Expanding Approvals Rule

The Expanding Approvals Rule (EAR), can be thought of as a sequential PR version of Bucklin Voting which can use the Droop quota. It works by first examining the 1st rank; any candidate with a Droop quota of 1st choices is elected, and a Droop quota of the ballots ranking them 1st are spent. Then, it examines both the 1st and 2nd ranks; any candidate ranked either 1st or 2nd by a Droop quota of ballots is elected, and a Droop quota of the ballots ranking them either 1st or 2nd are spent. And so on, until all ranks have been examined.

By analogy to the single transferable vote, the Expanding Approvals Rule is also known as Bucklin transferable vote (BTV).

There is a great deal of flexibility in how EAR is implemented (e.g., the specific quota applied, how voter weights are spent, how exhausted ballots are interpreted, and how candidates are elected when there are "ties"). Therefore, EAR may be considered as a family of voting rules rather than one specific voting.

The authors propose a specific EAR voting rule. Suppose $m$ is the number of candidates, $k$ is the number of candidates to be elected, and $n$ is the number of voters. The authors propose that


 * 1) EAR be applied with the quota displayed in Figure 1, which is essentially the Droop quota;
 * 2) Voter weights are "spent" using a uniform and fractional reweighting scheme (i.e., whenever a candidate is elected, every voter with a ballot supporting this candidate at this stage will have their current weight reduced by $\frac{W-\bar{q}}{W}$, where $W$ is the total weight of voters casting a ballot for the elected candidate);
 * 3) If a voter's ballot is exhausted at any stage, then, in all future stages, the voter is assumed to support all candidates;
 * 4) When multiple candidates can be elected at a given stage, then the candidate with the highest rank-maximal order is elected (where the rank-maximal order is calculated with respect to the voters' original ballots and, hence, this ordering remains fixed throughout the EAR process). Rank-maximal ordering is constructed as follows: Given 2 candidates $a$ and $b$, if candidate $a$ is the first preference of (strictly) more voters than candidate $b$, then candidate $a$ is strictly higher in the rank-maximal ordering. If candidate $a$ and $b$ have equal numbers of voter holding them in first preference, then we consider the number of voters that place $a$ and $b$ as their second preference and so on.

Unlike many other voting rules, EAR can be applied to ballots that include "indifferences" between candidates, i.e., voters are not forced to rank candidates as they can incorporate "ties" into their ballots.

Properties
The authors' proposed EAR voting rule satisfies a number of properties:


 * Generalized Proportionality for Solid Coalitions (PSC)
 * Generalized weak-PSC
 * Proportional Justified Representation
 * PSC
 * Weak-PSC
 * Candidate monotonicity when voters have dichotomous (or approval) ballots
 * Candidate monotonicity if only one candidate is to be elected
 * Rank respecting candidate monotonicity
 * Non-crossing candidate monotonicity.

Semi-solid coalitions
Semi-solid coalitions are groups of voters who rank some candidates above all others, but some voters in the group may rank other candidates above the candidates the group ranked above all others. EAR is claimed by its creators to always elect from Droop semi-solid coalitions.

According to the paper, (4) EAR addresses a criticism of Tideman (2006): “Suppose there are voters who would be members of a solid coalition except that they included an “extraneous” candidate, which is quickly eliminated among their top choices. These voters’ nearly solid support for the coalition counts for nothing which seems to me inappropriate.” We demonstrate the last flaw of QBS pointed out by Tideman in the explicit example below. EAR does not have this flaw.

Example 7 Consider the profile with 9 voters and where k [number of winners] = 3.

1 : c1, c2, c3, e1, e2, e3, e4, d1

2 : c2, c3, c1, e1, e2, e3, e4, d1

3 : c3, c1, d1, c2, e1, e2, e3, e4

4 : e1, e2, e3, e4, c1, c2, c3, d1

5 : e1, e2, e3, e4, c1, c2, c3, d1

6 : e1, e2, e3, e4, c1, c2, c3, d1

7 : e1, e2, e3, e4, c1, c2, c3, d1

8 : e1, e2, e3, e4, c1, c2, c3, d1

9 : e1, e2, e3, e4, c1, c2, c3, d1

In the example, {e1, e2, e3} is the outcome of QBS. Although PSC is not violated for voters in {1, 2, 3} but the outcome appears to be unfair to them because they almost have a solid coalition. Since they form one-third of the electorate they may feel that they deserve that at least one candidate such as c1, c2 or c3 should be selected. In contrast, it was shown in Example 5 that EAR does not have this flaw and instead produces the outcome {e1, e2, c1}.

Monotonicity
EAR fails the monotonicity criterion.

The example has two factions support two candidates each (e.g. one faction supports candidates X and Y, and another candidates W and Z), where X and W are almost exactly tied. In the original two-seat election, X wins and then the other winner must come from the WZ coalition, so Z wins. Then Z is raised on a X>Z ballot, which makes W win the first seat instead. Now the second winner must come from the other coalition, and so Y wins: raising Z makes Z lose.