Maximum Majority Voting

Improving the Roots of Democracy through Election Reform

A. Problem

Voter participation and majority rule are often considered the heart of democracy. However, the most common form of single-winner voting in the United States — “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 (versus a plurality), but voters are faced with the dilemma between voting ‘strategically’ (for the lesser of two evils) vs. ‘sincerely’ (for whom they feel is the ‘best’ candidate). This dilemma tends to promote a two-party system, which despite its many merits 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. Solution

To address these problems, we recommend an alternate election system we call Maximum Majority Voting, or MMV [1]. Maximum Majority Voting is based on the latest research into election reform, but is still designed to be as simple as possible to use and understand. Each voter simply fills out a ranked ballot, listing the candidates in order of preference. The winner is the candidate with the largest majority voting for them over other candidates. By specifying a complete set of preferences, you never have to worry about “wasting your vote”; e.g., if your most-preferred candidate doesn’t win, your vote still helps your second choice to beat your third (or worse) choice. Thus, you can still vote your conscience without giving up the ability to influence the election.

C. Features

While there are several forms of ranked voting[2], Maximum Majority Voting is one of the best at reducing the need for strategic voting[3]. The reason you generally don’t need to vote strategically under MMV is that each election is broken down into a series of one-on-one matchups, like a round-robin tournament. For each pairing of candidates A and B, MMV compares the number of voters who prefer A over B (often written “A > B”) to those who prefer B over A (“B > A”). Each voter’s list of preferences is interpreted in terms of these pairwise contests, so a vote of “A > B > C” implies A > B, A > C, and B > C. Even if the top two candidates are B and C, it still doesn’t hurt for me to vote for A first, since my vote for B > C counts just as much as my vote for A > B. Thus, all of each voter’s preferences are used to determine the final ranking, rather than treating higher or lower-preferences separately.

D. Benefits

In addition to reducing the need for strategic voting, MMV provides a number of other benefits:

  • MMV ensures that the winning candidate is preferred by a majority of the voters over any other given alternative, no matter how many candidates are running.
  • MMV system allows voters to fully express their preferences among all the available candidates.
  • MMV 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.
  • MMV allows primary losers, third-parties, and other non-traditional candidates to run without fear of becoming spoilers, increasing the range of meaningful choices available to voters.

Thus, in contrast to traditional Plurality voting, MMV actually becomes more effective — rather than more polarized — with more candidates and greater citizen involvement. In the place of a splintered and disenfranchised electorate, it can actually help us find and elect candidates who reflect our underlying shared values.

E. Process

The formal definition of Maximum Majority Voting involves six 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 “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 it is redundant).

2. Preferences

Each ballot defines preferences based on one cycle of pairwise matchups between each of the candidates.

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 > A

3. Counting

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 > E

Note how some matchups add up to less than 9, because some candidates were given equal preference by some voters. Also, the total number of matchups for N candidates is always N * (N-1)/2, or 10 for N = 5.

4. Sorting

Next, the matchups are sorted from the largest win to the smallest. If two matchups have the same number of winning votes, the one with the largest margin (weakest loser) is listed first. It is also possible for two matchups to have the exact same score; while such “same-size majorities” are extremely unlikely in public elections, they could well occur in small committees.

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 > D

5. Candidates ordered

This list of matchups is used to rank the candidates, starting from the largest win on down. The order is important, because in rare cases [5]a later matchup may conflict with an earlier one:

(i) If a matchup later in the list conflicts with the previously-determined order, the latter matchup is superseded (ignored).

(ii) In the even unlikelier case where several matchups with same-size majorities conflict with each other, all such conflicting matchups are ignored (though any non-conflicting matchups of that size are still included).

Stepping through the ordered list of matchups above, we find (using [X,Y] for unordered candidates):

 1: C > D                (C > D)
 2: A > B, C > D         (A > B)
 3: A > B, [A,C] > D     (A > D)
 4: A > [B,E], [A,C] > D (A > E)
 5: A > B, [A,C] > [D,E] (C > E)
 6: C > A > [B,D,E]      (C > A + A > B => C > B)
 7: C > A > [B,D,E]      (C > B already assumed)
 8: C > A > [D > B, E]   (D > B)
 9: C > A > [D,E] > B    (E > B)
 10: C > A > E > D > B   (E > D)

6. Winner selection

Based on the above, we see that MMV typically generates a strict ranking of candidates in the order they were preferred by the most voters (unless all the pairwise matchups for multiple candidate are precisely identical). The winner is thus the top candidate, i.e. the one with the Maximum Majority. In the extraordinary case there really is a complete tie among all the top candidates, then the winner would need to be chosen by some external mechanism.[7].

F. Conclusions

Since Maximum Majority Voting uses all the information available, and weights larger majorities over lesser ones (in case of conflict), it will always reflect the Maximum Majority of the electorate to the maximum extent possible – no matter how many candidates. This should allow for greater choice among candidates, and greater involvement from voters. While electoral reform may not solve all our political problems, by increasing competition it allows us to encouarge higher-quality candidates, thus opening the door to a more representative and accountable democracy.

Ernest N. Prabhakar, Ph.D.
February, 2004 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



MMV can be considered a deterministic variation of Steve Eppley‘s Maximize Affirmed Majorities (MAM) system, which in turn is based of Tideman’s well-studied Ranked Pairs algorithm for finding the pairwise winner (also known as the
Condorcet winner
). This particular Condorcet-compatible variant was apparently first proposed by Mike Ossipoff, and recommended to me by Eric Gorr
For example, another popular way of evaluating ranked ballots is called Instant Runoff Voting While a slight improvement over Plurality, IRV is not as good as MMV at using all the voters information and reducing the need for strategic voting. It also tends to choose the extremist candidates with strong support, rather than balanced candidtes with Maximum Majority (that is, the Condorcet winner).
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. Even with MAM it is theoretically possible for one party to attempt to vote ‘insincerely’ to prevent the election the true consensus winner; however, not only does this require significant public coordination and the risk of electing an even more undesireable candidate, but there are counter-strategies that can be used by other parties to defuse their impact. The deterministic variant used in MMV (somemtimes called MAM-d) is not as well studied as MAM, but so far appears to possess the same desireable properties.

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	-

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.
For example, say that the current ordering is “A > B, C > D”, and there is a same-size majority 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. However, a third matchup of that same size with “D > E” would be included, not discarded. Again, this is an extremely unnatural occurence, but is included here for theoretical completeness.
For example, in the U.S. Presidential Election ties are resolved via the House of Representatives. While such exact ties are extraordinarily unlikely in public elections, statistical ties (where the difference is within the margin of error) have been known to occur, and voting systems should provide some objective means of resolving them. For MMV, the default system is known as ‘Random Dictator’: one ballot in chosen at random, and its preferences are used to break the tie. If two or more top candidates are ranked equally by that ballot, then one of those doubly-tied candidates are picked at random. While such non-determinism may not be acceptable in public elections, this can be useful, e.g. in small committees, where such exact ties are more likely and there is no other way to resolve a deadlock. The reason for a random ballot, rather than simply picking among the top candidates, is to reduce the incentive for one faction to ‘stuff’ the ballot with multiple ‘clone’ candidates that would be equally ranked.

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