[EM] Description of "Median," my Approval variant
Forest Simmons
fsimmons at pcc.edu
Thu Mar 6 09:55:03 PST 2003
I finally had a chance to look this over. I think there are some good
ideas here.
The first matrix (of which you have shown the sub diagonal elements) is
the symmetric matrix whose entry in the i_th row and J_th column is the
number of voters who approved both candidate i and candidate j. If the
diagonal elements were included, they would be the total approvals for the
respective candidates (whether respectable or not).
I call this matrix the moment of inertia matrix, because (in the three
candidate case) it is precisely the matrix of moments about the coordinate
axes for a distribution of masses corresponding to the ballots at the
corners of the approval cube (assuming each ballot represents a unit point
mass).
The next matrix, derived from the first by subtracting it from a matrix
whose columns are constant vectors filled with the total approvals for the
respective candidates, can also be calculated directly as the transpose of
the pairwise matrix.
The pairwise matrix is the matrix whose entry in the i_th row and j_th
column is the number of ballots on which a distinct preference for
candidate i over candidate j is indicated.
So your method of minimizing the max row elements amounts to finding the
candidate with the minimal maximum number of votes against him/her in head
to head contests (as far as can be detected with the crude resolution of
the approval ballot).
If you subtracted the pairwise matrix from its transpose before doing this
MinMax count, you would be minimizing the maximum margin of defeat. In
other words, you would have the method called MinMax (margins) applied to
approval ballots.
If you deleted all of the losing votes from the (transpose of the
pairwise) matrix before doing the MinMax, you would have MinMax (winning
votes) applied to approval ballots.
But MinMax (in either form) applied to approval ballots will always yield
the same winner as plain Approval, because a pairwise matrix constructed
from approval ballots always has a beats all winner (when there is no tie)
and both versions of MinMax will always pick the beats all winner when
there is one.
But your method, by not deleting losing votes or going over to margins,
doesn't always pick the beats all winner. (This isn't necessarily bad, but
it is something to be aware of.) Note that in your first example, B is the
beats all winner as we would expect.
Here's another way to approach (and perhaps modify) your method:
Consider some of the possibilities for deciding the winning team in a
round robin basketball tournament:
(1) Take the team that gave up the fewest points per game on average (i.e.
had the fewest points scored against them per game on average). This
might be considered the team with the best defense.
(2) Take the team for which the highest score against them was minimal.
This is analogous to your method, and (arguably) could also be considered
as picking the team with the best defense.
(3) Take the team for which the maximum margin against them was minimal.
This is analogous to MinMax (margins). If there is a beats all team, this
method will pick it.
(4) Take the team for which the maximum score against them was minimal.
This is analogous to MinMax (winning votes), and also will detect a beats
all winner.
All of these methods pick teams based on measurements of defensive
prowess.
But what about offensive prowess?
(5) Take the team that scored the highest point average per game.
(6) Take the team with the maximal minimum number of points per game. In
other words, if team A scored at least 25 points in every game, and every
other team scored fewer than 25 points in at least one of their games,
then A should be the winner according to this criterion.
etc.
But why not use a combination of defensive and offensive prowess to
decide the winner?
(7) We could combine (1) and (5) to get the team whose difference in total
points scored and points given up (i.e. scored against them) was the
largest. This means subtracting the column sums from the row sums of the
pairwise matrix, which always yields the Borda winner in the voting
context (assuming that truncated ballots are scored correctly).
(8) We could combine your method (2) with method (6), by finding the max
of each row in your (transpose of pairwise) matrix and then subtracting
the min of the corresponding column. The candidate corresponding to the
minimum such difference would be the winner.
[Note that this is similar to MinMax (margins) but with the subtraction
done after the row and column max and min operations.]
This combination method (8) would take care of the anomalous results at
the end of your message.
Note for example, that disaster associated with
1000: A, 1000: B, 1: AC, and 1: BC
is avoided by taking into account the offensive prowess of candidates A
and B in comparison to C who is relatively strong defensively but weak
offensively.
In general method (8) doesn't always pick the beats all winner. I don't
know if it does when restricted to ballots of resolution two (i.e.
approval ballots). I suspect there may be some cases where it doesn't
pick the approval/beats-all winner (which is not necessarily a defect in
this context).
However it does satisfy the majority criterion, since a unique majority
candidate would have a row max of less than 50% and a column min of more
than 50% so the difference would be negative, which can happen for only
one candidate.
I believe that these ideas are worth exploring further.
Your idea of minimum maximum distance as locating the central point among
many is worth exploring with other measures of distance, as well.
Forest
On Wed, 5 Mar 2003, [iso-8859-1] Venzke Kevin wrote:
> Kevin Venzke
> stepjak at yahoo.fr
> 3/5/03
>
> I said in my first post that I was designing an
> Approval variant aimed at more often producing the
> Condorcet winner. Here it is. I hope it's of some
> interest, that it's novel, and that I don't come
> across as too pretentious.
>
> "MEDIAN" ELECTION METHOD
>
> This is a proposal for a single-winner election method
> I call "Median." Votes are cast as under Approval,
> and a method, justified by a geometry analogy, is used
> to determine the "median candidate." More precisely,
> this is the candidate most likely to be supported by
> the "median voter" as drawn on a political spectrum.
> The number of dimensions of the spectrum is
> unimportant, and the method will not be broken if
> there is in fact no spectrum behind the voters'
> choices.
>
> METHOD
>
> The voter votes for as many candidates as he wishes.
> The total number of voters "approving" each candidate
> is counted, as under Approval. However, a
> two-dimensional table has also to be maintained,
> recording the number of voters that approved each pair
> of candidates.
>
> For instance, if the ballots read:
> 12: AB
> 15: BC
> 10: AD
> 9: D
>
> After counting the votes, we would have counted not
> just that A, B, C, and D's total votes received were
> 22, 27, 15, and 19 respectively. We would also have
> maintained the following table of the "overlap" of
> support between each pair of candidates:
>
> A B C D
> A .
> B 12 .
> C 0 15 .
> D 10 0 0 .
>
> Next, for each of the candidates, we determine which
> of the other candidates had the most supporters who
> did not also vote for the candidate we're talking
> about. This is why the overlap information is needed:
> If we are determining A's supporters' disapproval for
> B, for example, we must subtract the A-B overlap from
> A's vote count. For the above scenario, we could make
> the following table. Each row is the disapproval for
> a candidate, while each column is the disapproval
> expressed by a candidate's supporters.
>
> A B C D greatest
> A 0 15 15 9 15 (from B or C)
> B 10 0 0 19 19 (from D)
> C 22 12 0 19 22 (from A)
> D 12 27 15 0 27 (from B)
>
> The "greatest" column is what matters. It tells us
> the "greatest constructive disapproval" for each
> candidate. In other words, for candidate A, there is
> no group of voters larger than 15 people who did not
> vote for A and who were agreed on a different
> candidate. Since 15 is the smallest "GCD," A is
> judged to be the median candidate and would, by this
> method, be the winner. (A tie could be broken simply
> by number of votes received.)
>
> You can also imagine the method this way: If you
> removed all of the ballots approving candidate X, how
> many votes would the top-scoring candidate then have?
> Elect X if the answer is the smallest.
>
> (You wouldn't have to use this method alone. You
> could elect the candidate who maximizes (votes
> received / GCD). Approval elects B, of course. If
> you do this division, A gets 1.467 while B gets 1.42.
> A would still win, but it's much closer.)
>
> I won't consider the question of whether the "median
> candidate," however defined, should be the winner. I
> will only use a geometrical analogy to argue that the
> candidate with the smallest GCD is probably the median
> candidate.
>
> Consider the following very simple scenario:
> 26: A
> 25: AB
> 25: BC
> 27: C
>
> Clearly we have a left, right, and center candidate.
> At least, it's possible to draw it that way. The
> Approval scores are 51 A, 50 B, 52 C, with C winning.
> Under Median, the scores are 52 A, 27 B, 51 C, with B
> winning. The spectrum basically looks like this, with
> B obviously being the median:
> AAAAAAAA
> BBBBBBBB
> CCCCCCCC
>
> Here is the analogy: Think of the candidates as points
> on a line or plane, and the "constructive disapproval"
> of (for instance) candidate A's supporters towards
> candidate B, as the distance (measured in voters) that
> B would need to "travel" in order to win over A's
> supporters. (Note that the distance is not
> reciprocal. If it were, then two-candidate elections
> would always be ties.)
>
> We can think of the "greatest constructive
> disapproval" for a given candidate as the greatest
> distance he would have to travel in one direction to
> obtain the approval of all the voters. Candidates A
> and C above would each have to win over 50 voters in a
> single direction, while B is only 27 voters away from
> the furthest candidate. It's unnecessary to be able
> to draw the candidates on a line or plane precisely.
> It's intuitive to suppose that the point with the
> least greatest distance to another point, is the one
> which is closest to the center.
>
> STRATEGY
>
> I think that it would be difficult to vote
> strategically under Median. This is because votes
> aren't counted directly. For instance, in order to
> "hurt" a certain candidate, it's not sufficient to not
> vote for him. You need to vote for a candidate who is
> likely to provide the GCD against him. In other
> words, if you want to "hurt" your second-favorite who
> is similar to your favorite, you're probably out of
> luck since the latter's supporters are probably not
> the largest bloc of constructive disapproval against
> the candidate you don't like. A candidate's fate is
> essentially decided by the voters who dislike him
> sincerely.
>
> Unlike in Approval, in Median there is reason for
> centrist voters to vote for second-favorite,
> off-center candidates in addition to their favorites.
> So doing can hurt a least favored candidate. It can
> never cause the favorite to lose, unless the favorite
> is otherwise the "most distant" candidate from the
> second-favorite. If anyone is interested, I have a
> scenario where if some centrist B voters had also
> approved off-center C, then B would have been judged
> the median instead of opposite off-center A.
>
> FLUKES
>
> Consider this:
> 3: A
> 1: AB
> 2: BC
> 3: C
>
> Approval scores are A 4, B 3, C 5. Median scores are
> A 5, B 3, C 4. B is the winner even though C has
> majority approval. Defensible? Is it necessary to
> claim that B was everyone's second choice, at worst?
> Or is "median voter's candidate" a valid criterion on
> its own?
>
> Two of the five C voters said B was acceptable. If C
> wins, 4 B supporters are unhappy. If B wins, 6 people
> are unhappy, but they're divided 3 and 3 on who
> would've been better. You could say that if B wins,
> no more than 3 people will attend the same protest.
> You could say that only 3 people are alienated by B in
> the same way. Possibly one could claim that it's
> impractical to give power to whichever majority
> manages to assemble itself for a given election.
>
> And consider this disaster:
> 1000: A
> 1000: B
> 1: AC
> 1: BC
>
> Approval scores are A 1001, B 1001, C 2. Median
> scores are A 1001, B 1001, C 1000. C wins with only
> two votes. Of course, this is not very likely; A and
> B have to be exactly tied for C to win like this.
> Specifically, for every A-only voter we add, C needs
> one more voter overlap with A in order to still win.
> Additional overlap with B will not help C in this
> case. Example:
>
> 1050: A
> 1: AC
> 1001: BC
> Median scores are A 1001, B 1051, C 1050. A is deemed
> to be the pick of the median voter.
>
>
> I'm interested in comments and criticism, especially
> regarding philosophy, voter strategy, and election
> method criteria. I suspect that Median meets similar
> criteria to Approval, although Median doesn't meet
> Independence from Irrelevant Alternatives. It is very
> possible for a non-winning candidate to be
> "kingmaker." I would actually propose that, assuming
> perfect information, the more candidates there are,
> the "better" the result will be.
>
> Thanks to any who may read this.
>
> Kevin Venzke
> Stepjak
>
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