Hi all,
I've been sitting on this for a while, but I'm thinking I'll post it now:
Here are some results from the simulation I recently wrote about:
(Description of it:)
1. It generates randomly-sized factions, and their sincere utilities for every
candidate.
2. The sincere Schulze winner is found. The complete rankings are derived from the
utilities.
(Sincere ties in utility are very rare.)
3. The Approval winner is found according to several methods for generating odds.
Every voter
uses "better than expectation" strategy. This means that they approve every candidate
for whom
their utility is greater than their expectation for the election. A voter's
expectation is the
sum of (voter's utility for J)*(odds of J winning) for every candidate J.
4. Both Schulze and Approval ties are resolved in favor of candidates with earlier
letters.
5. The simulation reports which of Schulze and Approval elected a candidate with
higher average
utility, or whether they picked the same candidate. It keeps track of this
information, as
well as the average absolute gain in utility over the Schulze result.
Here are the methods for generating Approval ballots:
"Two Evils": Every voter believes that candidates A and B have 50% odds each of
winning, and that
every other candidate has 0%.
"First Preference Proportion": Every voter knows every other voter's sincere favorite,
and
believes that each candidate's odds of winning are equal to the proportion of voters
whose
favorite they are.
"Utility Maximizer Known": Every voter knows which candidate has the highest average
utility, and
believes that this candidate has a 90% chance of winning, with the other 10% divided
evenly among
the other candidates.
"Schulze Winner Known": Every voter knows which candidate is the sincere Schulze
winner, and
believes that this candidate has a 90% chance of winning, with the other 10% divided
evenly among
the others.
"Zero-Info": Every voter believes that every candidate has an equal chance of winning.
"Acceptables": No odds. Every voter votes as though their expectation is equal to 50
(the maximum
being 100). This means they could approve all or none of the candidates.
"Bullet-Voting": Every voter bullet-votes for their favorite. (Donald Davison's
recommended
Approval strategy.)
"Random Candidate": A candidate wins at random. (Not really Approval.)
For the following results, there were always five factions and 10,000 trials. I did
five and
seven candidates with each of the above.
The format is:
Method: Number of candidates:
percentage of scenarios where Approval winner had higher average utility than the
Schulze winner,
percentage where Schulze's winner had higher than Approval's,
percentage where the two agreed on a candidate,
average absolute utility gain of Approval's result over Schulze's.
(Again, utility ranges from 0 to 100.)
Two Evils: 5: 17.65, 26.51, 55.84, -1.43
Two Evils: 7: 20.93, 32.67, 46.40, -1.895
First Pref Proportion: 5: 16.64, 15.25, 68.11, +.095
First Pref Proportion: 7: 21.46, 21.72, 56.82, -.042
Util Maxer Known: 5: 25.38, 4.81, 69.81, +1.55
Util Maxer Known: 7: 31.03, 4.49, 64.48, +2.07
Schulze Winner Known: 5: 5.86, 4.64, 89.50, +.102
Schulze Winner Known: 7: 6.76, 5.44, 87.80, +.165
Zero-Info: 5: 22.90, 21.83, 55.27, +.203
Zero-Info: 7: 26.79, 27.48, 45.73, -.004
Acceptables: 5: 23.17, 24.43, 52.40, -.06
Acceptables: 7: 26.23, 26.93, 46.84, -.09
Bullet-Voting: 5: 7.07, 18.86, 74.07, -1.78
Bullet-Voting: 7: 7.78, 25.80, 66.42, -2.86
Random Candidate: 5: 8.96, 70.49, 20.55, -14.83
Random Candidate: 7: 8.55, 77.12, 14.32, -17.04
Based on these absolute utility gain figures, it seems I haven't included an Approval
scenario
where the "greater utility" argument for Approval looks compelling. My original
reason for
writing this simulation was to look into that. (It's probably worth noting the two
Approval
scenarios where the utility gain actually increased with the addition of two
candidates: Schulze
Winner Known and Util Maxer Known.)
It seems more interesting to look at the "Schulze efficiency" of these Approval
scenarios. To
sort them:
Five candidates:
Schulze (100%), Schulze Winner Known (89.5%), Bullet-Voting (74%), Util Maxer Known
(70%), FP
Proportion (68%), Two Evils (55.8%), Zero-Info (55.3%), Acceptables (52.4%), Random
Candidate
(20%).
Seven candidates:
Schulze (100%), Schulze Winner Known (88%), Bullet-Voting (66%), Util Maxer Known
(64%), FP
Proportion (57%), Acceptables (46.8%), Two Evils (46.4%), Zero-Info (45.7%), Random
Candidate
(14%).
It's surely a fluke that "Two Evils" outperforms "Zero-Info" here. I have to doubt
that random
information could be better than none at all.
Any thoughts?
Kevin Venzke
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