Editing Approval voting (section)

== Strategic voting ==
{{See also|Strategic voting#Cardinal single-winner voting}}

=== Overview ===
{{more citations needed section|date=June 2019}}
Approval voting allows voters to select all the candidates whom they consider to be reasonable choices.

''Strategic approval'' differs from [[ranked voting]] (aka preferential voting) methods where voters are generally forced to ''reverse'' the preference order of two options, which if done on a larger scale can cause an unpopular candidate to win. Strategic approval, with more than two options, involves the voter changing their approval threshold. The voter decides which options to give the ''same'' rating, even if they were to have a preference order between them. This leaves a tactical concern any voter has for approving their second-favorite candidate, in the case that there are three or more candidates. Approving their second-favorite means the voter harms their favorite candidate's chance to win. Not approving their second-favorite means the voter helps the candidate they least desire to beat their second-favorite and perhaps win.

Approval technically allows for but is strategically immune to [[Tactical voting#Push-over|push-over]] and [[Tactical voting#Burying|burying]].

Bullet voting occurs when a voter approves ''only'' candidate "a" instead of ''both'' "a" and "b" for the reason that voting for "b" can cause "a" to lose. The voter would be satisfied with either "a" or "b" but has a moderate preference for "a". Were "b" to win, this hypothetical voter would still be satisfied. If supporters of both "a" and "b" do this, it could cause candidate "c" to win. This creates the "[[chicken dilemma]]", as supporters of "a" and "b" are [[playing chicken]] as to which will stop strategic voting first, before both of these candidates lose.

Compromising occurs when a voter approves an ''additional'' candidate who is otherwise considered unacceptable to the voter to prevent an even worse alternative from winning.

=== Sincere voting ===
{{more citations needed section|date=June 2019}}
Approval experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely.{{'"}}{{sfn|Brams|Fishburn|1983|p=29}} They also give a specific definition of a sincere approval vote in terms of the voter's [[preference|ordinal preferences]] as being any vote that, if it votes for one candidate, it also votes for any more preferred candidate. This definition allows a sincere vote to treat strictly preferred candidates the same, ensuring that every voter has at least one sincere vote. The definition also allows a sincere vote to treat equally preferred candidates differently. When there are two or more candidates, every voter has at least three sincere approval votes to choose from. Two of those sincere approval votes do not distinguish between any of the candidates: vote for none of the candidates and vote for all of the candidates. When there are three or more candidates, every voter has more than one sincere approval vote that distinguishes between the candidates.

==== Examples ====

Based on the definition above, if there are four candidates, A, B, C, and D, and a voter has a strict preference order, preferring A to B to C to D, then the following are the voter's possible sincere approval votes:
*vote for A, B, C, and D
*vote for A, B, and C
*vote for A and B
*vote for A
*vote for no candidates

If the voter instead equally prefers B and C, while A is still the most preferred candidate and D is the least preferred candidate, then all of the above votes are sincere and the following combination is also a sincere vote:
*vote for A and C

The decision between the above ballots is equivalent to deciding an arbitrary "approval cutoff." All candidates preferred to the cutoff are approved, all candidates less preferred are not approved, and any candidates equal to the cutoff may be approved or not arbitrarily.

=== 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>

=== Strategy with cardinal utilities ===

Voting strategy under approval is guided by two competing features of approval. On the one hand, approval fails the [[later-no-harm criterion]], so voting for a candidate can cause that candidate to win instead of a candidate more preferred by that voter. On the other hand, approval satisfies the [[monotonicity criterion]], so not voting for a candidate can never help that candidate win, but can cause that candidate to lose to a less preferred candidate. Either way, the voter can risk getting a less preferred election winner. A voter can balance the risk-benefit trade-offs by considering the voter's cardinal utilities, particularly via the [[von Neumann–Morgenstern utility theorem]], and the probabilities of how others vote.

A [[Tactical voting#Myerson-Weber strategy|rational voter model]] described by [[Roger Myerson|Myerson]] and Weber specifies an approval strategy that votes for those candidates that have a positive prospective rating.<ref name=":2">{{Cite journal | doi = 10.2307/2938959 |last1=Myerson |first1=R. |last2=Weber |first2=R. J. | year = 1993 | title = A theory of Voting Equilibria | jstor = 2938959| journal = American Political Science Review | volume = 87 | issue = 1| pages = 102–114 |url=http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |archive-date=October 9, 2022 |url-status=live | hdl = 10419/221141 |s2cid=143828854 | hdl-access = free }}</ref> This strategy is optimal in the sense that it maximizes the voter's [[Utility#Expected utility|expected utility]], subject to the constraints of the model and provided the number of other voters is sufficiently large.

An optimal approval vote always votes for the most preferred candidate and not for the least preferred candidate, which is a [[dominant strategy]]. An optimal vote can require supporting one candidate and not voting for a more preferred candidate if there 4 candidates or more, e.g. the third and fourth choices are correlated to gain or lose decisive votes together; however, such situations are inherently unstable, suggesting such strategy should be rare.<ref>{{Cite journal | last1=Dutta |first1=B |last2=De Sinopoli |first2=F. | last3=Laslier |first3= J.-F.| year = 2006 | title = Approval voting: three examples | journal = International Journal of Game Theory | volume = 35 | pages = 27–38 |doi=10.1007/s00182-006-0053-2 |s2cid=801286 | citeseerx=10.1.1.365.8090 }}</ref>

Other strategies are also available and coincide with the optimal strategy in special situations. For example:
* Vote for the candidates that have above average utility. This strategy coincides with the optimal strategy if the voter thinks that all pairwise ties are equally likely.{{sfn|Brams|Fishburn|1983|p=85}}
* Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities. This strategy, if used by all voters, implies at equilibrium the election of the Condorcet winner whenever it exists.<ref name=":3">{{Cite journal  | last1=Laslier |first1= J.-F.| year = 2009 | title = The Leader rule: a model of strategic approval voting in a large electorate | journal = Journal of Theoretical Politics | volume = 21 | issue=1 | pages = 113–136 |doi= 10.1177/0951629808097286|s2cid= 153790214}}</ref>
*Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when the best candidate is either much better than all others (i.e. is the only one with a positive expected value).{{sfn|Brams|Fishburn|1983|p=74, 81}}
*If all voters are rational and cast a strategically optimal vote based on a common knowledge of how all other voters vote except for small-probability, statistically independent errors, then the winner will be the Condorcet winner, if one exists.<ref name=":4">Laslier, J.-F. (2006) [http://halshs.archives-ouvertes.fr/docs/00/12/17/51/PDF/stratapproval4.pdf "Strategic approval voting in a large electorate,"] ''IDEP Working Papers'' No. 405 (Marseille, France: Institut D'Economie Publique)</ref>

=== Strategy examples ===
{{more citations needed section|date=June 2019}}In the example election described [[Condorcet method#Example: Voting on the location of Tennessee's capital|here]], assume that the voters in each faction share the following [[Von Neumann–Morgenstern utility theorem|von Neumann–Morgenstern utilities]], fitted to the interval between 0 and 100. The utilities are consistent with the rankings given earlier and reflect a strong preference each faction has for choosing its city, compared to weaker preferences for other factors such as the distance to the other cities.<!-- This seems like a very weird choice of utilities. Wouldn't it be better to make the disutility equal to the distance between the two cities? -->

{| class="wikitable" style="text-align: center"
|+Voter utilities for each candidate city
|-
! scope="col" rowspan="2" | Fraction of voters <small>(living close to)</small>
! scope="colgroup" colspan="4" | Candidates
! scope="col" rowspan=2 | Average
|-
! scope="col" style="width:6em;"| Memphis
! scope="col" style="width:6em;"| Nashville
! scope="col" style="width:6em;"| Chattanooga
! scope="col" style="width:6em;"| Knoxville
|-
| style="text-align:left;"| Memphis (42%)
| 100 || 15 || 10 || 0 || 31.25
|-
| style="text-align:left;"| Nashville (26%)
| 0 || 100 || 20 || 15 || 33.75
|-
| style="text-align:left;"| Chattanooga (15%)
| 0 || 15 || 100 || 35 || 37.5
|-
| style="text-align:left;"| Knoxville (17%)
| 0 || 15 || 40 || 100 || 38.75
|}

Using these utilities, voters choose their optimal strategic votes based on what they think the various pivot probabilities are for pairwise ties. In each of the scenarios summarized below, all voters share a common set of pivot probabilities.

{| class="wikitable" style="text-align: center"
|+Approval Voting results for scenarios using optimal strategic voting
|-
! scope="col" rowspan="2" | Strategy scenario
! scope="col" rowspan="2" | Winner
! scope="col" rowspan="2" | Runner-up
! scope="colgroup" colspan="4" | Candidate vote totals
|-
! scope="col" style="width:6em" | Memphis
! scope="col" style="width:6em" | Nashville
! scope="col" style="width:6em" | Chattanooga
! scope="col" style="width:6em" | Knoxville
|-
| style="text-align: left" | Zero-info
| Memphis || Chattanooga || 42 || 26 || 32 || 17
|-
| style="text-align: left" | Memphis leading Chattanooga
| colspan="2" | Three-way tie || 42 || 58 || 58 || 58
|-
| style="text-align: left" | Chattanooga leading Knoxville
| Chattanooga || Nashville || 42 || 68 || 83 || 17
|-
| style="text-align: left" | Chattanooga leading Nashville
| Nashville || Memphis || 42 || 68 || 32 || 17
|-
| style="text-align: left" | Nashville leading Memphis
| Nashville || Memphis || 42 || 58 || 32 || 32
|}

In the first scenario, voters all choose their votes based on the assumption that all pairwise ties are equally likely. As a result, they vote for any candidate with an above-average utility. Most voters vote for only their first choice. Only the Knoxville faction also votes for its second choice, Chattanooga. As a result, the winner is Memphis, the Condorcet loser, with Chattanooga coming in second place. In this scenario, the winner has minority approval (more voters disapproved than approved) and all the others had even less support, reflecting the position that no choice gave an above-average utility to a majority of voters.

In the second scenario, all of the voters expect that Memphis is the likely winner, that Chattanooga is the likely runner-up, and that the pivot probability for a Memphis-Chattanooga tie is much larger than the pivot probabilities of any other pair-wise ties. As a result, each voter votes for any candidate they prefer more than the leading candidate, and also vote for the leading candidate if they prefer that candidate more than the expected runner-up. Each remaining scenario follows a similar pattern of expectations and voting strategies.

In the second scenario, there is a three-way tie for first place. This happens because the expected winner, Memphis, was the Condorcet loser and was also ranked last by any voter that did not rank it first.

Only in the last scenario does the actual winner and runner-up match the expected winner and runner-up. As a result, this can be considered a stable strategic voting scenario. In the language of [[game theory]], this is an "equilibrium." In this scenario, the winner is also the Condorcet winner.

===Dichotomous cutoff===
{{more citations needed section|date=June 2019}}
Modeling voters with a 'dichotomous cutoff' assumes a voter has an immovable approval cutoff, while having meaningful cardinal preferences. This means that rather than voting for their top 3 candidates, or all candidates above the average approval, they instead vote for all candidates above a certain approval 'cutoff' that they have decided. This cutoff does not change, regardless of which and how many candidates are running, so when all available alternatives are either above or below the cutoff, the voter votes for all or none of the candidates, despite preferring some over others. This could be imagined to reflect a case where many voters become disenfranchised and apathetic if they see no candidates they approve of. In a case such as this, many voters may have an internal cutoff, and would not simply vote for their top 3, or the above average candidates.

For example, in this scenario, voters are voting for candidates with approval above 50% (bold signifies that the voters voted for the candidate):

{| class="wikitable" style="text-align:center; width:600px;"
|-
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| '''90%''' || '''60%''' || 40% || 10% || ''50%''
|-
! 35%
| 10% || '''90%''' || '''60%''' || 40% || ''50%''
|-
! 30%
| 40% || 10% || '''90%''' || '''60%''' || ''50%''
|-
! 10%
| '''60%''' || 40% || 10% || '''90%''' || ''50%''
|}

C wins with 65% of the voters' approval, beating B with 60%, D with 40% and A with 35%

If voters' threshold for receiving a vote is that the candidate has an above average approval, or they vote for their two most approved of candidates, this is not a dichotomous cutoff, as this can change if candidates drop out. On the other hand, if voters' threshold for receiving a vote is fixed (say 50%), this is a dichotomous cutoff, and satisfies IIA as shown below:

{| class="wikitable" style="text-align:center; width:600px;"
|+ A drops out, candidates voting for above average approval
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| – || '''60%''' || '''40%''' || 10% || ''37%''
|-
! 35%
| – || '''90%''' || 60% || 40% || ''63%''
|-
! 30%
| – || 10% || '''90%''' || '''60%''' || ''53%''
|-
! 10%
| – || 40% || 10% || '''90%''' || ''47%''
|}
B now wins with 60%, beating C with 55% and D with 40%
{| class="wikitable" style="text-align:center; width:600px;"
|+ A drops out, candidates voting for approval > 50%
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| – || '''60%''' || 40% || 10% || ''37%''
|-
! 35%
| – || '''90%''' || '''60%''' || 40% || ''63%''
|-
! 30%
| – || 10% || '''90%''' || '''60%''' || ''53%''
|-
! 10%
| – || 40% || 10% || '''90%''' || ''47%''
|}
With dichotomous cutoff, C still wins.

{| class="wikitable" style="text-align:center; width:600px;"
|+ D drops out, candidates voting for top 2 candidates
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| '''90%''' || '''60%''' || 40% || – || ''63%''
|-
! 35%
| 10% || '''90%''' || '''60%''' || – || ''53%''
|-
! 30%
| '''40%''' || 10% || '''90%''' || – || ''47%''
|-
! 10%
| '''60%''' || '''40%''' || 10% || – || ''37%''
|}
B now wins with 70%, beating C and A with 65%
{| class="wikitable" style="text-align:center; width:600px;"
|+ D drops out, candidates voting for approval > 50%
! Proportion of electorate
! Approval of Candidate A
! Approval of Candidate B
! Approval of Candidate C
! Approval of Candidate D
! Average approval
|-
! 25%
| '''90%''' || '''60%''' || 40% || – || ''63%''
|-
! 35%
| 10% || '''90%''' || '''60%''' || – || ''53%''
|-
! 30%
| 40% || 10% || '''90%''' || – || ''47%''
|-
! 10%
| '''60%''' || 40% || 10% || – || ''37%''
|}
With dichotomous cutoff, C still wins.