[EM] Improving DYN

[Jameson Quinn]

I think that fsimmons's "uniformly better" idea is a great benchmark. Any
significant step down from plurality - even if it's more than balanced by
improvements - will be attacked by opponents. (Well, actually, opponents
will not limit themselves to the truth, but the point is that it's a lot
easier to respond with "that's a lie" than with "well yes, that's
technically true, but on the other hand...").

The question, then, is what represents a "significant" step down. I'd argue
that the extra ballot complexity of DYN is not something that would tangle
us up in "yes, but" defensiveness. When opponents said "the ballot is too
complex", we can easily say "just forget about the yes/no part if you feel
that way". Unlike a bullet vote in other systems, a bullet vote in DYN has
just as much strategic voting power as any other vote; the ONLY reason you'd
use the Y/N is if you don't fully trust your favorite candidate's
preferences to align with your own.

In a practical sense, I think that DYN would choose the pairwise champion
(CW) in the large majority of cases, and also the utility-maximizing
candidate. In most real-world elections, both exist, and are the same
person.

---

There is, however, one case where DYN could still fall down. I call it the
"near-clone" scenario, and the simplest example is three non-majority
candidates, two of whom are mutually second choices (let's call them X1 and
X2, versus Y opposing both of them). One of the two X's will then be the
condorcet (and probably utility) winner. But there is a "game of chicken"
between the two voting blocks. An X1 voter can improve X1's chances by not
approving X2; but if too many voters on both sides do that, then Y will win.

The vote-delegation stage helps with this problem; X1 and X2 could mutually
approve each other, and prevent Y from winning. However, if this happens,
there's no particular reason that the winner between the two would be the
true Condorcet winner. Moreover, if one of them didn't cooperate for some
reason, they could win thereby. This reward for non-cooperation is a
dangerous feature; although it would rarely matter, the effects when it did
would be exactly the wrong incentives.

I've been trying to figure out a way to prevent this breakdown. I've come up
with what I believe is the strongest fix, which I discuss below. However, it
requires runoffs, and so no longer is uniformly better than plurality. So
I'd make a simpler suggestion, which I call SDYN - the S is for "Safe". The
winner is the DYN winner or the candidate who is approved by the DYN winner
who beats them pairwise. This means that if two candidates mutually approve
each other, the pairwise winner of those two will win; so it's "safe" for a
candidate to support another with fewer core voters than them. It's a "for
grownups" fix; it prevents problems if the two candidates mutually approve
each other, but does not attempt to resolve anything if one of them
"betrays". The hope is that iterated prisoner's dilemma considerations -
that is, maturity rather than ultimately-self-defeating treachery - would
ensure the mutual approval.

Below is the more-complex fix - one which would improve results overall -
but which I don't think is ultimately worth the extra complexity. So the
discussion below is something I consider interesting, but not a practical
suggestion.

----

I'm pretty sure that the only way to do it reliably is by having runoffs in
some circumstances. Any one-round tricks either open up new strategic
loopholes as quickly as they close others (as with DYN-balloted Condorcet
hybrids) or are vulnerable to center squeeze (as with only-delegate-upwards
elimination-style sytems). Meanwhile, with runoffs, you have some assurance
that blatantly gaming the system could be punished by voters.

Still, runoffs are clearly a "step down" in fsimmons's sense. While I think
democracy is worth voting twice occasionally, I understand that any
suggestions that follow make a system that can't be considered
pareto-dominant over plurality.

The best runoff-based DYN fix I have come up with — in that it would have
the most-infrequent runoffs, while still solving the problem — is: if there
is a candidate who beats the DYN winner pairwise (that is, has a higher
approval including all the delegated votes except those from the other
candidate.in the pair under consideration), then there is a runoff between
those two. (In the incredibly-unlikely event that there are two who beat the
DYN-winner pairwise, you'd take the one with the highest winning votes).

To analyze the effectiveness of this fix to DYN, I think we should break up
three-candidate elections into 9 exhaustive and exclusive possibilities. The
first is that one candidate has a majority of first preferences. This is
boring because any sane system elects them. For the other 8, label the
candidates A, B, and C in order of their number of first preferences. (This
tells you nothing about the ideological relationship between the three; any
two of them could end up being near-clones in the sense above). Assuming
that all voters bullet vote, and that all candidates have some preference
between the other two, each candidate has a two-way choice of who else to
prefer. That leaves 2^3 = 8 possibilities.

First, lets consider the 6 possibilities where one candidate is a pairwise
champion (CW).

ACB BAC CAB: in these two cases, A is the CW. They are also the DYN winner
even before any vote delegation happens. There is no potential strategies or
trickery, so A wins. Success for both DYN and my "fixed" version.

ABC BAC CAB: As above, A is the CW. Things could play out just the same as
the above. However, in my "fixed" version, B could try to be tricky and
support C, hoping to win the resulting runoff. A could respond by supporting
B to avoid the runoff, and also to force C to support A. Then there will be
a runoff between A and B, which A will win. So B's attempt at underhanded
strategy will probably not be successful, and thus they won't try it. I'd
say that both DYN and my "fix" are successful here, but DYN is more so.

ABC BAC CBA: B is the CW. C, seeing that A will win without delegation,
delegates their votes to B. B wins. Success for both DYN and my "fixed"
version.

ABC BCA CBA: As above, B is the CW. In normal circumstances, C, seeing that
A will win without delegation, delegates their votes to B. B can then, out
of pure goodwill, delegate their votes to C; that would be a purely symbolic
gesture which would not affect the result. So this would appear to work for
both DYN and my "fix". However, what if C, either stubbornly or
treacherously, refuses to support B? Under DYN, If B knows that this threat
is real, they must rationally choose between rewarding this behavior by
supporting C, or allowing the dreaded A to win. Under my fix, B can  support
C, knowing that there will then be a runoff, which B will win. When A
realizes that a runoff between B and C is coming, they can support B and
avoid that runoff. So my "fix" works: even if C plays chicken, the pairwise
champion B will win, and could do so without a runoff.

ACB BCA CBA: C is the CW. B sees that A will win, but waits for C to support
them - "I have more votes, you support me first". C supports B. A sees that
B will now win, and supports C. Now, C wins. This is a success for both DYN
and my fix. However, my fix might help to break the initial stalemate
between B and C of who will support whom. Since B's support for C will just
lead to a runoff, it is clear that C must support B; this makes A's support
for C a clearer necessity.

ACB BCA CAB: C is the CW. As above, with A and B reversed.

ABC BCA CAB: A condorcet cycle. A would win, so B supports C, so A supports
B, so C supports A. Now B is winning. In DYN, that means that A withdraws
support for B and wins - the minimax result. In my "fix", A supports B, so
there's an A/B runoff, which A wins. DYN is marginally better here, in that
it avoids the runoff, but on the down side, it does require a longer chain
of strategy to reach a stable result.

ACB BAC CBA: A cycle in the other direction. A would win, so C supports B,
so A supports C, so B supports A. A is winning. In DYN, A wins. In my "fix",
that means a runoff between A and B, which B wins. B is the minimax winner.

So, out of 8 scenarios, my "fix" avoids a potentially-successful strategy
once, makes correct and socially-optimal strategy more obvious twice, and
gets an arguably-better result once. On the downside, it risks an
ultimately-unsuccessful strategy once, and causes a runoff election (under
optimal strategy) twice (in the cases where there is a true condorcet
cycle).

The SDYN suggestion above is better than the runoff fix. It avoids runoffs
in the cases where it's clear what to do (mutual approval). It does not
short-circuit runoffs in any other case, because of risks of tricky
strategies such as the one discussed in the ABC BAC CAB case above.

JQ
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