Abstract
In this paper, we present a new result for axiomatic characterization of approval voting. Having defined a model in which the actual set of alternatives becomes known only after a vote has been taken, we characterize approval voting as the only voting procedure (to be precise, “family of ballot aggregation functions”) that satisfies faithfulness (F), consistency (C), stability on selected alternatives (SSA), and independence of dropped alternatives (IDA). SSA, which is a version of the property introduced by Arrow (Economica 16:121–127, 1959), states that if the actual set of alternatives 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 other hand, IDA suggests that we should select alternatives based on the outcome of the second vote. Therefore, given F and C, approval voting is the only voting procedure that selects the same set of alternatives irrespective of which vote counts, that is, the first or second vote.
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Notes
Faithfulness states that if there is only one voter, all, and only those alternatives approved by the voter should be selected. Consistency requires that all the alternatives chosen by two separate groups of voters, if any, should also be selected by the union of the two groups, and in that case, only such alternatives should be selected by the union.
In fact, our formulation is more similar to that of Massó and Vorsatz (2008), in which a generalized notion of approval voting is considered.
Neutrality states that the names of alternatives should not affect the outcome. Alcalde-Unzu and Vorsatz (2013) call the property “symmetry across alternatives.”
They also offer as the weakest possible property the no-disapproval voting axiom, which eliminates disapproval voting from the domain of possible voting procedures.
Thus, in our setting, a nonempty subset of K may have two interpretations: a ballot or a set of feasible alternatives.
The following equivalent statement will be useful: if for all \(\bar{K} \in \mathcal {K},\) and all \(\pi \in \varPi ,\,f^{\bar{K}}(\pi ) = f^{\bar{K}}(\pi ^{\bar{K}}).\)
This supposition is crucial to the arguments that follow.
We denote by \(\pi ^{\bar{K}}|_{2^{\bar{K}}\setminus \{\emptyset \}}\) the restriction of \(\pi ^{\bar{K}}\) on \(2^{\bar{K}}\setminus \{\emptyset \}.\)
In addition, the function satisfying (1) is unique.
I am grateful to one of the anonymous referees for pointing this out.
Xu (2010) refers to this property as “equal treatment.”
In the first version of this paper, the possibility of \(m^{*}=0\) was omitted. The author is most grateful to one of the anonymous referees for pointing out this omission.
More concretely, \(n(x,\,\pi ^{\bar{K}})=n(x,\,\pi ^{*})=0\) for all \(x\notin \bar{K},\) and for all \(x\in \bar{K}\) with \(n(x,\,\pi )=0.\) (Note that in both cases, \(x\notin A_{m}\) for all \(m=1,\ldots ,m^{*}.\)) For \(x\in \bar{K}\) with \(n(x,\,\pi )=\bar{m}\le m^{*}\) and \(\bar{m}\ge 1,\) we have \(n(x,\,\pi ^{\bar{K}})=n(x,\,\pi ^{*})=\bar{m}.\) (Note that \(x\in A_{m}\) for all \(m=m^{*}+1-\bar{m}, \ldots , m^{*},\) and the number of such \(A_{m}\) is equal to \(\bar{m}.\))
This remark was suggested by one of the anonymous referees. The author is most grateful for the suggestion.
The work of Alcalde-Unzu and Vorsatz (2013) has been developed independently of this paper. Their model differs from ours in several aspects. For the details, see Footnote 4 (p. 15) of Alcalde-Unzu and Vorsatz (2013). As mentioned in Footnote 4 of the current paper, Alcalde-Unzu and Vorsatz (2011), which is the first version of Alcalde-Unzu and Vorsatz (2013), had considered the property corresponding to IDA prior to this paper (although Theorem 2 of Alcalde-Unzu and Vorsatz 2013 mentioned below did not appear in the first version of their paper).
In addition, the proof techniques used in Alcalde-Unzu and Vorsatz (2013) and the current paper are quite different.
More technically, this property can be expressed as follows. For each \(\bar{K}\in \mathcal {K}\) and \(\pi \in \varPi \) with \(f^{K}(\pi )\cap \bar{K}\ne \emptyset :\)
$$\begin{aligned} f^{K}(\pi )\cap \bar{K} = f^{\bar{K}}\left( \pi ^{\bar{K}}\right) . \end{aligned}$$Recall that we can regard \(\pi ^{\bar{K}}\) as the outcome of the second vote.
We can eliminate this counterexample by adding N.
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The author is grateful to the Associate Editor and two anonymous referees for their insightful comments and useful suggestions.
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Sato, N. A characterization result for approval voting with a variable set of alternatives. Soc Choice Welf 43, 809–825 (2014). https://doi.org/10.1007/s00355-014-0811-1
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DOI: https://doi.org/10.1007/s00355-014-0811-1