# How is Approval Voting providing more information than an ordinal method?

Kenneth Arrow says in an interview that Approval Voting provides more information than ordinal methods subject to his theorem:

CES: So it seems like one of the points that you’re stressing within the theorem is that even though none of the ranking methods can fulfill the criteria, that’s really not speaking to the degree that they fail nor the frequency they fail–even if at some point it’s possible for them to fail.

Dr. Arrow: That’s correct. Yes. Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good. Or this is very good. And this is bad. So I have three or four classes. You have two classes is what you call Approval Voting. Just say some measures are satisfactory, and some aren’t. This gives more structure. And, in effect, say I approve and you approve, we sort of should count equally. So this gives more information than simply what I have asked for. This changes the nature of voting. We don’t just rank the candidates. We say something like they’re good or bad or something. This again, this method does not necessarily avoid paradoxes. But it seems empirically to minimize their importance.

CES: Now, you mention that your theorem applies to preferential systems or ranking systems.

Dr. Arrow: Yes

CES: But the system that you’re just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems.

Dr. Arrow: And as I said, that in effect implies more information.

But it looks to me like Approval Voting provides a partial order (trivially, all approved are "greater" than all non-approved), which is a weaker level of information than a total order (asked for in ordinal methods subject to Arrow's theorem). So how comes Arrow himself says that AV provides more information than a total order? Is it because the labels approved/not-approved are assumed to have absolute (instead of relative) relevance? Or is it because it allows ties? Or to disapprove (or approve) of everyone on the list? And how relevant are these features practice in comparison to ordinal rankings? (Arrow hints that there's apparently empirical evidence that might be some.)

• Approval gives quantized ratings, not rankings. Ratings can be converted to rankings. Approval gives less 'data' than many other methods, but it avoids some pitfalls of any ranked method.
– amI
Oct 2, 2018 at 17:17
• Should have somehow added Trump so this excellent Q&A gets votes it deserves far more than most questions on this site. :( Oct 3, 2018 at 9:56

The "more information" indeed comes from the assumption of a "common language" (i.e. a common & absolute scale in minds of all the voters). I should stress however that this point (which Arrow conveys) is being stressed not by the better-known proponents of Approval Voting (e.g. Brams & Fishburn), but rather by those of majority judgement (Balanski & Laraki). At a pure technical level, subsumming Approval Voting to majority judgement methods seems correct (see e.g. free PNAS article by Balanski and Laraki (2007) which summarizes their main ideas). The more subtle catch is that Approval Voting doesn't seem to depend on this notion of common language, while majority judgment methods with more than two grades for each candidates seem to need it to avoid some paradoxical interpretations. (As an aside, Arrow mentions by name Balanksi, but not Brams in that interview.)

In the absence of such an assumption, some cardinal methods can produce really counterintuitive results. This is not illustratable with Approval Voting (as far as I know), but with more grades it's easily shown; e.g. a book review by Edelman of Balinski (who is one of the main proponents of cardinal voting) & Laraki Majority Judgement (2011) gives the following example:

where the candidates are in Roman numerals (I-III) and the voters in Arabic (1-5), and there's American-school-like grading (A is best, F is worst). Under the assumption of common language, Balinski's method of picking II as the winner makes sense (he use the median, not the average of votes for the selection, and voter #3 is the median). On the other hand, if we don't assume a common [grading] language, then I is preferred to II by 4 of 5 voters, under the traditional social choice theory.

The interpretation(s) of approve/disapprove in Approval Voting have been at least 3 (in other words, that highlighted by Arrow from Balinski is not the only one), and these entail a complex discussion of strategic voting:

With regard to the meaning of “approval” there appears to be a fundamental split among points-of-view. Some presume that each voter actually has an underlying ranking (possibly weak) of the alternatives, and in choosing an approved ballot must somehow compress several distinct levels of approval into exactly two levels. Others view the dichotomous ballot as a direct reflection of a dichotomous primitive: each voter either likes or dislikes each alternative, and is indifferent among those within either group. A third view presumes that a voter has a ranking together with a line dividing those alternatives she likes from those she dislikes; we will refer to such a line as a true zero. An assignment of cardinal utilities might underlie the first view, or the third (if utilities can be negative). [footnote: The authors of Approval Voting take two views, referring sometimes to an underlying ranking and at other times speaking of the approved alternatives as those “acceptable” to the voter. The third view appears in Brams and Sanver (2009), which proposes voting rules that use a ballot consisting of a ranking and a dividing line both, and in Sanver (2010); see also the related Bucklin voting, fallback voting, and majoritarian compromise discussed in Hoag and Hallett (1926) and Brams (2008).]

A strategic analysis of approval voting cannot easily be disentangled from this more philosophical matter of what it means to approve an alternative. If approval is a primitive, then each voter has only one sincere ballot, but lacks any incentive to vote insincerely. For a voter with an underlying ranking, it is clearly never strategically advantageous to approve an alternative without also approving all others that you like as well or better, so deciding on an approval ballot amounts to choosing “where to draw the line.” If that line has no intrinsic meaning, there is no basis on which to discriminate between a sincere ballot and an insincere one; one might argue that all ballots are strategic, or that none are. If a voter has both a ranking and a line with intrinsic meaning as a true zero, then any ballot drawing the line somewhere else might be classified as insincere—and such a voter might have a strategic incentive to cast such a ballot. [footnote: Marking a voter’s true zero is not the only way to ascribe intrinsic meaning to the location of “the line.” For a voter who assigns a cardinal utility to each alternative, the mean utility value serves as a sort of relative zero (quite possibly different from that voter’s “true” zero, if she has one) and arguments have been made (in Brams and Fishburn, 2007) for drawing the line there. Duddy et al. (2013) show that drawing the line at the mean maximizes a measure of total separation between the approved and unapproved groups. Laslier (2009) argues that it is strategically advantageous to draw the line near the utility (to the individual voter) of the expected winner, and that voters tend to behave this way.]

And another interesting observation:

Moreover, if we relax the SCF [social choice function] notion by allowing voters to express indifference among alternatives, approval voting becomes an SCF—in fact it coincides both with Borda and with all Condorcet extensions, when these rules are suitably adapted to handle ballots with many indifferences.

Both quotes from Zwicker's Introduction to the Theory of Voting (2016).

And to be fair to Balanski & Laraki, in their book they have a section (18.3) that discusses how approval voting may be cast "approval judgement", in terms of formulating the questions on the ballot as to suggest a common grading scale etc.

When approval voting is practiced as a majority judgment, a language of two words must be formulated that makes clear the evaluations are absolute grades. To distinguish it from its traditional practice we call it approval judgment. [...]

So to nitpick, Arrow was basically talking of the latter.