[EM] Truncation-resistant MCA method: MCA-Asset

[Jameson Quinn]

Discussing truncation/burial resistance and MCA got me thinking. The system
I'll propose below is not great for mathematical criteria compliance, but I
think it would work well in the real world.

MCA (aka rated Bucklin) methods never obey either LNH, so they cannot be
strictly burial-proof. But the only kind of burial necessary is truncation.
Consider the typical real-world case where burial, via truncation, would be
a factor:

Honest ratings:
40: A1=A2>...>...>B
7: A1>A2>...>B
6: A2>A1>...>B
47: B>...>...>A2=A1

With honest ballots, A1 and A2 both get a second-round majority, which any
sane tiebreaker would give to A1. But if the A2 voters truncate A1, then A2
wins. If the A1 voters also defensively truncate A2, then there is a failed
majority. Under any tiebreaker without a possible runoff, B then wins - a
socially-suboptimal result.

You can think of this as a truncation arms race. At the beginning, with pure
honest ballots, you're in a multi-majority domain, and the right tiebreaker
could remove the incentive for a small, marginal amount of truncation. At
the end of the arms race, there is a failed majority. Either the tiebreaker,
through runoffs, saves the voters from themselves (but this cannot always
work in an N-candidate case); or the voters must regret their excessive
truncation. In neither case are they glad they truncated.

So at the beginning and the end of the arms race, marginal truncation is
useless or counterproductive. But in the middle, there is an unavoidable
domain where A2's truncation is effective, and they have the only majority.

What tiebreaker minimizes that problem? Ideally, it would be one which
doesn't reward truncation in the honest case, and minimizes its impact in
the fully-strategic case. This can't altogether prevent the possibility of a
sneaky, truncated A2 win; but hopefully, it can make it negligible.

Before this line of thought, I'd never seriously considered IRV as an MCA
tiebreaker. Aside from the known problems of IRV, such a hybrid would be too
complex for real-world use. But IRV does have excellent resistance to
burial, so it's worth considering in this context. The problem is, in the
case of a strategic failed majority, ballots are too truncated for IRV to
redeem the situation.

However, what if the system could somehow know that A1 and A2 are allied
candidates, even though the truncated ballots fail to tell us that? That
made me think of Asset Voting.

So, here's the proposal, which I call MCA-Asset:
(MCA base) Voters rank candidates into N rating categories, including the
default bottom rating category. Any candidate who is the only one with a
majority at or above a given rank wins.
(Tiebreaker specification) If there are multiple simultaneous majorities at
a given rank (including a failed majority, which means that all candidates
are unanimous at or above bottom rank) then the winner is the member of that
set with the most "transferred votes", defined as follows:
1. Initialize each candidate with their share of the top-ranked ballots.
Each ballot is divided evenly between the candidates it ranks at highest
rank.
2. Eliminate all candidates without a majority at the highest rank where any
candidate has a majority.
3. Eliminated candidates may unconditionally give their ballots to
non-eliminated candidates.
4. Eliminate the candidate with the fewest votes, allowing transfers.
5. Repeat 4 until some candidate has a majority of the remaining votes.

In the example given, if all voters strategically truncated, then nobody
would get a majority until everyone did at the bottom rank. A1 would have 27
votes, A2 would have 26, and B would have 47. A2 would be eliminated in step
4 above, and would presumably opt to transfer their votes to A1. Thus, the
truncation strategy would ultimately have had no effect. Hopefully, this
fact would prevent pathological over-truncation, and thus cases like this
would be rare.

This system is complex to specify (and could be further "improved" by more
complications), but to me it is intuitively clear. If nobody gets a
majority, let the candidates build coalitions, prodding them along with
IRV-style elimination, until somebody does. It is clone-proof, it gives no
incentive to dishonest strategy, and it is reasonably resistant to
truncation strategy. Unlike Range, it is majoritarian, without the need for
voters to exaggerate. If there is a ballot CW, that candidate must make it
at least to step 4 of the tiebreaker, and if candidates follow their voters'
will, the CW will almost certainly win.

What do y'all think?

Jameson
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