# [EM] Arrow's theorem and cardinal voting systems

Kristofer Munsterhjelm km_elmet at t-online.de
Tue Jan 14 14:35:43 PST 2020

```On 11/01/2020 06.24, fdpk69p6uq at snkmail.com wrote:
> On Thu, Jan 9, 2020 at 11:46 PM robert bristow-johnson wrote:

>>    my question that i have asked the Score Voting or Approval Voting
>>    advocates years ago remains: "How much should I score my second choice?"
>
>
> If asked to rank three ice cream flavors, my preference would be
> Strawberry > Chocolate > Garlic.
>
> If then asked to choose between:
>
> 1. Chocolate
> 2. A mystery box with a 75% chance of containing Strawberry and a 25%
> chance of containing Garlic
>
> I would choose #1, which shows that:
>
> A. My preference for Chocolate > Garlic is significantly stronger than
> my preference for Strawberry > Chocolate.
> B. If voting honestly, I should give Chocolate at least a 4 out of 5 on
> a Score ballot.
>
> The odds can then be varied, to narrow in on a more precise rating,
> which is essentially what we all do internally when we rate a movie or
> restaurant or product or student or respond to a Likert scale survey, etc.
>
> Of course, this is imprecise, but so is forcing voters to rank many
> candidates when they are indifferent between some of them.

That sounds like it's still a normalized ballot, so that "honest Range"
with this calibration would fail Steve Eppley's independence. It doesn't
solve the problem of Range having many sincere ways to express the same
ballot, either, though it does narrow down which ballots are honest/sincere.

Suppose we have an ice cream election and (for the sake of the argument)
you're exactly indifferent between a 75:25 lottery between Strawberry
and Garlic, and a certain choice of Chocolate. (Suppose also that you're
risk neutral, because risk aversion is not the point.)

Then what we know is that
0.75 * utility(Strawberry) + 0.25 * utility(Garlic) = utility(Chocolate)

This is a linear equation with three unknowns. We need two more to
unambiguously determine the values. A standard zero and a standard unit
will do.

A standard unit is reasonable to have, because multiplying every unknown
by some constant preserves all the lottery-based equations, so someone
who likes exaggerating his scale (e.g. by saying "OMG, this is the best
thing ever!" every time he sees something good) will have more influence
than someone who likes to be economical with his values.

But it's hardly clear how to *find* that standard zero and unit. For
e.g. a pizza election, you could say the standard zero is no pizza at
all and the standard unit is a Margherita (say). But for a political
election? And strictly speaking, the scale would have to be unbounded so
that it can both accommodate people who don't particularly like pizza
and people who have lived their whole lives for the purpose of getting a
pizza.

So the point is that the lack of a single reference honest ballot for
Range is a due to cardinal utilities being very hard to calibrate
between people. And if you can't say what the one honest ballot is, then
there will still be ambiguity in any cardinal system as to what
constitutes sincerity, and how you should rate your choices.

You may try to define a single honest ballot for a semi-cardinal method
that automatically normalizes the endpoints to max and min value, so
that the two unknowns are given and the voter can just use the lottery
method to fill in the remaining data. But if you do so, then a
two-candidate election becomes a majority election and you're back to
Eppley's independence failure example. So in some sense, the
impossibility is "tight" - if you want IIA, the ballots must be
independently calibrated. If you let them be relatively calibrated even
a little, IIA goes away.

That doesn't mean that rankings are better than ratings, period. But a
ranked ballot makes it possible to have a single honest ballot without
needing to standardize it -- at the expense of ranked methods failing
IIA. And the ambiguity of rated ballots makes honesty and strategy blur
together. Since no election method can know if you've chosen the right
scale, it seems like honesty and strategy will always be blurred
somewhat together, no matter the cardinal method.

> If Vanilla and French Vanilla were both on the same ballot, I would be
> indifferent between them.  Forcing me to choose between them and then
> arbitrarily assigning the same weight to this very weak preference that
> was applied to my Chocolate > Garlic preference would be rather
> undemocratic, no?
> http://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html

To some degree, what I said above also holds for truncation and
equal-rank, but there it doesn't seem to be as serious a problem. It
would be interesting to find out why.

Perhaps truncation and equal rank are convenience features, so the voter
says "determining who wins of these is worth less to me than the effort
it is to rank those candidates, so I'll let someone else decide". That
might be a decision that a voter can do without needing to do any
absolute calibration. But if so, the problem with cardinal ballots is
then not that there are many honest ballots *as such*, but rather that
the voter is required to make a strategic effort.

> Organisms don't have ordered lists of equal-strength preferences in
> their brains.  They have fuzzy estimates of utility that they then
> convert to rankings when necessary.

(I have a suspicion that what we really have are utility vectors, and
what we call "utility" is more like a norm of these. But I have no
proof, and it's sort of besides the point.)

>     that tactical question faces the voter in a Score or Approval
>     election the second he/she steps into the voting booth.  but not so
>     for the ordinal Ranked ballot.
>
>
> From what I've been told (though I haven't read and understood it
> myself), Gibbard's theorem proves that ALL voting systems require voters
> to make tactical decisions, no matter whether they are ranked or rated
> or otherwise.

All deterministic ones, to be precise :-) And Gibbard doesn't say voters
*need* to do it - it only says that a voter who wants to maximize the
impact of his vote needs to do so.
```