[EM] "Unmanipulable Majority" strategy criterion

Chris Benham cbenhamau at yahoo.com.au
Sat Dec 6 08:56:44 PST 2008

```Kristofer,
"You have some examples showing that RP/Schulze/"etc" fail the criterion."

By my lazy "etc." I just meant  'and the other Condorcet methods that are
all equivalent to MinMax when there are just 3 candidates and Smith//Minmax
when there are not more than 3 candidates in the Smith set'.

"Do they show that Condorcet and UM is incompatible? Or have they just
been constructed on basis of some Condorcet methods, with differing
methods for each?"

My intention was to show that all the methods that take account of more than
one possible voter preference-level (i.e. not Approval or FPP) (and are
well-known and/or advocated by anyone on EM) are vulnerable to UM except
SMD,TP.

"I think I remember that you said Condorcet implies some vulnerability to
burial. Is that sufficient to make it fail UM?"

Probably yes, but I haven't  tried to prove as much.

Returning to this demonstration:

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

B>A  101-95,  B>C 87-20,  A>C 102-20.
All Condorcet methods, plus MDD,X  and  MAMPO and  ICA elect B.

B has a majority-strength pairwise win against A, but say 82 of the 93A change to
A>C  thus:

82: A>C
11: A
09: B>A
78: B
14: C>B
02: C>A
04: C

B>A  101-95,  C>B 102-87,  A>C 102-20
Approvals: A104, B101, C102
TR scores: A93,   B87,   C 20

Now MDD,A and MDD,TR and MAMPO and ICA and  Schulze/RP/MinMax etc. using
WV or Margins elect A.  So all those methods fail the UM criterion.

Working in exactly the same way as ICA (because no ballots have voted more than one candidate
top), this also applies to  Condorcet//Approval and Smith//Approval and Schwartz//Approval.
So those methods also fail UM.

"I did a bit of calculation and it seems my FPC (first preference
Copeland) variant elects B here, as should plain FPC. Since it's
nonmonotonic, it's vulnerable to Pushover, though, and I'm not sure
whether that can be fixed at all."

My impression is/was that in 3-candidates-in-a-cycle examples that method behaves just like IRV.
The demonstration that I gave of  IRV failing UM certainly also applies to it.

Chris Benham

Kristofer Munsterhjelm  wrote (Thurs.Dec.4):
Chris Benham wrote:
>Regarding my proposed Unmanipulable Majority criterion:
>
>*If (assuming there are more than two candidates) the ballot
>rules don't constrain voters to expressing fewer than three
>preference-levels, and A wins being voted above B on more
>than half the ballots, then it must not be possible to make B>the winner by altering any of the ballots on which B is voted
>above A without raising their ranking or rating of B.*
>
>To have any point a criterion must be met by some method.
>
>It is met by my recently proposed SMD,TR method, which I introduced
>as "3-slot SMD,FPP(w)":
>
>*Voters fill out 3-slot ratings ballots, default rating is bottom-most
>(indicating least preferred and not approved).
>
>Interpreting top and middle rating as approval, disqualify all candidates
>with an approval score lower than their maximum approval-opposition
>(MAO) score.
>(X's  MAO score is the approval score of the most approved candidate on
>ballots that don't approve X).
>
>Elect the undisqualified candidate with the highest top-ratings score.*
>
[snip examples of methods failing the criterion]

You have some examples showing that RP/Schulze/"etc" fail the criterion.
Do they show that Condorcet and UM is incompatible? Or have they just
been constructed on basis of some Condorcet methods, with differing
methods for each?

I think I remember that you said Condorcet implies some vulnerability to
burial. Is that sufficient to make it fail UM? I wouldn't be surprised
if it is, seeing that you have examples for a very broad range of
election methods.

>93: A
>09: B>A
>78: B
>14: C>B
>02: C>A
>04: C
>200 ballots
>
>B>A  101-95,  B>C 87-20,  A>C 102-20.
>All Condorcet methods, plus MDD,X  and  MAMPO and  ICA elect B.
>
>B has a majority-strength pairwise win against A, but say 82 of the 93A
>change to
>A>C  thus:
>
>82: A>C
>11: A
>09: B>A
>78: B
>14: C>B
>02: C>A
>04: C
>
>B>A  101-95,  C>B 102-87,  A>C 102-20
>Approvals: A104, B101, C102
>TR scores: A93,   B87,   C 20
>
>Now MDD,A and MDD,TR and MAMPO and ICA and  Schulze/RP/MinMax etc. using
>WV or Margins elect A.  So all those methods fail the UM criterion.

I did a bit of calculation and it seems my FPC (first preference
Copeland) variant elects B here, as should plain FPC. Since it's
nonmonotonic, it's vulnerable to Pushover, though, and I'm not sure
whether that can be fixed at all.

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