[EM] Approval Voting vs Instant Runoff Voting:

[Richard Moore]

I previously wrote, "If my utility scores for the candidates are spread evenly
(e.g., 100-75-50-25), I might consider data from the polls secondarily."

There is one problem with that statement which I'll address at the end of my
reply
to Craig, below.

 -- Richard

LAYTON Craig wrote:

> I'm not necessarily looking for a mathematical solution.  The point is that,
> even given a significant amount of information on which to base one's vote,
> it is still difficult to divine how to vote in approval.  So far, two
> people, both highly intelligent and with a thorough understanding of voting
> systems, have decided to vote in two separate ways, given the same
> information, preferences and expected utility outcomes.  I suspect the rest
> of the list would be similarly divided, if not the populace as a whole.

I maintain that I would prefer to use a zero-information strategy. Others may
not make
the same choices and that is OK with me. But for the reasons I stated
previously,
I reject the premise that I can place enough confidence in the poll results to
change my
strategy. Accepting the premise may lead to a different conclusion.

If I accept the premise that the poll is useful, then I would first want to
calculate the
initial probability of wins by each of the four candidate. Statistics provides
the means
to do this. The assumption is that any deviation of election results from the
poll is due
to random effects. Next I would determine the relative effects of voting A, AB,
and
ABC on those probabilities. I don't need the absolute effect of each vote, I
only need
to know, for each candidate, the ratio of the effect of voting AB on that
candidate's
chances to the effect of voting A only. I could then calculate the incremental
utility of
each possible vote as a dot product of the relative probabilities and the
utilities. I
would then select the combination for which the incremental utility is the
highest. A
lot of work but it seems feasible. Actually, just knowing that a solution exists
is enough
for most folks; they will most likely use a neural network to find their answer.

> If people, given exactly the same utilities, cast totally different approval
> ballots, then I don't see how Approval can maximise voter utility in any
> even handed manner.  Any utility (in a broader sense) advantage Approval has
> over Condorcet is more than countenanced by the fact that it doesn't give
> votes equal power.  This is also true, of course, if someone votes
> insincerely (and non-strategically) or truncates their vote in a Condorcet
> election, but this inequality is the fault of the voter.  The inequality in
> Approval is the fault of the voting system.

Approval tends to maximize aggregate voter utility. It cannot do so without
forcing some compromises. The only inequality built into the system that I
can see is that it does tend to help the voters who have the most at stake.
But this is arguably not anymore unfair than the fact that those who are
apathetic won't vote at all.

> If A is your favourite, B is your lesser of two evils candidate and C is the
> greater of two evils candidate, you cannot express your A>B preference as
> well as your B>C preference.  You have to pick one, and if you make the
> wrong choice, your candidate can lose.  In a Condorcet election, there can
> be strategies involved, but they are complex and diffuse enough so that,
> without highly detailed information, it is safest to number all your
> preferences in sincere order.  The only choice you have to make in making
> your vote as effective as is possible, is who you like the best.

No, you cannot simultaneously express all preferences, but I was challenging
your
statement that you can only express one preference. Either voting A over both B
and C, or voting both A and B over C, expresses two.

Now my earlier mistake: A zero-info (ZI) strategy for approval voting does not
break even
when utilities are equally-spaced, as I had assumed. By that I mean that I do
not find all
my choices (A, AB, or ABC) to be equal-valued; there still is an optimum ZI
strategy:
Vote for the top half of the candidates. So I might not switch from ZI under
these conditions
after all.

So in general, what is the optimum ZI strategy?

Given that I will vote for A no matter what, what is the additional utility of a
vote for B?
The vote for B increase the probability of B getting elected by a small delta,
and decreases
the probabilites for A, C, and D, each by delta/3 (remember, outside information
is
disregarded). Thus the utility gained by adding a B vote to my A vote is

    ub - (ua+uc+ud)/3

where ua, ub, uc, and ud are the utilities of the four candidates. The
break-even point
occurs when ub is equal to (ua+uc+ud)/3. For the values in your example, keeping

ua, uc, and ud constant, I break even by adding a B vote when ub = 40.

This means I would vote AB for utilities 100, 41, 20, and 0, even though the
utility gap between A and B is greater than the utility gap between B and C.
Since B's
utility in your example is only 25, my ZI strategy still dictates a vote for A
only.

But a threshold of 40 indicates that I should be more willing to vote a second
choice
than I had previously thought. This of course contradicts assertions that people
will
have a strong tendency to cast bullet votes in an approval election. What they
will
do depends on the utility distribution. If there is one really bad candidate and
three
good to excellent candidates, they will cast an anti-bullet vote against the bad
candidate
by selecting all three good ones.

The result of such a strategy would be that if my second choice is more likely
to win
than my first choice, the break-even point shifts upward. Thus, I would be more
likely to vote for a front runner who was not my first choice if I didn't use
ZI.

Note that this disproves Don's assertion that approval "subsidizes lower
candidates".
If the ZI strategy were applied by all voters, then the outcome of the election
would not
be biased toward either "front runners" or "lower candidates". It would be based
on
utilities only. Applying a non-ZI strategy to approval does shift the process
from
popularity-neutral toward the front runners. Don had it backwards.

Here's one more thought. Suppose instead of approval we were using a variation
of Borda
that allows truncated votes. Might it be possible to find a set of weights for
this modified
Borda count, such that the break-even point for ZI strategy occurs when utility
gaps are
equal? Such a system might provide voters with a more intuitive ZI strategy. The
weights
on the ordered votes for each voter would have to shift based on how many
choices are
voted by that voter (e.g., 1-0-0-0 for a single choice, 1.4-0.6-0-0 for two
choices, and so
on). Allowing truncation in a non-Condorcet ranked voting system strikes me as
an
improvement over the non-truncated version of the system, because by truncating
I don't
have to worry about one of my lower rankings of a candidate I don't like being
aliased into
a vote for that candidate. (This truncated adaptive weighting system has one big
drawback:
voters using a non-ZI strategy might have even more motivation to vote
insincerely than in
straight Borda).

 -- Richard