[EM] Re: reccomendations (Schwartz // SC-WMA)

[Chris Benham]

Schwartz // Symetrically Completed - Weighted Median Approval  (Schwartz 
// SC-WMA).

Hello,
I deduce from an off-list enquiry that, in my recent posts, I may not 
have made the exact rules of this method completely clear.
I  shall now attempt to spell it out very clearly:

The method is a plain ranked-ballot method, that allows truncation and 
can accomodate equal non-last preferences.

It is a  Condorcet-completion method. The first step is to eliminate all 
non-members of  the Schwartz set. Thereafter the eliminated
candidates are ignored, that is the count proceeds as if though they 
hadn't stood.
(The "Schwartz set" is the smallest possible non-empty set of candidates 
with none of the members having any pairwise losses to
any non-member candidates. If there are no pairwise ties, it is the same 
as the  Smith set.)

Next the ballots are  "symetrically completed". That means that each 
ballot that doesn't strictly rank all the candidates is replaced by
a number of ballots, one for each alternative way of strictly ordering 
the candidates that were not strictly ordered on the original ballot,
and the total weight of these new ballots will sum to one.
(To take a simple example, if  non-last equal-ranking is allowed and 
there are three candidates ABC, then a  A=B>C ballot will be
"replaced" by two ballots, each with a weight of  1/2,  one A>B>C  and 
one B>A>C).

Lastly, on these "completed" ballots, the "Weighted Median Approval" 
(WMA) method is used to pick the winner, thus:
The ballots are converted to "Approval" ballots, with each ballot 
 wholly  "approving" or "not approving" each candidate.
(There are no partial approvals).
Each candidate is considered to have a "weight" that is equal to the 
number of  ("symetrically completed") ballots on which it is ranked
in first-place (of course disregarding eliminated candidates).
Each ballot approves the candidate it ranks first. Is the "weight" of 
this candidate less than half the total weight of  all the candidates?
If yes, then the ballot also approves the candidate ranked second. Is 
the combined weight of these two candidates less than half
the total weight of all the candidates. If no, then this ballot approves 
no other candidate. If yes, then this ballot also approves the
candidate it ranks third, and so on, until each ballot has approved  at 
least  "half" the candidates, measured  "by weight".
The candidate with the highest (thus derived)  "approval" score wins. 
 If  there is tie, elect the tied candidare with the higher "weight".

In practice there will often be easy shortcuts. For example, if  there 
are three candidates in the Schwartz set, then  (without bothering
with the "symetric completion") every ballot will approve the two 
candidates it  ranks non-last, and those that only rank one candidate
will approve that candidate and  half-approve the other two.

The justification for the method is  that unlike Winning Votes, it meets 
the Sincere Expectation Criterion, which I now regard as
essential. It is a weaker form of  No Zero-Information Strategy, and it 
says that  with every possible valid by the other voters being
equally likely, then regarding the categories:(1) the voter's favourite 
candidates, (2) the voter's first and second favourites, etc. down
to  all the candidates the voter ranks non-last; then there should be no 
insincere vote that gives  the voter a better chance in  all the 
categories than voting sincerely. The sincere vote must do better in at 
least one, or  as well in all of them.

This standard is met by Margins, but unlike Margins, Schwartz // SC-WMA 
meets Steve Eppley's  "Minimal Defense" and
"Truncation Resistance" defensive strategy criteria.
(I've been wrong before. In a later post I might get around to 
justifying these claims.)

The cost of this is that it fails his "Immunity from Majority 
Complaints" criterion, and (in my view fairly benignly) it can fail
Mono-raise  (aka monotonicity) when there are more than three candidates 
in the Schwartz set.
It may be more vulnerable to uncountered Burying than Winning Votes. It 
gives more Bucklin-like results which are probably
(with sincere voting) higher  "social utility".

Here, without the ratings, is  an example James G-A gave to illustrate 
the need for his  "Weighted Pairwise" method.

26: B>D>K
22:B>K>D
26:D>K>B
01:D>B>K
21:K>D>B
04>K>B>D

100 ballots. B>D 52-48,  D>K 53-47,  K>B 51-49.  All the candidates are 
in the Schwartz set.

All the ballots give a full  "strict" ranking (no equal preferenes), so 
there is no "symetric completing" to do.
Each ballot simply approves the two candidates they rank non-last, to 
give these scores:
B:53,   D:74,   K:73.
D  has the highest score and so wins.  Ranked Pairs, Beat Path, MinMax 
etc. all elect B

Chris Benham



Sincere Expectation Standard
Given that a voter has no knowledge about how others will vote, a
sincere vote must be at least as likely as any insincere vote to
give results that are in some way better in the eyes of the voter.

Or expressed as a more rigid criterion:
-----
Sincere Expectation Criterion (SEC)
Consider a voter with a preference order between the possible
outcomes of the election.  Let us call his sincere ballot, X.  Now,
assuming that every possible legal ballot is equally likely for every
other voter, there must be some justification for the vote X over any
other way to fill out the ballot, which I will call Y.
This justification is given by the following comparisons:

The probability of X electing one of the voter's first choices vs.
the probability of Y electing one of these choices

The probability of X electing one of the voter's first or second
choices vs. the probability of Y electing one of these.

The probability of X electing one of the voter's first, second or
third choices vs. the probability of Y electing one of these.
... And so on through all the voter's choices

X must either do better in one of these comparisons than Y, or equal
in all.  Otherwise the sincere vote can not be justified.
----
In other words, there must be some justification for voting sincerely
even if the voter does not know how any one else is voting.