[EM] "Comprehensive 3-slot Ratings Winner" suggested 2 criteria

[Chris Benham]

The examples I used to demonstrate my suggested "Unmanipulable Majority"
criterion have inspired me to suggest  another two  (very similar to each
other, and not including my recent "Push-over Invulnerability" suggestion):

"Comprehensive 3-slot Ratings Winner":

*If no voter expresses more than three preference-levels and the ballot
rules allow the expression of 3 preference-levels when there are 3 (or 
more) candidates, then (interpreting candidates that are voted above one
or more candidates and below none as "top-rated", those voted above
one or more candidates but below all the top-rated candidates as 
"middle-rated" and those not voted above any other candidate and below
at least one other candidate as "bottom-rated", and interpreting above-
bottom rating as approval) it must not be possible for candidate X to
win if there is some candidate Y with simultaneously a higher Top-Ratings
score, a higher Approval score and a lower  Maximum Approval-Opposition
score.*

(Recall that a candidate Y's MAO score is the approval score of the most
approved candidate only on ballots that don't approve Y.)

This may  seem a bit long-winded and a bit arbitrary, but the standard it
seeks to capture is that in relatively simple situations the obviously strongest
candidate shouldn't lose, such as C in the following example:

25: A>B
26: B>C
23: C>A
26: C

TR scores:      C49,   B26,    A25
App. scores:   C75,   B51,    A48
MAO scores: C25,   B49,    A52

I think all determinist methods that don't elect C here must be vulnerable to
Push-over strategy. As noted in my last posting, this is true of  IRV and
Winning Votes and MAMPO and AWP which all elect B.

Unfortunately this criterion is incompatible with Condorcet, as demonstrated
by another example from my UM post:

93: A
09: B>A
78: B
14: C>B
02: C>A
04: C

TR scores:      A93,     B87,    C20
App. scores:   A104,   B101,  C20
MAO scores: A92,     B95,    C102

The "C3RW" criterion says that A must win, but B is the Condorcet winner
(B>A 101-95, B>C 87-20).

But I'm still interested in Condorcet methods, so I suggest "Smith-C3RW",
which specifies that the candidate Y disallowing the election of candidate X
must have a beatpath to X.

*If no voter expresses more than three preference-levels and the ballot 
rules allow the expression of 3 preference-levels when there are 3 (or 
more) candidates, then (interpreting candidates that are voted above one
or more candidates and below none as "top-rated", those voted above
one or more candidates but below all the top-rated candidates as 
"middle-rated" and those not voted above any other candidate and below
at least one other candidate as "bottom-rated", and interpreting above-
bottom rating as approval) it must not be possible for candidate X to
win if there is some candidate Y which has a beat-path to X and  
simultaneously higher Top-Ratings and Approval scores and a lower  
Maximum Approval-Opposition score.*

This clearly failed by Winning Votes and met by Schwartz//Approval.
(In the first example, C has a beatpath to B.)

As with "Push-over Invulnerability" I strongly suspect that it is met by
Margins. 


Chris Benham


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