Iterated Bucklin

Iterated Bucklin (alternatively Bucklin-IRV, or BIRV) is a variant of Bucklin voting whereby the number of top preferences an individual ballot reveals at any point is a function of the rank in which the ballot places the winner of the count so far. This modifies the standard Bucklin voting procedure by which all ballots reveal the same, progressively increasing number of top preferences until a quota of support is filled by at least one candidate.

Iterated Bucklin was proposed by Etjon Basha in December 2020 as an attempt to produce a count that exhibits few practical violations of voting criteria (elicits the greatest degree of sincerity for voters) and the least degree of count complexity, at the cost of not formally meeting many criteria. Iterated Bucklin violates Condorcet, Later-No-Harm, Later-No-Help and is not precinct summable. A preliminary simulation of the count’s propensity to violate various criteria was conducted by Kevin Venzke.

Purpose
Iterated Bucklin attempts to mimic the behavior of a rational Bucklin voter under conditions of an iterated count (hence the name), and specifically an iterated Approval Count, where after every round a chance is given to amend one’s vote once the winner until that point is presented, and with the count only ending once no voter wishes to further amend their vote. Whilst a true iterated approval count would be impractical under conditions of all but the smallest elections, Iterated Bucklin attempts to formalize a simulation of this process. It is hoped that a count that already produces a reasonable strategy on the voter’s behalf would hopefully lead to voters presenting relatively sincere rankings as in input.

Single-winner procedure
Voters present ranked ballots which allow for truncation and equal ranking. As in standard Bucklin voting, all preferences revealed by a ballot count as approvals for those candidates.

At the start of the count, only the top-ranked preferences of each ballot are counted (only the top preferences are revealed). If a candidate receives a majority of preferences at this stage, they are declared elected and the procedure ends.

Otherwise, the candidate currently enjoying a plurality of preferences is provisionally declared the Placeholder Winner (PW) and the count proceeds by every individual ballot revealing N of their top preferences as below:


 * Ballots that have revealed preferences as far down as the current PW but no further, are stationary and reveal no further preferences;
 * Ballots whose revealed preferences do not include the PW reveal one more preference; and
 * Ballots whose revealed preferences include the PW but extend to other candidates ranked further below the PW, revert to revealing preferences only as far down as the PW.

Once all ballots complete one of these moves as above, the Placeholder Winner is recalculated. The next round of moves is carried out, and the PW is again recalculated, and so on until no more moves are possible: at the end of the count all ballots either reveal preferences as far down as the Placeholder Winner or are entirely exhausted. At this stage the Placeholder Winner is declared elected.

Multi-winner generalization
The same procedure can be generalized to the election of a number of winning candidates. At the first stage, all candidates with at least a quota of support (first preferences) are declared elected, and a Hare or Droop quota of ballot weight is allocated to the winners from the relevant ballots. The remaining places (should there be any) are filled by applying the same iterative procedure to the remaining (de-weighted) ballots to elect the next winner. Ballots are again de-weighted (with a quota of support being subtracted from all ballots having approved of the PW as of the count’s end) and the count begins again, until all eligible places are filled.

In keeping with the method’s mimicking of a sound strategy, support from an elected candidate would first be removed from those votes that have no more candidates left to elect (have already elected all those they approve of), and only afterward (if a quota of support has not been filled yet) from other voters who still approve of as yet unelected candidates.

It is noteworthy to highlight that once the iterative procedure begins, it is not possible for a candidate to be elected whilst some of their votes have not being counted yet, an eventuality which can happen in most other ranked counts such as Bucklin or the Single Transferable Vote. The avoidance of the issues raised by such possibility has led in the latter system to the development of the complex Meek and Warren counts. Nevertheless, when applied to a multi-winner scenario, Iterated Bucklin can still elect a candidate before all of their votes are counted before the iterative procedure starts (i.e. whenever a candidate is elected based on receiving a quota of support of first preferences only.).

Surplus Handling
If viewed as a Sequential Cardinal Method then  Fractional Surplus Handling can be used. Other mechanisms like those used in Single transferable vote are also possible.

Ties and Cycles
If, at any point of the count, two or more candidates receive the most support, both (or all) are considered Placeholder Winners.

The rules are then amended such that those ballots that approve of both (or all) PWs revert to revealing only their top-ranked PW.

Ties can - but not necessarily do - lead to cycles, where two successive rounds cycle back and forth between one-another. The count will not end in such a circumstance.

The method itself cannot resolve such cycles, and a fall-back method is called for. Options may include restarting the entire count and using simple plurality, IRV or Bucklin. Another option could be the random breaking of the tie or the “branching” of the count by resolving the tie in each contender’s favor, and then running the count to its completion, with whichever branch leading to the most lopsided ultimate victory (absolute difference between winner and runner up) being adopted. Branching may introduce a significant degree of complexity and could potentially lead to dozens of branches.

In any election with a significant number of voters, ties (and thus cycles) should be rare.

Cycles are a separate eventuality from the method producing two or more ultimate winners. This case could be resolved by random choice, or plurality of preferences at the most recent (latest) stage of the count where one of the tied candidates had a plurality.

Potential for Hand Count
It may be feasible to hand-count an election conducted by Iterated Bucklin, especially if precincts are few or reduced to one. Limiting the number of allowable preferences to a given number (ex. no more than five) could allow for a speedier resolution.

In practice, even simple elections in Iterated Bucklin may require a relatively large number of rounds until final resolution is achieved (see example) and the count may be especially susceptible to small errors at any point of the count which could produce a completely altered succession of Placeholder Winners and, thus, ultimate winner.

Example
The example below presents a single-winner election contested by four candidates and attended by 7 voters: 3 ABC, 2 BDC, 2CDA.

In this example, A wins but standard Bucklin would have elected B in round two with 5 approvals, due to A’s votes helping B at A’s expense.