[EM] DMC,AWP,AM

[Forest Simmons]

Chris,

I wonder if the following Approval Margins Sort (AMS) is equivalent to 
your Approval Margins method:

1. List the alternatives in order of approval with highest approval at the 
top of the list.

2. While any adjacent pair of alternatives is out of order pairwise ...
    among all such pairs swap the members of the pair that differ
    the least in approval.

This method is clone independent and monotonic, and yields a social order 
that reverses exactly when the ballots are reversed.

If AMS and AM are the same, it might be useful to have this alternative 
description.

If they are not the same, it would be interesting to see how they compare 
in properties and performance.

I haven't had the time to run AMS on your examples below, but I will soon 
if you don't beat me to it.

Forest



Chris' message follows:

Date: Fri, 25 Mar 2005 02:29:39 +1030
From: Chris Benham <chrisbenham at bigpond.com>
Subject: [EM]Definite Majority Choice, AWP, AM

Ted, Russ, Forest, James,Juho and others,

I think that Ted's draft public "Definite Majority Choice" proposal is
excellent, in the sense that anything that might be slightly better
would be more complicated and/or less intuitive.

Two contending methods that use the same style of ballot are  James
G-A's Approval-Weighted Pairwise and my Approval Margins.
I've found a couple of examples that illustrate differences between the
three methods.
The first is copied from a Sep.22,04 James G-A post.

3 candidates: Kerry, Dean, and Bush. 100 voters.
 	Sincere preferences
19: K>D>>B
5: K>>D>B
4: K>>B>D
18: D>K>>B
5: D>>K>B
1: D>>B>K
25: B>>K>D
23: B>>D>K
 	Kerry is a Condorcet winner.

 	Altered preferences
19: K>D>>B
5: K>>D>B
4: K>>B>D
18: D>K>>B
5: D>>K>B
1: D>>B>K
21: B>>K>D
23: B>>D>K
4: B>D>>K (these are sincerely B>>K>D)
 	There is a cycle now, K>B>D>K

On the "sincere preferences" ballots, the approval scores are B48, K46, D43, while
on the "altered preferences" ballots, the approval scores are B48, D47, K46.

Approval Margins uses a "defeat-dropper" method, measuring the strengths of the defeats by the margins
between the approval scores (but like AWP,determines their "directions" purely by the rankings.)

Approval Margins:
D>K   47-46  (m+1)
K>B   46-48  (m-2)
B>D   48-47  (m+1)

B's defeat, with an approval margin of -2, is the weakest and so is "dropped". B, the Buriers' favourite
but the sincere (and voted) Approval winner,wins.
DMC gets the same result by eliminating D and K.

AWP differs from AM in the way that it weighs defeats. Quoting James:

>For a given defeat A over B, the magnitude of the defeat is defined by
>the number of voters who place A above their approval cutoff and B below
>their approval cutoff.

Approval-Weighted Pairwise:
D>K   06
K>B   46
B>D   44

AWP elects the sincere CW, K!

I used to think that electing the voted approval loser was absurd if we assume that the votes are sincere,
but by that logic we should resolve all top cycles by electing the Approval winner. From that point of view,
sometimes electing the approval loser is only a degree "worse" than not always electing the approval winner!

Still, I don't see this example being a great advertisement for AWP versus AM because the winner is the sincere
Approval winner.

My next example is the one I used in my last post on AM.


An old example given by Adam Tarr.
Sincere rankings:
49 R>C>L
12 C>R>L
12 C>L>R
27 L>C>R
C is the CW.

Suppose there is pre-polling and so the L supporters decide to approve
C, while the C supporters sincerely divide their approvals.
Further suppose that the R supporters all decide to completely Bury C.
Then we might get:

49 R>L>>C
06 C>R>>L
06 C>>R>L
06 C>>L>R
06 C>L>>R
27 L>C>>R

Now all the candidates are in the top cycle: L>C>R>L.  The approval
scores are  L82,  R55, C51.

Approval Margins:
L>C  82-51 = +31
C>R  51-55 =  -4
R>L  55-82 = -27

AM elects L, backfiring on the Buriers!

Unfortunately this time DMC eliminates C, and then the Buriers' candidate R wins.

Approval-Weighted Pairwise:
L>C  49
C>R  45
R>L  06

AWP gives the same good result as AM!


Chris Benham