[EM] Favorite Betrayal and Condorcet, and LNHarm

[Kevin Venzke]

Hi Kristofer/Forest/all,

Kristofer wrote:
> Kevin's simulations of
> http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-June/114476.html
> seem to indicate that Condorcet methods (at least "advanced" ones like
> Schulze) have a low rate of FBC failure.

Not so advanced: I have MinMax(WV) performing about the same as Schulze(WV) and
better than both River and RP(WV). If anything Smith compliance could probably
be guessed to be a liability since no known FBC method does any path-tracing.

> The "Improved Condorcet"
> methods would presumably be the flipside of this coin, passing FBC
> absolutely but having some (low?) rate of Condorcet failure.

I've been thinking about this lately. Experimentally ICA gives results less
resembling MinMax(WV) etc. than MAMPO does, which is odd since ICA is at least
trying to satisfy Condorcet.

It seems that every FBC method is composed of one or more "layers" of logic,
with results of the combined whole determined basically DSC-style.

The layers have some properties:
1. Each one is calculated independently with no awareness of another layer.
2. Each one returns an ordering of the candidates, not necessarily strict. (As
to use multiple layers there should be some indecision at the top.)
3. Each satisfies FBC, according to a definition that makes sense with
orderings as opposed to candidate win odds.
4. A layer is used only to break ties on any layers already applied.

So layer examples would include the Bucklin(ERW) mechanism, FBC-compatible ways
of Borda scoring, implicit approval, a majority approval filter, the MMPO score,
Majority Defeat Disqualification, whatever MajBTP is doing, top rankings, and
Improved Condorcet, including the IC-modified MinMax(WV) score (which I call
tMMWV).

(IC usually uses a "tied at the top" rule; I've considered whether "tied and
approved" would better match voters' desires, but this would clearly make IC
less like Condorcet, so I won't consider that anymore.)

These layers seemingly can be applied in any order, and we can make them less
decisive if we want (such as the difference between approval and majority
approval).

So ICA is IC then approval. MDDA is MDD then approval. MAMPO is actually
majority approval, then MMPO, then approval (as a tiebreaker). MAMPOA really.

Since two of the most Condorcet-like rules are probably IC and MMPO, can we just
mix those for an "ICMPO" method? Probably not, because it fails Plurality.
That's an issue with a number of these rules, and a reason why MAMPO uses a
majority approval filter before MMPO.

ICMAMPO (or ICMAMPOA), though, does seem to be an improvement on MAMPO, at least
from the standpoint of resembling MinMax and maximizing Condorcet efficiency.
(And it satisfies Plurality.)

FBC-compatible layers that ensure Plurality seem to be possible.

Consider FPF ("FBC-compatible Plurality filter"): A candidate X is disqualified
(meaning: returned in the bottom rank of the layer's output ranking) if for some
other candidate Y, Y's top rankings minus the X-Y tied-at-the-top count exceeds
X's implicit approval.

That apparently isn't monotone. But this appears to be:

AC ("Approval check"): A candidate X is disqualified if their implicit approval
score is below the max PO against them.

Methods like AC-MPO-A and AC-tMMWV-MPO-A (using hyphens for readability) seem to
be very slightly better than MAMPO, but definitely not as good as ICMAMPO. If
one doesn't want to mess with tied-at-the-top or a majority approval threshold,
though, maybe this "ACMPO" or "ACMPOA" method could be attractive.

An adjacent issue that occurs to me is whether we can use any similar pattern to
make a new Later-no-harm method. There is a definite similarity between weak FBC
and LNHarm as they both can be conceived of as carving out a new ranking for one
of multiple candidates at either the top or bottom ranking.

A big problem is that there aren't as many known options for LNHarm "layers,"
and the ones that do exist are very hard for me to wrap my head around in order
to learn some general patterns. The MMPO and FPTP principles are pretty clear.
Chain Runoff could be seen as a hybrid of those two. The IRV and DSC principles
seem to not offer many variations.

Another problem is how to enforce Plurality. We can't use implicit approval in a
LNHarm method. Only MMPO really runs any risk of violating Plurality, but MMPO
seems like one of the more promising tools here.

And another issue is that for even three candidates it's clear that Plurality,
LNHarm, and minimal defense are incompatible. MD is usually a lower-hanging
fruit, but here it's impossible. Instead we have to ask for something "more like
Condorcet," a "weak Condorcet," but I don't know what that might look like.
"Elect a candidate with full majorities over everyone," i.e. Woodall's
Condorcet(gross), is not doable either.

Kevin