Improved Condorcet Approval

From electowiki

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. It uses the tied at the top rule. Despite the name, it isn't actually a Condorcet method.

Definition

  1. 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.
  2. Identify all of the pairwise losses.
  3. 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.
  4. If any candidates do not have any pairwise loss, disqualify all the candidates who do have some pairwise loss.
  5. Elect the (non-disqualified) candidate approved by the greatest number of voters.

A more precise definition:

  1. The voter submits a ranked ballot, with equal-ranking and truncation permitted.
  2. A voter is deemed to have "approved" every candidate whom he explicitly ranks.
  3. 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).
  4. 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].
  5. If S is empty, then let S contain all the candidates.
  6. 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.

Satisfying this criterion is desirable because it means that, unlike in ordinary Condorcet//Approval, voters in an ICA election can always vote at least as sincerely as under Approval voting (by ranking some candidates equally at the top, and the other candidates not at all), without forfeiting a preferable outcome that might be obtained by voting less sincerely than this.

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.

Here is an example: 

40 A>B
35 A=B
25 B

The Condorcet winner is A, but ICA elects B. Both A and B make it into the set of finalists. The Condorcet winner is always a finalist, and B is a finalist because although A defeats B pairwise, A would not be able to do so if the 35 A=B voters were to side with B, so that this defeat isn't counted. Then B has greater approval than A and is elected.

In ordinary Condorcet//Approval, A's win over B is counted. This creates problems with the 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
20 A=B
15 B>C
45 C

There is a 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.

This is problematic because it could be that the 20 A=B voters actually prefer A to B, so that if they can benefit from voting B>A, they have favorite betrayal incentive. If these voters don't believe that A is likely to win the election, anyway, they may vote B>A just to be cautious. This could cause A to lose an election solely because A's supporters didn't believe A could win it.

Improved Condorcet Approval fixes this problem by anticipating the A=B voters' dilemma. In this scenario, ICA already elects B when the 20 voters vote A=B. A's pairwise win over B is disregarded, and B is considered the decisive winner.

When no voter uses equal ranking in the first position, ICA is equivalent to ordinary Condorcet//Approval.

Variants

Variant: "Tied and approved" rather than "Tied at the top"

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 in the first definition.

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 clone independence (though it is likely not too difficult to modify it to satisfy the criterion). Example:

3 A

1 B1>B2>B3

1 B2>B3>B1

1 B3>B1>B2

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:

3 A 3 B1

Now both A and B1 pairwise tie, and thus each has a 50% chance of winning.

Links

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