[EM] D2MAC can be much more efficient than Range Voting

[Abd ul-Rahman Lomax]

At 05:06 AM 3/7/2007, Jobst Heitzig wrote:
>it is frequently claimed that methods which involve randomness may be
>fairer than other methods but will give "worse" results.

Given that it couold appear that I have made that claim, let me be 
explicit that I have not. I have claimed something which could be 
misread as that claim, and the misreading shows, if this is what 
happened, that my actual claim has not been understood.

Of course, Jobst may simply be making a general statement and, in 
general, he will be correct that such claims, as he has stated them, are false.

My actual claim is that *ordinarily*, where the use of random choice 
allows a minority to prevail over a majority *without the consent of 
the majority*, introducing this randomness is introducing noise.

And "noise" is precisely the correct term. If we have an electronic 
decision-making system that depends on logic and/or pattern 
recognition to make choices, and we introduce into that system 
electronic noise that causes the built-in choice functions to be 
ignored, *under most conditions* this will degrade the performance of 
the system.

There *are* conditions where this is not true. For example, suppose 
there is a missile guidance system that is designed to vary the path 
of the missile in order to avoid attacks against it. If the path of 
this missile can be predicted, the avoidance can be defeated. But if 
the path is random, to a certain degree, it becomes much more 
difficult to predict where the missile will be when the intercepting 
missile reaches the incoming one.

But, of course, if *too much* randomness is introduced to the missile 
path, it will begin to lose enough fuel efficiency both through the 
fuel used for making corrections and the extra fuel consumed through 
increased flight time. There would be an optimum level of randomness.

Strictly speaking, this is not noise, because it is an intended part 
of the process. But typically noise generators will be used to make 
the random choices.

Looking at this missile system, there are two processes going on. 
There is a process which determines the optimum path to the target. 
And there is a process which randomly deviates from this in order to 
make the path sufficiently unpredictable. If not for this need to 
confuse countermeasures, randomness introduces into the process would 
be seen as noise and as undesirable if the missile is to fulfill its 
purpose in the best manner.

Put it another way: in the example, noise is counter-intelligent. It 
deviates from the normal intelligent path in order to defeat an enemy 
which has that normal intelligence.

Any society must be able to make decisions. What we call individuals 
are even tightly organized societies of cells. How do individuals 
make decisions?

In a situation where a sane individual would find it useful to toss a 
coin, then in analogous situations a sane society could introduce 
randomness to the process.

Note that there is already a level of noise. Factors which are 
irrelevant to the decision being made can nevertheless influence it, 
and there are host of these. All of these distractions are noise. If 
we could somehow filter them out without harm, we would. I hasten to 
add that such filters could be quite dangerous. FA/DP *does* filter 
out such noise, to a degree, but in a particular way that is not 
likely to be hazardous.


>Here's evidence for just the contrary:
>
>A typical voting situation is
>
>55%:  A>C>B
>45%:  B>C>A

Consider this election as a sincere Range election in a sane society. 
What has happened is that out of the infinite possible range of 
choices, these three choices have been preselected. Usually in such a 
situation truly bad choices have already been eliminated, and the 
difference between A and B, on some absolute scale, could be seen to 
be quite minor. What we have here could simply be a difference of 
opinion on, say, the optimum tax rate to set, not so low as to 
provide insufficient funding for necessary or desirable projects, and 
not so high as to injure the economy. In this example, the choices 
can be represented as numbers.

And we would not be surprised to note that numerically, A>C>B. The 
55% group generally leans toward higher social spending and thus 
higher taxes, and the 45% group towards the reverse.

The society, collectively, wants to set the tax rate at the optimum 
level such that the sum of the various effects of the tax rate is 
maximized (let's presume that all the effects can be quantified as 
dollar figures).

What is the rate to choose?

As the problem has been stated, there are only three possible rates. 
It can easily be that none of these would be optimal; from collective 
intelligence theory we would expect that the optimum rate is most 
likely to be the average of A and B, assuming that A would be the 
free choice of one group and B the free choice of the other. In fact, 
were this a Range election, A would be the maximum rate reasonably 
acceptable, and B the minimum rate necessary to fund reasonably 
desirable services. Some might set the rate at zero. A few, 
collectivists or true communists, might set the rate at 100%. So, in 
fact, the Range might be from 0% to 100%.

What is likely to be the ideal rate? My suggestion would be that the 
average of Range votes on this question would be optimal, though 
there may be more sophisticated methods of analysis.

None of them would involve the introduction of noise.

Let's see what Jobst does with this:

>with C being considered a good compromise by all voters
>(in the sense that all voters would definitely prefer C strongly
>to tossing a coin between A and B).

That would not necessarily be a normal condition, by the way. If C is 
located very close to A, those voters who prefer A would dislike a 
coin toss, we could expect, but the B supporters would prefer the 
coin toss. The condition described only holds where the sincere 
rating of C is the average of the sincere ratings of A and B by the 
respective A and B supporters.

What would make sense of the preference for C over a coin toss? If C 
is the average of A and B, a coin toss should have the same expected 
utility as C. Under these conditions, the "definite" preference 
described could not be likely. When would it be a reasonable 
condition? When A and B are sufficiently different that A is not only 
undesirable to the B supporters but actually unacceptable, and vice-versa.

We have a tendency to think of election methods in highly polarized 
situations, but those are situations where society is in trouble 
almost no matter what election method is used (though some may be 
likely to make things worse and others may tend to ameliorate the 
situation). We need to remember that election methods, in general, 
are merely methods for aggregating expressions of opinion or 
preference into a choice or set of choices.

So to understand the situation presented, it is polarized and 
charged, and it is generally recognized, in fact, that there is great 
danger in merely going ahead and implementing the preference of the 
majority. This situation is one in which the Majority Criterion will 
properly be violated. (And it is situations like this where the 
Majority Criterion will frequently suggest suboptimal results.)

If the decision is a minor one, a fully sincere Range election would 
show the difference between A and B as being small enough that, with 
the numbers given, a coin toss would, in fact, be reasonably 
acceptable to most.

However, that still leaves the question of what is the best choice. I 
don't think that tossing a coin is, in fact, optimal given the 
preferences described. It is introducing noise. Again let's see what 
Jobst does with this. (I'm writing this, by the way, as I read it, I 
have not read ahead. Readers are following, if they are following, my 
understanding process.)

>At first, it seems that this is exactly the situation where
>methods which claim to maximize "social utility" or "social benefit"
>should lead to the "right" answer C.

C is not necessarily the right answer. In fact, the example shows the 
deficiency of methods that don't collect preference strength 
information. I can presume from the stated conditions that C is the 
right answer, in general, but I also assume that circumstances can be 
shown where this is not true. It may be necessary to specify those 
circumstances.

>If all voters were sincere, indeed both Approval Voting and
>Range Voting will elect C, since the ballots would then
>look like this:
>
>Approval Voting:
>   55%: A,C
>   45%: B,C
>Winner: C

That is correct given the conditions stated.

>Range Voting:
>   55%: A 100, C 50+whatever, B 0
>   45%: B 100, C 50+whatever, A 0
>Winner: C

Remember, the votes have been normalized. But the conditions in fact 
would be most likely to represent a situation which is like this. 
Such votes would *not* be the general case in real elections with the 
preference order described. We must keep in mind that this is 
actually a special case, even though Jobst asserted that it was common.

The tipoff that it is a special case is the restriction regarding 
random choice between A and B, presumably with equal probability of 
each. And the rationality of that restriction (i.e., the consensus 
preference) is dependent upon more information about C than we have.

>The problem is, rational voters just *won't* vote sincere in this
>situation.

This is asserted. I claim that it is false. Voters afflicted with 
what I consider the disease of polarization and zero-sum political 
games would behave as Jobst expects. Voters in a non-coerced society, 
where the general expectation is that collective intelligence is 
superior to individual intelligence (*for collective choices, not 
necessarily for individual ones*), where the unity of society is 
valued and it is understand that peak expectation of utility is 
likely to be associated with actual realization of benefit (the goal 
of the whole process!), voters will simply vote what they think best, 
and they will not modify this in order to "get what they want," 
because what they actually want is the best choice *and they 
understand that their own opinion will generally deviate from this*.

The irrationality underneath what Jobst considers rational is an 
assumption that *my* opinions are superior to those of others, and, 
in particular, my opinions are superior to an efficiently and fairly 
aggregated opinion of the society collectively.

There is a legend that large groups of people can fairly accurately 
estimate the number of pennies in a large jar by each guessing and 
the averaging the guesses. I'm not sure that this legend is true in 
detail, but I would suggest that the average error of the averaged 
guess, over many such trials, would be less than the average error of 
the individual guesses.

And, in fact, for the large majority of people, they will see a more 
accurate result if the average guess of all participants is used 
rather than their personal guess.

Now, if the people want an accurate result, they will simply vote 
their personal estimate, using whatever intuition or method is 
available to them.

There will be a few of them who are "penny-guessing experts," and for 
this group the individual estimate might be more accurate than the 
averaged guess. Perhaps the expert is autistic. Problem is, it can be 
quite difficult to discover who these experts are, and, in 
democracies, we have decided to assume that people are equally 
capable of making choices accurately.

And, in fact, the results of the average guess and the expert guess 
are quite likely to be close to each other. So even the expert would 
not rationally be disturbed by the use of the average guess unless 
for some reason the problem is crucial and a very slight error 
disastrous. Perhaps this is a lottery being conducted by invading and 
powerful aliens....

Not exactly common!

I really think this needs to be understood. What I see as an error of 
personal arrogance is a very common assumption, I have seen it 
asserted many times that this behavior is "rational." It is not. It 
is irrational. Being irrational does not mean that it won't happen! 
It will, in a society which is structured to encourage such behavior.

The strategic behavior would be the equivalent of someone who 
considers himself an expert, and who expects the guess of the 
generality to be X, while he thinks it to be Y, voting, instead of Y, 
whatever value is optimal for this voter to produce an average as 
close to Y as possible. Usually this would be maximum in the opposite 
direction of the perceived error.

Yet for this behavior to be rational, this voter must expect not only 
to be correct about the pennies (in that example) but to be correct 
about the guess of the generality, which might be a set of abilities 
not likely to coincide in the same person! (The autistic expert may 
be quite likely to accomplish the first task and quite unlikely to 
accomplish the second.) If there is error in the second guess, this 
expert, who could have added weight to his expert choice, will 
instead cause increased error.

However, I'd leave the judgement of the two relative abilities to the 
individual voter. Voters *can* distort their votes like this where 
they deem it advisable. *It is all part of the process by which a 
society aggregates opinions.*


>  And since Approval and Range Voting are *majoritarian*
>methods, the real outcome will rather look like this:
>
>Approval Voting:
>   55%: A
>   45%: B,C
>Winner: A

If voters behave in that way, which is known to be irrational. 
Remember, I've noted that if there is an irrational majority, society 
is in deep trouble and it could be very difficult to get out of it. 
The majority can change the rules!

(It ain't easy being paranoid!)



>Range Voting:
>   55%: A 100, C 0, B 0
>   45%: B 100, C 99, A 0
>Winner: A
>
>So, both these methods *fail* to do just what they were apparently
>constructed to do!

No, what has happened is that a majority has exercised its weight and 
has refused to *use* Range Voting, instead they have clearly 
abandoned any attempt to find consensus. They have the power to do 
this, if they are organized, no matter what election method is set 
up, because they can simply ignore it, change the rules, whatever.

And the conditions stated are contradictory. Remember, it was said 
that all voters agreed that C was superior to a coin toss. Yet a coin 
toss in the context described will produce a better expected outcome 
than no coin toss. So an irrational condition has been imposed. 
Garbage in, garbage out.

>Now let us look how D2MAC performs in this typical situation,
>a democratic, non-majoritarian method:
>
>Recall that in D2MAC you specify a favourite and as many "also approved"
>options as you want. Then two ballots are drawn and the winner is the
>most approved option amoung those that are approved on both ballots
>(if such an option exists), or else the favourite option of the first
>ballot.
>
>If voters are sincere, the result will be this:
>   55%: favourite A, also approved C
>   45%: favourite B, also approved C
>Winner: C

Look, in contrast to a situation where insincere votes are presumed, 
we now are going to see how D2MAC performs with sincere votes. Is 
this a fair comparison?

Set up the same condition with majoritarian voters voting 
strategically, the same results. The majority prevails with its 
preference. Sauce for the goose is sauce for the gander.

>Can the A-faction improve their result upon this by voting differently?
>
>If they switch to
>   55%: favourite A, none also approved
>then A will win whenever an A-ballot is drawn first, i.e., with 55%
>probability. However, B will win in the remaining cases, i.e., with 45%
>probability. For the A-supporters, this "almost coin tossing" is not
>preferable to C, so the strategy won't help them.
>
>Thus, the "obvious" A-strategy cannot destroy the compromise under D2MAC!

But the A voters have the power to alter the rules. This is something 
that is totally ignored in analyses like this. It is as if election 
methods arise, lotus-born, or are imposed by some benevolent 
dictator. And as if they cannot be changed.

Remember, the conditions were that all voters agreed that C was 
preferable to a coin toss. Yet here, the method is, to some degree, a 
coin toss. Therefore all voters would agree that this election method 
sucks big. And if all voters agree on a change, even if the rule is 
written into the constitution, what constitution in the world would 
not allow the change?

Given the conflicting assumptions, I'm not taking this one further.

With two of the assumptions, Jobst wrote his desired outcome. Those 
two were the coin-toss rejection by consensus, and the assumption 
that all voters will vote insincerely (incorporating an assumption 
that this is rational even though it clearly leads to a suboptimal result).

There are conditions where such a choice to vote strategically may be 
rational. They are not the conditions of the example, though, they 
conflict with the coin-toss agreement.