# Difference between revisions of "Improved Condorcet Approval"

(→Comments) |
(→Definition) |
||

Line 4: | Line 4: | ||

#Optionally define a proportion of the votes ''q'' as the minimum necessary on the winning side of a pairwise comparison for this win to be counted. Set it to zero to do without ''q''. | #Optionally define a proportion of the votes ''q'' as the minimum necessary on the winning side of a pairwise comparison for this win to be counted. Set it to zero to do without ''q''. | ||

#The voter submits a ranked ballot, with equal-ranking and truncation permitted. Let ''v'' signify the total number of voters. | #The voter submits a ranked ballot, with equal-ranking and truncation permitted. Let ''v'' signify the total number of voters. | ||

− | #A voter ''approves'' every candidate whom he explicitly ranks. | + | #A voter implicitly ''approves'' every candidate whom he explicitly ranks. |

#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). | #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] ''and'' v[y,x]>''qv''. | #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] ''and'' v[y,x]>''qv''. |

## Revision as of 09:26, 17 August 2005

**Improved Condorcet Approval** or **ICA** or **tCA** is a variant of Condorcet//Approval devised by Kevin Venzke to satisfy the Favorite Betrayal criterion. It uses the tied at the top rule.

## Definition

- Optionally define a proportion of the votes
*q*as the minimum necessary on the winning side of a pairwise comparison for this win to be counted. Set it to zero to do without*q*. - The voter submits a ranked ballot, with equal-ranking and truncation permitted. Let
*v*signify the total number of voters. - A voter implicitly
*approves*every candidate whom he explicitly ranks. - 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]*and*v[y,x]>*qv*. - If
*S*is empty, then let*S*contain all the candidates. - Elect the candidate in
*S*with the greatest approval.

## Comments

**ICA** satisfies the Favorite Betrayal criterion by treating voters ranking *x* and *y* equally at the top as attempting to create a pairwise tie between the two candidates. Then instead of looking first for a candidate with only pairwise wins (the Condorcet winner), ICA selects as finalists every candidate with no pairwise losses.

As a result of this tweaking, ICA does not strictly satisfy the Condorcet criterion. It is possible that the voted Condorcet winner could lose to another candidate, due to voters tying both candidates at the top, and the Condorcet winner having lower approval.

When *q* is set to 50%, then the method is equivalent to Majority Defeat Disqualification Approval, and all values of t[a,b] (for any candidates *a* and *b*) can be assumed to be zero without affecting the result.

When *q* is left at 0%, and no voter uses equal ranking in the top position, then the method is equivalent to ordinary Condorcet//Approval.