This is a very interesting idea.
A couple of thoughts:
Suppose that f and g were the same. Then each voter could be asked to
compare each candidate to herself. This could be very appropriate in
a representative democracy where the representatives are supposed to serve
as proxies for the citizens that they represent.
The ballot could be worded as follows: Check the YES box next to each
candidate that you believe would do a better job in the position to which
they aspire than you yourself would if you had the appropriate technical
competency and stomach for that kind of work.
>From this vantage point it is clear that your hypothetical standard of
comparison method is strategically equivalent to ordinary Approval in the
case where f = g , the most democratic case.
Also, don't just use normal and uniform distributions. Politics isn't very
interesting until you get into bimodal (and better) distributions. Any
simple minded method will do OK with a single peaked symmetrical
distribution, though some will do better than others.
Forest
On Tue, 8 May 2001, Richard Moore wrote:
> [EMAIL PROTECTED] 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
>
>
>
