[EM] DAMC (was: MDDA vs UncAAO, ASM & DMC)

Jobst Heitzig heitzig-j at web.de
Mon Mar 12 08:16:16 PDT 2007

Dear Mike,

you wrote:
> Offensive order-reversal works in DMC too. If there’s less incentive
> for it, it’s because truncation works just as well in DMC.  DMC is
> vulnerable to truncation in a sense that MDDA, MAMPO and wv Condorcet
> are not, as examples below will demonstrate.

I like MDDA much, but I think it can still be improved. Some 
disadvantages are in my opinion the lacking clone-proofness and 
immunity from absolute majority complaints. 

By the latter I mean that some absolute may complain that they all 
prefer some option Y to the MDDA-winner X without us being able to 
point out a sequence of "stronger" majority defeats leading back from Y 
to X.

Both these problems with MDDA arise from the distinction whether all 
options are defeated by an absolute majority or not.

Consider these examples:
(1) absolute majority defeats: A>B, approval: B>A. 
MDDA-winner: A
(2) absolute majority defeats: A1>B,A2>B,A3>B,A1>A2>A3, approval: 
MDDA-winner: B

It seems to me that in (2), B is at least as defeated as in (1) and that 
in (2), A1 should win.

Also, MDDA is not immune from 2nd place complaints, that is, when the 
winner is removed, the new winner may have had an absolute majority 
size defeat against the original winner. And the MDDA-winner may also 
have by majority size defeated by her most approved contender.

I want to suggest a modification of DMC which is motivated by MDDA and 
does neither have these problems MDDA has, nor the one you pointed out 
DMC has:

Recall that DMC can be described in at least three different ways:

Either: Define defeat strength as the approval score of the defeating 
option and then apply an immune cycle-resolution method (e.g., River, 
Ranked Pairs, or Beatpath, all give the same in this case).

Or: Sort options by descending approval; then, as long as some option 
defeats its upper neighbour, exchange the topmost such pair of options.

Or: Remove every option which is defeated by a more approved 
(="definitively defeated") one and then elect the CW of the remaining 

In order to make DMC immune to your kind of example, we take the first 
of the above descriptions but use only defeats which are supported by 
an absolute majority:

Def. DAMC (Definite Absolute Majority Choice):
Make a list of absolute majority size pairwise defeats. 
  Process this list in order of descending approval score of the 
defeating option. Keep the defeat at hand iff (i) the defeated option 
is not already defeated by the kept defeats and (ii) the new defeat 
does not build a cycle with those defeats already kept. 
  From those options not defeated in the end, elect the most approved 

In other words: We use River with 
  defeat := absolute majority size defeat
  defeat strength := approval score of defeating option
and resolve the remaining ambiguity by Approval.

I'm pretty sure that this method has the following properties:
- monotonicity
- clone-proofness
- immunity from absulute majority complaints (in the above sense)
- immunity from 2nd place complaints
- the winner is never defeated with absolute majority by a more approved 
option or by the most approved contender.

What I'm not sure about so far is whether using Beatpath or Ranked Pairs 
instead of River gives the same winner, and what would happen when we 
used the "resorting" or the "definitively defaeted" version of DMC with 
absolute majority size defeats only.

Yours, Jobst
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