[EM] Adjusted Condorcet Plurality, an interesting new LNHarm+LNHelp method

[Kevin Venzke]

Hello,

Here is the procedure for "Adjusted Condorcet Plurality":

1. From the submitted rankings, identify the first preference winner (FPW).
2. Edit all the ballots so that all preferences below the FPW are removed/truncated.
3. On the revised ballot set, check for a Condorcet winner. Elect them if there is one.
4. Else elect the FPW.

What's interesting about this is that, like IRV, you can't hurt *or* help a candidate by
adding additional lower preferences. But this is done without eliminating any candidates.
The candidate with the fewest first preferences can win!

26 A
25 B>D
25 C>D
24 D

D wins, no problem. Among other LNHarm methods, MMPO and "LNH Borda" will elect D as well,
but neither of those satisfies the Plurality criterion. This "ACP" method does.

In other cases IRV does do better:
25 A>B>D
20 B>C>D
20 C>D>A>B>E
18 D>C
17 E>D

The CW is D, and IRV elects D. But the "adjusted" CW is C, and wins in ACP.

Another:
25 A>E>B>C>D
21 B>D>C
20 C
19 D>C>A
15 E>C

The CW is C, which IRV elects. But on the adjusted rankings C no longer beats D; there is
no adjusted CW, and so ACP defaults to electing the FPW, A.

With three candidates, ACP gives the same result as IRV. As the number of candidates
increases, the Condorcet efficiency and compromise incentive seem to improve over IRV (i.e.
in simulations), but it's very slight. I guess that if you generate ballots randomly, then
no matter how many candidates there are, roughly the same percentage of preferences will
get truncated in the adjustment.

On the downside, ACP loses IRV's satisfaction of Condorcet Loser, mutual majority, and
Mono-add-top. Mono-raise performance is also worse.

I have added ACP to my LNHarm calculator, so if you like you can experiment and see how
similar ACP is to IRV. You can click on the "Random scenario" button, so no need to type
out any ballots:
https://www.votingmethods.net/lnharm

I found two ways to vary this to create a total of four methods, but I decided the others
are more complicated while giving less interesting results. One thing you can do is find
the top two instead of the FPW, runoff between them, and cut all preferences below *either*
of those two. When there is no "adjusted CW" then elect the runoff winner. This method
further distorts the "adjusted" form of the ballots since we now truncate under two
candidates instead of one.

The other thing you can do is produce a LNHarm method without LNHelp, by not simply
truncating the preferences below the FPW. Instead, for example if your vote is FPW>X>Y,
then your X>Y preference cannot help X beat Y when considering whether X is the adjusted
CW, but *does* help X beat Y when considering whether Y is the adjusted CW. While it seems
like it might (?) be desirable to allow voters to benefit somewhat from these lower
preferences, all it can do is further help the FPW, which probably makes the method less
interesting. Example:

40 A>C
35 B>C
25 C>B

In the LNHarm-only version, the A>C voters are able to block B from beating C, and C can't
defeat B, so A wins. Not great: FPP is probably the only other method that picks A here.
(The normal version of ACP with both LNHs elects B.)

Kevin
votingmethods.net