Approval voting under dichotomous preferences: a catalogue of characterizations
The authors give eight characterizations of approval voting. All their results involve the consistency axiom, which requires choices to be consistent across different electorates. In addition, the authors consider strategy-proofness, agreement with majority opinions, independence of cloned alternatives, and invariance under removing inferior alternatives. They prove their results by reducing them to a single base theorem, Theorem 1: approval voting (AV) is the only ballot aggregation function satisfying consistency, disjoint equality and faithfulness. Let \(X\) be a finite set of alternatives. A ballot \(A\) is a non-empty set of (approved) alternatives and let \(\mathcal{A}\) be the set of all ballots. A ballot profile \(P\) is a function which assigns to each ballot \(A\) the number of voters whose ballot is \(A\). The approval score of an alternative \(a\), denoted by \(P[a]\), is the number of voters whose ballot includes \(a\). \(P + A\) denotes the profile resulting from \(P\) by adding one voter with ballot \(A\) and the profile \(P + k A\) is obtained from \(P\) by adding \(k\) voters with ballot \(A\) to \(P\). For a permutation \(\pi\) on \(X\), the profile \(\pi(P)\) has \(P(A)\) voters with ballot \(\pi(A)\) for every ballot \(A\). A ballot aggregation function \(f\) maps each profile \(P\) to a set \(f(P)\) of winning alternatives; typically, \(f(P)\) will be a singleton set. \(f\) is \textit{neutral} iff \(f(\pi(P)) = \pi(f(P))\). \(f\) is \textit{consistent} iff \(f(P + P') = f(P) \cap f(P')\) whenever \(f(P) \cap f(P') \neq \emptyset\). \(f\) is \textit{faithful} iff \(f(A) = A\). \(f\) is \textit{continuous} iff whenever \(f(P) = \{a\}\), there is a natural number \(k\) such that \(f(P' + kP) = \{a\}\). \(f\) satisfies \textit{disjoint equality} iff \(f(A + B) = A \cup B\) whenever \(A \cap B = \emptyset\). And \(f\) satisfies \textit{cancellation} iff \(f(P) = X\) whenever \(P[a] = P[b]\) for all \(a, b \in X\). Examples in Appendix B show that no axiom can be dropped from any of the eight characterizations of approval voting (AV). In the discussion at the end of their paper the authors argue that their eight characterizations of AV are based on the consistency axiom in combination with other appealing properties and that these results are stong arguments for using approval voting, once we accept the premiss that voters have dichotomous preferences. It is this premiss which I am doubtful about. As pointed out by Balinski and Laraki, what does one mean with approval? At least acceptable (\(ac\)) or at least good (\(go\))? It may make a huge difference, as the following example shows. Suppose 5 voters evaluate two candidates \(A\) and \(B\) as follows, where \(vg\) stands for `very good' and \(po\) for `poor': \begin{center} \begin{tabular}{c|ccccc} \(A\) \ & \ \(ac\) \ & \ \(ac\) \ & \ \(ac\) \ & \ \(go\) \ & \ \(go\) \\ \(B\) \ & \ \(po\) \ & \ \(ac\) \ & \ \(go\) \ & \ \(go\) \ & \ \(vg\) \\ \end{tabular} \end{center} If approval means `at least acceptable', then \(A\) is the winner under AV, but if approval means `at least good', then \(B\) is the winner under AV. So, it seems fair to say that Approval Voting gives arbitrary outcomes. Notice that both Majority Rule and Majority Judgment make \(B\) the winner. As argued by Balinski and Laraki, with only two evaluations (approval or non-approval) the voters cannot properly express their evaluations of candidates they want to distinguish and a richer language than \(\{\text{approval, non-approval}\}\) is needed. For instance, the last voter clearly distingiuishes between \(A\) and \(B\), but cannot express this in the language of approval voting. One may expect that the more information an election rule uses, the better the outcome shall reflect the will of the voters.
scientific article; zbMATH DE number 7614323
scientific article