[EM] Pizza, Utilities, and Majority Criterion

David Cary dcarysysb at yahoo.com
Wed Mar 7 09:22:24 PST 2007

The pizza example is worth reviewing with Jobst's descriptions of
individual and social utilities in mind.

Due to the length of this post, here is an outline:

        1.1. Original Conclusions
        1.2. Alternative Lessons
        3.1. Base Circumstances
        3.2. Additional Stipulations
        4.1. ZSI vs. CSI
        4.2. Alternative CSI Preferences, Ideal Social Utilities
        4.3. Was Range Voting Rigged?
        4.4. Maximizing Social Utility


1.1. Original Conclusions

The pizza example was originally in support of the following

C1.  The Majority Criterion is clearly flawed because it requires bad
election results.
C2.  Range voting is superior to every Majority Criterion  (MC)
compliant election method, primarily a result of Range voting
considering individual preference strength and aggregating those
strengths across voters.

1.2. Alternative Lessons

However, a careful analysis of the example offers some alternative
lessons, including:

A1.  Sometimes, even Plurality Voting delivers better election
results than Range Voting.
A2.  The example mostly demonstrates the value of having a
well-informed electorate rather than distinguishing the purported
relative merits of one election method over another, or one election
criterion over another.  
A3. Analysis of the pizza example highlights the results of two
extremes: preferences formed with zero social information (ZSI) and
preferences formed with complete social information (CSI).
A4.  The outcome of the Range vote may be artificially contrived and
might not necessarily reflect the realistic results based on the
other circumstances.
A5.  The pizza example is, if not rare, at least exceptional:  for a
specific set of determining circumstances, there is a (nearly)
universal or at least unanimous agreement about what is the ideally
best social choice.  Finding and justifying an “ideal” utility
function in other circumstances may be an issue that keeps social
utility analysis from being universally and definitively applied.
A6.  Anyone who uses social utility analysis to evaluate election
methods against a standard or ideal utility function, carries the
burden of specifying the standard as well as justifying its use.


Jobst's recent essay described a kind of utility that Wikipedia calls
von Neumann-Morgenstern utilities.  That is the kind of utility
discussed here.  Every set of  individual preferences are assumed to
be representable by a utility function.

Social utility analysis evaluates various options against some
standard or ideal utility function.  In the pizza example, a
stipulated universal or near universal agreement on CSI  preferences
is used to justify using those preferences as the standard.  At least
for the time being, I'm ignoring any distinction between an
individual's CSI  preferences and the preferences an individual
considers to be the ideal social preferences.


3.1 Base Circumstances

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.

3.2 Additional Stipulations

It is further stipulated that:
  S1. Choosing the pizza flavor with a Majority Criterion compliant
election method, such as simple plurality, will result in choosing
  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.
  S3. Everyone (at least nearly every reasonable person) agrees that
choosing vegetarian is the choice that ought to be made.
  S4. Everyone (at least nearly every reasonable person) agrees that
choosing vegetarian is the choice with the higher ideal social


4.1. ZSI vs. CSI

If everyone agrees that choosing vegetarian is the better ideal
social choice, that highlights that there are really two distinct
sets of preferences involved:  the zero social information (ZSI)
preferences used for voting and the complete social information
preferences (CSI) used to establish an ideal social choice.  

The ZSI preferences, or socially isolated preferences, for
individuals reflect the choice each person would make without any
information about the preferences of others, the direct or indirect
consequences on others, or indirect consequences to the individual by
others.  The CSI preferences, or fully socially aware preferences,
for individuals reflect the choices each person would make while
rationally taking into account complete information about the
preferences of others and about the direct and indirect consequences
on everyone and everything.

ZSI social preferences correspond to the choices the group would make
with its members acting on their individual ZSI preferences. 
Similarly for CSI social preferences.  Of course the ZSI and CSI
social preferences do not depend solely on the respective individual
preferences in the group.  They can also depend on the actual
decision making process the group uses.

4.2. Alternative CSI Preferences, Ideal Social Utilities

S3 and S4 really have to be stipulations.  They are not reasonable
conclusions from the base circumstances, S1, and S2.  In particular,
the base circumstances, S1, and S2 can be augmented in different ways
so either pepperoni or vegetarian is the individual CSI preferred
choice on a (near) universal basis, or at least unanimously within
the group.  After all, the group choice is about what flavor to
purchase, not whether to feed it, by force if necessary, to every
member.  As with many elections, there are options for each
individual to adapt to and mitigate any adverse impact arising from
the group choice.

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; 
  -- 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.

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.

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.  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.
 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.  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.

4.4. Maximizing Social Utility

Maximizing social utility really means selecting the option that has
the highest utility value for some “ideal” or standard utility
function, not the actual social utility function.  Of course, when
explicitly stated this way, it highlights the dilemma of identifying
and justifying a particular ideal or standard utility function.

In the pizza example there is a stipulated unanimous set of
individual CSI preferences.  This unanimity is sufficient for many
people to accept those preferences as the ideal or standard.  In that
case, most commonly considered election methods using individual CSI
preferences will also pick the option with the highest ideal utility.
 But election methods need to be evaluated in other circumstances,
where there is a diversity of individual CSI preferences.  To the
extent that no utility function can be justified as the standard,
social utility analysis falters, just when there is actually a job to
be done.

-- David Cary

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