Title
On the Normative Uniqueness of Majority Rule: Generalizing May’s Theorem for Arrovian Social Welfare Functions1
Abstract
May's theorem demonstrated that when restricted to two alternatives and a particular ballot domain, majority rule is uniquely characterized by four normative conditions. The theorem is
widely used as a normative defense of majority preference as "the" democratic decision-making
norm. It was later shown that May's theorem is a two-alternatives special case of a multiple-alternatives characterization of plurality rule. Left open was the question: "Are there other voting systems that could be uniquely characterized by the four conditions on their ballot domains?" While
a very large number of voting systems satisfy the four conditions for multiple alternatives, we
identify all Arrovian social welfare functions that uniquely satisfy the four conditions on their
respective ballot domains for multiple alternatives. Our results suggest that while majority preference may have some normative appeal, it is perhaps not so unique in the ways in which it is appealing.
1
The main result of this paper is a special case of a more general result I have arrived at, which generalizes May’s
theorem beyond Arrovian social welfare functions to cardinal social welfare functions. A rough draft of that result
is here.
1
Introduction
May’s theorem shows that for a particular ballot domain with three ballots and two alternatives, majority rule is uniquely characterized by four conditions: decisiveness, anonymity, neutrality, and positive responsiveness (May 1952). Known as May’s theorem, this is a significant
result discussed in standard overviews of social choice theory (Arrow 1963; Barry and Hardin
1982; List 2013; Mueller 2003; Nitzan 2010; Patty and Penn 2014; Sen 1970). But the theorem’s
importance goes well beyond social choice, and has become a significant argument for majority
rule, and more generally majority preference,2 as the democratic group decision-making norm in
political theory and philosophy (Ackerman 1980; Beitz 1989; Chapman and Wertheimer 1990;
Christiano 1996; Christiano 2008; Coleman and Ferejohn 1986; Copp, Hampton and Roemer
1993; Dahl 1989; Riker 1982; Risse 2004; Saward 1998; Vermeule 2014; Waldron 1999). Given
its importance, there have been several excellent extensions of May’s theorem building on May’s
original work (Cantillon and Rangel 2002; Dasgupta and Maskin 2008; Fey 2004; Fishburn
1978; Freixas and Zwicker 2009; Goodin and List 2006; McMorris and Powers 2008; Murakami
1966; Murakami 1968; Pattanaik 1971; Sato 2015; Sen and Pattanaik 1969; Surekha and Rao
2010; Vorsatz 2007). We build on these multiple traditions.
2
A more rigorous distinction between majority rule and majority preference will be made later,
but for now, it suffices to say, majority rule is a voting system that can only be applied in cases
where there are exactly two alternatives, while majority preference is a norm that can be applied
with two or more alternatives.
2
In a 2006 AJPS paper, Robert Goodin and Christian List demonstrated that May’s theorem is a special case of a characterization of plurality rule. Specifically, for a specific ballot domain, plurality rule is uniquely characterized by generalized versions of the four conditions; and
when the number of alternatives is reduced to two, May’s theorem is a special case of the
Goodin-List characterization.
The Goodin-List paper left open the question of whether or not other voting systems were
uniquely characterized by the appropriately generalized four conditions on the voting systems’
respective ballot domains. In this paper we identify all Arrovian social welfare functions that are
uniquely characterized by the four conditions on their respective ballot domains. In the appendix,
using a different generalization of May’s positive responsiveness condition, we identify all Arrovian social welfare functions which are uniquely characterized on their respective ballot domains
by decisiveness, anonymity, neutrality, and this different version of positive responsiveness.
Definitions
Let the set of n alternatives, namely a1, a2, …, an, be A, where n > 1 and an integer. A
ballot, b, is a complete and transitive rank ordering of the alternatives in A.3 If ax is ranked above
ay in a given ballot, then when we express this ballot, there will be at least one vertical bar, “|”,
3
A ballot is complete if all n alternatives in A occur in the ballot. A ballot is transitive if it fol-
lows these four rules: If in the ballot, ax is ranked above ay, and ay is ranked above az, then ax is
ranked above az in that ballot. If in the ballot, ax is ranked above ay, and ay is ranked equally with
az, then ax is ranked above az in that ballot. If in the ballot, ax is ranked equally with ay, and ay is
ranked above az, then ax is ranked above az in that ballot. If in the ballot, ax is ranked equally
with ay, and ay is ranked equally with az, then ax is ranked equally with az in that ballot.
3
between ax and ay. If ax and ay are ranked equally on a given ballot, then when we express this
ballot, there will be no vertical bar between ax and ay. For example, the ballot ax|ayaz|aw means
“ax is ranked above ay and az which are ranked above aw, while ay and az are ranked equally.” Let
the set of m voters be V (where m > 1 and a positive integer), with the corresponding ballots b1,
b2, …, bm. Those m ballots are the profile p.
A part of a ballot consists of all alternatives that are equally ranked. For example,
ax|ayaz|aw consists of three parts: ax is the first part, ayaz is the second part, and aw is the third
part. The example ballot is polychotomous, because a polychotomous ballot is a ballot with more
than two parts. (A dichotomous ballot has exactly two parts. A trivial ballot has exactly one
part.) The number of alternatives in the first part of the example ballot is n1 = 1 (since ax is the
only alternative in the first part), in the second part n2 = 2, and in the third part n3 = 1. Thus the
example ballot is a 1-2-1 composition of four alternatives. When there are n alternatives, there
are 2n-1 possible compositions. For example, with four alternatives, there are eight possible compositions: 4, 3-1, 2-2, 1-3, 2-1-1, 1-2-1, 1-1-2, and 1-1-1-1.
But keep in mind, for any given composition, if it consists of d parts, there are
n!/[(n1!)(n2!)…(nd!)] logically possible ballots that have that composition given A. For example,
there are twelve different possible ballots with the composition 1-2-1 where A = {w, x, y, z}:
w|xy|z,
x|wy|z,
y|wx|z,
z|wx|y,
w|xz|y,
x|wz|y,
y|wz|x,
z|wy|x,
w|yz|x,
x|yz|w,
y|xz|w,
z|xy|w
Those 12 ballots compose a bundle; a bundle is the set of all ballots of a given composition given A.
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A ballot domain, B, lists all possible ballots which voters are allowed to submit to the Arrovian social welfare function f given A. Let c be the number of ballots in B. For example, if f is
plurality voting, B contains exactly c = n + 1 ballots. One of those ballots is the trivial ballot. The
remaining n ballots are all n ballots of the composition 1-(n - 1) given A.
Plurality voting has what is called a bundled ballot domain. A bundled ballot domain is a
ballot domain where if a ballot b occurs in the ballot domain, then every ballot in the bundle to
which b belongs occurs in the ballot domain. For example, for every ballot in a plurality voting
ballot domain, every other member of that ballot’s bundle is also in the ballot domain; only two
compositions occur in a plurality voting ballot domain: the trivial ballot and the 1-(n - 1) composition. The trivial ballot is only member of its composition given A. And every possible ballot of
the 1-(n - 1) composition given A occurs in the plurality voting ballot domain.
An Arrovian social welfare function (SWF), f, takes a profile p and outputs f(p), which is
a complete and transitive rank ordering of A or the empty set.4
May’s Four Conditions Generalized to SWFs
A SWF is decisive if for each of the cm possible profiles, f(p) is some complete and transitive preference ordering of A.
Let p′ be identical to p, except that ballots in p have been permuted among the voters in
p′. A SWF is anonymous if for any such p and p′, f(p) = f(p′).
4
Admittedly, it is unusual that both rank orderings and sets are included in the codomain of f(p).
This is done to accommodate May’s notion of decisiveness. In this context, the output of an
empty set by f(p) means that f has no output for p.
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Let p* be identical to p, except that alternatives in p have been permuted in p*. A SWF is
neutral if for any such p and p*, f(p*) always permutes alternatives in f(p) in the same manner as
p* permutes alternatives in p.
Consider two profiles p and p†, each with m ballots ranking all n alternatives in A. Profile
p and p† consist of the ballots b1, …, bm, and b1†, …, bm†, respectively. Suppose that these profiles are identical except for one ballot; more precisely suppose that bi = bi† except for a single
index j for which bj ≠ bj†. Further suppose that bj and bj† only differ with regards to a particular
alternative ax in A, and in a particular manner; precisely, suppose that for any alternatives ay ≠ ax
and az ≠ ax, the preference relations between ay and az are identical to each other in bj and bj†.
The only difference between bj and bj† is that there exists at least one alternative a where the
change from bj to bj† raises the preference of ax relative to a.5 We say that a change from such a
p to p† is a positive flip with respect to ax.
A SWF is positively responsive if it is always the case that for any positive flip from p
and p†, with respect to alternative ax, that if f(p) ranks ax above ay or f(p) ranks ax and ay equally,
then f(p†) ranks ax above ay.
Some Conditions for Ballot Domains
We have already discussed the bundled ballot domain condition. Here, we will discuss
some more ballot domain conditions. A ballot domain is non-polychotomous if it contains no
polychotomous ballots. If all non-trivial ballots in a ballot domain are of the same composition,
5
By raising the preference of ax relative to a, we mean that [if in bj a is ranked above ax, then (ax
is ranked above a in bj†) or (ax is ranked equally with a in bj†)] or [if in bj a is ranked equally
with ax, then ax is ranked above a in bj†].
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then the ballot domain satisfies compositional dictatorship. If a ballot domain contains the trivial
ballot, then the ballot domain satisfies trivial allowance. If all non-trivial ballots in the ballot domain are such that exactly one alternative is ranked above all other n - 1 alternatives or exactly
one alternative is ranked below all other n - 1 alternatives, then such a ballot domain is a resolute
ballot domain. A ballot domain is said to be unrestricted if the ballot domain contains all logically possible complete and transitive preference orders given the n alternatives in A.
If a ballot domain is such that it is impossible for any positive flip to occur between any
two profiles, then such a ballot domain is non-flippable. For example, consider a compositionally
dictatorial ballot domain where n > 2 and all ballots are dichotomous. Under such circumstances,
raising some ax in a ballot requires lowering some ay, which changes the preference relations between such an ay and some az. This of course makes such a ballot domain non-flippable.
Some SWFs
We will need some SWFs to demonstrate our results. At this juncture, I will define the
SWFs, while examples, which clarify the nature of the SWFs, will be provided in the proofs
when the SWFs are applied. It is worth mentioning a few things at this point. First, in order to
satisfy neutrality, it is necessary that the ballot domain of the SWF is a bundled ballot domain.
Second, if a ballot domain is non-flippable, then any SWF on that ballot domain will necessarily
satisfy positive responsiveness. Third, it is easy to verify that all of the SWFs described here satisfy decisiveness, anonymity, neutrality, and positive responsiveness.
Majority Rule: By definition, n = 2 and the ballot domain contains all three logically possible
ballots. (This is the majority rule ballot domain.) If in the profile p, the number of voters that
place ax in the first part of their ballots is greater than that for ay, then ax is ranked above ay. If an
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equal number of voters place ax in the first parts of their ballots as those that do that for ay, then
the two alternatives are socially ranked equally.
Maximum Borda Mark Maximization (MaxBMM): The ballot domain is always some bundled
ballot domain with some polychotomous ballots. The Borda mark for an alternative on a given
ballot is the number of alternatives ranked below that alternative on that ballot minus the number
of alternatives ranked above that alternative on that ballot. The m Borda marks in the profile for
each alternative are arrayed from largest to smallest. In the first iteration, the highest Borda mark
for each alternative is compared. If ax has a larger highest mark than ay, then ax is ranked higher.
If two or more alternatives have the same highest mark, this tentative tie is broken by examining
the second highest marks of the tentatively tied alternatives. In this second iteration, applied to
tentatively tied alternatives (say ax and ay), if the second highest mark of ax is greater than the
second highest mark for ay, then the tentative tie is broken and ax is ranked above ay. If such an
ax and ay have the same second highest marks, we iteratively proceed thru the third highest,
fourth highest, and so on, marks in the same manner until the tie is broken or all m Borda marks
are exhausted. If they are exhausted, then tentatively tied alternatives are tied in the social rank
ordering of alternatives.
Minimum Borda Mark Maximization (MinBMM): The ballot domain is always some bundled ballot domain with some polychotomous ballots. The aggregation procedure is the same as that for
MaxBMM, but instead of maximizing the maximum Borda mark, we maximize the minimum
Borda mark, where tentative ties are broken by iteratively examining the second lowest, then
third lowest, and so forth until all m Borda marks are exhausted. Alternatives that remain tentatively tied after all m Borda marks have been exhausted are socially ranked equally.
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Unrestricted Borda Count: The ballot domain consists of all possible complete and transitive
preference orders of the n alternatives. The Borda score of an alternative ax is the sum of all m
Borda marks for ax in p. If the Borda score of ax is greater than the Borda score of ay, then rank
ax above ay. If they have equal Borda scores, then rank them equally.
Dichotomous Voting (DV): The ballot domain is always some non-polychotomous bundled ballot
domain. If an alternative occurs on a trivial ballot, then that alternative is assigned a zero DV
mark for that ballot. If an alternative occurs on a ballot with two parts, but occurs on the second
part of the ballot, then the alternative is assigned a zero DV mark for that ballot. But if the alternative occurs on a ballot with two parts, and is in the first part of the ballot, then the alternative is
assigned a DV mark of one for that ballot. Sum an alternative’s DV marks across all m ballots in
the profile to find its DV score. If the DV score of ax is greater than the DV score of ay, then rank
ax above ay. If they have equal DV scores, then rank them equally.
Dichotomous Composition Weighted Voting Systems (DCWV): The ballot domain is always some
non-polychotomous bundled ballot domain with at least two different bundles of dichotomous
ballots. Specifically, say that the ballot domain contains exactly h bundles of dichotomous ballots, namely the bundles g1, …, gh. If an alternative occurs in the last part of a dichotomous ballot
or in a trivial ballot, then that alternative is given a zero DCWV mark for that ballot in the profile. If an alternative is placed in the first part of a ballot from the bundle g1, then that alternative
is assigned a DCWV mark equal to k1, for that ballot in the profile, where k1 > 0. If an alternative
is placed in the first part of a ballot from the bundle g2, then that alternative is assigned a DCWV
mark equal to k2, for that ballot in the profile, where k2 > 0. And so forth. The m DCWV marks
for an alternative in a profile are summed up to produce the DCWV score for that alternative. If
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the DCWV score of ax is greater than the DCWV score of ay, then rank ax above ay. If they have
equal DCWV scores, then rank them equally.
Bloc Voting: The ballot domain is bundled and consists of exactly one dichotomous bundle and
the trivial ballot. If ax occurs on the first part of more ballots in p than does ay, then ax is socially
ranked above ay. If ax and ay occur in the same number of first parts of ballots in p, then ax and ay
are socially ranked equally.
Pythagorean Voting (PV): The ballot domain is always some non-polychotomous, compositionally dictatorial, and non-resolute bundled ballot domain with dichotomous ballots. We say there
are s ballots with exactly two parts in the ballot domain, and that l is the number of alternatives
in the first part of any dichotomous ballot from the ballot domain. We can label each of these s
ballots b(1), b(2),…, b(s). Now we can examine the profile p and count how many voters, for example, submitted b(1). Let the number of voters that submitted b(1) in p be m1; the number that
submitted b(2) in p be m2, and so forth. Note that for the ballot domain, any given alternative in
A will occur in the first part of exactly r of the ballots that have exactly two parts, where ls/n = r.
For an alternative ax in A, let us call the r ballots where ax is in the first part: bx(1), bx(2), …,
bx(r). Let mx(1) be the number of times that bx(1) occurs in p, mx(2) be the number of times that
bx(2) occurs in p, and so forth. If that is the case, then the Pythagorean score of ax for p is
[mx(1)]2+[mx(2)]2+…+[mx(r)]2. (Thus the intuition behind calling the SWF, Pythagorean voting).
If the Pythagorean score of ax is greater than the Pythagorean score of ay, then ax is ranked above
ay. If they have the same Pythagorean scores, then they are ranked equally.
Plurality Voting: The ballot domain contains the trivial ballot and the n ballots in the 1-(n - 1)
composition bundle. (That is called the plurality voting ballot domain). The plurality score for an
alternative ax is constructed by counting the number of ballots in p where ax occurs in the first
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part of the ballot. If the plurality score of ax is greater than that for ay, then ax is ranked above ay.
If they have the same plurality score, they are ranked equally.
Anti-Plurality Voting: The ballot domain contains the trivial ballot and the n ballots in the (n - 1)1 composition bundle. (That is called the anti-plurality voting ballot domain). The anti-plurality
score for an alternative ax is constructed by counting the number of ballots in p where ax occurs
in the first part of the ballot. If the anti-plurality score of ax is greater than that for ay, then ax is
ranked above ay. If they have the same anti-plurality score, they are ranked equally.
Absolute Anti-Plurality Voting, Absolute Bloc Voting, Absolute Majority Rule, and Absolute Plurality Voting: These four SWFs are respectively identical to anti-plurality voting, bloc voting, absolute majority rule, and plurality voting, except they lack the trivial ballot.
Non-Flippable Trivial Voting (NTV): The ballot domain is any non-flippable bundled ballot domain. (But note, when the ballot domain contains only the trivial ballot and no other ballots, this
is called the trivial ballot domain). The output for each of the cm possible profiles is always a
rank ordering of alternatives where all alternatives are equally ranked.
Theorem 1 (Generalized May’s Theorem for Arrovian SWFs)
Statement: Define the collection of decisive, anonymous, neutral, and positively responsive Arrovian SWFs as the admissible class of SWFs. Fix a nonempty set of alternatives A, a set of at least
two voters V, and a ballot domain B. Then [an SWF f is the unique admissible SWF on B] iff [f is
plurality voting and B is the plurality voting ballot domain, or f is anti-plurality voting and B is
the anti-plurality voting ballot domain, or f is NTV and B is the trivial ballot domain, or f is absolute majority rule and B is the absolute majority rule ballot domain.]
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Proof Sketch: The proof is completed by demonstrating two lemmas. In the first lemma, we identify all SWFs that are unique members of the admissible class on their respective ballot domains,
if n = 2. In the second lemma, we do the same, but for n > 2.
Given that if a ballot domain is not bundled then no SWF on that ballot can satisfy neutrality, proof of the second lemma requires examination of seven cases. First, we show that if a
ballot domain is not non-polychotomous, then a SWF on that ballot domain cannot be uniquely
characterized by the four conditions. Second, we show that if a non-polychotomous ballot domain does not satisfy compositional dictatorship, then a SWF cannot be uniquely characterized
by the four conditions on that ballot domain. Third, if a non-polychotomous and compositionally
dictatorial ballot domain is not resolute, then there is not a SWF that is uniquely characterized by
the four conditions on that ballot domain. Fourth, if n > 2 and a non-polychotomous, compositionally dictatorial, resolute ballot domain does not satisfy trivial allowance, then no SWF is
uniquely characterized by the four conditions on that ballot domain. Fifth, if the ballot domain is
the anti-plurality voting ballot domain, then the four conditions are uniquely characterized by
anti-plurality voting. Sixth, if the ballot domain is the plurality voting ballot domain, then the
four conditions are uniquely characterized by plurality voting. Seventh, if the ballot domain consists of one ballot, namely the trivial ballot, then the four conditions are uniquely characterized
on that ballot domain by NTV. This examination exhausts all possible Arrovian ballot domains.
Lemma 1 (n = 2): If n = 2, then (a ballot domain is bundled) iff (some SWF uniquely satisfies decisiveness, anonymity, neutrality, and positive responsiveness on that ballot domain).
First note, if a ballot domain is not bundled, it is impossible for any SWF using that ballot
domain to satisfy neutrality. If n = 2, there are exactly three bundled ballot domains: trivial, majority rule, and absolute majority rule. Clearly, NTV uniquely satisfies decisiveness, anonymity,
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and neutrality on the trivial ballot domain. Since this ballot domain is non-flippable, NTV vacuously satisfies positive responsiveness on the trivial ballot domain. Therefore, NTV is the unique
member of the admissible class on the trivial ballot domain. May’s theorem clearly demonstrates
that majority rule is the unique member of the admissible class for the majority rule ballot domain (May 1952). And it is easy to go from May’s theorem, and show that absolute majority rule
is the unique admissible member of the absolute majority rule ballot domain.
Lemma 2 (n > 2): If n > 2, then (a ballot domain is bundled, non-polychotomous, compositionally dictatorial, resolute, and trivial allowing) iff (some SWF uniquely satisfies decisiveness, anonymity, neutrality, and positive responsiveness on that ballot domain)
Case 1 (Non-polychotomy): If a ballot domain contains any polychotomous ballots, it will fail to
have a unique member in the admissible class.
Consider an election with m voters. Suppose m - 1 voters submit exactly the same polychotomous ballot. The remaining voter submits a different ballot, but one from the same bundle.
Since these are polychotomous ballots, they will all have a first part, second part, and last part.
All voters place ax in the second parts of their ballots. The m - 1 voters place ay in the first part of
their ballot, while the remaining voter places ay in the last part of her ballot.
With MaxBMM, the first thing we do is array the Borda marks from largest to smallest
for each alternative. For ax, since all voters have ballots from the same composition, and all voters placed ax in the second part of their respective ballots, the Borda marks for ax for all voters
will be the same. The maximum Borda mark for ax is therefore clearly the number of alternatives
ranked below the second part minus the number of alternatives ranked in the first part of a ballot
in the profile (since all ballots in the profile are of the same composition). For ay, m - 1 of the
Borda marks are equal to the number of alternatives ranked below the first part; and exactly one
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Borda mark for ay is negative one times the number of alternatives ranked above the last part of a
ballot in the profile. Clearly, the maximum Borda mark for ay is the number of alternatives
ranked below the first part. Therefore, the maximum Borda mark for ay is larger than the maximum Borda mark for ax, so ay is ranked above ax. Had their maximum Borda marks been the
same, we would have iteratively examined their second largest, third largest, and so on Borda
marks until we break the tentative tie or all m marks are exhausted.
Now suppose that instead of MaxBMM, we had used MinBMM on the same profile. The
first step is to array each alternative’s Borda marks from largest to smallest, which we have already done. The smallest Borda mark for ax is the number of alternatives ranked below the second part minus the number of alternatives in the first part of a ballot in the profile. The smallest
Borda mark for ay is negative one times the number of alternatives ranked above the last part of a
ballot in the profile. Clearly, the smallest Borda mark for ax is larger than the smallest Borda
mark for ay; therefore, MinBMM ranks ax above ay. Had ax and ay had identical smallest Borda
marks, we would have examined their second smallest, third smallest, and so on Borda marks until we broke the tentative tie or exhausted all m marks.
Since both MaxBMM and MinBMM are in the admissible class for any bundled ballot
domain with a polychotomous ballot, yet they produce different social rankings, there can be no
unique member of the admissible class if the ballot domain contains a polychotomous ballot.
Case 2 (Compositional Dictatorship): If a non-polychotomous ballot domain is not compositionally dictatorial, it does not have a unique member in its admissible class.
Consider an election with m voters where the ballot domain is bundled and non-polychotomous, but contains at least two bundles of dichotomous ballots. Let us call two of those bundles
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with dichotomous ballots g1 and g2. Define m(g1) and m(g2) to be the number of voters that submit a ballot from g1 and g2, respectively. Assume that in this profile, m(g1) > m(g2) > 0 and that
m(g1) + m(g2) = m. Now suppose that the m(g1) voters that submit a ballot from the bundle g1 are
all submitting the same ballot, where ax is in the first part and ay is in the second part. Similarly
suppose that the m(g2) voters that submit a ballot from the bundle g2 are all submitting the same
ballot, where ay is in the first part and ax is in the second part. Clearly, for any such profile, there
exists some k1 and k2 where k1m(g1) = k2m(g2) such that some DCWV system will rank ax and ay
equally, but dichotomous voting will rank ax above ay.6 Given that dichotomous voting and
DCWV systems are in the admissible class, there can be no unique member of the admissible
class if a ballot domain is non-polychotomous but not compositionally dictatorial.
Case 3 (Resoluteness): If a ballot domain, which contains dichotomous ballots, is bundled, nonpolychotomous, and compositionally dictatorial, but not resolute, then there is no unique member
in the admissible class for that ballot domain.
A simple proof for this case involves a profile with m voters, where more voters place ax
in the first part of their ballot than voters place ay in the first part of their ballot. Dichotomous
voting socially ranks ax above ay for such profiles, but NTV ranks them equally.
6
This example leaves out the small subcase where m = 2. However, it is easy to construct an ex-
ample with two voters where the ballot domain is bundled and non-polychotomous, but contains
at least two bundles of dichotomous ballots. In such an example, which is left to the reader, dichotomous voting produces a tie between ax and ay, while some DCWV system ranks ax above
ay.
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That said, it is helpful to show that something similar can be demonstrated with a less
vacuous SWF than NTV. Consider an election where m > 3 and is even, and three of the alternatives under consideration are ax, ay, and az. Suppose the ballot domain, which contains dichotomous ballots, is bundled, non-polychotomous, and compositionally dictatorial, but not resolute.
Suppose that in the profile, exactly m/2 voters submit the same ballot, where ax is in the first part
and ay is in the last part. The remaining m/2 voters each have one of two ballots: at least one
voter has a ballot where ay and az are in the first part and ax is in the last part, and at least one
voter has a ballot where ay is in the first part and ax and az are in the last part. By dichotomous
voting, ax and ay are ranked equally, but by Pythagorean voting, due to superadditivity, ax is socially ranked above ay.7
Case 4 (Trivial Allowance): If n > 2 and the ballot domain is bundled, non-polychotomous, compositionally dictatorial, and resolute, but there is no trivial ballot, then there is not a unique member of the admissible class on that ballot domain.
For such a ballot domain, it is clear it is non-flippable. Dichotomous voting and NTV
both clearly are members of the admissible class, so there can be no unique member for this case.
Case 5 (Anti-Plurality Voting): If the ballot domain is the anti-plurality voting ballot domain,
then the unique member of the admissible class is anti-plurality voting.
Suppose ax and ay are alternatives in A, where the ballot domain B consists of a trivial
ballot and the (n - 1)-1 composition bundle. Let the x-anti-plurality ballot be the one where ax is
7
If m > 3 and odd, all we need to do to the profile is add a ballot where both ax and ay are in the
last part. Dichotomous voting and NTV cover the subcases where m is two or three.
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ranked below all other alternatives. Let mx(p) be the number of x-anti-plurality ballots in the profile p. An x-positive flip occurs to a ballot when the preference of ax is raised while the preference relations between any other two alternatives on the ballot remains the same. Likewise for
ay, y-anti-plurality ballot, my(p), and y-positive flip.
Note that if mx(p) = my(p), then if f satisfies the four conditions, then f(p) must rank ax and
ay equally. This is because when we permute the alternatives ax and ay in p, the only ballots that
can change are the x-anti-plurality and y-anti-plurality ballots. This occurs since in all other ballots that might exist in p, ax and ay are in the same part. So when we permute ax and ay in p, the
x-anti plurality ballots become y-anti-plurality ballots and vice versa. Thus, neutrality, along with
decisiveness and anonymity, require that f(p) rank ax and ay equally if mx(p) = my(p).
Now suppose that mx(p) > my(p). Then this means that if mx(p) - my(p) of the x-anti-plurality ballots are converted to trivial ballots, then the number of x-anti-plurality and y-anti-plurality ballots would be the same on the resulting profile. Such a conversion can be interpreted as a
series of mx(p) - my(p) number of positive flips, where ax is raised on mx(p) - my(p) ballots, one by
one. But these mx(p) - my(p) x-positive flips create a new profile, say p†, where mx(p†) = my(p†).
Therefore, for p†, f(p†) must output a tie between ax and ay. That said, f(p) cannot rank ax above
or equal to ay, because a series of x-positive flips from p to p† leading to a tie between ax and ay
in f(p†) would violate positive responsiveness. Therefore, if mx(p) > my(p) (i.e. more voters rank
ay above ax than rank ax above ay), then f(p) must rank ay above ax if f satisfies the four conditions.
Using appropriate symmetries, we can similarly show that if my(p) > mx(p), then f(p) must
rank ax above ay in order for f to satisfy the four conditions. Therefore, if the ballot domain is the
17
trivial ballot and the m ballots in the (n - 1)-1 bundle, then the four conditions are uniquely satisfied by anti-plurality voting.
Cases 6 & 7 (Plurality Voting and NTV): Plurality voting and NTV are the unique members of
the admissible class on their ballot domains, the plurality voting and trivial ballot domains, respectively.
Applying appropriate symmetries to the proof from case 5, we can demonstrate that plurality voting is the unique member of the admissible class for the plurality voting ballot domain.
And it should be clear that NTV is the only member of the admissible class for the trivial ballot
domain. ■
Corollary 1 (Goodin-List Plurality Voting Characterization): Plurality voting is the unique social
choice function that satisfies decisiveness, anonymity, neutrality, and positive responsiveness on
the plurality voting ballot domain.8
Corollary 2 (May’s Theorem): A SWF satisfies decisiveness, anonymity, neutrality, and positive
responsiveness on the majority rule ballot domain iff the SWF is majority rule.
Necessity or Sufficiency
8
Technically speaking, the Goodin-List characterization from (Goodin and List 2006) is done in
terms of social choice functions (SCFs), which only choose a non-empty subset of alternatives as
winners instead of ranking all alternatives, like SWFs. (So roughly speaking, the plurality rule
SCF only outputs the first part of the social rank orderings outputted by the plurality voting
SWF.) This technical distinction does not arise for May’s theorem and majority rule since SCFs
and SWFs are the same thing when n = 2.
18
Most experts agree that May’s conditions have great normative appeal (Dahl 1989; List
2013), though this opinion is not universal (Sen 1970). If we take their normative appeal for
granted (given that issue is discussed extensively elsewhere), do we appreciate them for their necessity or their sufficiency. More specifically, are the four conditions desirable when they are
sufficient to specify a unique SWF on a given ballot domain, or are they normatively desirable
conditions that basically any “good” SWF should satisfy?
If the four conditions are just normatively desirable conditions that “good” SWFs should
satisfy, then we have shown that there are several SWFs that satisfy all four conditions. For example, if we wanted to allow each voter the opportunity to submit any complete and transitive
ballot given A (i.e. unrestricted ballot domain), then MaxBMM, MinBMM, and unrestricted
Borda count can all satisfy the four conditions and unrestricted ballot domain.9 And as noted be-
9
In the process of constructing a social rank ordering, MaxBMM and MinBMM can be under-
stood as both arraying the m Borda marks for each alternative from lowest to highest. MaxBMM
uses the largest order statistic first to break ties between alternatives, then the second largest order statistic if necessary, and so forth until the first order statistic if necessary. MinBMM uses the
first order statistic at first to break ties, then the second order statistic if necessary, and so forth
until the largest order statistic if necessary. With m voters (and thus m Borda marks for each alternative), there are m! ways to order the order statistics to break ties, which means that MaxBMM and MinBMM are just two SWFs out of m! SWFs (Balinski and Laraki 2010), which can
satisfy the four conditions and unrestricted ballot domain.
19
fore, all SWFs in the “Some SWFs” section satisfy the four conditions. If satisfying the four conditions are necessary to be a “good” SWF, even without the SWF being unique on the ballot domain, then clearly majority preference is not so unique when n > 2.
If on the other hand, the normative desirability of the four conditions is that they specify
a unique SWF on a ballot domain, then (putting aside absolute majority rule, which does not apply for n > 2 given our definition of positive responsiveness, and trivial voting for practical purposes), the four conditions identify two special SWFs: plurality voting and anti-plurality voting.
And when n = 2, plurality and anti-plurality voting are majority rule. But is there any normative
basis for why we would want a SWF uniquely characterized by the four conditions on a given
ballot domain, or is this akin to seeking we restrict the set of integers to those below five to ensure that three is the unique odd prime number?
When (Goodin and List 2006) demonstrated that plurality rule (i.e. plurality voting as a
social choice function), was uniquely characterized on its ballot domain, their normative defense
of plurality rule was a conditional one. Specifically, they stated, “To emphasize, we do not unconditionally defend plurality rule. In particular, we do not defend single-vote balloting procedures [i.e. plurality rule ballot domains]; we only make a conditional claim: If single-vote balloting is used—as it often is, in practice—then plurality rule is the way to go (Goodin and List
2006, 941).”
Usually, when we speak of May’s theorem, we think of it as having four conditions. But
what (Goodin and List 2006) suggests is that we should really think of it as having at least five
conditions: the traditional four and a ballot domain condition. But if we are going to impose a
ballot domain condition, which restricts possible ballots, we need to have good normative reasons for such a restriction, because as Goodin and List state themselves, “It would undeniably be
20
ideal in many cases to collect voters’ full preferences, top to bottom, over all available options. It
would at least be an improvement to collect more (even short of “full”) information about these
preferences…. Such richer informational environments would allow us to use more sophisticated
aggregation procedures than plurality rule (Goodin and List 2006, 941).”
As noted before, in order to for a SWF to uniquely satisfy the four conditions, it is necessary that the SWF’s ballot domain satisfy some ballot domain conditions. In other words, we
may have to justify those ballot domain conditions if we wish to claim that uniquely satisfying
the four conditions on a ballot domain is normatively desirable. But the normative case for those
ballot domain conditions does not appear to be clear cut, as the following SWF condition demonstrates.
First, some definitions. A voter vj’s sincere preferences on A is a ballot from the unrestricted ballot domain for A, where for any ax and ay in A (she ranks ax above ay iff her utility
from ax being enacted is greater than that for ay) and (she ranks ax and ay equally iff her utility
from ax being enacted is equal to that for ay).10 Note, a given voter’s sincere preferences may not
occur within the ballot domain used by the SWF being implemented if the ballot domain is not
unrestricted.
10
There are many well evidenced arguments from philosophy and behavioral economics that
preferences are not derived from utility. For such arguments, one can contend that utilities are a
post-facto description of the real cause of preferences. In other words, the ontology and epistemology of utilities is irrelevant for our current concerns.
21
Define the general betrayal condition as follows:11 Suppose that in voter vj’s sincere and
complete preference order, ax is the sole alternative in the first part, az is the sole alternative in
the last part, and ay is neither in the first nor last part, where all three alternatives occur in A. We
say that a SWF satisfies the general betrayal condition if there exists a profile where [(vj must not
place ax in the first part of her ballot to ensure az is not socially ranked above ay) or (vj must not
place az in the last part of her ballot to ensure ay is not socially ranked above ax)].
Clearly, anti-plurality voting, bloc voting, plurality voting, and their absolute versions,12
(all six of which satisfy compositional dictatorship), satisfy the general betrayal condition; while
dichotomous voting, which does not satisfy compositional dictatorship but does satisfy the four
conditions, does not satisfy the general betrayal condition. If satisfaction of the general betrayal
condition is normatively undesirable,13 then the normative desirability of a SWF being uniquely
characterized by the four conditions on its ballot domain is questionable.
11
This condition is similar to the favorite betrayal criterion of (Ossipoff and Smith n.d.).
12
In the appendix, when we use another version of positive responsiveness, bloc voting and ab-
solute versions of anti-plurality, bloc, and plurality voting also uniquely satisfy the four conditions in particular situations, which makes it appropriate to include them in this discussion.
13
The point being made here is not that a “good” Arrovian SWF must avoid all possibility of in-
centivizing the submission of insincere preferences; that is clearly impossible given the GibbardSatterthwaite theorem and related results. The point being made is that if a SWF that satisfies the
four conditions (but not unique in the admissible class given its ballot domain) is better in respect
to some normatively desirable property than the SWFs in the admissible class (that are unique
given their respective ballot domains), then there has to be a normative argument as to why must
22
If there is no strong normative or practical reason to restrict ourselves to the ballot domains of SWFs that are uniquely characterized by the four conditions, then it would appear that
we have many SWFs available to us which satisfy the four conditions. It’s just when n = 2, many
of these SWFs (e.g. dichotomous voting, unrestricted Borda count, and the m! BMM SWFs (of
which MaxBMM and MinBMM are just two examples)) reduce to majority rule. This reduction
suggests that we may have been looking at May’s theorem the wrong way.
When we generalize majority rule to more than two alternatives, there seem to be many
ways to do it. One way is to conceptualize majority rule as a two-alternatives version of dichotomous voting, another is to view it as a two-alternatives version of unrestricted Borda count, and
so on. But traditionally, the way it has been generalized is as a two-alternatives version of a Condorcet method. A Condorcet method is a voting system that guarantees that a Condorcet winner
is socially ranked above all other alternatives, when a Condorcet winner exists. A Condorcet
winner is an alternative which defeats all other alternatives in A, one-on-one, via majority rule.
Of course, Condorcet winners do not always exist, since majority rule, when extended to more
than two alternatives, is intransitive due to the famous Condorcet paradox. But Condorcet methods represent a prevalent democratic decision making norm, namely majority preference: if a
Condorcet winner exists in a profile, then the Condorcet winner should be socially ranked above
all other alternatives in the profile.
What the results in this article suggest is that looking at May’s theorem as a justification
of majority preference as the democratic decision making norm is a wrong way to normatively
the SWF satisfy the four conditions and be unique on its ballot domain if we are to hold the
“unique” SWFs above those that merely non-uniquely satisfy the four conditions.
23
interpret the theorem. In the context of Arrovian social welfare functions, we should not make
the normative inference: majority rule uniquely satisfies four normative conditions when n = 2
and there is an unrestricted ballot domain; therefore, since Condorcet methods can be reduced to
majority rule when n = 2, Condorcet methods and majority preference are normatively desirable.
Rather, the following inference seems more appropriate: Given that several SWFs (e.g. dichotomous voting, unrestricted Borda count, etc.) that satisfy the four conditions reduce to majority
rule when n = 2, majority rule should have more normative appeal when n = 2 and we are restricted to Arrovian social welfare functions.
Is Majority Preference Really That Normatively Unique?
Several reasons have been given for the normative uniqueness of majority preference,
only one of which is May’s theorem. Two others to consider include maximization of self-determination and Condorcet’s jury theorem.
With respect to maximizing self-determination, the basic argument is that when majority
rule occurs, more people get what they prefer than if minority rule were to occur. Of course, as
those who have noted this argument have said (Dahl 1989, 144-146), majority rule can be intransitive when we allow more than two alternatives. With this intransitivity, it may not be clear that
we are actually maximizing self-determination, even with a Condorcet method. Furthermore, we
could conceive of dichotomous voting in a manner that could be interpreted as transitively maximizing self-determination. Specifically, if we assume voters individually rank alternatives they
individually consent to above alternatives they do not individually consent, then we could ask
voters to place alternatives they consent to in the first part of the ballot, and alternatives which
they do not consent to in the second part of the ballot. (If a voter consents to all alternatives, or to
none of them, they could be asked to submit a trivial ballot.) Using dichotomous voting on such
24
ballots, alternatives are ranked based on maximizing the number of voters that consent to them.14
Furthermore, even when there are more than two alternatives, dichotomous voting is transitive.
This should not be interpreted as a way to herald dichotomous voting, but rather to demonstrate
that there are reasonable ways to normatively assert that majority preference is not unique in
terms of maximizing self-determination.
Condorcet’s jury theorem has also been used to assert that majority rule is more likely to
produce correct decisions. Specifically, Nicholas de Condorcet had demonstrated that if there
was a statement, and each voter has a probability π of correctly determining whether the statement is true or not, where π > ½ and each voter’s determination is independent of other voters’
determinations, then the majority of voters is more likely correct about the veracity of the statement than the minority. Peyton Young extended the jury theorem to multiple alternatives and
showed that under particular conditions, there exists a Condorcet method that chooses a rank ordering of alternatives that is most likely correct (Young 1988). But beginning with an important
generalization of Condorcet’s jury theorem by (List and Goodin 2001), there have been many
works showing that Condorcet’s theorem can be extended to multiple voting systems, including
the plurality voting, dichotomous voting, and unrestricted Borda count (Ben-Yashar and Kraus
2002; Prasad 2012; Brams and Kilgour 2014). Majority preference and Condorcet methods no
longer seem so unique in this respect.
Roughly speaking, while majority rule seems to still be relatively unique with respect to
May’s theorem, Condorcet’s jury theorem, and maximization of self-determination, in the
14
This is similar to approval voting, but not the same because for example, dichotomous voting
makes no distinction between consenting to all alternatives and consenting to none.
25
context of two alternatives and Arrovian SWFs; this seems to be the case because there are
multiple complying Arrovian SWFs that reduce to majority rule when there are only two
alternatives. It does not seem to be the case that majority rule uniquely generalizes to majority
preference in these respects when there are more than two alternatives. This is important because
May’s theorem, Condorcet’s jury theorem, and maximization are common major normative
defenses of majority rule. For example, Robert Dahl listed these three as three of the four major
defenses for majority rule (Dahl 1989). If these defenses generalize beyond majority preference
when there are more than two alternatives, then the special place majority preference is given
among democratic decision making procedures may not be so justified. This does not mean that
majority preference is necessarily undesirable as a norm; rather it just may not be so unique.
Appendix: Proof of Theorem 2
Theorem 2 identifies all Arrovian SWFs that are uniquely characterized by decisiveness,
anonymity, neutrality, and pairwise positive responsiveness on their respective ballot domains.
Pairwise positive responsiveness is a different generalization of May’s original positive responsiveness condition for two alternatives, and leads to slightly different results than the main result
(i.e. theorem 1). We will more formally state theorem 2 after defining some additional terms.
Pairwise Positive Responsiveness
Consider two profiles p and p†, each with m ballots ranking all n alternatives in A. The
profiles p and p† consist of the ballots b1, …, bm, and b1†, …, bm†, respectively. Suppose that
these profiles are identical except for one ballot; more precisely suppose that bi = bi† except for a
single index j for which bj ≠ bj†. Specifically, bj† is some complete and transitive preference order
of A, where the change from bj to bj† raises the preference of ax relative to ay. We say that a
change from such a p to p† is a pairwise positive flip with respect to ax relative to ay.
26
A SWF is pairwise positively responsive if it is always the case, for any pair of alternatives ax and ay in A, that for any pairwise positive flip from p and p† with respect to ax relative to
ay, that if f(p) ranks ax above ay or f(p) ranks ax and ay equally, then f(p†) ranks ax above ay.
Theorem 2: The only SWFs that are uniquely characterized by decisiveness, anonymity, neutrality, and pairwise positive responsiveness are plurality voting, absolute plurality voting, anti-plurality voting, absolute anti-plurality voting, trivial voting, (and bloc voting and absolute bloc voting when m < 4)
Borda Class SWF
A SWF is a Borda Class SWF (BCSWF) if it meets the following six requirements. First,
if a voter’s ballot ranks ax above ay, then for that voter’s ballot, the SWF gives ax more marks
than ay. Second, if a voter’s ballot ranks ax and ay equally, then for that voter’s ballot, the SWF
gives ax and ay the same number of marks. Third, for any given alternative, its score is equal to
the sum of its marks across all m ballots. Fourth, if the score of ax is greater than the score of ay,
then the SWF socially ranks ax over ay. Fifth, if ax and ay have the same score, then the SWF
ranks those two alternatives equally. Finally, the SWF must be anonymous and neutral.
Let Δi(x, y) = (the number of marks ax receives for bi) – (the number of marks ay receives
for bi). Note, if the SWF is a BCSWF, then (if Δi(x, y) > 0, then for bi ax|ay) and (if Δi(x, y) = 0,
then for bi axay) and (if Δi(x, y) < 0, then for bi ay|ax). Furthermore, if the SWF is a BCSWF, then
𝑚
(if [∑𝑚
𝑖=1 Δi(x, y)] > 0, then ax is socially ranked above ay) and (if [∑𝑖=1 Δi(x, y)] = 0, then ax is so-
cially ranked equally with ay) and (if [∑𝑚
𝑖=1 Δi(x, y)] < 0, then ay is socially ranked above ax).
Suppose f is a BCSWF. If f(p) is such that ax is socially ranked above or equal to ay, then
any pairwise positive flip from p and p† with respect to ax relative to ay should produce a f(p†)
where ax is socially ranked above ay. The reason is that only one ballot, say bj changed to bj†, is
27
different between the two profiles. When the pairwise positive flip occurs, we go from Δj(x, y) <
†
0 to Δj†(x, y) ≥ 0, or from Δj(x, y) = 0 to Δj†(x, y) > 0. This clearly ensures that [∑𝑚
𝑖=1 Δi(x, y)] >
0 and ax is socially ranked above ay for f(p†).
So it should be clear that every BCSWF is decisive, anonymous, neutral, and pairwise
positively responsive. With that we can demonstrate there is no polychotomous bundled ballot
domain that has a SWF that is uniquely characterized by decisiveness, anonymity, neutrality, and
pairwise positive responsiveness.
Case 1 (Non-polychotomy): If a ballot domain is polychotomous, then it is impossible for a SWF
to be uniquely characterized by decisiveness, anonymity, neutrality, and pairwise positive responsiveness.
Consider the following profile. Suppose every voter has a polychotomous ballot of the
same composition. Suppose that one voter has a ballot where ax is in the first part, and ay is in the
next to last part. Further suppose that m – 1 voters have the same ballot, where ay is in the next to
last part and ax is in the last part. If f is a BCSWF where alternatives in the last part of a ballot
get zero marks, alternatives in the next to last part of a ballot get one mark, and alternatives in
the first part of the ballot get a particular number of marks that happens to be greater than m, then
ax will be socially ranked above ay. However there could exist another BCSWF that is identical,
except that alternatives in the first part of the ballot happen to get exactly m marks, such that ax
and ay would be socially ranked equally. Therefore, there can be no SWF uniquely characterized
by decisiveness, anonymity, neutrality, and pairwise positive responsiveness when the ballot domain contains a polychotomous ballot.
Case 2 (Compositional Dictatorship): Case 2 of Lemma 2 for the proof of theorem 1 covers this
case.
28
Case 3 (Resoluteness): For m > 3, Case 3 of Lemma 2 of the main result covers this case. The
claim here is that if m = 2 or m = 3, and the ballot domain is bundled, non-polychotomous, and
compositionally dictatorial, then decisiveness, anonymity, neutrality, and pairwise positive responsiveness uniquely characterize bloc voting and absolute bloc voting on their respective ballot domains.
The proof given here is in terms of absolute bloc voting and its ballot domain from which
it is easy for the reader to extend the proof to bloc voting and its ballot domain. So suppose voters submit ballots from an absolute bloc voting ballot domain. First note that if all voters put ax
and ay in the first part of their respective ballots, or all voters put ax and ay in the last part of their
respective ballots, then decisiveness, anonymity, and neutrality require the SWF to output a social tie between the two alternatives. This implies two things. One, if no voters place ay in the
first part of their respective ballots, and at least one voter places ax in the first part of their ballot,
then due to pairwise positive responsiveness, the SWF must rank ax above ay. Two, if all m voters place ax in the first part of their respective ballots, and not all m voters place ay in the first
part of their respective ballots, then the SWF must socially rank ax above ay due to pairwise positive responsiveness.
Subcase 3.1: m = 2
Now if there are only two voters in the profile and their ballot domain is the absolute bloc
voting ballot domain, there only remains one more subcase; the subcase where exactly one voter
places ax in the first part of their ballot, and exactly one voter places ay in the first part of their
29
ballot.15 Note that both of these ballots will have the same number of alternatives in their first
parts, and they will have the same number of alternatives in common and differ on the same
number of alternatives in their first parts. For example, consider the two ballots axa1a2|aya3a4a5
and aya1a3|axa2a4a5; both have the same number of alternatives in the first part (namely three),
both have the same number of alternatives in common in the first part (namely one, the alternative a1), and both have the same number of alternatives that differ in their first parts (two alternatives for both ballots, ax and a2 for one ballot and ay and a3 for the other ballot).
Define a swap as a permutation where for any pair of alternatives in A, aw and az, aw replaces az iff az replaces aw. Now consider two dichotomous ballots of the same composition, one
where ax is in the first part but ay is not, and one where ay is in the first part but ax is not. Let the
number of alternatives that differ in their first parts be u. (For example, if the two ballots are
axa1a2|aya3a4a5 and aya1a3|axa2a4a5, then u = 2.) Whatever u is, suppose we commit u swaps on
the 2u alternatives that occur in the first part of only one of the two ballots. One of these u swaps
should be between ax and ay. Note that after we commit these swaps, the first ballot becomes the
second ballot and the second ballot becomes the first ballot. Due to anonymity, the profile before
the swap(s) and the profile after the swap(s) must output the same rank ordering. Given that ax
and ay were swapped, the outputs for both profiles must socially rank ax and ay equally. Therefore, if the ballot domain is the absolute bloc voting ballot domain, and the SWF is decisive,
anonymous, neutral, pairwise positively responsive, and there are exactly two voters, then the
SWF must be absolute bloc voting.
15
There are two ways this subcase can occur. In one, one voter places both alternatives in the
first part and another voter places both voters in the second part. To satisfy decisiveness, anonymity, and neutrality, the SWF must clearly output a tie between both alternatives. Outside this
footnote, subcase 3.1 addresses the other way this subcase can occur.
30
Subcase 3.2: m = 3
Note, we know that with respect to ax and ay, the SWF must behave like absolute bloc
voting if either ax or ay occur in the first part of all three ballots or none of the ballots, if the SWF
satisfies decisiveness, anonymity, neutrality, and pairwise positive responsiveness. Therefore, we
only have four situations left to examine: (1) where exactly one voter places ax in the first part
and exactly one places ay in the first part, (2) where exactly two voters place ax in the first part
and exactly two place ay in the first part, (3) where exactly two voters place ax in the first part
and exactly one places ay in the first part, and (4) where exactly one voter places ax in the first
part and exactly two place ay in the first part. Note that due to neutrality, (3) and (4) can be
treated identically, so we only really need to proceed through three situations.
Subcase 3.2.1: Exactly one voter places ax in the first part and exactly one places ay in the first
part
This can only occur in one of two ways. In the first way, one voter places both ax and ay
in the first part, while the other two voters place both alternatives in the second part; therefore,
due to neutrality and decisiveness, the SWF must output a social tie between those two alternatives. The second way this can occur is if one voter places ax in the first part and ay in the second
part (for shorthand ax|ay), one voter submits ay|ax, and another voter submits |axay. Now suppose
we perform a swap between ax and ay, and swaps on any of the remaining 2u – 2 alternatives in
the first parts of the ballots ax|ay and ay|ax that differ. Note that after the swaps, the ax|ay and ay|ax
ballots in the old profile will become ay|ax and ax|ay respectively of the new profile. After taking
anonymity and neutrality into account, the only differences between the old and new profiles will
potentially be the ballot where both ax and ay are in the same part. Now recall that neutrality requires that alternatives be permuted in the outputs in the same manner they were permuted in the
31
inputted profiles. Note that after taking anonymity and neutrality into account, ax and ay appear
not permuted between the old and new profile; this would imply that the outputs of the old and
new profiles must be identical with respect to ax and ay, and this can only occur if they are
equally ranked.
Subcase 3.2.2: Exactly two voters place ax in the first part and exactly two place ay in the first
part
There are exactly two ways this can occur. In one, two voters place both alternatives in
the first part, and the remaining voter places both in their last part. Clearly, this must result in a
social tie between the two alternatives due to decisiveness and neutrality. In the other way, one
voter places ax in the first part and ay in the last part, another voter places ay in the first part and
ax in the last part, and yet another voter submits a ballot with both alternatives in the first part.
Applying appropriate symmetries to the proof for the second way from subcase 3.2.1, we can
show that this second way for subcase 3.2.2 must behave like absolute bloc voting if the SWF is
decisive, anonymous, neutral, and pairwise positively responsive.
Subcase 3.2.3: Exactly two voters place ax in the first part and exactly one places ay in the first
part
When we take anonymity and neutrality into account, there are only two ways this can
occur. One is that one voter places both alternatives in the first part, another voter places ax in the
first part but ay in the last part, and yet another voter places both in the last part of their ballot.
The second way is that two voters place ax in the first part of their ballot while ay is in the second
part, while the remaining voter places ay in the first part and ax in the second part. Note that for
any profile that can be described in one of these two ways, such a profile can be constructed by a
pairwise positive flip (with respect to ax relative to ay) from some profile where ax occurs in the
32
first part of exactly one ballot and ay occurs in the first part of exactly one ballot. Because it is a
pairwise positive flip from a profile that must produce a social tie between ax and ay, it is necessary that any profile (with exactly two voters placing ax in the first part and exactly one placing
ay in the first part) that is inputted into a SWF that is decisive, anonymous, neutral, and pairwise
positively responsive must socially rank ax and ay in the same manner as absolute bloc voting.
Summary of Subcase 3.2: Therefore, if the ballot domain is the absolute bloc voting ballot domain, and the SWF is decisive, anonymous, neutral, pairwise positively responsive, and there are
exactly three voters, then the SWF must be absolute bloc voting.
Remaining Cases: Using information for the proofs for cases 5 thru 7 of theorem 1, it is easy to
demonstrate that the only SWFs that are uniquely characterized by decisiveness, anonymity, neutrality, and pairwise positive responsiveness are plurality voting, absolute plurality voting, antiplurality voting, absolute anti-plurality voting, trivial voting, (and bloc voting and absolute bloc
voting when m < 4). ■
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