Improved Condorcet Approval: Difference between revisions
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'''Improved Condorcet Approval''' or '''ICA''' or '''tCA''' is a variant of [[Condorcet//Approval]] devised by [[Kevin Venzke]]
==Definition==
#The voter submits a ranked ballot. She is permitted to give more than one candidate the same ranking, and is not obliged to rank every candidate.
#Identify all of the pairwise losses.
#Disregard any pairwise loss that can be reversed or turned into a pairwise tie if the voters ranking both candidates equal in first place (possibly with other candidates) are counted in favor of the pairwise loser.
#If any candidates do not have any pairwise loss, disqualify all the candidates who do have some pairwise loss.
#Elect the (non-disqualified) candidate approved by the greatest number of voters.
A more precise definition:
#The voter submits a ranked ballot, with equal-ranking and truncation permitted.
#A voter
#Let v[a,b] signify the number of voters ranking candidate ''a'' above candidate ''b'', and let t[a,b] signify the number of voters ranking ''a'' and ''b'' equally at the top of the ranking (possibly tied with other candidates).
#Define a set ''S'' of candidates, which contains every candidate ''x'' for whom there is no other candidate ''y'' such that v[x,y]+t[x,y]<v[y,x].
#If ''S'' is empty, then let ''S'' contain all the candidates.
#Elect the candidate in ''S'' with the greatest approval.
==Comments==
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In ordinary [[Condorcet//Approval]], A's win over B is counted. This creates problems with the [[Favorite Betrayal criterion|favorite betrayal]] criterion (or the [[Sincere Favorite criterion]]) since it could happen that the 35 A=B voters are preventing either A or B from being the decisive winner, and that in trying to support both equally, the win is instead moved to the approval winner, who might be someone worse. Here is an example:
20 A=B
15 B>C
There is a [[Condorcet method|Condorcet cycle]], and ordinary [[Condorcet//Approval]] elects C as the approval winner. But if the 20 A=B voters change their vote to B>A, this turns B into the decisive, Condorcet winner, which is a result that these voters prefer.
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==Variants==
===Variant:
The above definition defines t[a,b] to be the number of voters tying ''a'' and ''b'' in the top position. This is the most conservative change from [[Condorcet//Approval]], since it's only in this case that we can be sure the voter would like to do whatever is necessary to ensure that the winner is either ''a'' or ''b''.▼
However, it might be more intuitive, and preferred, if t[a,b] were defined rather as the number of voters ranking ''a'' equal to ''b'' and explicitly voting for both. This requires a simple change to step #3
For the second definition, replace step 3 with:
:Disregard any pairwise loss that can be reversed or turned into a pairwise tie if the voters approving both candidates and ranking them equal are counted in favor of the pairwise loser.
The definitions can only differ in practice when there are more than three candidates (unless it is allowed to approve all the candidates).
== Notes ==
As formulated, ICA fails [[Cloneproof|clone independence]] (though it is likely not too difficult to modify it to satisfy the criterion).{{Clarify|reason=How?|date=April 2024}} Example:<blockquote>3 A
1 B1>B2>B3
▲The above definition defines t[a,b] to be the number of voters tying ''a'' and ''b'' in the top position. This is the most conservative change from [[Condorcet//Approval]], since it's only in this case that we can be sure the voter would like to do whatever is necessary to ensure that the winner is either ''a'' or ''b''.
1 B2>B3>B1
1 B3>B1>B2</blockquote>B1, B2, and B3 all have a pairwise defeat (they are in a cycle with each other), so A is elected for being an unbeaten candidate. But if B2 and B3 drop out:<blockquote>3 A
3 B1</blockquote>Now both A and B1 pairwise tie, and thus each has a 50% chance of winning.
▲However, it might be more intuitive, and preferred, if t[a,b] were defined rather as the number of voters ranking ''a'' equal to ''b'' and explicitly voting for both. This requires a change to step #3 only.
==Links==
*[http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-May/
[[Category:Single-winner voting
[[Category:No-favorite-betrayal electoral systems]]
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