Hi folks,

I'm going to send a similar email here to the EM list that I recently
sent to several folks who hang out in academic circles.  The answer I
received from the academic circles was valuable, but I also think that
folks on this mailing list can provide a different (and useful)
perspective.

I've long taken it for granted that impossibility theorems like
Arrow's theorem and Gibbard's theorem mathematically prove that there
are always going to be important electoral criteria that will be
mutually exclusive in ANY credible electoral system.  I've been at
peace with that for a long time, much in the same way that I'm at
peace with mutually exclusive criteria for my transportation needs
(e.g. I should take something with more carrying capacity than a
bicycle to go shopping for large furniture, no matter how good the
bike is).  The physics of electoral systems and the physics of the
real world have certain mathematical rules that are tough to get
around.

Since the Center for Election Science (https://electionscience.org)
started getting momentum and having some electoral success in the late
2010s, there's been a push to distinguish between "cardinal voting"
and "ordinal voting" as the top of the hierarchy distinguishing all
voting systems.  Since the ballot is what people see, that's
understandable, I suppose.  However, in my mind, the ballots don't
matter as much as the tallying method, and moreover, it's possible to
use cardinal voting ballots and then tally them using systems that
some folks classify as "ordinal" systems.

In discussions with electoral reform folks over the past few years,
I've been learning about Arrow, Gibbard, and Satterthwaite, and trying
to document what I've learned on Wikipedia and electowiki.

In editing Wikipedia articles related to election methods in the past
few years, it seems there are three theorems that have made the rounds
with regards to impossibility theorems:

1. Arrow's impossibility theorem (published in 1951): basically the
   granddaddy of impossibility theorems, which seemingly only applies to
   ordinal voting methods.
   https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem
2. Gibbard's theorem (published in 1973): generalizes Arrow's theorem
   to apply to pretty much every social choice function
   https://en.wikipedia.org/wiki/Gibbard%27s_theorem
3. The Gibbard–Satterthwaite theorem (published in 1978): a more
   specific version of Gibbard's theorem which apparently only applies to
   ordinal systems, and focuses on strategic voting
   https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem

What bothers me of late is a recent change that's been made to the
Gibbard-Satterthwaite article.  I'll quote the most bothersome
addition/replacement that's in the "Gibbard–Satterthwaite theorem"
article as of this writing:

It seems disingenuous to say that all of these voting systems don't
apply to cardinal systems if there is some way to vote "honestly"
(whatever that means).  Strategy and honesty are not mutually
exclusive, and cardinal systems like "score voting" require voters to
be very strategic as part of their voting calculus.  As noted above,
Condorcet tallying methods can be used to tally "cardinal ballots" and
"ordinal ballots", since both express the preferences.

I'll quote what one of the folks in the academic circles stated:

This assertion more-or-less comports with my opinion.  While I don't
think that systems that insist on ranking-based ballots (ordinal
ballots) are ALWAYS superior to systems that rely on simple addition
of rating-based ballots (cardinal ballots), I think the implicit
rankings are at least as important as the explicit ratings.  I
generally think of STAR voting as "Condorcet lite", because, for two
finalists "candA" and "candB", the final runoff doesn't pay attention
to whether:
scenario 1 ) "candA" has 5 stars and "candB" has 4 stars on ballot #1234
or
scenario 2) "candA" has 1 star, and "candB" has 0 stars on ballot #1234

In the end, in both scenarios, ballot #1234 counts in full for
"candA", which seems fair to me.  Regardless, I've frequently found
myself distrusting hardcore cardinal advocates when I see changes like
the one made to English Wikipedia's "Gibbard–Satterthwaite theorem"
article.

Are cardinal voting advocates correct to continually claim that
Arrow's, Gibbard's, and Satterthwaite's theorems don't apply to their
favorite voting methods?  Is there a useful distinction to be drawn
between Gibbard's 1973 theorem and the "Gibbard-Satterthwaite theorem"
published in 1978?  Is the distinction I draw above correct?

Rob

Whether you are right or no, there is no conensus on the matter. Theorem
Arrow is like a Cold War between electoral systems. It acknowledges
ordinal votes as a basis for elections, in a denigratory sort of way,
but over-looks the count beyond crude plurality. Ever since,
mathematicians have demonstrated it takes more than mathematics to have
a good understanding of elections. Formerly, it was not so, when they
acted freely as enthusiasts, but the institutionalisation of election
studies appears to have robbed them of any independent critical sense.

That understanding, I gather from his parliamentary speeches on "Mr
Hare's system," is what John Stuart Mill had, the greatest philosopher
of science in the 19th century. The Hare-Mill tradition, that has
continued to the present day, has been by-passed. In so doing, social
choice theory reveals its provincialism. A Nobel prize or so, to give
away, is not a proof. A theorem is only as good as the assumptions on
which it is based. And theorem Arrow compares to a critique of a bicycle
on the basis of the short-comings of a unicycle. It does not deal with
the democratic necessity of a proportional count as well as an ordinal
vote. Simple plurality is "maiorocracy" or the tyranny of the majority,
as Mill and Lani Guinier said.

It is not apparent what decisive argument the social choice school have
that they can take to the voters, for whom elections are supposed to be
meant, and has not been so for 70 years. It is not even apparent that,
after 70 years, they have any idea of, or even belief in, a standard
model of democratic election.

Richard Lung.

On 15/02/2024 07:13, Rob Lanphier wrote:

On 2024-02-15 08:13, Rob Lanphier wrote:

My understanding and opinion is this:

You have three different "main" impossibility theorems:

• Arrow's says that no deterministic reasonable ordinal voting method
  can pass IIA. This means that regardless of whether voters are honest or
  strategic, it's possible that A's win over B depends on not just how
  many voters prefer A to B, but also how many prefer X to Y. This is only
  for cardinal methods.

• Gibbard-Satterthwaite says that no deterministic reasonable ordinal
  method is strategy-proof: there always exists at least one election
  where at least one voter has an incentive to adjust his preferences
  based on how others are voting.

• Gibbard's theorem extends this to a much broader class of election
  methods that includes cardinal methods, thus also implying that no
  deterministic cardinal voting method is strategy-proof.

(In particular, both of Gibbard's theorems are about strategy.)

The part about the revelation principle is incorrect and seems to be
using a very specific definition of honesty, inspired by Warren Smith.
To my knowledge, what the revelation principle says is this:

• Say that a participant in a mechanism employs strategy if the
  information he submits to the mechanism depends on the actions of the
  other participants or his belief about them.[1]

• Say that a mechanism only uses honesty if no participant has an
  incentive to employ strategy.

• Then, if there exists a mechanism where people employ strategy to
  drive it to an optimal or equilibrium outcome, then there also exists a
  mechanism that only uses honesty that reaches the same outcome or
  equilibrium.

Consider it like this: suppose you're involved in a court case, and you
hire a lawyer. You tell the lawyer the truth and the lawyer comes up
with whatever strategy that will advance your interests, given the
evidence and information about the other party. That's a system where
you or your lawyer use strategy to drive the system to a particular state.

But consider a hypothetical legal system using an AI judge. This AI has
a lawyer interface to each party; upon hearing the truth from each
party, it then simulates a court case the way it would proceed with
virtual lawyers who employs strategy based on the honest information.
The AI comes with a proof that your virtual lawyer won't incriminate
you. You would then interact honestly with the system, represented by
the judge, and it would strategize internally.

Thus for any system that requires strategy to get an equilibrium, there
exists another system where you can be honest. It just absorbs the
"lawyer component" into itself.

The problem is that there is no such equilibrium for deterministic
voting methods. That's implied by Gibbard, because otherwise, you could
create a strategy-proof method by first creating a method that invites a
particular type of strategy, and then embodying a "lawyer component"
into it to perform that strategy.

If you try to do this, as far as I understand it, you get a
nondeterministic method since the equilibrium is a mixed strategy. And,
as we know, there exist strategy-proof nondeterministic methods, so that
fits.

This would be like: sometimes, your lawyer says "if we do this, then no
matter what the other party does, we win". But other times he says "if
we focus on these elements and the other party focuses on those, then we
win". Like rock-paper-scissors, which strategy you should play depends
on the strategy the other guy is going to use. So you play them at
random in such a way that the other guy can't guess what you're doing.
That gives a nondeterministic method.

Okay.

So now about IIA.

The seeming advantage that cardinal methods have over ordinal ones is
that they pass IIA. In Range, if A has score 100 and B has score 50,
then A will continue to beat B even if we remove every other candidate.

But I've always been of the opinion that this IIA compliance is either
illusory or a distraction, because it doesn't answer what we really care
about. And that is whether the presence of a candidate who doesn't win
changes the outcome.

Some cardinal proponents say that methods like Range have multiple
honest ballots: there are many ways to vote that are all consistent with
your preference ordering. But a voter has to choose which honest ballot
to cast, and there's no externally fixed scale (what exactly does ten
points mean? What does zero mean?).[2] Thus the selection of candidates
who run will affect the scale, which means that some voters would change
their ratings based on who's running - even if those additional
candidates don't win. So these methods fail what we could call "de facto
IIA", for lack of a better term.

For instance, suppose the election starts off with two pro-democracy
candidates. Then a number of authoritarians enter the race. It's likely
that in Approval, some voters who would've approved of one of the
pro-democracy candidates but not the other, would now approve both to
mark their distaste for authoritarianism and keep the authoritarians
from winning.

So in my opinion, ordinal methods are merely honest about their
limitations. Their logic says: "okay, I can't know if his 10/10 is the
same as her 10/10 or her 5/10. I'll accept that this means I must fail
IIA, instead of seeming to pass it by passing the buck to the voters
that they must use a fixed scale that's not affected by who's in the race."

So ordinal methods clearly fail IIA, and aren't strategy-proof (by
Gibbard-Satterthwaite). Cardinal methods pass IIA (but it doesn't mean
what one may think it means) and aren't strategy-proof (by Gibbard's
theorem).

Both ordinal and cardinal methods may pass de facto IIA for subsets of
elections. E.g. Condorcet methods pass IIA as long as there is a CW,
because adding a candidate either makes that candidate the new CW (hence
he's not irrelevant) or the current winner stays a CW. Similarly,
Approval passes de facto IIA with dichotomous preferences where every
voter has a class of OK candidates and a class of not-OK candidates, and
the boundary between OK and not OK doesn't depend on who's in the race.
(Such voters may sometimes approve everybody or nobody.)

But in general: both cardinal and ordinal methods are susceptible to
strategy. And both cardinal and ordinal methods may have the winner
change from A to B as a consequence of C entering the race.

-km

[1] Strictly speaking we would also want "honesty" to have some
connotation of "being the actual information being asked for". Say
you're dealing with a system that asks you what you like the least and
then gives it to you. You would answer its question with a thing that
you want to be given, so that answer stays the same no matter what other
people interacting with it would say. So by my definition that wouldn't
be strategy, but common-sense would say that it is not honest either.

Warren's definition of honesty is somewhat based on this idea, but it
goes too far in the other direction. It doesn't consider ballots where
your ranking stays the same as under honesty but your scores don't, as
being strategic. I think in part that's due to the difficulty in
comparing utilities, but we can hold rated voting to a higher
standard. Ask and I'll elaborate - this post is long enough :-)

[2] As a side note: we probably can make some observations of other
people's utilities even if we don't have a fixed scale. For instance, I
can probably reason that a candidate who would put you in a prison camp
would be a much worse choice from your perspective than one who would
arrange a party; and that the difference in utilities would be much
greater than say, between a candidate who holds a week-long party and
one who holds a two-week long party.

Some cardinal proponents also refer to von Neumann-Morgenstern utilities
as a way of making comparisons between people's strength of preference:
basically using lotteries to determine how much more a voter prefers one
choice to another. But such scales still need to be normalized because
they always have two unknown variables per voter. Methods that do the
normalization so as to give each voter the same strength fail IIA. Not
doing such renormalization can make the method pass IIA but they still
don't pass "de facto IIA".

See e.g.
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#Incomparability_between_agents
for the need for normalization.

Hello Rob,

The real significance of theorem Arrow is that it refuted a false view
of democraxcy as majority counting. A majority of half the votes or over
is only a first approximation to democracy, not democracy itself.
A comparable instance would be that the arithmetic mean is generally
considered as the standard average. The Michelson-Morley calculation was
based on an arithmetic mean of a light beam return journey, whereas the
geometric mean would have agreed with the M-M experiment. The AM is only
a first approximation to the necessary use of the geometric mean in
exponential rates of kinetic change, such as occur in the high energy
physics of special relativity, as distinct from (low energy) classical
physics. Indeed the supposed determinism of classical physics is merely
an artifact of finding variables which are not precise, as thought, but
only averages. Special relativity reveals that these supposedly
deterministic variables are really averages, because at high energies
they turn out to have ranges.

Like quantum physics, and indeed special relativity, if the truth were
known, election method is a matter of finding the most representative
averages to ranges of preference votes. I illustrated this with FAB STV:
Four Averages Binomial Single Transferable Vote.

https://www.smashwords.com/books/view/806030

Regards,

Richard Lung.

On 15/02/2024 07:13, Rob Lanphier wrote: