[EM] Conditional Approval (was "Does this method have a name?")

[Forest W Simmons]

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

>From: Forest W Simmons <fsimmons at pcc.edu>
>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
>
>
>
>>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