Well, after digesting Eric Gorr's deterministic variant of MAM (if he, and I, understand it correctly), I have rewritten my IMV proposal to use that instead of Smith PC, under the new name "Multiple Matchup Voting" - since it is based on ranking the results of multiple matchups. I've also added Adam's suggestion of hiding the Pairwise Matrix in favor of simplicity.
The terminology is probably a little rough (since I'm writing this late at night), but I still welcome your comments. I think I have a good idea how to implement this, but I want to make sure I understand the basic algorithm first.
-- Ernie P.
Multiple Matchup Voting: Improving the Roots of Democracy
http://RadicalCentrism.org/mm_voting.html
A. The Problem
Voter participation and majority rule are often considered the heart of democracy. However, the most common form of voting in the U.S. -- "one person, one vote" (technically known as 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' (what their conscience feel is the 'best' candidate). 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.
B. Our Solution
To address these problems, we recommend an alternate election system we call Multiple Matchup Voting [1], where voter rank the candidates in order of preference. Multiple Matchup Voting, or MMV, treats the ballots as the result 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 results of the matchups are used to compile an ordered list of candidates. This guarantees[2] that the winner is the majority preference under all circumstances, regardless of the number of candidates.
C. 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 for you as 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.
D. Multiple Matchup Voting The formal procedure for MMV has five phases:
1. Voting
Each voter votes for all the candidates they like, indicating order of preference if any.
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 > D = E". This can be shortened to "A > B > C" since unranked candidates are considered to be at the bottom and equivalent. Similarly, if another voter liked E most, considered D and C tied for second, and ranked B over A, their ballot would be "E > D = C > B > A" (with the last "> A" being optional, since its redundant).
2.Matchups Defined
Each ballot defines the result of one round 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 > E while "E > D = C > B > A" would be:
E > D, E > C, E > B, E > A
D > B, D > A
C > B, C > A
B > A3. Matchups Counted
When all the balots are counted, this gives a final score for each matchup. [4].
Say we have nine voters, who voted as follows 4: A > B > C 3: E > D = C > B 2: C > A > D
this gives:
6/3: A > B
4/5: A > C
6/3: A > D
6/3: A > E
4/5: B > C
4/5: B > D
4/3: B > E
6/0: C > D
6/3: C > E
2/3: D > ENote how some matchups add up to less than 9, because some candidates were ranked equal by some voters. Also, the total number of matchups for N candidates is always N * (N-1)/2, or 10 for N = 5.
4. Matchups Ranked
Next, the matchups are ranked in order of the largest winner. If two matchups have the same winner, the one with the smallest loser (largest margin) is listed first. If the matchups are completely identical, it is called a pairwise tie; while common in small committees, such ties are extremely unlikely in public elections.
The ballots above would thus be reordered to give:
6/0: C > D
6/3: A > B
6/3: A > D
6/3: A > E
6/3: C > E
5/4: C > A
5/4: C > B
5/4: D > B
5/4: E > B
3/2: E > D5. Candidates ordered
This list of matchups is traversed in order, from the largest win down. The candidates are sorted into a list that is consistent with that order, with the following caveats:
(i) In the unlikely event that a matchup later in the list conflicts with the previously-determined order[5], that matchup is discarded (this is why the ordering of matchups is significant).
(ii) In the even unlikelier case where several members of a pairwise tie might conflict with each other, all such conflicting matchups are discarded. [6].
Stepping through the ordered list of matchups, we find (using (X,Y) for unordered candidates):
1: C > D
2: A > B, C > D
3: A > B, (A,C) > D
4: A > (B,E), (A,C) > D
5: A > B, (A,C) > (D,E)
6: C > A > (B,D,E) [implies C > B]
7: C > A > (B,D,E) [thus unchanged]
8: C > A > D > (B,E)
9: C > A > D > E > B
10: C > A > D > E > B [E > D discarded as inconsistent]
6. Winner selection
Thus, with MMV will always have a strict ordering of candidates (unless all the pairwise matchups for multiple candidate are precisely identical). The top candidate, obviously, is the winner. If there really is a true tie among the top candidates, then the winner would need to be chosen from the top candidates by random draw, or some external mechanism.
E. Conclusions
Since Multiple Matchup Voting uses all the information available, and weights larger majorities over lesser ones (in case of conflict), it is always gauranteed to reflect the majority preference of the electorate. While this sort of electoral reform may not solve all our political problems, it will help involve more voters -- and candidates -- in the electoral process, which can only strengthen our democratic institutions.
Ernest N. Prabhakar, Ph.D.
Founder, RadicalCentrism.org
February 8, 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]
MMV could also stand for Majority Maximization Voting, as it is based on a deterministic form of Steve Eppley's Maximize Affirmed Majorities (MAM) system, which in turn is a variant of Tideman's well-studied Ranked Pairs algorithm. See http://www.wikiepedia.org/wiki?condorcet for more details. This deterministic variant was first suggested to me by Eric Gorr of http://www.condorcet.org/.
[2]
Under a suitable definition of majority preference. See Note 5.
[3]
To be precise, it is mathematically impossible to have a perfect voting system free of any strategic considerations. However, with Maximize Affirmed Majorities sincere voting is usually the optimal strategy. It is theoretically possible for one party to attempt to vote 'insincerely' (at the risk of electing a non-preferred candidate) to avoid electing the true consensus winner, but in that case there are defensive strategies that other parties can use to preserve the intended winner. The deterministic variant(somemtimes called MAM-d) is not quite as well studied as MAM, but so far appears to possess all the same desireable properties.
[4]
This tabulation is usually done via what is called a 'pairwise matrix', where the rows indicate votes -for- a candidate, and the columns indicate votes -against- a candidate. This is often used in voting systems associated with the Condorcet Criteria, which states that any candidate which is unanimously preferred to each other candidate on the basis of pairiwse matchups should win the election.
For example, 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 (sometimes called the Condorcet 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 -[5]
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.
[6]
For example, say that the current ordering is "A > B, C > D", and there is a pairwise tie between the next two items, "B > C" and "D > A". Since the former would imply "A > D" and the latter would imply "C > B", they are inconsistent with each other, and would both be discarded. Again, this is an extremely unnatural occurence, but is included here for theoretical completeness.
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