Approval discussion

[Richard Moore]

MIKE OSSIPOFF wrote:
> Sometimes a number of same-voting voters change their strategy
> in the same way, or in a few same ways. Writing definitions that
> way, then, doesn't seem unrealistic or unreasonable. After all,
> when only one voter changes his strategy, there's rarely any result.

I'm not disagreeing with the idea of defining concepts in terms of
deterministic strategies and/or block strategies, even though actual
voters will not use those strategies. But then we are no longer
talking about what the best strategy is based on information known
to an individual at voting time, but rather about what would have
been the best strategy for a block of voters had they known in
advance the actual ballots that would be cast by the other blocks.

Such definitions might be useful for purposes of determining a
method's potential for causing regret. They also give us the
tie-in to Condorcet.

 > As you said, a group of same-voting, same-utilities voters can
 > sometimes benefit by dividing their vote between 2 or more strategies.
 > But that could be considered just one strategy--one probabilistic
 > strategy that they all use. "Flip a coin; if it's heads, use strategy1,
 > and if it's tails use strategy2." Of course it's been pointed out that
 > it won't be so easy to get people to vote based on flipping a coin.

It is interesting that your equilibrium-based definitions don't
really require voter utilities but only the ranked values, since
these definitions relate to whether a group of voters could have
gotten a better (for them) result than the actual result, which
only requires a knowledge of relative rankings. Probabilistic
strategies for Approval necessarily require utilities.

> , and
> counting actual Approval votes cast by individual voters with their
> assorted fine-grained utilities and probability estimates.
> 
> I reply:
> 
> It would be interesting to consider a game in which the voters
> have probability-info about eachother's preferences, or maybe about
> eachother's votes, or ties & near-ties. Maybe someone here
> is familiar with game theory enough to study Approval, IRV, wv,
> Margins, etc. in that way.

I thought about this some but don't have the time to study it in
depth. Plus I'm not an expert in probability and statistics.

> Richard continues:
> 
> So
> real-world Approval voting likely won't always converge to a stable
> Nash equilibrium.

Well, to be precise, I should have said that real-world Approval
voting won't have the same equilibria that are found modeling the
election with the block-vote, deterministic method. So if it finds
an equilibrium, it won't be what that model predicts.

However, if there are multiple equilibrium points, then unless
the voters (or voter blocks) know ahead of time which point the
method will converge on, they do not know which strategy to play.
Which incidentally leads back to probabilistic strategies. BTW,
unless I'm mistaken, those dice-rolling strategies also have
Nash equilibria, in the sense that each player has an optimum
way of setting the probabilities of selecting each of his
possible strategies.

> Richard continues;

<snip>

> I reply:
> 
> Yes, the delta-p are difficult to know, and if they're difficult enough
> to know, that would make that method the less useful one.
> 
> You say, as you said when this was discussed before, that yours is
> the only exact optimum strategy, because the others are approximations.
> 
> But when I asked you how you'd determine the delta-p values, you
> either derived them from the Pij, or assumed that all ties are
> 2-way ties. Your way of determining your delta-p values relied on
> the same approximations as the methods that you say are more approximate.

In theory they could be derived if the probability distribution functions
are known. One might assume a particular pdf, and work backwords from the
poll data to determine the parameters (mean and standard deviation) that
would give the exact curves for the assumed pdf. The probabilites could
then be found from these curves with a bit of calculus. Not an easy
task, and the actual pdf might differ from the assumed one (influenced
by such things as the sampling methodology used in the poll, I suppose).
But every strategy method I listed here requires some assumptions. So I
was only making a statement about what could be done in principle, rather
than making a practical recommendation.

> This is illustrated by the fact that you yourself agreed that none
> of us have a few-voters strategy even for the relatively simple
> 0-info case. You described several rough approximations for 0-info
> strategy. 0-info allows us to regard many terms as equal, and that's a 
> simplification. If you
> don't have a precise 0-info strategy, you don't have a precise
> probability-info strategy either. And you said that you don't have
> a precise 0-info strategy when voters are few.

Again, in principle, the few-voters strategy should be possible to
calculate. It would make an interesting paper if somebody (with a
stronger statistics background than I have) would write it. I doubt
anybody would use it in an actual election, but it could be useful
for modeling purposes.

> But if you're sure that more useful results can be gotten by
> your strategy than by the Pij strategy, then I hope that you'll
> notify professors Weber & Merrill.

As I noted, the difference in large elections is too small to even
consider.

> By the way, would you re-post your suggestion for estimating the
> delta-p values?

I don't have it written down but you're welcome to search the
archives (roughly a year ago, if my memory serves me). It was
based on the voter being able to estimate the total number of
"approve" votes cast for all candidates (excluding that voter's
ballot), and noting that if this number is zero, then once that
voter approves his favorite he need not include any more
candidates. I then chose a fraction (with a linear expression
in numerator and denominator) that would give this result, as
well as give the result of the above-mean strategy when the
number of "approve" votes is large. I.e., for a utility range
from 0 to 100, with a mean utility for all the candidates of M,
if we estimate that there will be N "yes" votes,

	f(N)/g(N) = 100 if N = 0,
	
	lim(N->infinity) [f(N)/g(N)] = M

so if f(N) = aN + b, and g(N) = cN + d, then b/d = 100 and
a/c = M. Our fraction then equals

	(cMN + 100d)/(cN + d)

Unless we can assign a ratio c/d, this doesn't completely
specify the threshold. I don't remember what I assumed for
that ratio before but for example if we use c/d = 1 then
this becomes

	(MN + 100)/(N + 1)

which seems close to the formula I remember posting, but
it might not be exactly the same. (It might also have been
something like M + (100-M)/(N + 1), which has similar
properties.)

If you ask me to justify a particular ratio for c/d, sorry,
I can't. If we assume there is one other "yes" vote besides
ours, and solve that case for the missing ratio, we would
probably find that the threshold varies according to how our
utilities are distributed, which means there really should
be more terms in the fraction.

Regarding the above-expected utility strategy:

> That approach seems convincing, but, unless I made an error (I
> didn't recheck all the calculation), it gives an Approval cutoff-utility
> that doesn't look anything like the one that can be derived from
> the Weber-Tideman approach, using the same winning probabilities and
> candidate utilities.

It should match the delta-P method exactly *if* the ballot
distribution is such that, if my vote for X increases X's chances
by some amount, the resulting decreases in the other candidates'
chances is distributed in a way that is proportional to those
candidates' winning probabilities. That represents a very special
condition, however.

I don't expect it to match the geometric mean approach. Both make
simplifying assumptions in order to arrive at an approximation
but the assumptions aren't the same.

> I'm not claiming that one of those 2 procedures is better, only
> that they seem to disagree. Does the expected utility approach have
> the obvious motivating principle that the Pij has? Just a question,
> not a judgement.

It has the merit of being very intuitive. If you expect very low
utility to result from an election, you will approve more compromise
choices. If you expect very high utility from the outcome, then you
will have less need of compromise.

Its mathematical justification can be paraphrased by saying that
if your statistical expectation of the outcome (however you arrive
at that number) is E, then any candidate higher than E would
represent an improvement for you, so you can improve your expectation
(which is what probabilistic strategies are all about) by helping
that candidate. The pitfall is that if votes for that candidate are
strongly correlated with votes for your favorite, then you might
be helping that candidate defeat your favorite, lowering your
expected outcome utility instead of raising it. So it is an inexact
strategy. The geometric mean approximation falls into the same trap;
I can't say which of these two methods would fall into that trap
more often.

> Richard continues:
> 
> Like strategy #3, strategy #4 has underlying assumptions about the
> distribution of ballots. Those assumptions can sometimes lead to
> problems. Here is a (very contrived) example:
> 
> Suppose, in a four-way election, your utilites are A(100), B(60), C(40),
> D(0). Strategies 3 and 4 would both fail in the following scenario.
> 
> x1: AB
> x2: CD
> x3: A
> x4: B
> x5: C
> x6: D
> 
> I reply:
> 
> What do x1, x2, etc. stand for? Which strategy method is that?

Those are the numbers of voters casting each type of ballot. We don't
know the exact numbers so we represent them with variables. We do
know that A and B voters have a strong tendency to align with each
other, and the same goes for the C and D voters. So there are no
AC, AD, BC, or BD voters in this example.

> It's known that combinations of Pij and Ui can be written for which
> the voter's utility-expectation-maximizing strategy involves
> "skipping", not voting for a candidate though you vote for someone
> whom you like less.
> 
> But of course that never means not voting for one's favorite.

Of course, you are right about that. You only skip candidates if a
"yes" for that candidate can lower your outcome expectation, and a
"yes" for your favorite can't ever do that.

> And I believe it was Merrill who quoted another author as pointing
> out that situations that cause skipping are so implausible and
> unlikely that the possibility can be ignored for practical purposes.
> Bart, do you have that quote?

In fact I had a different example that had uniform distributions for
x1 and x2, but it occurred to me that if x1 and x2 are far apart, it
means a tie is less likely than when x1 and x2 are closer than the
differences between x3, x4, x5, and x6, so I couldn't determine if
that case was indeed a legitimate example -- maybe ties between
A and C or B and D would be likely (relative to the likelihood of
AB or CD ties). That's why I went to this example, where x1 would
take one of two point values that straddle a single point value for
x2 by a wide margin. A contrived case, as I pointed out. There are
surely other contrivances that would lead to skipping.

I don't know about the likelihood of this happening in the real
world, but would be interested in what Merrill, or whoever made the
quote you are referencing, has to say.

  -- Richard