[EM] Re: Tyranny of the Majority

[Richard Moore]

DEMOREP1 at aol.com wrote:

>> Mr. Harper wrote in part-
>>
>> 100 A >> B > C
>> 100 C >> B > A
>> 1 B > A = C
>>
<snip>

> Since none of them gets a YES majority, then none of them should be
> elected
> (even if there was a Condorcet Winner).
>
While I don't agree with Demorep's statement, it did get me thinking
along the following
lines:

Suppose the above election is thrown out, in accordance with Demorep's
wishes.
A new election is held. This one is between X, Y, and Z. Let's say X
wins a "YES
majority".

Can we test whether this choice represents the will of the people?
Suppose after
the election, we ask the voters whether they would have preferred X to
A, B, or C.
It is actually somewhat likely that a majority will prefer at least one
of the original
three candidates to X. It's even possible that all three will be
preferred to X by a
majority. So much for majority rule.

Now, carry this thinking a little farther. What if X had been drafted at
random
from the field of potential qualified candidates, instead of elected
over Y and Z?
Suppose, for every such potential draft candidate "X", we ask the
voters:
"Whom do you prefer between A and X? B and X? C and X?" Since this is
just
a thought experiment, we can assume the voters are told enough about A,
B, C,
and X to make an informed decision.

Unless A, B, and C are truly bottom-of-the-barrel candidates, there will

be some X over which A is preferred by a majority, and likewise for B
and
C. Suppose we could repeat this exhaustively (as is only possible in a
thought
experiment); that is, for every possible choice of X. We could then
count the
number of majority wins against the random candidates for each of the
three
real candidates. The real candidate with the most majority wins against
random
candidates would be the most democratic choice. (Admittedly, the truth
of
this statement depends on the exact definition of democracy, but I don't

think it would be inconsistent with popular views on the subject. Though
it
certainly is more abstract.)

One thing I like about this method is that it seems more meaningful than

social utilities, because it does not have the scaling problem that SUs
have.
SUs cannot be evaluated using absolute utility scales without
diminishing
the weight of some votes relative to others. SU does at least attempt
to represent relative preference intensities, something this new
approach
does not try to do.

Another thing I like is that it provides a meaningful interpretation of
"majority rule". Simply to say that the winner of an election is
preferred
to the other candidates in the election by a majority provides little
comfort if it is easy to find a random candidate outside the original
field who, if he or she had run, would have been preferred by a
majority over the actual winner. Of course, since the random candidate
didn't run, we can't say that the actual winner wasn't the best
democratic choice on the ballot; but now we have a more meaningful
way to apply a "majority rule" standard -- and one that doesn't admit
non-transitive defeat cycles.

Of course, it would be impractical to apply this technique in real life.

But I've still got one more step to go: Suppose we use this technique
to evaluate election methods. A mathematical method of measuring
the democratic potential of EMs would result. The rating methodology
would go something like this (if this gets complicated, it's because I'm

trying to state the methodology in the most general form possible):

1. Assume a k-dimensional policy space, filled with potential candidates

according to some probability density function f and potential voters
according to some probability density function g. Assume the number
of potential candidates and the number of voters are both large.
2. Select N candidates from the set of potential candidates. Use a
random
selection obeying the probability density function f.
3. Hold an election among the N candidates using election method M.
(Assume all voters apply the same strategy S, and each rates the
candidates
according to distance of the candidate from the voter).
4. For every member of the set of potential candidates (or a
representative
sample), determine which of the N candidates would have won pairwise
against that member of the set. Then determine which of the N candidates

would have the most pairwise wins of all such matches.
5. If the result of step 3 matches the result of step 4, count this as a

success for method M.
6. Repeat steps 2 through 5 for a very large number of trials.
7. Determine the ratio (R) of successes to the total number of trials.
Then R gives the rating of method M for the case of k policy dimensions,

N candidates, probability density functions f (for the candidates) and g

(for the voters), and uniform strategy S.

No, I'm not going to attempt to derive any results for this method! But
it
should be feasible to apply a computer simulation to the problem. Maybe
someday I will tackle that one. Anyway, here are a few preliminary
thoughts:

First, anticipating Mike's response: This is a mathematical expression
of
something of great concern to voters: How democratic is a given method?
Which of two methods is more democratic?

On choice of strategy (S): For zero-info elections, the best strategy is

well-known for all the major methods. But for non-zero-info elections,
what strategies should be used to compare EMs? I suspect that, for some
methods, it may not be computationally feasible to calculate the best
NZI
strategy. And even if it were, how do we determine the outcome
probabilities to use for the simulation? I suspect my methodology is
limited in practice to ZI strategies. That means that strategic concerns

and cloning concerns still need to be considered separately.

The rating of a method will probably vary with the choices of k, N, f,
and
g. These are probably not as problematic as the strategy issue. For the
pdfs, f and g, I suspect that a uniform distribution would be a
reasonable
simplification. However, things will get more interesting if f and g are

different, especially if they are skewed in different directions.

For N=2, all methods will score equally well (since they all pick the
same winners). As N increases, differences should emerge. I predict
that Approval, Condorcet, and Borda will hold up well, ignoring the
strategic considerations.

One thing I don't like about my methodology is that it doesn't measure
how badly the methods fail when they don't pick the "most democratic"
choice. I would rather have a method that picks the second best
candidate all the time than one that picks the best candidate 90% of the

time and the worst candidate 10% of the time. A slight modification
could be made to take this into account: Give an election method
partial credit for picking the second highest candidate in step 4,
smaller partial credit for picking the third highest, and so on.

Richard