Approval discussion

[MIKE OSSIPOFF]

Richard said:

However, there is a difference between
extrapolating Approval voting Nash equilibria when all voted
preferences are known (and assuming voters decide in blocks)

I reply:

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.

As for all voted preferences being known, the equilibria that we've
been discussing are useful no matter what we assume voters know.
If a strategy-configuration & outcome can be improved on by some
voters, or if it can't be, that's relevant in itself.

Of course often voters would notice that they could improve on
the most recent outcome, by a different strategy, and would likely use
the different strategy next time.

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.

Richard continued:

, 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.

Richard continues:

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


I reply:

No doubt it's premature to say for sure what would happen. But
the fact that Approval & wv always have equilibria in which the CW
wins, and no one order-reverses, and the fact that Margins often
has only equilibria in which defensive order-reversal is used--
that says something about the methods. One thing for sure is, a
strategy-configuration and outcome that someone can improve on
is unstable, compared to an equilibrium.

On the next topic, if voters know what they're doing, the'll elect
the candidate at the voter-median, if there is a candidate there.

Richard continues;

Now on to voting strategy.

The following Approval strategies have some valid mathematical reasoning
behind them. Let u(i) be your utility for candidate i.

1. delta-p strategy: If delta-p(i,j) is the increase (positive or
negative) in the probability of candidate i winning caused by your
vote approving candidate j, then the strategic value of approving
candidate j is

s(j) = sum_over_i( delta-p(i,j)*u(i) )

Approve candidate j if s(j) > 0; disapprove if s(j) < 0; if s(j) = 0
then it doesn't matter. This is the only exact optimum strategy formula
I know of. The others are approximations. Of course, the delta-p values
are hard to know, which makes an approximate strategy desirable.

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.

This became evident when we began discussing the strategy difficulties
when there are few voters. I thought that then you knew that your
method suffers from the same problems (unless you ask an oracle what the 
delts-p values are--while you're at it, ask him what the best
candidate is :-)  ).

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.

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.

Richard continues:

2. p(i,j) strategy: If p(i,j) is the probability, given a two-way
first-place tie, that it will be between candidates i and j, then the
utility of approving j is

s(j) = sum_over_i( p(i,j)*(u(i)-u(j)) )

In http://groups.yahoo.com/group/election-methods-list/message/6706,
I showed this to be almost equivalent to the delta-p strategy. The
source of error (I think) is that this strategy only considers two-way
ties. For a large electorate, that error term will be exceedingly
small. Again, we have some values -- p(i,j) in this case -- that are
hard to know.

I reply

Again, saying that the delta-p approach is more error-free depends
on telling us how you can get your delta-p values without approximations, or 
in ways that are less approximate than the Pij
method's assumptions and its Pij estimates.

Richard continues:

3. geometric mean strategy: This is the approximation discussed by Mike
at http://www.barnsdle.demon.co.uk/vote/strat.html, wherein the p(i,j)
values are each replaced with sqrt( p(i)*p(j) ). So this method is an
approximation of an approximation.

I reply:

In public elections the assumption that ties are 2-way is much
more reliable than any of our probability assumptions, or even
our candidate-ratings guesses. In few-voter elections, your way
of determining your delta-p values used the same approximating
assumptions that the Pij approach uses.

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

As for Tideman's geometric mean suggestion for estimating the
Pij, of course it's an approximation. So is whatever way you'd
determine your delte-p values. Approximations will be needed for
any of these methods.

But geometric mean isn't really a separate strategy method. It's
a way of estimating the Pij for the Pij method. Tideman's Pij estimate
for Weber's strategy method.

Richard continues:

4. above-expected-utility strategy: Forest Simmons, Joe Weinstein, and
I have all mentioned this method in the last couple of months. The
expected utility of the election is

EU = sum_over_i( p(i)*u(i) )

and we vote for candidate j if u(j) > EU. This strategy can easily
translate into an intuitive strategy. In just about every election,
we all have some expectation going into the election about how much
we will like the outcome.

I reply:

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.

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.

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?

Richard continues:
Richard continues:

This of course goes against the conventional wisdom, that voters
should never reverse a preference in Approval.

I reply:

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.

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?

Mike Ossipoff



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