Thanks for all your input. I've rewritten my IMV proposal to reflect as much of the comments as I agreed with. :-) I hope to post this on my site next week, along with a Python implementation. The audience of this document is primarily people who do *not* have experience in either set theory or electoral reform. Thus, I have tried to carefully define my terms, and leave much of the jargon in endnotes. However, I do also want to be correct in the terminology I do use, so experts (such as yourselves :-) don't get unnecessarily annoyed.
I would welcome any further comments, as well as additional URLs to recommend for people who want to learn more. The name, however, is fixed. :-)
Yours, Ernest "I want my IMV" Prabhakar
Instant Matchup Voting: Improving the Roots of Democracy
The Problem
Voter participation and majority rule are often considered the heart of democracy. However, the most common form of "one person, one vote" elections in the U.S. (usually called Plurality or First Past the Post) implicitly assumes there are only two candidates. When there are more than two candidates, not only is there a risk that no candidate will get an absolute majority, but voters are faced with the dilemma between voting 'strategically' (for the lesser of two evils) vs. voting 'sincerely' (with their conscience). This also tends to promote a two-party system, which despite its many merits (such as the ability to ensure governable coalitions), is vulnerable to systemic bias (as evidenced by low voter turnout, due to a perception that neither party offers a meaningful choice). The end result is that candidates lack a true majoritarian mandate, due both to low voter turnout and the possibility of a split vote.
Our Solution
To address these problems, we recommend an alternate election system called Instant Matchup Voting [1]. Instant Matchup Voting, or IMV, treats an election as a series of simultaneous one-on-one matchups, like a round-robin tournament. Each voter's ballot describes the results of one complete round of matchups between each pair of candidates. The candidate who gets the majority of votes in all their matchups is the winner. In the unlikely event of a tie (where no candidate wins all their matches) the tied candidate with the fewest votes against them wins.
The Benefits
This system allows voters to fully express their preferences among the available candidates, and generally makes it possible for them to vote sincerely without having to worry about strategy [3]. It also tends to discourage mudslinging in multi-candidate elections, since there is an incentive to have the other candidate's supporters vote you for second or third place. Perhaps more importantly, it allows non-traditional and third-party candidates to run without fear of becoming spoilers, increasing the range of meaningful choices available to voters.
Instant Matchup Voting
The formal procedure for IMV has five standard phases, plus three tiebreaking phases:
1. Votes
Each voter votes for all the candidates they like, indicating order of preference.
Consider an example in a five-candidate election, where a voter likes A most, B next, and C even less, but doesn't care at all for D or E. In that case, their ballot would be ranked "A > B > C"
2. Matchups
Each ballot defines the result of matchups between all the candidates in a election.
Thus, the ballot A > B > C would be interpreted as:
A > B, A > C, A > D, A > E
B > C, B > D, B > E
C > D, C > E3. Pairwise Matrix
The results from all the ballots is summed up in what is called a 'pairwise matrix', where the rows indicate votes -for- a candidate, and the columns indicate votes -against- a candidate.
So, given the following results from 9 voters:
4 votes of A > B > C (over D and E)
3 votes of D > C > B (over A and E)
2 votes of B > A (over C, D and E)The pairwise matrix would be:
A B C D E
A - 4 6 6 6
B 5 - 6 6 9
C 3 3 - 5 7
D 3 3 3 - 3
E 0 0 0 0 -
4. Win tabulation Using the matrix, tabulate the wins for each candidate.
For the case above, we have:
A> C:6/3, D:6/3, E:6/0
B> A:5/4, C:6/3, D:6/3, E:9/0
C> D:5/3, E:7/0
D> E:3/0
E> -nobody-Note: "A> D:6/3" means 6 votes for A > D vs. 3 votes for D > A
5. Majority winner The candidate who wins all their matchups is the majority winner
In the case above, B wins all their matchups. In particular, B beats A, since a majority of voters indicated that they prefer B over A. True, some of those voters would ideally have preferred D or C, but since they can't have that they're still happier with B than A. By using all the information from every voter, IMV is one of the most efficient forms of voting available, in terms of maximizing the wishes of the majority.
6. First-round ties
In rare cases there may be no majority winner. [4]. In this case, we declare a 'first round tie' (also known as a Smith Set[2]) between the candidate with the least losses [5], and all the candidates which don't lose to that candidate or another member of the Smith Set.
For example, consider 9 ballots of the form:
2 votes of A > B > C (over D and E)
4 votes of D > C > B (over A and E)
3 votes of E > A (over B, C and D)This gives a win table of:
A> B:5/4, C:5/4, D:5/4
B> E:6/2
C> B:4/2, E:6/2
D> B:4/2, C:4/2, E:4/3
E> A:3/2or, equivalently, a loss table of:
A< E:2\3
B< A:4\5, C:2\4, D:2\4
C< A:4\5, D:2\4
D< A:4\5
E < B:2\6, C:2\6, D:3\4Thus, A < E, E < (B, C, D), so the Smith Set is [A,B,C,D,E] - a five way tie!
7. Tiebreaker
The tiebreaker is the member of the Smith Set who had the least number of people voting against them.
For the example in (6), the most votes against each candidate is:
A:3, B:5, C:5, D:5, E:6
Thus, A is the tiebreaking winner with only 3 votes against them -- though in this exceptional case they not the majority winner.
8. Second-round ties
If two or more candidates have the least number of people voting against them (within statistical uncertainties[6]), these then form a second-round tie. If there is no alternative mechanism available[7], the winner is picked at random from within the second-round tie.
Ernest N. Prabhakar, Ph.D. Founder, RadicalCentrism.org January 30, 2004
RadicalCentrism.org is an anti-partisan think tank based near Sacramento, California, which is seeking to develop a new paradigm of civil society encompassing politics, economics, psychology, and philosophy. We are dedicated to developing and promoting the ideals of Reality, Character, Community & Humility as expressed in our Radical Centrist Manifesto: The Ground Rules of Civil Society.
Notes
[1]
IMV is a variant of what is called Smith PC(wv) -- short for Plain Condorcet using winning-votes within the Smith Set[2]. See http://www.wikiepedia.org/wiki?condorcet for more details.
[2]
The Smith Set is those candidates which beat all the candidates outside the set, but beat or tie (do not lose to) each other.
[3]
To be precise, it is mathematically impossible to have a perfect voting system free of any strategic considerations. However, we believe Smith PC (wv) is the simplest system that meets most of the important criteria. More complicated tiebreaking formulas might resist strategic manipulation better on certain occasions, but only under what we consider extraordinary and unlikely circumstances (at least for large public elections).
[4]
This can only happen if we have a 'rock-paper-scissors' situation (also called a circular tie), where A beats B, and B beats C, but C beat A. This is very unlikely in normal public elections -- since each individual ballot requires a strict ranking among candidates -- but is possible if, for example, a significant fraction of the population casts ballots that don't reflect a linear Left-Right political spectrum.
[5]
If several candidates have the maximum number of wins, pick the one which beats the other such candidates. If there is a circular tie among them, include them all in the Smith Set.
[6]
Voting machines and processes, like all human activities, are not infinitely accurate. If the difference between two votes is less than the achievable precision, it is statistically insignificant, and therefore random. In such cases, we believe it is better to be explicitly random than to use arbitrary criteria to create a false level of precision. Of course, this requires having a well-established error rate (level of uncertainty) before the election, to avoid uncertainty about the uncertainty!
[7]
In some environments, there is an alternate mechanism available for 'true' tiebreakers. Either an alternate body (e.g., the House of Representatives) can break the tie, or (for small groups) it may be feasible to re-run the election and hope that the results will shift. However, in other cases no alternate mechanism is available, but a decision is still necessary. For these cases, we specify random choice in order that IMV may be completely specified, especially for computer analysis of voting methods.
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