[EM] monotonicity and summability criteria

[Russ Paielli]

I have an "off the wall" question that some of the math geniuses on this 
list might find interesting.

Before its recent modification, the ElectionMethods.org website had a 
page called "Technical Evaluation of Election Methods." Two of the 
criteria listed on that page were monotonicity and summability. Most of 
you are familiar with the former, but the latter was my own idea. I have 
cut some of the text from the definition of summability and included it 
below for reference.

It occurred to me a while back that the two criteria may be equivalent. 
That is, if a method passes monotonicity, perhaps it must also pass 
summability, and vice versa. That's just a hunch. Can anyone prove (or 
disprove) it?

--Russ


Summability Criterion

Statement of Criterion

Each vote should map onto a summable array, where the summation
operation is associative and commutative, and the winner should be
determined from the array sum for all votes cast.

Complying Methods

All of the methods in the compliance table above comply with the
summability criterion except Instant Runoff Voting (IRV).

Commentary

<p>The summability criterion is the only criteria discussed on this
webpage that addresses implementation logistics. Election methods that
comply with the summability criterion are substantially easier to
implement with integrity than those that do not. All the election
methods listed in Table 1 comply except Instant Runoff Voting (IRV).</p>

<p>In plurality voting, each vote is equivalent to a one-dimensional
array with a 1 in the element for the selected candidate, and a 0 for
each of the other candidates. The sum of the arrays for all the votes
cast is simply a list of vote counts for each candidate.</p>

<p>Approval voting is the same as plurality voting except that more than
one candidate can get a 1 in the array for each vote. Each of the
selected or "approved" candidates gets a 1, and the others get a 0.</p>

<p>In Condorcet voting, each vote is equivalent to a two-dimensional
array referred to as a pairwise matrix. If candidate A is ranked above
candidate B, then the element in the A row and B column gets a 1, while
the element in the B row and A column gets a 0. The pairwise matrices
for all the votes are summed, and the winner is determined from the
resulting pairwise matrix sum.</p>

<p>IRV does not comply with the summability criterion. In the IRV
system, a count can be maintained of identical votes, but votes do not
correspond to a summable array. The total possible number of unique
votes grows factorially with the number of candidates. The larger the
number of candidates, the more error-prone and less practical it becomes
to maintain counts of each possible unique vote. It becomes impractical
with more than about six candidates.</p>

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