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Forest,<br>
I had a quick look at this. It seems that when there are just three
candidates and they are in<br>
a cycle it elects the Approval winner. Is that right?<br>
<br>
In that situation I prefer ASM. I think my favourite method that uses
the same type of ballots<br>
as your "Conditional Approval" would be a version of ASM Elimination
that uses the voters'<br>
original approval cutoffs while they make some distinction among
remaining candidates and <br>
thereafter interprets the voters' ballots as approving all but the
lowest ranked of the remaining<br>
candidates.<br>
<br>
Chris Benham<br>
<br>
<br>
<br>
<br>
Forest W Simmons wrote:
<blockquote cite="mid4676015.1179345546042.JavaMail.fsimmons@pcc.edu"
type="cite">
<pre wrap="">In probability theory when partial info about a random variable is
given, the resulting updated expections and probabilities are called
conditional expectations and probabilities.
In that spirit, instead of calling the updated approvals based on
partial info "reactive," from now on I'm going to call them
"conditional."
So the (i,j) element of the conditional approval matrix approximates
the approval that candidate i would get, given only that candidate j is
the poll front runner.
Now for an update on how to use this conditional approval matrix to
choose an election winner.
Previously, I suggested circling the highest number in each column,
removing each row not having a circled number as well as the
corresponding column, repeating the process until every row has exactly
one circled number (not worrying about ties for now), and finally
electing the candidate with the largest row minimum in the remaining
matrix.
For this update I would like to change the final step.
By the time each row has exactly one circled number the set of
candidates is partitioned into cycles of one or more candidates each of
the type
x0, x1, x2, ... x0
where candidate i follows candidate j in the cycle if and only if
element i of column j is circled.
Let's use X~Y to denote that X and Y are members of the same cycle.
In the revised final step, elect the candidate X with the largest
minimum conditional approval given Y over all candidates Y such that
Y~X.
In other words X maximizes
Min over Y~X of CA(X,Y)
where CA(X,Y) is the (X,Y) entry of the remaining conditional approval
matrix.
Ideally, each of the remaining candidates (after iteratively crossing
out the rows and columnns of the conditional approval losers) would be
a conditional approval equilibrium candidate, which would make each
cycle consist of exactly one candidate. In that ideal case, the
equilibrium candidate with the greatest approval would be the winner.
But since the ideal case is too much to expect, we think of equilibrium
cycles instead of equilibrium candidates, and go with the winner of the
cycle that maximizes the min conditional approval of its cycle winner.
Forest
</pre>
<blockquote type="cite">
<pre wrap="">From: Forest W Simmons <a class="moz-txt-link-rfc2396E" href="mailto:fsimmons@pcc.edu"><fsimmons@pcc.edu></a>
Subject: Re: [EM] Does this method have a name?
The "reactive approval" of candidate X relative to Y as defined below
is supposed to approximate the approval that X would get given only
that Y was ahead of all the other candidates in the polls.
In other words, if there were zero info up until someone reveals that Y
is the front runner, would you approve X or not?
Suppose that under zero info your approval cutoff was below Y. Given
the information that Y is the frontrunner, wouldn't it make sense to
move your cutoff up to just below Y?
On the other hand, suppose that under zero info you disapprove Y.
Given the info that Y is the frontrunner wouldn't it make sense to move
your cutoff down to just above Y?
The move would be in reaction to the given information, hence the term
"reactive."
So we define the reactive approval of X relative to Y as the number of
ballots on which X would be approved if the voted approval cutoff were
moved adjacent to (but not past) Y on each and every ballot.
Let RA be the matrix whose entry in row i and column j is the reactive
approval of candidate i relative to candidate j.
Let's call this matrix the reactive approval matrix.
Below I suggested one way of using this matrix to determine a winner.
Here's a more interesting one:
1. Circle the largest number in each column of the RA matrix.
2. Cross out each row that has no circled element.
3. Cross out the columns that correspond to the rows that were crossed
out. [These rows and columns represent straw men or false alarms, so to
speak, since any poll indicating that they were ahead would be
misleading.]
4. Repeat steps 2 and 3 until each remaining row has exactly one
remaining circled element.
5. The winner is the candidate who has the largest row minimum in the
remaining matrix.
What do you think?
Forest
</pre>
<blockquote type="cite">
<pre wrap="">Here's an example that might clear up some questions:
Suppose that the original ballot is
A=B>C=D>E=F|G=H>I=J>K=L
where "|" is the voter's marked approval cutoff.
Then in calculating reactive approvals relative to C we move the
approval cutoff adjacent to but not past the position shared by C and D:
A=B>C=D|E=F>G=H>I=J>K=L
Note that this ballot gives A, B, C, and D reactive approval relative
to C. The reactive approvals relative to D are exactly the same.
Going in the other direction, let's see which candidates receive
reactive approval relative to either I or J. Starting at the original
approval position and moving to (but not past) the position shared by I
and J we get
A=B>C=D>E=F>G=H|I=J>K=L
All of the candidates except I, J, K , and L get reactive approval
relative to I or J from this ballot.
Note that in every case, the reactive approval of candidate X relative
to candidate X is just its original approval, since the cutoff does not
move past X.
Furthermore, if a voter wants all of the reactive approvals to be the
same as his original approvals, all he has to do is rank all of his
approved candidates equal top and truncate the rest.
Here's the nitty gritty of deciding an election by this method:
Form a square array in which the number in row i and column j is the
total reactive approval of i relative to j.
To the right of each row in the array write the smallest number in that
row. Then circle the largest of these row minima. The winner is the
candidate whose row is to the left of the circled number.
Note that I have started using "reactive" instead of "reactionary"
because of the negative political connotation of the latter term (which
I used formerly).
Forest
</pre>
</blockquote>
</blockquote>
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