[EM] Simulations (non-uniform voter distribution)

[Richard Moore]

Here are the results from the non-uniform voter distribution
simulations. There are a few key differences between the
uniform voter distribution sims and these sims.

1. The background pool size for these sims was 50 candidates
instead of 49. As before, the candidates have a uniform
probability distribution, but this time the distribution
was random rather than spaced at equal intervals.

2. The voters are randomly distributed as follows: 300
voters covering the entire square region, plus two
rectangular clusters: one of 260 voters in the upper left
quadrant, and one of 240 voters in the lower right quadrant.
The two clusters are not quite symmetrical in relation to
the center of the region.

3. This time there were 10000 trials instead of 4000.

RESULTS AND COMMENTS:

Random Choice Histogram:
	1: 2117	2: 2039	3: 1966	4: 1915	5: 1963	
Random Ballot Histogram:
	1: 2415	2: 2463	3: 2162	4: 1787	5: 1173	
Plurality Histogram:
	1: 2824	2: 2722	3: 2166	4: 1544	5: 744	
Plurality/NZI Histogram:
	1: 3546	2: 2912	3: 1913	4: 1167	5: 462	
IRV Histogram:
	1: 4160	2: 3478	3: 1757	4: 545	5: 60	
Plain Condorcet Histogram:
	1: 8844	2: 958	3: 182	4: 14	5: 2	
SSD Histogram:
	1: 8848	2: 955	3: 182	4: 15	5: 0	
Borda Histogram:
	1: 6590	2: 2327	3: 840	4: 241	5: 2	
Borda/NZI Histogram:
	1: 6338	2: 2524	3: 897	4: 235	5: 6	
Approval Histogram:
	1: 6910	2: 2243	3: 692	4: 151	5: 4	
Approval/NZI Histogram:
	1: 6804	2: 2314	3: 727	4: 148	5: 7	

Random Choice: This produced a flatter histogram this time
because exact MP ties are less common with the random
distribution.

Plurality: First-place success rate fell from 65% to 28%
for non-strategic voting, and from 50% to 35% for strategic
voting. I don't know why the NZI result is better than the
ZI result in this case, but both are very poor.

IRV: Fell from 82% to 42%.

PC and SSD: Minor differences -- less than 0.1% -- have
emerged between these two methods with this voter
distribution. Performance fell from 95% to 88%. Much less
sensitive to voter clustering than the other methods here.

Borda: Fell from 92% to 66% (ZI case) and from 87% to 64%
(non-ZI case). Less stable than Approval and Condorcet,
but more stable than IRV and Plurality, when voter
distribution varies.

Approval: Fell from 91% to 69% (ZI) and from 89% to 68%
(NZI). Approval appears to be less susceptible to
non-uniform voter distributions than Plurality and Borda,
and remains less susceptible to strategy when compared to 
Borda. It also appears to be less susceptible to voter
clustering than IRV.

I would like to post the code (written in PERL) for this
simulation to the list but it's around 1200 lines/35 KB.
Maybe I'll do it in 3 or 4 parts. Suggestions?

  -- Richard