[EM] A proxy for Monroe's "strategic election of universally loathed candidates" criterion

[Closed Limelike Curves]

Thank you so much for all of your hard work+analysis Kris :)
>
> Smith passes DH2, but (surprisingly) Schwartz fails!
>
Could you clarify what you mean by Smith/Schwartz passing/failing?

————
That said, I'd like to offer some clarification. My concern isn't actually
with DH2 or DH3 scenarios. My point in our earlier discussion was I reject
two common frameworks for strategic analysis.

First, I reject the group strategy framework. In elections, hidden
communication is infeasible because of how many people you'd need to get in
on the conspiracy, and the secret ballot makes collusion impossible to
enforce. If group strategy was a reasonable model of voters, it wouldn't
matter which electoral system we picked, because the outcome would *always*
be a maximal lottery. What I find more meaningful is *individual*
 incentive-compatibility.

Second, I reject the idea of looking at individual scenarios like DH3 or
Burr, then judging if a voting system fails or succeeds in such a scenario.
(Or rather, using this to rule electoral systems in, not out). This feels
like declaring a theorem proven after trying a few numbers and not finding
a counterexample.

(The Burr dilemma should be the poster boy for this kind of bad strategic
analysis: it focuses on a single scenario, *and* on group strategy, *and* it
completely ignores the possibility of correlated equilibria and mixed
strategies...)

What I generally believe in is starting with a reasonable model
characterizing voting behavior, then identifying how the method performs
across all elections in this model. The two models I've gotten a lot of
mileage out of reading and studying are:
1. Myerson-Weber-style decisive vote models, where voters only care about
deciding an uncertain election.
2. Balinski and Laraki's *grading* model, where voters look to optimize the
score of candidates they support (because votes send signals about
popularity of different policies).

I do wonder if it's possible to get a system where the optimal strategy in
most elections is min-maxing, whereas with zero information the best
strategy is strict ranking (a real advantage of most ordinal methods). I
know there are incentive-compatible cardinal mechanisms in the
zero-information case (http://dx.doi.org/10.1016/j.geb.2017.04.012
<https://www.researchgate.net/publication/316805816_Ordinal_Versus_Cardinal_Voting_Rules_A_Mechanism_Design_Approach>).
Later-no-help+no favorite betrayal would be enough to guarantee min-maxing
with perfect information.

On Sat, Jul 20, 2024 at 11:18 AM Kristofer Munsterhjelm <
km-elmet at munsterhjelm.no> wrote:

> So I'm gonna try this list thing again. We'll see how the plonking goes.
> But I'm kind of tired of debate, (particularly the sort that ends in
> back-and-forth insults,) so let's try something a bit closer to research.
>
> Before I left, there was a mention of Monroe's Nonelection of Irrelevant
> Alternatives as a measure of burial resistance, so I tried to play with
> it a bit. But I found it very hard because you have to come up with
> separate Myerson-Weber strategies depending on the method and their
> state spaces. (So hard that Monroe might actually have got some of the
> failure results in his draft paper wrong!) So I thought I'd try to find
> a more... mechanically applicable criterion that would imply NIA - i.e.
> if a method passes this, it passes NIA.
>
> And I think I've found two. One that's still pretty complex but that can
> be checked purely mechanically (at least for ranked methods), and
> another that's much easier but limited to majoritarian unrestricted
> domain methods. Both criteria only properly cover/imply NIA if the
> method also passes a (pretty straightforward) mono-add-plump-related
> criterion.
>
> (One might seriously question the relevance or realism of the knife's
> edge election that follows, but such a knife's edge is what Monroe based
> his reasoning on, so...)
>
> Both criteria start with a three-candidate honest election of the form:
>
>         N: A>B>C
>         N: B>A>C
>         1: C>A>B
>         1: C>B>A
>
> with N being a very large number (or equivalently, replace the C-faction
> weights with some infinitesimal epsilon). The idea is that most methods
> will give this as an A=B tie. Then there might exist some risky
> strategic ballot that the A faction can cast, and some risky strategic
> ballot that the B faction can cast, so that if one of them does so, then
> their candidate wins, but if both do it, then C wins.
>
> The A and B factions are balanced so that there's initially a tie. And
> the incredibly weak C faction exists to stop a method from passing the
> criteria by refusing to elect candidates who have no first preferences.
> The C-faction also helps restore results that Monroe might have got
> wrong, thus being closer to the spirit of his criterion.
>
> So let's go to the criteria.
>
> The complex criterion is the "strong non-collapse tie criterion" or
> SNCTC for short. I'm going to limit myself to deterministic methods
> because handling nondeterministic ones seems to come with a ton of
> corner cases.
>
> I'll also call the A-faction just "A" and similarly for the other factions.
>
> Here's the SNCTC definition:
>
> ===
>
> If C is elected (wins or ties) in the honest election, then the method
> fails.
>
> Otherwise, define an effective ballot s for A as one that:
>         1. when A casts s instead of his honest ballot, A becomes the only
> winner, and
>         2. no matter what ballot t B casts, if A casts s and B casts t, B
> does
> not become the unique winner.
> If A has no such effective ballots, then let every ballot that satisfies
> the first point above be an effective ballot for A. (If A still has no
> effective ballots, that's okay.)
>
> B's effective ballots are defined similarly. Note that the honest ballot
> may be effective if the faction's favorite wins outright.
>
> If only one side has any effective ballots, then we pass (because the
> side without effective ballots can't unilaterally make their candidate
> win, so the collapse can't occur).
>
> If there exists an effective ballot s for A so that no matter what
> effective ballot t that B chooses, if A casts s and B casats t, then C
> doesn't win; and the same is true (with the faction labels flipped) of
> the B faction; then we pass.
>
> Otherwise, we fail.
>
> For methods that are more expressive than ranked ones, the above must
> hold for every election whose rank data correspond to the honest ranked
> election. E.g. for a rated method, every rated ballot where the A
> faction rates A higher than B and B higher than C, the B faction rates B
> higher than A and A higher than C, etc., must pass. The criterion is
> inapplicable to methods that are *less* expressive (like Approval).
>
> ===
>
> The point is that an effective ballot by A is a strategy (either safe or
> risky) that can't be exploited by B: i.e. A casting it improves A's
> winning chances without B being able to exploit this by casting some
> other ballot that makes B the winner.
>
> By reasoning this way, when A and B strategize, they'll limit themselves
> to effective ballots.
>
> Then we pass if A and B can't step on each other's toes no matter what
> effective ballots they use. No matter what they do, C is not going to win.
>
>
> This is still pretty cumbersome, but for ranked methods with truncation
> and/or equal-rank, we can at least test every possible ballot to
> determine which ballots are effective. Rated methods require more
> general reasoning and can't be done mechanically.
>
> So, let's say we want to simplify it even more. Here's a criterion for
> majoritarian unrestricted domain methods (i.e. no equal-rank or
> truncation):
>
> DH2:
>
> Let election e1 be:
>
>                 N: A>B>C
>                 N: B>A>C
>                 1: C>A>B
>                 1: C>B>A
>
> Let election e2 be:
>
>                 N: A>C>B
>                 N: B>A>C
>                 1: C>A>B
>                 1: C>B>A
>
> Let election e3 be:
>
>                 N: A>B>C
>                 N: B>C>A
>                 1: C>A>B
>                 1: C>B>A
>
> And let election e4 be:
>
>                 N: A>C>B
>                 N: B>C>A
>                 1: C>A>B
>                 1: C>B>A
>
> 1. If C wins or ties for first in e1, the method fails.
> 2. If A doesn't uniquely win in e2, the method passes.
> 3. If B doesn't uniquely win in e3, the method passes.
> 4. If C doesn't win or tie for first in e4, the method passes.
> Otherwise, the method fails.
>
> The criterion is named after Warren Smith's DH3 ("dark horse plus
> three") because, in a sense, this is "dark horse plus two".
>
> The explanation is:
>         For 1.: if C wins in the first election, then C wins even without
> strategy, which is a failure. (I don't expect that to happen, but why
> not cover our bases?)
>
>         For 2. and 3.: if unilateral burial doesn't break the tie in favor
> of
> the buriers, then the buriers have no reason to do it. (Particularly not
> if it makes C win too.) And if only one side (at most) has a reason to
> bury, then the combined effect of two sides burying can never happen,
> because only one side (at most) has any incentive to bury.
>
>         And for 4.: if they both have incentive to bury, then the two
> burying
> at once must not make C win.
>
> Why is this equivalent to SNCTC? Consider the effective ballots for A.
> By the majority criterion, ranking anybody but A first will make that
> candidate win. So the effective ballots must either be A>B>C or A>C>B.
> If it's A>B>C, then B can't unilaterally break the tie, hence B has no
> reason to strategize, and we pass. If it's A>C>B, then we get e2, e3,
> and e4.
>
> ===
>
> I've skipped my reasoning about how SNCTC plus a mono-add-plumpish
> criterion implies NIA, what criterion is required for the implication,
> and what methods pass SNCTC or DH2, because this post is long enough.
> Just ask if you'd like more info.
>
> But I can say that if method M passes DH2, then all of Condorcet,M and
> Resistant,M and Condorcet//M and Condorcet,M pass. Resistant//M pass for
> all M. Resistant,M may fail for some M (e.g. Borda) but there's a tweak
> that makes them pass for all M.
>
> Range with any sufficiently large discrete scale passes SNCTC, as does
> discrete lp-cumulative vote with any p-norm >= 1. (The continuous
> versions fail on a technicality.)
>
> Smith passes DH2, but (surprisingly) Schwartz fails!
>
> Ranked methods with no burial incentive immediately pass DH2 because the
> only possible strategic effective vote (as established) is A>C>B, but
> this buries B, and if there's no burial incentive, that has no effect.
>
> DH2 requires unrestricted domain. In one direction, consider a method
> that's like Plurality, but each faction may also indicate "I'm serious".
> If one faction does so, that faction's candidate wins, but if both do,
> then C wins. This fails SNCTC but its UD version, Plurality, passes DH2.
> In the other direction, if I got it right, Bucklin fails DH2 but passes
> SNCTC because truncation is just as good as burial and doesn't come with
> its risks.
>
> -km
> ----
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> info
>
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