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]] which preserves [[approval voting]]'s compliance with the [[Favorite Betrayal criterion|favorite betrayal criterion]]. It uses the [[tied at the top]] rule. It satisfies [[Condorcet criterion|majority-strength Condorcet]], but not the full [[Condorcet criterion]] which also requires relative-majority Condorcet.
==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.▼
#The voter submits a ranked ballot, with equal-ranking and truncation permitted.
#A voter is deemed to have "
#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.
▲An equivalent definition:
▲#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 with the greatest approval.
==Comments==
Line 54:
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
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.
==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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