[EM] Condorcet for public proposals - IMV/MMV & MAM-d

[Ernest Prabhakar]

Hi all,

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 > A


  3. 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 > E

  Note 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 > D


  5. 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.