[EM] Trying to get at the Condorcification proof

[Forest Simmons]

I think you are on to something with your DSV remarks.

You can do DSV where the voters' input is anything from their Von
Morgenstern- Neumann utilities to their ranked preference ballots.

The DSV is supposed to relieve the strategic burden from the voters because
of their lack of .... what?

Lack of information?

Lack of sophistication?

Not really ... when evaluating the theoretical effectiveness of a DSV
method, don't you assume worst case cleverness of potential rational voters
with complete information?

The voters are the potential manipulators ... theoritically (if not
practically) with just as much information about the other voters'
utilities or preferences as the DSV input provides.

Any theorem about the limitations or advantages of a DSV method, would have
to make some assumptions distinguishing unsophisticated voters from
sophisticated voters ... which really must include all voters ... at least
in the worst case scenarion.

I think the analysis is based on all rational voters with complete
information, but the reality in public elections is near complete
disinformation ... along with high levels of irrationality.

Which makes DSV design more of an art than a science.

We're talking of DSV in a general sense that is broad enough to treat
Instant Runoff as a DSV method for transferring votes in a  runoff, which
is already a DSV system for deciding where their one and only Plurality
vote will go, for example.

One takeaway for me is that Condorcet, M is a mild constraint on method M
manipulators, because under complete information, they supposedly already
know all of the sincere preferences including who the sincere CW is if
there is one. The constrained system pins them down to some deterministic
CW that (under game theoretic omniscience) is known to all of the
sophisticated players ... taking away from the advantage of stochastic
strategies with  less constrained entropy ... strategies that would
otherwise be open to them to get a little more advantage over their
unsophisticated compatriots.

Thanks for your thought provoking insights!

fws

On Sat, Jun 24, 2023, 12:08 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> James Green-Armytage (and independently Durand) showed that if M is a
> method that has a property that a coordinated majority can always force
> an outcome (the InfMC property), and neither equal-rank nor truncation
> is allowed, then there's no election E where the outcome can be
> strategically altered in Condorcet//M but not in M.
>
> Furthermore, Durand stated that the same result holds with equal rank
> and/or truncation as long as "Condorcet" is replaced with "AbsCondorcet"
> - absolute Condorcet, where the absolute Condorcet winner is a candidate
> who some absolute majority (50% + 1 or more) of the voters strictly
> prefer to any other candidate.
>
> I've tried to understand the proof myself as JGA and Durand's notes are
> kind of terse. Here's my attempt:
>
> I'll say "Election E is manipulable under M" if, when the winner of E
> according to M is W, then there exists some group who all prefer some
> other candidate X to W, and this group can alter their ballots so that X
> wins instead of W.
>
> 1. Consider an initial honest election E. We want to show that if E is
> manipulable under AbsCondorcet//M, then it's also manipulable under M.
>
> 2. If E has an absolute Condorcet winner C and M doesn't elect him, then
> by definition, a majority prefers C to M. Due to InfMC, this majority
> can force C to win. So E is manipulable under M; even if E is
> manipulable under AbsCondorcet//M, the latter is no worse than M.
>
> 3. Now suppose that M is the winner of election E according to M, and an
> absolute majority prefers some other candidate Y to X. Then E is
> manipulable under M. By the same reasoning, AbsCondorcet//M can do no
> worse than M.
>
> 4. Thus, if E is unmanipulable under M, there can be no majority
> pairwise preference X>W where W is the winner according to M. So M and
> AbsCondorcet//M agree that W is the winner (no matter if W is an
> absolute Condorcet winner or not). So the only way to make E manipulable
> under AbsCondorcet//M but not under M is to somehow make manipulators
> create an absolute CW, so that the post-manipulation election for
> AbsCondorcet//M changes but the election for M doesn't.
>
> 5. But that's impossible. Suppose we want candidate X to become the new
> absolute CW. To do so, a majority must prefer X to W. But by point 3, no
> such majority exists. The manipulators can't change this fact because E
> was initially honest, so they already maximally expressed their
> preference for X over W.
>
> 6. Thus the only way for an election to be manipulable under
> AbsCondorcet//M but not under M is impossible, which was what was wanted.
>
> Does that seem right?
>
> Essentially, the trick seems to be that absolute Condorcet is a sort of
> DSV for compromising within the constraints of InfMC. An absolute
> Condorcet winner C is someone who, if the winner was someone else, InfMC
> enablest a group of people who all prefer C to the current winner, to
> force the election of C by compromising for C. By electing C outright,
> the method removes the need to compromise for C in such a case.
>
> "If a majority prefers A to B, then B is not elected", as Robert would
> say.  ... in this case because otherwise, that majority could force the
> election of A by strategizing!
>
> Some more thoughts:
>
> I *think* the proof works for "Absolute Smith" too as long as we use
> ASmith,M (not //): if M elects someone not in the Absolute Smith set,
> then strategists can make any candidate in that set win (not the same
> group for each, of course).
> Manipulators trying to change the winner according to ASmith,M have two
> options: to keep the Smith set the same or expand it. In the former
> case, any strategy they use will also work on M since ASmith,M uses M's
> order to break the ASmith tie. In the latter case, no absolute majority
> prefers some X outside the set to any inside (same reasoning as for
> Condorcet). But to get X inside the ASmith set, we must make an absolute
> majority prefer X to someone in it, which is impossible (again, same
> reasoning).
>
> But it doesn't really help, because ASmith,M is manipulable whenever the
> absolute Smith set has more than one candidate in it, because if the
> majority preferring A to B (in an A>B>C.. cycle) unite, then they can
> force the election of A using InfMC. Still, at least we don't *lose*
> anything.
>
> Making a similar proof for relative majorities would require some
> "natural" property similar to InfMC but relating to relative majorities
> (e.g. something like "if everybody else is indifferent between A and B,
> then a majority of the remaining voters can force whether A or B wins").
>
> There may be clever things one can do with the DSV idea, e.g. consider
> something like: Let A ~> B in election E if either A has a higher Borda
> score than B, or people who prefer A to B can bury B to make A's Borda
> score higher than B's. Then let the Bury Top set be the maximal elements
> set of this ~> relation. Is "Bury Top,Borda" less susceptible to Burial
> than Borda? Maybe?
>
> Or how about: Let a candidate X be "tenable" in election E if it's
> possible to lower X in E to make X be eliminated no earlier under IRV;
> let a candidate Y be "untenable" if it's possible to raise Y in E to
> make Y be eliminated earlier under IRV. Let the net tenable set be every
> candidate in the tenable set but not in the untenable set, or all
> candidates if no such candidate exists. Elect the net tenable set
> candidate ranking highest in IRV's social ordering (i.e. eliminated
> last). Is this method monotone? Does it retain IRV's burial resistance?
>
> -km
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