[EM] Pizza, Utilities, and Majority Criterion

Abd ul-Rahman Lomax abd at lomaxdesign.com
Wed Mar 7 11:18:01 PST 2007

At 12:22 PM 3/7/2007, David Cary wrote:
>A small group of 3-8 people are ready to eat pizza and are faced with
>making a choice of exactly 1 flavor of pizza to collectively buy.
>For the sake of simplicity, I consider the case where there are only
>two flavors to choose from:  pepperoni and  vegetarian.  One person
>has a strong preference against pepperoni, either for reasons of
>personal taste, severe allergies, or religious prohibitions.
>Everyone else has a slight preference for pepperoni.
>   S2. Choosing the pizza flavor with Range Voting will result in
>choosing vegetarian. For example, the one person with the
>pepperoni/vegetarian preference will vote 0/99, while the others will
>vote their weaker preference, say 70/60.

This is a stipulation rather than an assured result of Range Voting, 
Mr. Cary has quite correctly presented it as a stipulation. If there 
were enough pepperoni preferers a Range poll would not select 
vegetarian. However, a deliberative process *might*. It depends on 
how important it is to the electorate to satisfy *everyone*. Some 
groups actually do value this, when it is possible.

Fortunately, with very large groups, it would be quite rare that such 
a narrowly restricted choice is necessary. If you've got thousands of 
people, one pizza isn't going to go very far anyway!

Another point that I'd like to make: the pizza example shows that 
there are election contexts where the Majority Criterion leads to 
results that would generally be agreed are poor. Pizza is a simple 
code word for this. But there can be very serious political 
situations where satisfying the Majority Criterion will lead to a 
poor, dangerous, or even disastrous result. *And the majority would 
agree to this.*

(If not, then the society is *really* in trouble and it may not be 
remediable. If I were a member of the minority in such a society in 
difficulty, I'd consider it an emergency to leave, as soon as possible.)

>S3 is consistent with the group having a high degree of cohesiveness
>and/or having goals that both take precedence over the choice of
>pizza flavor and are facilitated by inclusiveness.  But some other
>possible scenarios that allow an alternate stipulation include:
>   -- The group has been eating vegetarian pizza as a group every
>Wednesday night for the last 6 months and it's only fair try
>pepperoni this time;

Some groups would not consider this "fair." But, once again, 
remember, I argue that the majority properly has the right of 
decision. This is not the same thing as satisfying the Majority 
Criterion in a single-stage election method. The majority can decide 
that their preference for pepperoni is important enough to 
occasionally be satisfied. There will be a little pizza left over. 
Maybe they could sell it and give the proceeds to the poor 
vegetarian, who might otherwise go hungry that night.

The majority *can* decide. But, remember, I consider that "majority 
rule" properly applies to a single question presented to an 
electorate which has been informed through deliberation or other 
means. The exact form of the question is itself subject to majority 
rule, and there is standard democratic process for this.

Problems arise in trying to compress this into a single stage 
election method....

>   -- Everybody is rather selfish and/or insensitive to others, but
>also easily irritated.  Come what may, they are going to be spending
>the whole night in close quarters working on an project with a
>looming deadline;
>   -- The boss wants pepperoni, the boss gets pepperoni, and where
>possible, everyone else wants what the same thing the boss is having;
>   -- The overriding group ethic is that if people can't fit in, they
>can and should leave.  The pepperoni-hating person prefers to
>reinforce that ethic because it is advantageous in other more
>important situations.

By the way, in some of these scenarios, does this sound like a 
society that you would want to be a member of? If you were not coerced?

>So under the alternative stipulations, Range Voting based on ZSI
>would then select the ideally worse option, and any MC-compliant
>election method, even plurality, would select the ideally better
>option.  A broader lesson is that with individual ZSI preferences, no
>specific election method can always produce the ideal social choice.

That's correct. It is quite true, the incorrect statement sometimes 
made from Arrow's Theorem, that he proved there was no perfect 
election method. Range may be much better from a social utility point 
of view, but there are certainly scenarios where it will fail.

Now, how does a group intelligently make decisions? Does it use an 
election method at all (other than, perhaps, majority rule on Yes/No 
questions, after deliberation which includes determining the form of 
the question)?

Groups will use election methods when efficiency demands it, and 
efficiency is more important, in the context, than making the ideal 
choice. Election methods are a short-cut.

My claim is that Range is *very* good as general purpose *input* to 
deliberative process. But that deliberative process is generally 
superior in every respect except the time it can take to use it.

(And there are solutions to that efficiency problem! If you don't 
know what they could be, then you haven't been reading the tomes I've 
been writing here for the past years. Don't blame you! But you *have* 
been missing something. Perhaps someone more succinct than I will 
summarize it.)

>It appears that the discrepancies are not so much caused by one or
>the other election method or by the Majority Criterion, but by the
>fact that the elections were conducted with the same extreme deficit
>of information, ZSI.

Yes. Absolutely. If you have full deliberative process *before* an 
election, the Majority Criterion is entirely appropriate, in my 
opinion, though some would argue for Range. (Because the majority, if 
that is what they want, may prevail with Range.) It becomes much less 
important a question, what method to use.

I really want to thank Mr. Cary for bringing up this point, that the 
central problem is the deficit of information. It's another slant on 
what I've been saying.

>   What this pizza example really demonstrates is
>the problems created by conducting elections without the voters being
>well-informed, without them being informed about the preferences of
>others and about the consequences of the various options.


>4.3. Was Range Voting Rigged?
>But were the two elections really conducted with the same available
>set of information?
>We know that it is either arbitrary or nonsense to simply make
>interpersonal comparisons or summations of von Neumann-Morgenstern
>utility values for a given option.   Similarly, it is either
>arbitrary or nonsense to simply make interpersonal comparisons and
>summations of the difference in utility values for two options.
>That's because utility functions can be shifted by a value and scaled
>by a positive value and still remain equivalent but give conflicting
>results for interpersonal comparisons and summations.
>So how do some people know  to give their range vote a full 99 point
>range or a narrower 10 point range.  Under ZSI, a person can't know
>that their preferences are weak or strong compared to another person.

This problem is one that I've mentioned with respect to Range. We 
generally assume that voters will rationally vote extremes, but this 
is in the polarized context of elections. With pizza, would I?

No, I'd probably not give a zero to any pizza that I could enjoy, and 
I would probably give my favorite 100%. You would see, from me, 
however, one zero, pepperoni and anything with pork in it. I am, 
after all, a Muslim. I wouldn't eat the pizza. If I was starving, I'd 
take the pepperoni off and eat what was left with distaste. And if I 
was *really* starving, I'd eat the whole thing. Such is actually 
explicitly allowed in the Qur'an, and is actually obligatory.

But this is worth a zero.

But if the election were between pizzas that I could accept, I would 
not vote the minimum.

>  Why doesn't the anti-pepperoni person vote with a 5 point range and
>save the 99 point range for life and death choices?  One possibility
>is that group members aren't really basing their range votes simply
>on their ZSI preferences.  It is unclear, though, exactly what
>additional information they might be using, and how.

That's right. They do know something about pizza choices and general 
satisfaction. The vegetarian knows that there will be other choices 
acceptable (a single choice in this case) to others, knows that the 
pepperoni is really totally unacceptable, and votes quite sincerely 
at 0 and 100 even though it isn't a life or death choice. If the 
choice were between death, going hungry, and a good pizza, *then* 
reserving zero for death would make sense.

Range is normalized, but not necessarily completely. Depends.

>   Another
>possibility is that the Range vote ranges are a contrivance
>introduced by the originator of the example to produce a certain
>result, but which don't reflect a necessary, realistic outcome, given
>the other circumstances.  Either way, the comparison of the range
>election and the MC-compliant election, even under S3 and S4, are not
>fairly comparable.

It's tricky. I've seen real examples where the Majority Criterion 
would have indicated one choice, in an immediate election, Range 
might have come up with the same, though would have made another, 
better choice, more possible. Approval in the case I have in mind 
*did* select what was later unanimously agreed to be the optimum 
choice. But it easily could have been different. This was a group 
that particularly valued unity and unanimity and so it was inclined 
to be somewhat lenient in Approval Voting. In any case, the Approval 
poll stimulated the introduction of a motion to adopt the clear 
Approval winner (that winner had all but one vote), and the motion 
was adopted unanimously. Including the holdout on the Approval Vote.

This experience points out how Condorcet results, for example, don't 
predict what would happen with a top-two runoff. People can change 
their minds once they know how others feel. Range could have been 
used for that initial poll, and I think the same result would have resulted.

Once again, this has been my general recommendation: where possible, 
where a Range winner does not satisfy the Majority Criterion -- and 
possibly in other circumstances as well -- the result should be 
ratified. In fact, I'd recommend that *all* elections be ratified. No 
election result should be accepted without majority consent.

*That* is majority rule!

But it is crucial that this majority be informed, and it can't be 
done without an accurate poll *and* deliberation. Public debate in 
elections does substitute for this to a degree, but imperfectly.

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