Soc Choice WelfDOI 10.1007/s00355-014-0811-1

A characterization result for approval votingwith a variable set of alternatives

Norihisa Sato

Received: 27 April 2013 / Accepted: 11 February 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper, we present a new result for axiomatic characterization ofapproval voting. Having defined a model in which the actual set of alternatives becomesknown only after a vote has been taken, we characterize approval voting as the onlyvoting procedure (to be precise, “family of ballot aggregation functions”) that sat-isfies faithfulness (F), consistency (C), stability on selected alternatives (SSA), andindependence of dropped alternatives (IDA). SSA, which is a version of the propertyintroduced by Arrow (Economica 16:121–127, 1959), states that if the actual set ofalternatives is smaller than the original set, we should select those alternatives, if any,that would have been selected on the first vote and that are still feasible. On the otherhand, IDA suggests that we should select alternatives based on the outcome of thesecond vote. Therefore, given F and C, approval voting is the only voting procedurethat selects the same set of alternatives irrespective of which vote counts, that is, thefirst or second vote.

1 Introduction

In this paper, we present a new result for the axiomatic characterization of approvalvoting, a well known voting procedure introduced in Brams and Fishburn (1978).Contrary to most of the previous studies, we assume that a set of alternatives can vary,thereby allowing for the possibility that some alternatives may be found infeasibleafter the vote has been taken. A typical example of such a situation is a company’shiring process, where selected applicants often decline the job offer in favor of onefrom another company. Another example is an official election, in which candidates

N. Sato (B)Faculty of Economics, Nagoya Gakuin University,1-25 Atsuta-nishimachi, Atsuta, Nagoya, Aichi 456-8612, Japane-mail: nrsato@gmail.com

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may be disqualified even after having been elected if they have committed seriouselectoral fraud.

When some (winning) alternatives do indeed turn out to be infeasible, we haveat least two options: (a) to accept all the alternatives, if any, that would have beenselected on the first vote and are still feasible, or (b) to take another vote on theremaining alternatives. It is a known fact that as long as we adopt approval votingas the selection method, we obtain the same result in both cases; that is, the votingprocedure selects the same set of alternatives. It is also known that this is not the casewith many other voting procedures, including the plurality rule and Borda count.1

Based on the above, one may wonder whether the property of selecting the sameset of alternatives regardless of whether the first or second vote counts, is actually adefining feature of approval voting. We show that the answer is in the affirmative atleast within a certain class of voting procedures, where approval voting is the onlyvoting procedure having this property and two other reasonable properties, namely,faithfulness (F) and consistency (C).2

More specifically, we capture the above stated property using a combination of twoindependent properties, that is, stability on selected alternatives (SSA) and indepen-dence of dropped alternatives (IDA). If the set of feasible alternatives is smaller thanthe original set, any voting procedure satisfying SSA selects alternatives in accor-dance with option (a) above. On the other hand, if a voting procedure satisfies IDA,it selects alternatives, as suggested by option (b), based on the outcome of the secondvote. Therefore, a voting procedure satisfies the above stated property if and only if itsatisfies both SSA and IDA.

Since the publication of Fishburn (1978a,b), several authors, including Sertel(1988), Baigent and Xu (1991) and Alós-Ferrer (2006), have established characteriza-tion theorems for approval voting. While most of these treat the set of alternatives asbeing fixed, Vorsatz (2007) and Alcalde-Unzu and Vorsatz (2013) work with a variableset of alternatives in a similar setting to ours.

Defining a model that allows the set of alternatives to vary, Vorsatz (2007) char-acterizes approval voting by using strategy-proofness as the main axiom. Althoughour formulation partially follows that of Vorsatz (2007), it differs in some importantaspects.3 For example, while Vorsatz (2007) explicitly defines the set of potentialvoters with dichotomous preferences, we rely on “ballot response profiles” instead,each of which describes a voting outcome of a group of voters without revealing theiridentities. Moreover, in order to define strategy-proofness, Vorsatz (2007) assumesthat each voter has a complete preference preorder on the set of all nonempty subsetsof potential alternatives.

1 For the details, see, for example, Nurmi (1987, 1999) and Saari (2000a,b).2 Faithfulness states that if there is only one voter, all, and only those alternatives approved by the votershould be selected. Consistency requires that all the alternatives chosen by two separate groups of voters, ifany, should also be selected by the union of the two groups, and in that case, only such alternatives shouldbe selected by the union.3 In fact, our formulation is more similar to that of Massó and Vorsatz (2008), in which a generalized notionof approval voting is considered.

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A characterization result for approval voting

Alcalde-Unzu and Vorsatz (2013) consider a generalized voting procedure forapproval voting, called type-weighted approval voting. Based on the characteriza-tion result for the new voting procedure, they also present a characterization result forapproval voting that is very similar to the one presented in this paper. More precisely,they present the set of properties satisfied by a voting procedure if and only if thisprocedure is approval voting or disapproval voting. (Disapproval voting selects thosealternatives receiving the least votes.) The properties used in the result of Alcalde-Unzuand Vorsatz (2013) and in our result are almost the same; in particular, a property cor-responding to IDA is also used in Alcalde-Unzu and Vorsatz (2013).4 Nevertheless,the two results are certainly different. First, we rely on F in the characterization, whileAlcalde-Unzu and Vorsatz (2013) rely on neutrality,5 which is an independent prop-erty of F. Second, a property must be added to obtain the exact characterization ofapproval voting from the result of Alcalde-Unzu and Vorsatz (2013). Although theauthors use F as an example of the property to be added,6 our result suggests that if wedid in fact add F, depending on its definition, it would render neutrality superfluous.

The rest of this paper is organized as follows: in Sect. 2, we introduce variousnotations and definitions. We state and prove our main theorem in Sect. 3. Section 4is devoted to confirming the independence of the properties used in the theorem. InSect. 5, we compare our result with that of Alcalde-Unzu and Vorsatz (2013). Finally,concluding remarks are presented in Sect. 6.

2 Notations and definitions

Let K be a finite set of alternatives for selection by voting. We assume that K has atleast three elements, and denote its generic elements by x, y, z or a, b. Each voter isallowed to vote for (or “approve of”) any number of alternatives. A ballot is a listof alternatives for which a voter has voted (or plans to vote), and we represent eachpossible ballot by a nonempty subset of K.

A ballot response profile π represents the voting outcome of a group of voters.More specifically, π is a function from 2K \{∅} to N, which associates each nonemptysubset B of K with the number of voters whose ballot coincides with B. Let Π be theset of all ballot response profiles, which includes the ballot response profile π̂ definedby π̂(B) = 0 for all B ∈ 2K \ {∅}. Given π ∈ Π, the number of voters who cast anyvote is

∑B∈2K \{∅} π(B). For each π ∈ Π and each x ∈ K , let n(x, π) be the number

of voters who vote for alternative x, i.e., n(x, π) = ∑B∈2K \{∅},x∈B π(B).

For π, π ′ ∈ Π, we define the ballot response profile π + π ′ as (π + π ′)(B) =π(B) + π ′(B) for all B ∈ 2K \ {∅}. We interpret π + π ′ as the voting outcome thatarises if two separate groups of voters corresponding to π and π ′ get together. We

4 It should be noted that Alcalde-Unzu and Vorsatz (2011), which is the first version of Alcalde-Unzu andVorsatz (2013), considered this property prior to this paper. However, the above mentioned characterizationresult for approval voting did not appear in the first version.5 Neutrality states that the names of alternatives should not affect the outcome. Alcalde-Unzu and Vorsatz(2013) call the property “symmetry across alternatives.”6 They also offer as the weakest possible property the no-disapproval voting axiom, which eliminatesdisapproval voting from the domain of possible voting procedures.

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also use the following specific type of ballot response profile: for each B ∈ 2K \ {∅},let πB be the ballot response profile satisfying πB(B) = 1 and πB(A) = 0 for allA ∈ 2K \ {∅} with A �= B.

Here we introduce an additional structure into the model:7 there is the possibilitythat after a vote has been taken, some alternatives could turn out to be infeasible forvarious reasons. More specifically, we assume that the set of feasible alternatives israndomly drawn from the set K = {K̄ ∈ 2K \ {∅} : |K̄ | ≥ 2} after we observe aballot response profile π ∈ Π.8 Note that we exclude trivial cases in which only onealternative remains. Then, we are faced with the following problem: given a ballotresponse profile π ∈ Π, how do we select alternatives from each possible K̄ ∈ K?

To deal with this problem, we introduce the notion of families of ballot aggregationfunctions. First, given K̄ ∈ K, we define a ballot aggregation function f K̄ : Π →2K̄ \ {∅} as a set-valued function that associates each ballot response profile π witha nonempty subset of K̄ . We understand f K̄ (π) to be the set of alternatives selectedfrom K̄ when the voting outcome is π. Then, by specifying a ballot aggregationfunction for each K̄ ∈ K, we obtain a family of ballot aggregation functions { f K̄ :Π → 2K̄ \ {∅}}K̄∈K. In other words, { f K̄ }K̄∈K is a list of selection rules for feasiblealternatives covering all possible cases.

Our particular interest lies in the following family of ballot aggregation functions,called approval voting.

Definition 1 The family of ballot aggregation functions { f K̄ : Π → 2K̄ \ {∅}}K̄∈Kis approval voting if for all K̄ ∈ K and all π ∈ Π,

f K̄ (π) = arg maxx∈K̄

n(x, π).

In other words, for each K̄ , approval voting selects all alternatives with the highestvotes among K̄ .

Before proceeding, we need to define a type of ballot response profile: given K̄ ∈ Kand π ∈ Π, we define the ballot response profile π K̄ as

π K̄ (B) ={∑

A∈K̄(B) π(A) if B ∈ 2K̄ \ {∅},0 otherwise,

where K̄(B) = {A ∈ 2K \ {∅} : A ∩ K̄ = B}. Note that n(x, π) = n(x, π K̄ ) for allx ∈ K̄ . As explained later in this section, if π is given, we can regard π K̄ as the votingoutcome that arises if we conduct another vote with K̄ as the new set of alternatives.

We are now ready to introduce four properties for families of ballot aggregationfunctions, which are used to characterize approval voting. The family of ballot aggre-gation functions { f K̄ : Π → 2K̄ \ {∅}}K̄∈K satisfies:

7 The formulation that follows was inspired by those of Vorsatz (2007) and Massó and Vorsatz (2008).8 Thus, in our setting, a nonempty subset of K may have two interpretations: a ballot or a set of feasiblealternatives.

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Faithfulness (F) if for all B ∈ 2K \ {∅},

f K (πB) = B.

Consistency (C): if for all π, π ′ ∈ Π with f K (π) ∩ f K (π ′) �= ∅,

f K (π + π ′) = f K (π) ∩ f K (π ′).

Stability on selected alternatives (SSA) if for all K̄ ∈ K and all π ∈ Π withf K (π) ∩ K̄ �= ∅,

f K̄ (π) = f K (π) ∩ K̄ .

Independence of dropped alternatives (IDA) if for all K̄ ∈ K and all π, π ′ ∈ Π

with π K̄ = π ′K̄ ,9

f K̄ (π) = f K̄ (π ′).

Faithfulness and consistency, abbreviated as F and C, respectively, are straightfor-ward extensions of the corresponding properties utilized by Fishburn (1978a) in thesetting of the fixed set of alternatives.10

Stability on selected alternatives, or SSA for short, states that the alternativesselected from the original set K are also selected from any K̄ ∈ K that containsthem, and only such alternatives are selected from K̄ .11 Note that under SSA, theproperty of F holds for all f K̄ , that is, F together with SSA implies that f K̄ (πB) = Bfor all K̄ ∈ K and all B ∈ 2K̄ \ {∅}.

Independence of dropped alternatives, or IDA, requires slightly more explanation.This property is based on the following simple assumption about voters’ behaviorwhen the number of feasible alternatives has been reduced:

Suppose that there is a voter whose ballot is B ∈ 2K \ {∅}. If another vote isconducted with K̄ �= K as the new set of alternatives, the voter votes for thealternatives in B ∩ K̄ .

The following example shows a consequence of this assumption. Let K={x, y, z},and suppose that we observe the ballot response profile π with

π({x}) = 1, π({x, y}) = 1, π({x, z}) = 1,

π(B) = 0 for all B ∈ 2K \ {∅} with B �= {x}, {x, y}, {x, z}.

9 The following equivalent statement will be useful: if for all K̄ ∈ K, and all π ∈ Π, f K̄ (π) = f K̄ (π K̄ ).

10 See also Alós-Ferrer (2006) and Xu (2010).11 Arrow (1959) introduces this property for social choice functions. Vorsatz (2007) slightly strengthensthe property to characterize approval voting on dichotomous preferences with a variable set of alternatives.

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Suppose too that alternative z suddenly becomes infeasible, and another vote is takenwith K̄ = {x, y} as the new set of alternatives. We also suppose that the same groupof voters cast a ballot between the two votes.12 Then, based on the above assumption,we can infer the number of voters for each possible ballot on the second vote:

– two voters for {x},– one voter for {x, y},– no voters for {y}.

Note that π K̄ , or more precisely, its restriction on 2K̄ \ {∅}, summarizes the aboveresult.

Let us now consider another ballot response profile π ′ defined as

π ′({x}) = 2, π ′({x, y, z}) = 1,

π ′(B) = 0 for all B ∈ 2K \ {∅} with B �= {x}, K .

Supposing once again that the set of feasible alternatives is reduced to K̄ = {x, y},we can calculate the number of voters for each ballot on the second vote. It turns outthat we obtain exactly the same result as for π. (Note that this occurs if and only ifπ K̄ = π ′K̄ .) Therefore, when focusing on the outcome of the second vote, we shouldselect the same set of alternatives from K̄ for the two profiles π and π ′.

The meaning of IDA is now clear. Suppose that the observed ballot response profileis π, and the actual set of alternatives K̄ turns out to be smaller than the original setK. Then, IDA states that each f K̄ selects alternatives from K̄ based on the outcomeof the second vote π K̄ |2K̄ \{∅} rather than on π.13 This is the same as saying that there

is a function gK̄ from Π K̄ to 2K̄ \ {∅} such that

f K̄ (π) = gK̄(π K̄ |2K̄ \{∅}

)for all π ∈ Π, (1)

where Π K̄ is the set of all functions from 2K̄ \ {∅} to N. In fact, we can easily findsuch a gK̄ :14 Define a function gK̄ : Π K̄ → 2K̄ \ {∅} as

gK̄ (μ) = f K̄ (μ̄) for all μ ∈ Π K̄ ,

where μ̄ is the ballot response profile such that μ̄(B) = μ(B) if B ∈ 2K̄ \ {∅} andμ̄(B) = 0 otherwise. Note that μ coincides with the restriction of μ̄K̄ on 2K̄ \ {∅}.Then, for all π ∈ Π, we have π K̄ = (π K̄ )K̄ , and thus,

f K̄ (π) = f K̄(π K̄

)= gK̄

(π K̄ |2K̄ \{∅}

).

12 This supposition is crucial to the arguments that follow.13 We denote by π K̄ |

2K̄ \{∅} the restriction of π K̄ on 2K̄ \ {∅}.14 In addition, the function satisfying (1) is unique.

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We end this section by mentioning a possible controversy regarding IDA.15 UnderIDA, a voter whose ballot in the first vote is B ∈ 2K \ {∅} casts no vote in the secondvote if B ∩ K̄ = ∅. However, in practice, it is possible that the voter, perhaps out of asense of duty, may choose to vote for some alternatives in K̄ .

3 Main results

Our main result is the following theorem:

Theorem 1 The family of ballot aggregation functions { f K̄ : Π → 2K̄ \ {∅}}K̄∈K isapproval voting if and only if it satisfies F, C, SSA, and IDA.

We need several lemmas for the proof of this theorem.

Lemma 1 For each K̄ ∈ K,

f K̄(∑

a∈K̄

πa

)

= K̄ . (2)

Proof We first prove the assertion of the lemma for K̄ = K .

Let x ∈ f K (∑

a∈K πa) �= ∅. Since |K | ≥ 3, we can take another two elementsy, z from K. We prove that y, z ∈ f K (

∑a∈K πa).

First note that from SSA,

f {x, y}(∑

a∈K

πa

)

= {x} or {x, y},

f {x, z}(∑

a∈K

πa

)

= {x} or {x, z}.

Let us consider two ballot response profiles π1 = π{x,y} +πz and π2 = π{x,z} +πy .

It is easy to check that

π{x,z}1 = πx + πz =

(∑

a∈K

πa

){x,z}and π

{x,y}2 = πx + πy =

(∑

a∈K

πa

){x,y}.

Thus, IDA implies

f {x,z} (π1) = f {x,z}(∑

a∈K

πa

)

= {x} or {x, z}, (3)

f {x,y} (π2) = f {x,y}(∑

a∈K

πa

)

= {x} or {x, y}. (4)

15 I am grateful to one of the anonymous referees for pointing this out.

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Moreover, since π{x,y}1 = π

{x,y}{x,y} and π

{x,z}2 = π

{x,z}{x,z} , IDA together with F implies

f {x,y} (π1) = f {x,y} (π{x,y}

) = {x, y}, (5)

f {x,z} (π2) = f {x,z} (π{x,z}

) = {x, z}. (6)

From (3) to (6), we can deduce

x, y ∈ f K (π1) and x, z ∈ f K (π2) . (7)

To see this, we first show that f K (π1) ∩ {x, y, z} �= ∅. Suppose on the contrary thatf K (π1)∩ {x, y, z} = ∅. Take any b ∈ f K (π1), and let S = {b, x}. Then, on the onehand, SSA implies

f S(π1

) = f K (π1

) ∩ S = {b}.

On the other hand, since π S1 = π S

x , IDA implies

f S(π1

) = f S(πx

) = {x},

which is a contradiction. Thus, f K (π1)∩{x, y, z} �= ∅. Given this observation, SSAtogether with (3) and (5) implies x, y ∈ f K (π1). We obtain x, z ∈ f K (π2) in thesame manner.

From (7) and SSA, we deduce that

y ∈ f {y,z} (π1) and z ∈ f {y,z} (π2) .

Since π{y,z}1 = π

{y,z}2 , IDA implies

f {y,z} (π1) = f {y,z} (π2) = {y, z}.

Then, we deduce again from (7) and SSA that

{x, y, z} ⊂ f K (π1) and {x, y, z} ⊂ f K (π2) ,

and in turn, that f {x,z}(π1) = {x, z} and f {x,y}(π2) = {x, y}.The last two equations together with (3) and (4) imply

f {x,z}(∑

a∈K

πa

)

= {x, z} and f {x,y}(∑

a∈K

πa

)

= {x, y}.

From these equations and SSA, we conclude that y, z ∈ f K (∑

a∈K πa). Since y andz were arbitrarily chosen from K, we obtain f K (

∑a∈K πa) = K .

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Suppose then that K̄ �= K . On the one hand, SSA, together with the first part ofthe proof, implies

f K̄(∑

a∈K

πa

)

= f K(∑

a∈K

πa

)

∩ K̄ = K̄ .

On the other hand, since (∑

a∈K πa)K̄ = (∑

a∈K̄ πa)K̄ , IDA implies

f K̄(∑

a∈K

πa

)

= f K̄(∑

a∈K̄

πa

)

.

Therefore, f K̄ (∑

a∈K̄ πa) = K̄ . �

Lemma 2 Let {Bi }�i=1 be a partition of K , i.e., K = ⋃�i=1 Bi with Bi �= ∅ and

Bi ∩ B j = ∅ for all i, j. Then,

f K( �∑

i=1

πBi

)

=�⋃

i=1

Bi = K .

Proof First suppose that f K (∑�

i=1 πBi ) ∩ Bi = ∅ for some i. Taking any b ∈ Bi andx ∈ f K (

∑�i=1 πBi ), let S = {b, x}.

On the one hand, SSA implies

f S( �∑

i=1

πBi

)

= f K( �∑

i=1

πBi

)

∩ S = {x}.

On the other hand, since (∑�

i=1 πBi )S = (πb + πx )

S , IDA implies

f S( �∑

i=1

πBi

)

= f S (πb + πx ) .

Then, Lemma 1 implies f S(∑�

i=1 πBi ) = {b, x}, which is a contradiction.Next we show that Bτ ⊂ f K (

∑�i=1 πBi ) for all τ = 1, . . . , �. For each τ =

1, . . . , �, in view of the first part of the proof, SSA implies

f K( �∑

i=1

πBi

)

∩ Bτ = f Bτ

( �∑

i=1

πBi

)

. (8)

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N. Sato

Moreover, since (∑�

i=1 πBi )Bτ = π

Bτ

Bτ, IDA together with F implies

f Bτ

( �∑

i=1

πBi

)

= f Bτ(πBτ

) = Bτ . (9)

The assertion follows from (8) and (9).Finally, since f K (

∑�i=1 πBi ) ⊂ K , we obtain f K (

∑�i=1 πBi ) = K . �

Lemma 3 For all π ∈ Π and all B ∈ 2K \ {∅},

f K (π + πB) = f K(

π +∑

b∈B

πb

)

.

Proof By Lemma 2, for any π ∈ Π, and any B ∈ 2K \{∅}, the following two equationshold:

f K(∑

b∈B

πb + πK\B

)

= K and f K (πB + πK\B

) = K .

Then, C implies

f K(

π + πB +∑

b∈B

πb + πK\B

)

= f K (π + πB) ∩ K = f K (π + πB) ,

f K(

π +∑

b∈B

πb + πB + πK\B

)

= f K(

π +∑

b∈B

πb

)

∩ K = f K(

π +∑

b∈B

πb

)

.

Since the left-hand sides of the above two equations are equal, the assertion of thelemma follows. �

Lemma 4 For all π, π ′ ∈ Π,16

f K (π) = f K (π ′) if n(x, π) = n(x, π ′) for all x ∈ K .

Proof Note that π and π ′ can be “decomposed” as follows:

π =∑

B∈2K \{∅}π(B)πB and π ′ =

∑

B∈2K \{∅}π ′(B)πB .

16 Xu (2010) refers to this property as “equal treatment.”

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A characterization result for approval voting

Then, repeated application of Lemma 3 gives

f K (π) = f K( ∑

B∈2K \{∅}π(B)πB

)

= f K( ∑

B∈2K \{∅}π(B)

∑

b∈B

πb

)

= f K(∑

x∈K

n(x, π)πx

)

,

f K (π ′) = f K( ∑

B∈2K \{∅}π ′(B)πB

)

= f K( ∑

B∈2K \{∅}π ′(B)

∑

b∈B

πb

)

= f K(∑

x∈K

n(x, π ′)πx

)

.

Since n(x, π) = n(x, π ′) for all x ∈ K , we obtain f K (π) = f K (π ′). �

Lemma 5 For all K̄ ∈ K and all π ∈ Π with π K̄ = π̂ ,

f K̄ (π) = K̄ .

Proof We first prove that f K (π̂) = K . For this purpose, suppose that f K (π̂) �= K .

Then, from F and C,

f K (π̂ + πK

) = f K (π̂).

However, from the definition of π̂ , it follows that π̂ + πK = πK , and thus,

f K (π̂ + πK

) = f K (πK ) = K ,

which is a contradiction.Let π ∈ Π and K̄ ∈ K such that π K̄ = π̂ . Since π̂ = π̂ K̄ , IDA together with SSA

implies

f K̄ (π) = f K̄ (π̂) = f K (π̂) ∩ K̄ = K̄ ,

which completes the proof of Lemma 5. �

We are now ready to prove Theorem 1.

Proof of Theorem 1 It is easy to see that approval voting satisfies F, C, and SSA. Italso satisfies IDA since n(x, π) = n(x, π K̄ ) for all K̄ ∈ K and all x ∈ K̄ .

Suppose that a family of ballot aggregation functions { f K̄ : Π → 2K̄ \ {∅}}K̄∈Ksatisfies F, C, SSA, and IDA.

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N. Sato

We prove that for each K̄ ∈ K and each π ∈ Π,17

f K̄ (π) = arg maxx∈K̄

n(x, π).

Let m∗ = maxx∈K̄ n(x, π), which is well-defined since K̄ is finite.

We first consider the case m∗ = 0, which occurs only if π K̄ = π̂ .18 Then, byLemma 5, we obtain f K̄ (π) = K̄ = arg maxx∈K̄ n(x, π).

Now, suppose that m∗ ≥ 1. For each m = 1, . . . , m∗, define Am = {x ∈ K̄ :n(x, π) ≥ m∗ + 1 − m}. Note that

arg maxx∈K̄

n(x, π) = A1 ⊂ A2 ⊂ · · · ⊂ Am∗ ⊂ K̄ .

Let us now consider π∗ = ∑m∗m=1 πAm with π∗(B) �= 0 only if B = Am for some

m = 1, . . . , m∗. Then, by F and C,

f K (π∗) =m∗⋂

m=1

Am = A1.

We can easily verify that n(x, π K̄ ) = n(x, π∗) for all x ∈ K .19 Lemma 4 thus impliesf K (π K̄ ) = f K (π∗). Then, it follows from SSA that

f K̄(π K̄

)= f K

(π K̄

)∩ K̄ = f K (π∗) ∩ K̄ = A1.

Finally, by applying IDA we obtain

f K̄ (π) = f K̄(π K̄

)= A1 = arg max

x∈K̄

n(x, π),

which completes the proof. �Remark 1 To be precise, in Theorem 1, approval voting is characterized by five prop-erties. The fifth property is anonymity (A), that is, the names of voters do not affect theselection of alternatives. Note that in our setting, every family of ballot aggregationfunctions implicitly satisfies this property.

17 The method of proof that follows is based on those by Alós-Ferrer (2006) and Xu (2010).18 In the first version of this paper, the possibility of m∗ = 0 was omitted. The author is most grateful toone of the anonymous referees for pointing out this omission.19 More concretely, n(x, π K̄ ) = n(x, π∗) = 0 for all x /∈ K̄ , and for all x ∈ K̄ with n(x, π) = 0. (Notethat in both cases, x /∈ Am for all m = 1, . . . , m∗.) For x ∈ K̄ with n(x, π) = m̄ ≤ m∗ and m̄ ≥ 1, we

have n(x, π K̄ ) = n(x, π∗) = m̄. (Note that x ∈ Am for all m = m∗ + 1 − m̄, . . . , m∗, and the numberof such Am is equal to m̄.)

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Remark 2 Alós-Ferrer (2006) shows that when the set of alternatives is fixed, approvalvoting is characterized by three properties: F, C, and the property called cancellation.

Cancellation states that if all alternatives receive the same number of votes, allalternatives should be selected. In our setting, this can be stated as follows.

A ballot aggregation function f K : Π → 2K \ {∅} satisfies cancellation if forall K̄ ∈ K and all π ∈ Π,

f K̄ (π) = K̄ ,

whenever n(x, π) = n(y, π) for all x, y ∈ K̄ .

Since the properties stated in Lemmas 1, 2 and 5 are obvious implication of can-cellation, we could say that (a weak form of) cancellation is “hidden” in our results.20

4 Independence of the four properties

In this section, we show that each of the four properties in Theorem 1 is independentof the others.

4.1 Faithfulness

Define the family of ballot aggregation functions { f K̄1 : Π → 2K̄ \ {∅}}K̄∈K as

follows. For all K̄ ∈ K and all π ∈ Π,

f K̄1 (π) = K̄ .

It is obvious that { f K̄1 }K̄∈K satisfies C, SSA, and IDA, but not F.

4.2 Consistency

Define the family of ballot aggregation functions { f K̄2 : Π → 2K̄ \ {∅}}K̄∈K as

follows. For all K̄ ∈ K and all π ∈ Π,

f K̄2 (π) =

{{x ∈ K̄ : n(x, π) ≥ 1} if n(x, π) ≥ 1 for some x ∈ K̄ ,

K̄ otherwise.

It is easy to see that { f K̄2 }K̄∈K satisfies F and SSA. It also satisfies IDA since n(x, π) =

n(x, π ′) for all x ∈ K̄ whenever π K̄ = π ′K̄ . However, we can easily verify that{ f K̄

2 }K̄∈K fails to satisfy C.

20 This remark was suggested by one of the anonymous referees. The author is most grateful for thesuggestion.

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N. Sato

4.3 Stability on selected alternatives

Define the family of ballot aggregation functions { f K̄3 : Π → 2K̄ \ {∅}}K̄∈K as

follows. For all K̄ ∈ K and all π ∈ Π,

f K̄3 (π) = arg max

x∈K̄

w(

x, π K̄)

,

where

w(

x, π K̄)

=∑

B∈2K \{∅},x∈B

1

|B|πK̄ (B).

The family { f K̄3 }K̄∈K satisfies F, C, and IDA. However, as the following example

shows, it violates SSA.Let K = {x, y, z}, and consider the ballot response profile π̄ defined as follows.

π̄(B) ={

1 if B = {x} or {y, z},0 otherwise.

Then, f K3 (π̄) = {x}. However, for K̄ = {x, y},

π̄ K̄ (B) ={

1 if B = {x} or {y},0 otherwise,

and

f K̄3 (π̄) = {x, y}.

Therefore, f K̄3 (π̄) �= f K

3 (π̄) ∩ K̄ = {x}, which shows the violation of SSA.

4.4 Independence of dropped alternatives

Define the family of ballot aggregation functions { f K̄4 : Π → 2K̄ \ {∅}}K̄∈K as

follows. For all K̄ ∈ K and all π ∈ Π,

f K̄4 (π) = arg max

x∈K̄

w(x, π),

where

w(x, π) =∑

B∈2K \{∅},x∈B

1

|B|π(B).

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A characterization result for approval voting

The family { f K̄4 }K̄∈K satisfies F, C, and SSA. However, as the following example

shows, it fails to satisfy IDA.Let K = {x, y, z} and π̄ be the ballot response profile in the previous example.

Then, for K̄ = {x, y}, we have f K̄4 (π̄) = {x}. However, f K̄

4 (π̄ K̄ ) = {x, y} �=f K̄4 (π̄), which shows the violation of IDA.

5 Relation to Alcalde-Unzu and Vorsatz (2013)

In this section, we compare our result with that of Alcalde-Unzu and Vorsatz (2013),who analyze a generalized voting procedure for approval voting (type-weightedapproval voting) in a framework with a variable set of alternatives.21

For the comparison, we introduce some additional properties. A family of ballotaggregation functions { f K̄ : Π → 2K̄ \ {∅}}K̄∈K is:

Non-degenerate (ND) if f K̄ (π) �= K̄ for some K̄ ∈ K and π ∈ Π.

Neutral (N) if for all K̄ ∈ K, all π ∈ Π, and all permutation σ of K,

f K̄ (πσ

) = σ(

f K̄ (π)),

where πσ is a ballot response profile defined as πσ (B) = π(σ−1(B)) for allB ∈ 2K \ {∅}.

Non-degeneration (abbreviated by ND) is a property introduced in Alcalde-Unzu andVorsatz (2013), while neutrality (N) is a classical property in social choice theory.

One of the two main theorems of Alcalde-Unzu and Vorsatz (2013) (Theorem2) states that six properties A, ND, N, C, SSA, and IDA “essentially” characterizeapproval voting. “Essentially” means that although there is another voting proceduresatisfying the six properties, it is the same as approval voting for rational voters. Thisvoting procedure is disapproval voting, which is defined in our model as a family ofballot aggregation functions { f K̄ : Π → 2K̄ \ {∅}}K̄∈K satisfying for all K̄ ∈ K andall π ∈ Π,

f K̄ (π) = arg minx∈K̄

n(x, π).

Alcalde-Unzu and Vorsatz (2013) argued that if disapproval voting were implementedinstead of approval voting, rational voters would replace their ballot B with K \ B,

and thus, the two voting procedures always produce the same result. For the exactcharacterization of approval voting, the authors give some examples of the propertyto be added, such as (a weak form of) F.

21 The work of Alcalde-Unzu and Vorsatz (2013) has been developed independently of this paper. Theirmodel differs from ours in several aspects. For the details, see Footnote 4 (p. 15) of Alcalde-Unzu andVorsatz (2013). As mentioned in Footnote 4 of the current paper, Alcalde-Unzu and Vorsatz (2011), whichis the first version of Alcalde-Unzu and Vorsatz (2013), had considered the property corresponding to IDAprior to this paper (although Theorem 2 of Alcalde-Unzu and Vorsatz 2013 mentioned below did not appearin the first version of their paper).

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N. Sato

The result of Alcalde-Unzu and Vorsatz (2013) and our result are very similarsince five properties A, ND, C, SSA, and IDA are common to both results. (Notethat F implies ND.) Nevertheless, they are certainly different. For example, our resultsuggests that if we added F (in our definition) to the list of properties in Theorem 2of Alcalde-Unzu and Vorsatz (2013) in order to obtain an exact characterization ofapproval voting, N would be superfluous.22

6 Concluding remarks

In this paper, we showed that approval voting is the only family of ballot aggregationfunctions that satisfies F, C, SSA, and IDA. This result has the following implications.(1) Approval voting selects the same set of alternatives regardless of whether the first orsecond vote counts,23 and more importantly, (2) when F and C are given, only approvalvoting has this property among all the families of ballot aggregation functions. Whereasthe former implication is well known, the latter is new and should further increase thepractical significance of approval voting.

We conclude the paper with a remark on the importance of the cardinality of K forour results. We assume that |K | is greater than or equal to 3 throughout the paper.Unfortunately, we cannot extend Theorem 1 to the case of |K | = 2, since SSA andIDA are vacuous in this case, and as the following example shows, F and C are notsufficient to characterize approval voting.

Let K = {x, y}, and f K5 be the ballot aggregation function defined as follows. For

all π ∈ Π,

f K5 (π) =

{{x} if π({x}) = π({y}) ≥ 1,

arg maxx∈K n(x, π) otherwise.

We can verify that f K5 satisfies F and C, although this is not approval voting.24

Acknowledgments The author is grateful to the Associate Editor and two anonymous referees for theirinsightful comments and useful suggestions.

22 In addition, the proof techniques used in Alcalde-Unzu and Vorsatz (2013) and the current paper arequite different.23 More technically, this property can be expressed as follows. For each K̄ ∈ K and π ∈ Π with f K (π)∩K̄ �= ∅ :

f K (π) ∩ K̄ = f K̄(π K̄

).

Recall that we can regard π K̄ as the outcome of the second vote.24 We can eliminate this counterexample by adding N.

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