[EM] smith/schwartz/landau

Juho Laatu juho.laatu at gmail.com
Tue Mar 27 10:54:34 PDT 2018

> On 27 Mar 2018, at 18:11, Kristofer Munsterhjelm <km_elmet at t-online.de> wrote:
> On 03/26/2018 11:18 PM, Juho Laatu wrote:
>> In summary, I don't see how to make any conclusions on the relative preferences (strength) of the ABC group and d. Just like Condorcet Winners could be popular or not (strength), also Smith Set members and non Smith Set members could be popular or not (strength). The example shows a situation where preferences can be whatever or about the same (strength), or in favour of d (count). There are three candidates that beat d, but that should not carry much weight since number of pairwise victories is known to be a poor criterion (because of clone problems).
> Isn't this analogous to majority vs utility? Suppose you have a ranked election like this:

Yes, my count vs strength corresponds to majority vs utility. I should have already introduced new standard terms for "better if measured in utility" and "better if measured in majority/plurality".

> 51: A>B>C
> 48: B>C>A
> 2: C>B>A
> A is the majority winner, but it only takes two voters changing from A>B>C to B>A>C to make B the majority winner, analogous to how {ABC} is the Smith set but {d} is close to being the CW. A voting method that passes the majority criterion will always elect A, but a more consensus-focused method like Borda elects B.

Condorcet fans tend to emphasize the majority approach. One key reason is the strategy concerns.

> If we knew the utilities, then we could determine whether this is a
> 51: A: 100, B: 1, C: 0
> 48: B: 100, C: 1, A: 0
> 2: C: 100, B: 1, A: 0
> election (A should win if we're counting utilities), or a
> 51: A: 100, B: 99, C: 0
> 48: B: 100, C: 99, A: 0
> 2: C: 100, B: 99, A: 0
> election (B should win). But we don't, so we can't. Almost any situation where A is the superior candidate can be matched by a parallel situation where B is the superior candidate. Lacking utility information, we can't establish which is correct, and so there seem to be only two ways to get out of the problem.
> The first way is to say that we have a model of how the electorate behaves, which lets us determine which rated election is most likely given the ranked data. (E.g. in a very polarized electorate, it's more likely to be the former than the latter.)

This approach is good for studying the results (based on various assumptions on voter preferences). Not that easy if one would have to decide the winner based on an agreed model on what the preferences of the voters are.

> The second way is to say "we have no idea of which may be true, but we want to pass the majority criterion, so that settles the matter for us". In that case, the hypothetical "we" would choose a method that elects A.

I believe many Condorcet fans think that getting honest rankings is about as much honest information as we can get. That means that one is to some extent forced to follow the majority logic.

On the other hand people believe in the principle of one person one vote, and in the right on majorities to decide. That makes the majoritarian approach of Condorcet again very natural.

> If the Smith situation is analogous to majority, then it seems that the most consistent choice with respect to majority (as opposed to utility) is to elect from the Smith set.

If the Smith Set forms some sort of clear majority, then yes, but that need not be the case. If A, B and C would beat each others by max n votes, and they all would beat d by more than n votes, then one could say that someone from the Smith set should win to respect majority.

One can also study the examples assuming that some of the candidates are clones. In my example A, B and C certainly are not clones. But there could be also ballots where A, B and C would be next to each others in all ballots (technical clones). This does not yet guarantee that those candidates are also real life clones (candidates of one party, in friendly terms, under the assumption the utilities of the voters are about the same for all three).

In the example that I gave, we should probably assume that there are four parties that compete with each others. If we would elect A, there would be a 66:33 opposition saying that we should have elected C. If we elect d, there would be only a weak (majority) opposition saying that we should have elected one of the others. Although A, B and C form a Smith Set, they probably do not form any kind of ideological set that should be given the right to claim victory. I.e. defeats within the Smith Set are just as damaging as defeats of d. Being part of the (technical) Smith Set does not give any additional privileges.

> A method that doesn't elect from the Smith set may be superior in a utilitarian sense, but it may also not be. There's no way to know (short of taking the first approach, or by proving that e.g. for a very wide range of utility models, some non-Smith method has better utilitarian performance than some Smith method).

I think the utility uncertainties and arguments have no specific role in the given example. Condorcet Winners can have low or hight utility. Same with members of Smith Set, and candidates outside of the Smith Set. My arguments to consider d to be a reasonable winner are based on the majoritarian pairwise comparisons only. Shortly, d is two votes short of being a majoritarian Condorcet Winner (and the others are far behind in this kind of measurements).

I think human intuition is a problem here. The Smith Set with its three candidates seems to be between d and the winning position (that is drawn above the Smith Set). That makes d look bad. But I claim that this image hides all the defeats within the Smith Set, and it ignores the fact that d is actually very close to being a Condorcet Winner, not far from it.

> I suppose what I'm saying is that if we choose to have a method that passes majority on the grounds on the grounds that the majority is right, then the argument can be adapted pretty easily to Condorcet and then to Smith. Any counter of the sort that electing from Smith can get the wrong winner can be similarly "ported back" to the majority situation by showing that the majority criterion can also get it wrong.

Majority criterion is not a solution to all problems, but we can take it as granted in a traditional majoritarian political system. I would not say that the fact that Smith Set members beat all the other candidates is a valid majority based argument supporting election from the Smith Set. The reason is that this statement would ignore all the defeats within the Smith Set. We must see the Condorcet election as a competition between individual candidates, and in that competition all pairwise victories and defeats do count.

I note one more problem with human intuition. Humans have a tendency to force group opinions to form a linear preference order. It looks natural that d is behind the Smith Set, and that also all members of the Smith Set should be forced to form a linear preference order. I think this would be a big mistake since we know that group preferences are not linear. The winner should be chosen based on the ballot or matrix preferences only, not based on any imagined linear preference order.

In my example electing someone from the Smith Set is a violation of one of the 66:33 majorities.

> Of course, if there's a more indirect argument behind wanting majority, then that can break the symmetry. For instance, one could argue that passing majority is a sort of DSV; in strategic Range, everybody votes Approval style and so the majority winner wins anyway, so why not ease the burden of the voter? Or "a large majority has the backing to get what it wants, better let it have what it wants on the ballot than risk a riot". It's not obvious how such arguments would support Smith, whereas it is pretty clear that they would support the majority criterion.
> But then that argument has to be made, first; in the absence of such an argument, Smith seems a reasonable extrapolation from majority (by way of Condorcet).

I tried to avoid all more complex argumentation and stick to the pairwise majority comparisons (and potentially information from the ballots too).

I don't see Smith Set as an extension of majority. It would be if we would decide to see the Smith Set members as one unified block of candidates of one party, and their defeats of each others would be either minor, or we would decide that their defeats do not matter even if they are big (assuming a small difference in utility although there are many voters preferring one over the other).

BR, Juho

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