Log of Significant Revisions
This website was first established on 25 August 2012. All significant updates to content or presentation after this date are recorded in the table below. Minor revisions are those contained within the specified page while major ones are contained within the specified section of a chapter. Wider changes give rise to a new edition of the website. Revisions are listed in the table in chronological order with a recent entry appearing above an earlier one.
| Revision | ||||
|---|---|---|---|---|
| Nature and Scope | Date | Type | Number | |
| Comparisons: Positional Voting section entirely reworked. Bias indices, vector definitions and other matters have been revised in line with our peer-reviewed journal paper; see note for link to this open-access article. Other related pages have also been revised. | 10 Apr 2023 | Website revision [See note] | 2.00.00 | |
| Evaluations: Majority and Related Criteria section updated. Associated Maths Proofs and Table of Contents, Summary (RB), Conclusions (RB), Glossary, Site Map and links updated. | 25 Apr 2021 | Major revision [See note] | 1.11.00 | |
| Comparisons: Plurality section updated and Geometric Voting pages 9 & 10 added and 11 updated. Associated Maths Proof and Table of Contents, Summary (SW), Conclusions (RB), Glossary, About (Chapter Overviews), Site Map and links updated. | 01 Feb 2021 | Major revision [See note] | 1.10.00 | |
| Comparisons: Positional Voting pages 8, 9 & 10 updated. Summary (SW), Conclusions (RB), Site Map and links updated. | 20 Nov 2020 | Major revision [See note] | 1.09.00 | |
| Comparisons: Positional Voting section extended with insertion of new page 4. Associated Maths Proof, Notation & Formats and Table of Contents, Summary (SW), Site Map and links updated. | 18 Oct 2020 | Major revision [See note] | 1.08.00 | |
| Comparisons: Geometric Voting section extended with insertion of new page 2. Associated Maths Proof and Table of Contents, Summary (SW), Site Map and links updated. | 02 Oct 2020 | Major revision [See note] | 1.07.00 | |
| General: both Spreadsheet pages updated with the new option of a larger version for each. | 25 Aug 2020 | Minor revision | 1.06.01 | |
| Major revision of Comparisons: Positional Voting section; with three images removed. Associated Maths Proofs and Table of Contents, Summary (RB), Conclusions (RB), Chapter Overviews, Glossary, Site Map and links updated. | 25 May 2020 | Major revision [See note] | 1.06.00 | |
| Addition of Chapter Overviews page to About Website section. Website Guide, General: Table of Contents, Site Map and links updated. | 01 Jan 2018 | Minor revision | 1.05.02 | |
| Expansion of page 2 (Intensity of Differences ...) in Comparisons: Positional Voting section. | 01 Jan 2018 | Minor revision | 1.05.01 | |
| Revision of Comparisons: Positional Voting section and four related Maths Proofs. Summary (SW), Conclusions (RB), Maths Proofs (Table of Contents and Notations & Formats), Glossary, Site Map and links updated. | 25 Aug 2017 | Major revision [See note] | 1.05.00 | |
| Addition of Comparisons: Positional Voting section and four related Maths Proofs. Summary (SW), Conclusions (RB), Maths Proofs (Table of Contents), Glossary, Site Map and links updated. | 10 Oct 2016 | Major revision [See note] | 1.04.00 | |
| Expansion of Evaluations: Clones & Teaming, Comparisons: Geometric Voting and Comparisons: Plurality Rule Methods sections regarding Tied Voter Preferences, Exchanging Preferences and Instant Run-off CHPV respectively. Description: Voting, Description: Outcomes, Summary (RB), Summary (SW), Site Map and links updated. | 25 Jan 2016 | Major revision [See note and acknowledgement] | 1.03.00 | |
| Addition of an 'About Website' page with appropriate links and references to it. | 03 Nov 2015 | Minor revision | 1.02.02 | |
| Clarification of the range of the Least Squares Index (0 ≤ LSI ≤ 1): Initial page of Proportionality sections and Glossary. | 11 Jan 2015 | Minor revision | 1.02.01 | |
| Evaluations: General Criteria section expanded in relation to monotonicity criteria. Also, reversal symmetry criterion and proof EG3 added. Summary (RB), Conclusions (RB), Glossary, Site Map and links updated. | 25 Feb 2013 | Major revision [See note] | 1.02.00 | |
| New section on Plurality Rule Methods added to Comparisons (SW) chapter. Introduction (SW), Summary (SW), Conclusions (RB), Glossary, Site Map and links updated. | 31 Dec 2012 | Major revision | 1.01.00 | |
| Resolvability criterion added to Evaluations: General Criteria page 2 and Glossary. | 15 Dec 2012 | Minor revision | 1.00.01 | |
| First edition of the website. All pages are new. | 25 Aug 2012 | New edition | 1.00.00 | |
Acknowledgement
The author is delighted to acknowledge that Hal Switkay when Mathematics Professor at West Chester University, Pennsylvania, USA independently discovered the concept of Geometric Voting having appreciated the key significance of a GV(0 < r ≤ 1/2) voting system where, in his own words, "every element of the voting vector must be greater than the sum of all the following elements".
Revision Notes
Revision 2.00.00: A paper entitled Consecutively Halved Positional Voting: A Special Case of Geometric Voting and authored by myself and Hal Switkay was published in the online Social Sciences journal on 04 Mar 2023. The updated and revised analysis there is now reflected here in this website.
Revision 1.11.00: Pages 2 and 3 in the Majority and Related Criteria section and their associated maths proofs have been swapped over with minor edits. A new page 4 that addresses winning with strong versus broad support has been added.
Revision 1.10.00: Anti-plurality now included in the plurality section. The susceptibility of voting systems (such as CHPV, the Borda Count, plurality and anti-plurality) to the tactical voting options of burying and compromise lifting are now addressed in the geometric voting section.
Revision 1.09.00: New section added on page 8 regarding cloning maps and teaming thresholds for GV(r) anti-vectors. Remaining sections now appear on later pages.
Revision 1.08.00: The relationship between the weightings of a three-candidate positional voting system and the common ratio of the consistently equivalent geometric voting anti-vector [r = (w1-w2)/(w2-w3)] is now also incorporated.
Revision 1.07.00: The relationship between the weightings of a three-candidate positional voting system and the common ratio of the consistently equivalent geometric voting vector [r = (w2-w3)/(w1-w2)] is now incorporated.
Revision 1.06.00: The consensus and polarization indices have again been reworked; this time to simplify the justification for normalizing anti-vectors with non-positive weightings as opposed to the non-negative ones for vectors. Also, using truncation to raise the consensus index of a vector closer to its asymptote - the target value for a fixed system bias - is now included.
Revision 1.05.00: The consensus and polarization indices have been reworked with normalized rather than standardized vectors and the three types of positional voting vectors have been clarified.
Revision 1.04.00: Special thanks are due to Switkay for not only proposing the requirement for a consensus or polarization index for each vector in order that any two from the numerous positional voting systems that exist may be objectively compared against each other but also for his very constructive collaboration in the discussion and development of these two indices.
Revision 1.03.00: Several important issues were raised in a email from a co-discoverer of Geometric Voting; see acknowledgement above. Switkay suggested that there might be some merit in permitting tied voter preferences and wondered whether an Instant Run-off version of CHPV may have advantages. He also suggested employing hexagons as a visual aid for showing the effects of exchanging adjacent GV preferences. This revision reviews these three previously unaddressed topics.
Revision 1.02.00: The expansion of the monotonicity sub-section is in response to an email from R Hirth. Hirth pointed out that there is a variety of criteria relating to monotonicity in addition to the simple 'mono-raise' criterion. Hirth also spotted that CHPV and GV with r < 1/2 satisfy the 'mono-raise-random' and the 'mono-sub-top' criteria while GV with r > 1/2 fails them. This further highlights that CHPV is a critical threshold GV variant and that the choice of r = 1/2 is not arbitrary.
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