[EM] DAMC

[Jobst Heitzig]

Hello,

When proposing DAMC, I wrote:
> What I'm not sure about so far is whether using Beatpath or Ranked
> Pairs instead of River gives the same winner.

Now I'm sure all these would give different methods -- examples follow.

We define these methods:

DAMC(River), DAMC(Beatpath), DAMC(Ranked Pairs):
------------------------------------------------
Define 
  defeat := absolute majority size defeat
and 
  defeat strength := approval score of defeating option.
Then use the River/Beatpath/Ranked Pairs method of cycle resolution.
Of those options remaining undefeated, elect the most approved.

My original proposal was DAMC(River).

The following example shows DAMC(Ranked Pairs) to be different from the 
other two:

Defeats: E>A>D>C>B>D 
Approval scores: A>B>C>D>E
DAMC(Ranked Pairs) locks in E>A>D and C>B>D and hence elects C.
DAMC(River) locks in E>A>D>C>B and hence elects E.
In DAMC(Beatpath), E has beatpaths against A,B,C,D, hence E wins.

The following example shows that DAMC(Beatpath) is also different:

Defeats: A>B>D>A,C>D
Approval scores: A>B>C>D
DAMC(Ranked Pairs) locks in A>B>D and C>D and hence elects A.
DAMC(River) locks in A>B>D and also elects A.
In DAMC(Beatpath), C has beatpaths against A,B,D, hence C wins here.

The following example shows that DAMC(Ranked Pairs) violates IPDA and 
ISDA (just as ordinary Ranked Pairs does):

Defeats: C>D>E>C>F>B>A>D, A>E, B>D, and D is Pareto-dominated by A.
Approval scores: A>B>C>D>E>F
DAMC(Ranked Pairs) locks in A>D,A>E, B>D,B>A, C>D,C>F, D>E, F>B, which 
gives the social order C>F>B>A>D>E. 
But when D is removed, it locks in A>E, B>A, C>F, E>C, which gives the 
social order B>A>E>C>F.

DAMC(River) has not this problem: It locks in A>D,A>E, B>A, C>F, E>C 
from the beginning, which gives the same social order B>A>E>C>F as 
without D, with only the defeat A>D added to it. Also DAMC(Beatpath) 
elects B in both cases.

The following example shows that also DAMC(Beatpath) violates IPDA and 
ISDA (just as ordinary Beatpath does):

Defeats: C>D>E>A>B>D, A>C, B>E, and D is Pareto-dominated by C.
Approval scores: A>B>C>D>E
Since C has a beatpath against A,B,D,E, it wins. Without D, also A 
remains undefeated after the cycle resolution, hence A wins.

Unlike ordinary Beatpath, DAMC(Beatpath) at least fulfils "immunity from 
absolute majority 2nd place complaints". This is because all defeats by 
an option have the same strength (namely the approval score of that 
option). Assuming X wins and is absolute majority defeated by Y, there 
must be a beatpath from X against Y leading thru more-approved options 
than Y. In particular, Y is then defeated by a more approved option 
other than X, and hence cannot win after X is removed.

So, just as in the winning-votes case, River seems to be a slightly 
better cycle resolution method for DAMC than Ranked Pairs and Beatpath 
since it fulfils IPDA and ISDA and is therefore less vulnerable to the 
addition of weak options.

Yours, Jobst
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