[EM] Primaries

[Abel Stan]

I had more thoughts originating from thinking both on the 2 round Condorcet
of my last mail but also one of the replies on my challenge for an example
with the most different results.
Although I had one mixup with my example, you pointed out 2 methods that
actually elected a candidate I initially intended as a dictatorship example.
One of them was CrossMax by Kevin Venzke.

I otherwise have to assume this topic of primaries has been discussed
before somewhere but a quick search didn't lead me to results I was
interested in.
So here I am writing this in case my angle might point towards a new
approach.

A bit of context: In my country, primaries are not party affairs, but
mostly conducted by unofficial civil electoral commissions, with parties
participating in it and competing with one candidate each.
This is because primaries are a tool occasionally used by the opposition to
choose the best candidate against the government candidate (usually an
incumbent).
Official single winner elections are by plurality, and primaries were
either by plurality, or by modified two-round system (with candidates above
15% going into the second round and usually withdrawing in favour of each
other) or by instant-runoff (I think first used in this year).
Experts have published articles advocating for approval voting instead,
others have warned about drawbacks of primaries.

My line of thought is the following.
Primary voters and organizers want a system which elects the best candidate
to beat the government candidate, or at least have the best showing, not
only to show strength, but to decrease the amount of of transfer votes the
winning candidate gets (if government) and increase the amount of transfer
votes the opposition list gets (see my thought on transfer votes in another
of my previous mails).
herefore the stakes are in some sense higher than in other countries, where
it doesn't directly matter by how much the "electable" candidate wins or
loses.

The situation in most districts/races is almost always that is the
opposition would have run separately, the government candidate would have
won the district.
It was safe to assume the government candidate is the plurality winner
without sufficient tactical voting, but often a clear majority winner too
(but voters turn out more if the election seems competitive, therefore even
that cannot be assumed).
Therefore the opposition, on the surface level (without tactics) would
prefer the (general) voting method to be one that selects the candidate who
beats the plurality winner by the most, so Cross Max.
In some cases, where the government candidate would not be plurality
winner, they would prefer a method which specifically compares against the
government candidate.
But they cannot change the general system, only the primary system. It
seems they could put "government candidate" as an option on the primary
ballot, who may, of course, not win and see which candidate beats that
"candidate" with the highest margin.
My intuition is, this is equivalent to approval voting, since the
government candidate acts as an approval threshold.
However, the question is not "do you approve of this candidate"? but "do
you prefer this candidate over the government candidate?", or, one level
deeper? "do you think this candidate can beat the government candidate (the
most)?".

In any case, voters in the primary might vote not just to get the best
outcome for themselves as in a normal election, but might take into account
which candidate they consider the most electable, like in any primary or
two-round system.
I might prefer all three candidates A, B and C to the incumbent Z, in this
order, yet instead of approving of all 3 or or A and B, I might just
approve A and C since I think B is not going to do well against Z.
This is not only because I think B is not popular among non primary voters
and/or that government voters are trying to subvert the primary by voting B.
This can be also because I think there are many people who prefer C to Z to
B to A, but still vote C, B. Therefore, I want to support C over B, since I
know these voters are not serious (but not completely dishonest either)
about the primary in this case, since if B wins, they are voting Z in the
general, yet they still vote for B just so they can be sure A doesn't win.
But these are known forms of tactical voting, arising from two
rounds/primaries.

What do you think of this conundrum? My intuitions are, that
1) knowing which candidate (at least party, as a placeholder) to run
against makes it that neutrality is not necessarily desired when designing
the primary system (when quasi-including this candidate in the primary).
2) Condorcet is not necessarily desirable in the primary, since Condorcet
winners might have lower margins against the plurality winner or plurality
second than other candidates (it doesn't matter whether Condorcet is done
within the set of primary voters or all voters)
3) if the plurality election had only two candidates, the main opponent
would essentially be an approval cutoff of completely sincere voters of the
primary by modified CrossMax, but if there are 2 strong opponents, it will
be more complicated.
4) even with sincere ordering, since the method is approval voting, the
approval threshold of rational voters who prefer all primary candidates to
the incumbent should be above their last place (excluding incumbent). This
in itself could cause failures?

Another question: Is there an inherent (philosophical) difference between a
social welfare function like the election as a whole, where people vote
based on their (individualistic) preferences sincerely and tactically based
on other voters presumed or known preferences and a "social truth function"
(?) of a perfect primary, where people might not be voting per se, but
together searching for the best tactic for their coalition in a social
welfare function?
To me the potential difference be be that in the former, the "correct"
answer would be known from unbiased criteria (if only all these criteria
were never incompatible) set up, and whatever fulfills these is deemed
"correct"  (it is not a truth from outside, it's a tautology?), while in
the latter, a non-mathematical truth is presupposed (by one metric only,
too arbitrary and biased to be a logical criterion), like Condorcet's jury
theorem, that all (or most) people would prefer over any of their own
preferences, but they only have a guess. The "perfect primary" in this case
would be one where all voters prefer all primary candidates to their
opponents, but only one of the primary candidates can be beat their
opponent, yet they don't know which, but all have their preferences. From
the perspective of the system designer, the preferences are considered the
best proxy for truth, because similarly to Condorcet's jury, the primary is
a tool to maximize the probability of the correct decision with the
assumption that there is more truth in cast votes than falsehood.

Another way to think of it Condorcet is a majority vote, where the stable
optimum is found, if there is one, if all votes are sincere preferences.
But with sincere voting, approval voting is the method to find the ordering
of how likely something is true, if voter opinions aggregated are
considered to be best proxy. (they are supposedly evaluated individually).
The ordering derived from approval (/margin of beating the incumbent in the
primary) is the ordering of probability of each alternative being "true"

What do you think?
-Abel
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