Editing Approval voting (section)

===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.