Editing Approval voting (section)

=== Sincere strategy with ordinal preferences ===

A sincere voter with multiple options for voting sincerely still has to choose which sincere vote to use. [[tactical voting|Voting strategy]] is a way to make that choice, in which case strategic approval includes sincere voting, rather than being an alternative to it.<ref name=probstrat>{{Cite journal | doi = 10.2307/1955800 | last = Niemi |first=R. G. | year = 1984 | title = The Problem of Strategic Behavior under Approval Voting | journal = American Political Science Review | volume = 78 | issue = 4| pages = 952–958 | jstor = 1955800 | s2cid = 146976380 }}</ref> This differs from other voting systems that typically have a unique sincere vote for a voter.

When there are three or more candidates, the winner of an approval election can change, depending on which sincere votes are used. In some cases, approval can sincerely elect any one of the candidates, including a [[Condorcet winner]] and a [[Condorcet loser]], without the voter preferences changing. To the extent that electing a Condorcet winner and not electing a Condorcet loser is considered desirable outcomes for a voting system, approval can be considered vulnerable to sincere, strategic voting.<ref>{{Cite journal | last = Yilmaz | first=M. R. | year = 1999 | title = Can we improve upon approval voting? | doi = 10.1016/S0176-2680(98)00043-3 | journal = European Journal of Political Economy | volume = 15 | issue = 1| pages = 89–100 }}</ref> In one sense, conditions where this can happen are robust and are not isolated cases.<ref>{{Cite journal | doi = 10.1007/BF00054447 |last1=Saari |first1=Donald G.  |last2=Van Newenhizen |first2=Jill | year = 2004 | title = The problem of indeterminancy in approval, multiple, and truncated voting systems | journal = Public Choice | volume = 59 | issue = 2| pages = 101–120 |jstor=30024954 |s2cid=154705078 }}</ref> On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval, representing the flexibility and responsiveness of approval, not just to voter ordinal preferences, but cardinal utilities as well.<ref name=unmitigated>{{Cite journal | doi = 10.1007/BF00054449 |last1=Saari |first1=Donald G.  |last2=Van Newenhizen |first2=Jill | year = 2004 | title = Is approval voting an 'unmitigated evil?' A response to Brams, Fishburn, and Merrill | journal = Public Choice | volume = 59 | issue = 2| pages = 133–147 |jstor=30024956  |s2cid=154007278 }}</ref>

==== Dichotomous preferences ====

Approval avoids the issue of multiple sincere votes in special cases when voters have [[dichotomous preferences]]. For a voter with dichotomous preferences, approval is [[Strategyproofness|strategyproof]].{{sfn|Brams|Fishburn|1983|p=31}} When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval is guaranteed to elect a Condorcet winner.{{sfn|Brams|Fishburn|1983|p=38}} However, having dichotomous preferences when there are three or more candidates is not typical. It is an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters.<ref name=probstrat/>

Having dichotomous preferences means that a voter has bi-level preferences for the candidates. All of the candidates are divided into two groups such that the voter is indifferent between any two candidates in the same group and any candidate in the top-level group is preferred to any candidate in the bottom-level group.{{sfn|Brams|Fishburn|1983|p=16–17}} A voter that has strict preferences between three candidates—prefers A to B and B to C—does not have dichotomous preferences.

Being strategy-proof for a voter means that there is a unique way for the voter to vote that is a strategically best way to vote, regardless of how others vote. In approval, the strategy-proof vote, if it exists, is a sincere vote.{{sfn|Brams|Fishburn|1983|p=29}}

==== Approval threshold ====

Another way to deal with multiple sincere votes is to augment the ordinal preference model with an approval or acceptance threshold. An approval threshold divides all of the candidates into two sets, those the voter approves of and those the voter does not approve of. A voter can approve of more than one candidate and still prefer one approved candidate to another approved candidate. Acceptance thresholds are similar. With such a threshold, a voter simply votes for every candidate that meets or exceeds the threshold.<ref name=probstrat/>

With threshold voting, it is still possible to not elect the Condorcet winner and instead elect the Condorcet loser when they both exist. However, according to Steven Brams, this represents a strength rather than a weakness of approval. Without providing specifics, he argues that the pragmatic judgments of voters about which candidates are acceptable should take precedence over the [[Condorcet criterion]] and other social choice criteria.<ref name=critstrats>{{Cite journal |last1=Brams |first1=S. J.  |author2=Remzi Sanver, M. | year = 2005 | title = Critical strategies under approval voting: Who gets ruled in and ruled out | doi = 10.1016/j.electstud.2005.05.007 | journal = Electoral Studies | volume = 25 | issue = 2| pages = 287–305 }}</ref>