Geller-STV

Geller-STV, also called STV-B, is a proposed proportional representation system first proposed by Chris Geller in 2005 to address the quasi-chaotic nature of single transferable vote. It is similar to Quota Borda system, but using vote counts to elect.

Election procedure

Geller describes a theoretical modification of STV using the weighted Gregory method, stating that other variations of STV have their own merit, and that Gregory "is common for theoretical applications and does not substantially affect any conclusions in this paper." Other variations, such as Meek-Geller by recalculating quotas and adjusting Borda scores from fractional ballots, are possible by trivial modification; Geller only describes the principles. The base system does not matter in the special case of electing one winner (Geller-IRV), as only full votes transfer.

Voters rank candidates in order of preference. As this system uses vote count quotas, equal rankings are not allowed, except that a ballot need not rank all candidates.

When computing the quota, Borda scores are also computed. Candidates are elected by reaching a quota of votes, and votes are transferred as in the base variant of STV; rather than elimination by fewest votes, Geller's method eliminates the candidate with the lowest Borda score.

Properties

Vote transfers in STV and Geller-STV provide proportional representation of solid voting coalitions, as well as of minority voting groups who are not solid coalitions. Quota Borda system can only provide representation to solid coalitions.

Geller shows STV-B is less quasi-chaotic than STV due to the inclusion of all rankings at all stages of tabulation. Eliminations by vote count can cause large variations by small changes in the orderings of lower-ranked candidates, or little to no variations by large changes. Because Geller eliminates by the lowest Borda score, substantial changes in voter rankings are required to change the elimination order.

STV-B is also non-monotonic.

Representativeness

Assuming honest and fully-ranked ballots, Borda generally elects the Condorcet winner, and if not then a highly-preferred candidate. Geller demonstrates Borda eliminations reduce the tendency for STV to change outcomes from small changes in the number of votes, a "butterfly effect" referenced elsewhere in the literature; and the special case of a single winner between three candidates must elect the Condorcet candidate when one exists, as the Condorcet winner mathematically cannot have a lower Borda score than the Condorcet loser under the conditions when a Condorcet winner exists between only three candidates, or when all candidates are ranked on all ballots in general. Because Geller-STV elects by quota of votes, Tideman's proof of proportional representation is equally applicable.

Although "all voters have a voice through all stages of the selection process, including voters already fully represented among the winning candidates and those who belong to different solid coalitions than do the losing candidates," lower-ranked preferences have less influence on the Borda score, limiting their effects on elimination order proportionally to the number of candidates they rank above another coalition's candidates. Geller's assessment assumes weighted Gregory, and not a system in which Borda scores are recalculated on transfer, although such a system would only alter a voter's impact on Borda scores after a candidate on their ballot has been elected, and only for those candidates ranked below an elected candidate, so the order of elections and thus eliminations still has influence. Ballots truncated prior to ranking any candidate in a given coalition have no impact on elimination order in those coalitions.

These properties give Geller-STV similar properties to other STV systems, in that Borda elimination orders probabilistically approach independence between coalitions and behave as several separate instances of instant runoff voting. Voters ranking further candidates influence the Borda scores of those candidates, which may or may not change the elimination order of candidates in other coalitions, whereas STV only affects the elimination order by vote transfer. Geller suggests this influence may promote the election of moderate candidates because a majority coalition will prefer the more moderate candidate among those favored by an extreme coalition; however, as with STV, a quota-sized coalition will elect one of its top-preferred candidates. If the candidate is less-preferred by the minority coalition, then the majority coalition must contribute more to its Borda score with rankings contributing less per each ballot, minimizing this effect. If Borda counts are modified in proportion to partial vote transfers as candidates are elected, the majority coalition may lose influence over the Borda score before the elimination order affects the outcome.

While all of these give significant probability of each quota coalition electing independently, the elimination order under STV is affected only by the first ranking receiving votes from a ballot, and one coalition can affect another only after it has elected its own candidates; while elimination order under STV-B is affected by all ballots all the time, and a rule which recalculates Borda scores with each vote transfer only limits this to full impact before a candidate is elected, and partial impact after the election of a candidate. Geller-STV has similar representativeness to other forms of STV, except that it may be more "Borda-like" within a coalition in that it eliminates low-welfare candidates and may select a winner similar to the Condorcet candidate.

Strategic voting

The Gibbard–Satterthwaite theorem shows that any voting system with more than two alternatives and no dictator is vulnerable to tactical voting. The Borda rule is highly vulnerable to sophisticated voting and strategic nomination, which raises concerns for STV with Borda elimination. Sophisticated voting is only successful when the outcome changes to benefit the strategic voter.

Geller raises concerns about strategic voting based on Borda's weaknesses, including ranking popular opponents below opponents likely to lose nonetheless. With truncated ballots, this can mean ranking unlikely candidates at all. This concern is diminished in the context of electing by vote quota: ballots contribute less to the Borda score of lower-ranked candidates, while vote quota contributes to election. Ranking unlikely candidates above preferred candidates makes preferred candidates less likely to win by potentially transferring voting power to the minority coalition, helping them to elect multiple candidates; and by reducing the Borda score of preferred candidates, exercising less voting power over the coalition to which the strategic voter belongs. In the worst case, a coalition may lose so much voting power by strategic voting that both the popular candidate and the one they tried to elevate are elected, no harm is done to preferentially-adjacent coalitions, and the strategic voters gain no representation at all due to having insufficient voting power to elect one of the candidates they lowered on their ballots.

Because Borda achieves the same proportionality as STV, strategic voting can only affect the candidates elected within a quota coalition, and he impacts in terms of ideological similarity are diminished when electing more seats. Within a coalition, raising a less-preferred candidate over a more-preferred candidate is likely to harm the more-preferred candidate, both by contributing to its earlier elimination and by increasing the votes received by the less-preferred candidate after higher-ranked candidates are eliminated.

Geller's strongest arguments are for manipulation by parties and factions organizing their voters with agreements to vote strategically for the candidates the party dictates, so as to block manipulation by other parties. Organization of strategic voters is necessary due to the high degree of knowledge of other ballots required to carry out strategic voting.

Other properties

There are a number of formalized voting system criteria whose results are summarized in the following table. The special case of Geller-IRV is included.

Comparison of preferential electoral systems

Sys­tem	Mono­tonic	Condorcet winner	Majo­rity	Condorcet loser	Majority loser	Mutual majority	Smith	ISDA	LIIA	Independence of clones	Reversal symmetry	Participation, consistency	Later-no‑harm	Later-no‑help	Polynomial time	Resol­vability
Schulze	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No	No	No	Yes	Yes
Ranked pairs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	Yes	Yes
Split Cycle	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No	No	No	Yes	No
Tideman's Alternative	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No	Yes	Yes
Kemeny–Young	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No	Yes
Copeland	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No	Yes	No	No	No	Yes	No
Nanson	No	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	Yes	No	No	No	Yes	Yes
Black	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	Yes	No	No	No	Yes	Yes
Instant-runoff voting	No	No	Yes	Yes	Yes	Yes	No	No	No	Yes	No	No	Yes	Yes	Yes	Yes
Smith/IRV	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No	Yes	Yes
Borda	Yes	No	No	Yes	Yes	No	No	No	No	No	Yes	Yes	No	Yes	Yes	Yes
Geller-IRV	No	No	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Yes	Yes
Baldwin	No	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	Yes	Yes
Bucklin	Yes	No	Yes	No	Yes	Yes	No	No	No	No	No	No	No	Yes	Yes	Yes
Plurality	Yes	No	Yes	No	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes	Yes
Contingent voting	No	No	Yes	Yes	Yes	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes
Coombs	No	No	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No	No	Yes	Yes
MiniMax	Yes	Yes	Yes	No	No	No	No	No	No	No	No	No	No	No	Yes	Yes
Anti-plurality	Yes	No	No	No	Yes	No	No	No	No	No	No	Yes	No	No	Yes	Yes
Sri Lankan contingent voting	No	No	Yes	No	No	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes
Supplementary voting	No	No	Yes	No	No	No	No	No	No	No	No	No	Yes	Yes	Yes	Yes
Dodgson	No	Yes	Yes	No	No	No	No	No	No	No	No	No	No	No	No	Yes

References

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