuity, consistency,sovereignty, participation and continuity with respect to new members. However,it is not ordinal nor strictly responsive.
The proof is trivial from previous subsections as g is independent on theelectorate size, is continuous, etc. Definition 18 (Proportionality).
A voting rule ϕ : [0 , N → [0 , is propor-tional if ∀ X ∈ { , } N , ϕ ( X ) = P i X i N . Theorem 16 (Characterizing proportionality).
A voting function ϕ ∗ : [0 , N → [0 , is strategy-proof and proportional iff it is the linear median. This a direct consequence of the fact that a strategy proof function is com-pletely determined by the extreme votes i.e. its phantom function as proved intheorem 1.The linear median was first proposed and studied in Jenning (2010) PhDdissertation [9]. It was rediscovered independently in 2016 by Caragiannis et al.225] for its nice statistical properties. More recently, Freeman et al. (2019) [6]proved it to be the unique proportional, anonymous and continuous strategy-proof method. As one can observe, anonymity and continuity are not necessary.However, they use a different representation based on the Moulin phantom char-acterization. Namely, if there are n voters and Λ = [0 ,
1] then the linear mediancan be computed by the formulae ϕ ( r ) = med ( r , ..., r n , α , ..., α n ) , where α k = kn , ∀ k = 0 , ..., n , that’s the Moulin n +1 phantom voters are uniformlydistributed on the interval [0 , g ( x ) = x is independent on the size of the electorate.Here are the various characterizations of the linear median, depending on therepresentation, where n = N is the electorate size. Characterization FormulaPhantom Indentification Theorem ϕ ( r ) := kn if (cid:26) j | kn ≤ r j (cid:27) = kr i if k − n ≤ r i ≤ k N where { j | r i < r j } = k Moulin’s Inf-Sup ϕ ( r ) = sup S ⊂N min (cid:18) Sn , inf i ∈ S { r i } (cid:19) Phantom characterization ϕ ( r ) := med (cid:18) r , . . . , r n , n , . . . , n − n (cid:19) Functional characterization ϕ ( r ) := sup (cid:26) y | { r j ≥ y } n ≥ y (cid:27) Voting by issues ϕ ( r ) := a : { r i ≤ a } ≥ ⌈ (1 − a ) n ⌉ ∧ { r i ≥ a } ≥ ⌊ an ⌋ We can also be interested in voting rules where each voter is given a weight.For example the weight of a shareholder vote can be proportional to the numberof shares he owns. We may want to understand in a more general setup (andnot only under the proportionality axiom) the effect of attributing weights toelectors. This is the objective of this section.
Definition 19 (Integer Weighted Voters).
The voting function ϕ : Λ ∗ → Λ is integer weighted with weight sequence w if there is a homogeneous votingfunction φ : Λ ∗ → Λ such that: ∀ r , ϕ ( r ) = φ ( w z }| { r , . . . , r , . . . , w n z }| { r n , . . . , r n )23 here n denotes the last voter.Remark 9. According to the previous section if ϕ is strategy-proof and weightedwith weights w , there exists a grading curve g : [0 , → Λ such that for any X with at least one voter, we have: α X = g P i ∈ M ( X ) w i P i ∈N w i ! Lemma 6.
If a strategy-proof voting function ϕ : Λ ∗ → Λ is integer weightedthen it is consistent.Proof. Suppose that we have
X, Y ∈ Γ ∗ with α X ≤ α Y such that X and Y are associated to two disjoint set of voters. Let x , y be the respective weightvectors.By bycentric values: P i ∈ M ( X ) x i P i ∈ m ( X ) ∪ M ( X ) x i ≤ P i ∈ M ( X ) x i + P i ∈ M ( Y ) y i P i ∈ m ( X ) ∪ M ( X ) x i + P i ∈ m ( Y ) ∪ M ( Y ) y i ≤ P i ∈ M ( Y ) y i P i ∈ m ( Y ) ∪ M ( Y ) y i Therefore α X ≤ α merge ( X,Y ) ≤ α Y .It could be interesting to study the impact when voters weights can change. Definition 20 (Voting function with variable weights).
A voting functionwith variable weights ϕ : Λ ∗ × N E → Λ is a function such that there is anhomogeneous voting rule φ such that: ∀ w, ∀ r , ϕ ( r , w ) = φ ( w z }| { r , . . . , r , . . . , w n z }| { r n , . . . , r n ) Remark 10.
Our definition ensures that raising the weight of a voter by 1 is thesame as adding a voter of weight 1 with the same ballot.Using homogeneity, one can extend a voting function with integer weightsto a voting function with rational weights. The extension to real weights can beobtained using continuity.
Definition 21 (Real Weighted Voters).
A voting function ϕ : Λ ∗ × R E → Λ is a real weighted voting rule with variable weights if there exists a voting rulewith variable rational weights φ : Λ ∗ × Q E → Λ such that: ∀ r , lim w → w ′ φ ( r , w ) = ϕ ( r , w ) . Theorem 17 (Characterizing real weighted voting rules).
A strategy-proof voting function with variable weights ϕ : Λ ∗ × R E → Λ is well-defined forall real weights iff the grading curve g is continuous in all irrational values. emark 11. A continuous grading curve can therefore be used to define a votingfunction for any choice of weights.
Corollary 10 (Functional rewriting).
A voting function ϕ with variable weightsis strategy-proof iff there exists a grading curve g such that when there is at leastone voter: ∀ w r , ϕ ( r , w ) = sup (cid:26) y | g (cid:18) P i ∈N w i { r i ≥ y } P i ∈N w i (cid:19) ≥ y (cid:27) . Definition 22 (Proportionality with weights).
A voting function φ ∗ : Λ ∗ → Λ verifies proportionality with weights if Λ = [0 , and α X = P i ∈N w i { r i ≥ y } P i ∈N w i . Corollary 11.
The grading curve for the strategy-proof voting function that isproportional with weights is the identity.
We have introduced the notions of phantom functions and grading curves anddemonstrated their usefulness in (i) unifying a number of existing characteriza-tions of strategy-proof voting rules on the domain of single-peaked preferences,and (ii) obtaining insightful characterizations of special cases. As an importantexample, we have characterized the linear median as the unique strategy-proofvoting rule satisfying proportionality. It has been shown to possess further salientproperties such as consistency and participation. On the other hand, the linearmedian presupposes a cardinal scale. On the other hand, adding the naturalcondition of ordinality characterizes, in the anonymous case, the class of or-der (statistics) functions which play an important role in majority judgment–ordinal– method of voting (Balinski-Laraki [2] chapters 10-13). A particularlyappealing order function is the middlemost (chapter 13 in [2]).
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The main lemma which is used as a reference for the other more practical char-acterizations.
Theorem 18.
The voting function ϕ is strategy-proof iff there exists a phantomfunction α : Γ → Λ such that: ∀ r ; ϕ ( r ) := α X if ∀ j, M ( X ) = { j | α X ≤ r j } ∧ m ( X ) = { j | α X ≥ r j } r i if α X ≤ r i ≤ α Y where M ( X ) = { j | r i < r j } ∧ m ( Y ) = { j | r i > r j } (5) Lemma 7.
This can be rewritten as follows: ∀ r ; ϕ ( r ) := α m if ∀ j, r j ≤ α m α θ ( r ,r i ) if ( i ∈ N ) and r i = min { r j | r j ≥ α θ ( r ,r i ) } r i if ( i ∈ N ) and α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) (6)where θ is the same definition as seen in 4. Definition 23 (The θ function ). θ : R N × R → Γθ : r , x → X = θ ( r , x ) Such that ∀ i ; X i = m ⇔ r i < x .Proof. For any X ∈ Γ , and r ∈ Λ N such that: M ( X ) = { j | α X ≤ r j } ∧ m ( X ) = { j | α X ≥ r j } we have X = θ ( r , α X ) as such either X = m or X = θ ( r , r i ) where r i = min { r i | i ∈ M ( X ) } = min { r j | r j ≥ α θ ( r ,r i ) .For X and Y and i such that α X ≤ r i ≤ α Y where M ( X ) = { j | r i < r j } ∧ m ( Y ) = { j | r i > r j } we have X = θ ( r , r i + ǫ ) and Y = θ ( r , r i ).We will therefore show that the second characterization provides the strategy-proof functions.Throughout the proofs let x i be the i th smallest element the voting profile r . Definition 24.
For a given phantom function α , and a given voting profile r we define α r ,i := α ◦ θ ( r , x ) where x is the i th smallest element of ( r i ) i ∈N andwe define α r ,n +1 := α m = α ( m, ..., m ) . Observe that the definition of θ is asymmetric because it sends values strictly belowthe cut-off to m and greater than or equal to M . That’s why our characterizations“look” asymmetric. They are not. emma 8. Let us fix a phantom function α . For all r and ǫ small enough thereexists an i such that one of the following holds – r i = min { r j | r j ≥ α θ ( r ,r i ) } . – α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) . – ∀ i ∈ N, r i < α m .Proof. The sequence ( x i ) [ | ,n | ] is increasing with i and α r ,i is decreasing with i .Either x n < α m , α M ≤ x or ( x i ) n and α r ,i must cross. Suppose that theycross, there is an i such that x i − ≤ α r ,i ≤ x i or α r ,i +1 ≤ x i ≤ α r ,i .1. If x i − < α r ,i ≤ x i then α θ ( r ,x i ) = α r ,i ≤ x i and if r j ≥ α θ ( r ,x i ) then r j ≥ x i therefore x i = min { r j | r j ≥ α θ ( r ,x i ) } .2. If α r ,i +1 ≤ x i ≤ α r ,i then by definition α θ ( r ,x i +1 ) ≤ x i ≤ α θ ( r ,x i ) . For all j : θ ( r , x i +1 ) j = m ⇔ r j < x i +1 ⇒ r j ≤ x i ⇒ θ ( r , x i + ǫ ) j = m Therefore by monotonicity of α , α θ ( r ,x i + ǫ ) ≤ x i ≤ α θ ( r ,x i )
3. If α M < x then x = min { r j | r j ≥ α θ ( r ,x ) } . Lemma 9.
The function ϕ , defined from a fixed α as: ∀ r ; ϕ ( r ) := α m if ∀ j, r j ≤ α m α θ ( r ,r i ) if ( i ∈ N ) and r i = min { r j | r j ≥ α θ ( r ,r i ) } r i if ( i ∈ N ) and α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) is properly defined for all r when ǫ is always chosen small enoughProof. Given lemma 8 there is always an i that verifies one of the propertiesthat define ϕ . We must show that if different i satisfy the property this does notlead to an ambiguous definition. – If the first and second property are verified then α r ,i ≤ r i ≤ α m . Thereforeby monotonicity of α there is no contradiction. – If the first and third property are verified then α θ ( r ,r i + ǫ ) ≤ r i ≤ α m . There-fore by monotonicity of α , α m = r i . There is no contradiction. – If the second is verified for a voter i and the third for a voter k with r i ≤ r k ,then we have r i = min { r j | r j ≥ α θ ( r ,r i ) } and α θ ( r ,r k + ǫ ) ≤ r k ≤ α θ ( r ,r k ) . ∀ j ; θ ( r , r k ) j = M ⇔ r k ≤ r j ⇒ r i ≤ r j ⇒ θ ( r , r i ) j = M α , α θ ( r ,r k ) ≤ α θ ( r ,r i ) . r i ≤ r k ≤ α θ ( r ,r k ) ≤ α θ ( r ,r i ) ≤ r i This proves that α θ ( r ,r i ) = r k . This case does not lead to an ambiguousdefinition. – If the second is verified for a voter i and the third for a voter k with r k < r i ,then we have r i = min { r j | r j ≥ α θ ( r ,r i ) } and α θ ( r ,r k + ǫ ) ≤ r k ≤ α θ ( r ,r k ) . ∀ j ; θ ( r , r i ) j = M ⇔ r i ≤ r j ⇒ r k < r j ⇒ θ ( r , r k + ǫ ) j = M Therefore by monotonicity of α , α θ ( r ,r i ) ≤ α θ ( r ,r k + ǫ ) ≤ r k < r i This is absurd since r i = min { r j | r j ≥ α θ ( r ,r i ) } . Therefore this case neverhappens. – Suppose the second is true for two different voters i and k . Therefore r i =min { r j | r j ≥ α θ ( r ,r i ) } and r k = m ∈ { r j | r j ≥ α θ ( r ,r k ) } .Without loss of generality we can suppose that r i ≤ r k . Therefore by mono-tonicity of α , α θ ( r ,r k ) ≤ α θ ( r ,r i ) ≤ r i . Since r k is the min of { r j | r j ≥ α θ ( r ,r k ) } we have that r k ≤ r i . As such r k = r i . Therefore α θ ( r ,r k ) = α θ ( r ,r i ) . This case does not raise an ambiguous definition. – Suppose the third is true for two different voters i and k . Therefore α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) and α θ ( r ,r k + ǫ ) ≤ r k ≤ α θ ( r ,r k ) .Let us suppose that r i = r k . Without loss of generality, we will use r i < r k . ∀ j ; θ ( r , r k ) j = M ⇔ r j ≥ r k ⇒ r j > r i ⇒ θ ( r , r i + ǫ ) j = M Therefore by monotonicity of α : α θ ( r ,r i + ǫ ) ≤ r i < r k ≤ α θ ( r ,r k ) ≤ α θ ( r ,r i + ǫ ) This is absurd, which implies that r i = r k . This case does not cause anambiguous definition of ϕ . Lemma 10. If ϕ verifies strategy-proofness then there is a phantom function α such that : ∀ r ; ϕ ( r ) := α m if ∀ j, r j ≤ α m α θ ( r ,r i ) if ( i ∈ N ) and r i = min { r j | r j ≥ α θ ( r ,r i ) } r i if ( i ∈ N ) and α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) (7) where ǫ is small enough for r . roof. Let us use the phantom function such that α ( X ) := ϕ ( X )where we consider the Cauchy-extension of ϕ if necessary. We will show foreach case used in the equation 7 we obtain the desired value of ϕ ( r ). Lemma8 provides that we have studied all the cases and that therefore the proof iscomplete.1. Case ∀ j, r j ≤ α m :By definition of α m < ϕ ( m, . . . , m ). Strategy-proofness on each dimensiongives us that ϕ ( r ) = ϕ ( m, . . . , m ) = α m .2. Case ∃ i, r i = min { r j | r j ≥ α θ ( r ,r i ) } :Let X = θ ( r , r i ). Suppose that ϕ ( r ) < α X (resp. α X < ϕ ( r )). By strategy-proofness changing to M (resp. to m ) the ballot of a voter j such that X j = M (resp. X j = m ) does not change the outcome. Therefore by responsivenessof ϕ we have ϕ ( r ) < α X ≤ ϕ ( r ) (resp. ϕ ( r ) ≤ α X < ϕ ( r )). We havereached a contradiction.Therefore if ∃ i, r i = min { r j | r j ≥ α θ ( r ,r i ) } then ϕ ( r ) = α X .3. Case α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) :Suppose that ϕ ( r ) < r i (resp. r i < ϕ ( r )). Let X = α θ ( r ,r i ) (resp. X = θ ( r , r i + ǫ )). By strategy-proofness changing to M (resp. to m ) the ballot ofa voter j such that X i = M (resp. X i = m ) does not change the outcome.Therefore by responsiveness of ϕ we have ϕ ( r ) < r i ≤ α X ≤ ϕ ( r ) (resp. ϕ ( r ) ≤ α X ≤ r i < ϕ ( r )).We have reached a contradiction. Therefore if α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) ,then r i = ϕ ( r ). Lemma 11.
If there is a phantom function α such that : ∀ r ; ϕ ( r ) := α m if ∀ j, r j ≤ α m α θ ( r ,r k ) if ( k ∈ N ) and r k = min { r j | r j ≥ α θ ( r ,r k ) } r k if ( k ∈ N ) and α θ ( r ,r k + ǫ ) ≤ r k ≤ α θ ( r ,r k ) where ǫ is always chosen small enough for r . Then ϕ verifies strategy-proofnessProof. First we prove responsiveness then strategy-proofness. – Responsiveness: This proof is by induction. Let s that differs from r only indimension i and r i < s i .Let k be such that ϕ ( r ) ∈ { α θ ( r ,r k ) , r k } when ϕ ( r ) = α m .1. Case ϕ ( r ) = α m : α m is the minimum of ϕ therefore: ϕ ( r ) ≤ ϕ ( s ) .
30. Case s i ≤ ϕ ( r ):The situation where s i is strictly greater than r k is impossible.If s i = r k then r k ≤ α θ ( r ,r k ) . We have θ ( s , s k ) ≥ θ ( r , r k ) and θ ( s , s k + ǫ ) = θ ( r , r k + ǫ ). Therefore α θ ( s ,s k + ǫ ) ≤ s k ≤ α θ ( s ,s k ) .Else s i < r k implies θ ( s , s k ) = θ ( r , r k ) and θ ( s , s k + ǫ ) = θ ( r , r k + ǫ ).We can conclude that ϕ ( r ) = ϕ ( s ) .
3. Case ϕ ( r ) < r i :Then θ ( s , s k ) = θ ( r , r k ) and θ ( s , s k + ǫ ) = θ ( r , r k + ǫ ). ϕ ( r ) = ϕ ( s )4. Case ϕ ( r ) = r i :Let X be defined by X j = M iff j = i or r j > r i and α X < r i . We have θ ( r , r i + ǫ ) < X ≤ θ ( r , r i ).Let x = min { M } ∪ { r j | r j > r i } (a) Subcase α X < r i :If there where no k = i such that r k = r i then we would have X = θ ( r , r i ) so α θ ( r ,r i ) < r i . This is absurd. Therefore such a k exists. θ ( s , s k ) = θ ( r , r i ) and θ ( s , s k + ǫ ) = θ ( r , r i + ǫ ) ϕ ( s ) = s k = r i = ϕ ( r )(b) Subcase r i ≤ α X and s i ≤ x :We have X = θ ( s , s i ) and θ ( s , s i + ǫ ) ≤ θ ( r , r i + ǫ ). ϕ ( s ) = min ( α X , s i ) ≥ r i = ϕ ( r )(c) Subcase r i ≤ α X and s i > x :Let t differ from r only in dimension r i with t i = x .Show by induction that ϕ ( r ) ≤ ϕ ( t ) (4b) and ϕ ( t ) ≤ ϕ ( s ) (3) or (4)Transitivity gives: ϕ ( r ) ≤ ϕ ( s )5. Case r i < ϕ ( r ) < s i :Let t differ from r only in dimension r i with t i = ϕ ( r ).Show by induction that ϕ ( r ) ≤ ϕ ( t ) (2) and ϕ ( t ) ≤ ϕ ( s ) (4). Transitivitygives: ϕ ( r ) ≤ ϕ ( s )The induction finishes since there is a finite number values r k and α X be-tween r i and s i . – Strategy-proofness : Let s that differs from r only in dimension i .31 If r i ≤ min ( s i , ϕ ( r )) (resp. r i ≥ max ( s i , ϕ ( r ))) then by responsiveness ϕ ( r ) ≤ ϕ ( s ) (resp. ϕ ( r ) ≥ ϕ ( s )). • If ϕ ( r ) = α m and s i < r i (resp. ϕ ( r ) = α M and s i > r i ) then ϕ ( r ) = ϕ ( r ). • If s i < r i < ϕ ( r ) = α θ ( r ,r k ) then s k = min { s j | s j > α θ ( r ,s k ) } therefore ϕ ( r ) = ϕ ( s ) • If ϕ ( r ) = α θ ( r ,r k ) < r i < s i and k = i , then s k = min { s j | s j > α θ ( r ,s k ) } therefore ϕ ( r ) = ϕ ( s ). • If ϕ ( r ) = α θ ( r ,r i ) < r i < s i , let s k = min { s j | s j ≥ α θ ( r ,r i ) } . Then θ ( r , r i ) = ( s , s k ), therefore ϕ ( r ) = ϕ ( s ). • If s i < r i < ϕ ( r ) = r k (resp. s i > r i > ϕ ( r ) = r k ) then θ ( r , r k ) = θ ( s , s k ) and θ ( r , r k + ǫ ) = θ ( s , s k + ǫ ). Therefore ϕ ( r ) = ϕ ( s ). Lemma 12 (Phantom lemma).
The voting function ϕ is strategy-proof iffthere exists a phantom function α : Γ → Λ : ∀ r ; ϕ ( r ) := α m if ∀ j, r j ≤ α m α θ ( r ,r i ) if ( i ∈ N ) and r i = min { r j | r j ≥ α θ ( r ,r i ) } r i if ( i ∈ N ) and α θ ( r ,r i + ǫ ) ≤ r i ≤ α θ ( r ,r i ) (8) Where ǫ is always chosen small enough for r . The four cases given in thedefinition of the function are exhaustive.Proof. The proof is immediate by using lemma 9 11 and 10.
B Additional Properties
Properties Phantom function Winning coalitionsVoter sovereignty α ( m ) = m N ∈ W { y ≥ M } α ( M ) = M N ∈ W { y ≤ m } Strict Responsiveness α ( Γ ) = { m, M } ∀ r , ∃ r i , Ordinality { r j ≤ r i } ∈ W { y ≤ r j } { r j ≥ r i } ∈ W { y ≥ r j } Invisible voter i ( ∀ j = i, X j = Y j ) ∀ H ∈ H , W ∈ W H ⇒ ⇒ α ( X ) = α ( Y ) W − { i } ∈ W H Anonymity ∀ X, Y : P i X i = P i Y i ∀ H ∈ H , ∃ p, ⇒ W H = { W | W ≥ p } α ( X ) = α ( Y )Participation α is monotonous W ∈ W H ⇒ W ∪ { i } ∈ W H Consistency α X ≤ α Y ∧ ∀ j, ∅ ∈ { X j , Y j } ⇒ W , W ∈ W H α X ≤ α merge ( X,Y ) ≤ α Y W ∪ W ∈ W H Multi-dimension
We now wish to study the problem of sets of greater dimensions. Barber`a, Guland Stacchetti’s [3] model provides us with the results that allow us to do so.The proofs for the following are directly inspired from their work and can befound in the appendix.Let B = Λ × · · · × Λ p such that for all j , Λ j ⊆ R .Let u i : B → R + be the utility function of a voter i . We have that i prefersalternative x ∈ B to alternative y ∈ B iff u i ( x ) ≥ u i ( y ). Definition 25 (Single-peakness).
A utility provides a single-peaked prefer-ence on B with peak a if: ∀ x, y ∈ B ; k a − x k = k a − y k + k x − a k ⇒ u i ( x ) ≥ u i ( y ) Axiom 4 (Strategy-proofness) . For all voters i , let S that differs from R onlyfor voter i . Then if R i ∈ B is the peak of voter i then u i ( ϕ ( R )) ≥ u i ( ϕ ( S )) . Lemma 13. If a, x, y ∈ B and k a − x k < k a − y k + k y − x k then there is a utilityfunction u with peak a such that u i ( x ) > u i ( y ) . Lemma 14. If ϕ : B ∗ → B is a voting function, then the two following areequivalent: – ∃ , f , . . . , f n , ∀ R, ϕ ( R ) = ( f ( R , , . . . , R n, ) , . . . , f p ( R ,p , . . . , R ,p )) – For all j and i , if R and S differ only in dimension i and R i,j = S i,j then ϕ ( R ) j = ϕ ( S ) j . Theorem 19 (Separability). If ϕ : B ∗ → B is a strategy-proof voting functionthen there exists f , . . . , f p where f j : Λ nj ⇒ Λ j is strategy-proof such that: ∀ R, ϕ ( R ) = ( f ( R , , . . . , R n, ) , . . . , f p ( R ,p , . . . , R ,p ))Barber`a, Gul and Stacchetti’s [3] model and results have been extended byNehring and Puppe [14] [13] to a larger class of combinatorial domains (medianspaces) where preferences can be non-separable and the set of allocations con-strained. A very interesting particular case is the participative budgeting prob-lem in which voters can submit a proposal for how to divide a single divisibleresource among several possible alternatives (such as public projects or activ-ities). Under separability and convexity of the preferences, the social welfare-maximizing mechanism is efficient, anonymous and strategy-proof ([8] and [15]).However, this mechanism fails to be “proportional”, which leads Freeman, Pen-nock, Peters and Wortman-Vaughan [6] to propose a new class of moving phan-tom mechanisms that contains a proportional one. In the particular context ofbudgeting with only two alternatives, this reduces to the proportional medianwhich they characterize as the unique strategy-proof, anonymous and continuousvoting rule satisfying proportionality. As shown above, anonymity and continu-ity axioms are redundant and that function may be characterized by a linear33urve, electorate size independent, implying some of its salient properties suchas its consistency, and participation. The Freeman, Pennock, Peters and Wortman-Vaughan [6] paper goes muchfurther than the scope of our paper by proposing an extension of the proportionalmedian to budgeting problems with more than two alternatives. However, theydo not provide an axiomatization of their proportional method nor do they provethat their moving phantom mechanisms are the unique anonymous and strategy-proof methods. Those questions remain open and we hope our results will helpsolve them.
Lemma 15. If a, x, y ∈ B and k a − x k < k a − y k + k y − x k then there is a utilityfunction u with peak a such that u i ( x ) > u i ( y ) .Proof. Let u i be defined as u i ( r ) := − λ i | r i − a i | where we choose the λ i suchthat u i ( x ) = − u i ( y ) = 2. (this can be done by changing the basis). Lemma 16. If ϕ : B n → B is a voting rule, then the two following are equiva-lent: – ∃ , f , . . . , f n , ∀ R, ϕ ( R ) = ( f ( R , , . . . , R n, ) , . . . , f p ( R ,p , . . . , R ,p )) – For all j and i , if R and S differ only in dimension i and R i,j = S i,j then ϕ ( R ) j = ϕ ( S ) j .Proof. ⇒ : Since ϕ ( R ) j = f j ( R ,j , . . . , R ,j ) and S i,j = R i,j for all i we obtain ϕ ( R ) j = ϕ ( S ) j ⇐ : Suppose that for any voter i , for any j and any set of votes R and S , if R and S only differ in dimension i and R i,j = S i,j then ϕ ( R ) j = ϕ ( S ) j . Thenfor any T we can by using this equality on one voter at a time show that if R i,j = T i,j for all i then ϕ ( R ) j = ϕ ( T ) j thus the existence of f j Theorem 20 (Separability). If ϕ : B n → B is a strategy-proof voting rulethen there exists f , . . . , f p where f j : Λ nj ⇒ Λ j is strategy-proof such that: ∀ R, ϕ ( R ) = ( f ( R , , . . . , R n, ) , . . . , f p ( R ,p , . . . , R ,p )) Proof. ⇒ : We have that ∀ R, ϕ ( R ) = ( f ( R , , . . . , R n, ) , . . . , f p ( R ,p , . . . , R ,p ))is equivalent to for all k and i , if R and S differ only in dimension i and R i,k = S i,k = c then ϕ ( R ) k = ϕ ( S ) k .Let ϕ ( R ) = a and ϕ ( S ) = b . Assume by contradiction that b j = a j . If k R i − b k < k R i − a k + k a − b k then according to the lemma there is a utility function u such that R i is the peak and b is strictly preferred to a . This contradictsstrategy-proofness. Consistency and participation (or the absence of the no-show paradox) have beenproven by Freeman, Pennock, Peters and Wortman-Vaughan [6]) for the proportionalmedian rules. Our characterization allows understanding the reasons for these. k R i − b k = k R i − a k + k a − b k and k S i − a k = k S i − b k + k a − b k (symmetrical proof). It follows that | c − b j | = | c − a j | + | a j − b j | and | c − a j | = | c − b j | + | a j − b j | .Therefore a j = b j . This contradicts our assumption. Therefore we haveproven separability. ⇐ : If we are strategy-proof dimension by dimension we are strategy-prooffor the result. D Algorithm
Algorithm 1: ϕ with the lemma characterization Input:
The set of votes r , . . . , r n Output:
The value of ϕ ( r ) for X ∈ Γ doif M ( X ) = { r j ≥ α X } thenreturn α X endendfor i ∈ N do X := m ; Y := m ; for j ∈ N doif r j > r i then Y j := M ; X j := M endelse if r j = r i then Y j = M endendif α X ≤ r i ≤ α Y thenreturn r i endend lgorithm 2: ϕ with the Moulin characterization Input:
The set of votes r , . . . , r n Output:
The value of ϕ ( r ) l = [ α m ]; for i ∈ N do v i = ( r i , i ); l := add ( r i , l ); end ll := Sort { v i } according to natural order on r i ; X = m ; while ll = ∅ do v i = pop ( ll ); X. ( i ) ← M ; l := add ( α ( X ) , l ); endreturn med ( l ) lgorithm 3: ϕ with the functional characterization Input:
The set of votes r , . . . , r n Output:
The value of ϕ ( r ) if max { r i } ≤ α m thenreturn α m . end ll := Sort { ( r i , i ) } according to natural order on r i ; LL :=empty queue; X := m ; for ( r i , i ) ∈ ll do X. ( i ) ← M ; LL := add ( X, LL ); endwhile length ll > do p = [ length ( ll )2 ]; x = ll. ( p ); α x = α ( LL. ( p ); if α x < x then ll := ll [0 , p − LL := LL [0 , p − endelse ll := ll [ p, length ( ll ) − LL := LL [ p, length ( ll ) − endend a = pop ( ll ); α x = α ◦ pop ( LL ); if α x < a thenreturn α x endelsereturn a endend