[EM] Some utility results for resistant set methods

[Kristofer Munsterhjelm]

While I was away from the list, I implemented VSE and OUF simulators for 
my voting tool, quadelect. OUF is my term for the utility measure James 
Green-Armytage used in his papers; it's short for "Optimal Utility 
Fraction", i.e. the fraction of the time the method in question elects 
the highest utility winner.

(One benefit to OUF is that it converges faster and is more 
discriminating than VSE in the spatial model setting I used. But purists 
would probably prefer VSE.)

And then I thought I would check how much quality is lost by insisting 
on resistant set compliance. In line with JGA's Smith-IRV results, my 
simulations seem to indicate that there's not much of a loss. Which is 
fortunate if we want something that's highly strategy resistant.

Here are the results:

OUF:

Number of voters: 99, number of candidates: 3
Gaussian spatial distribution, 8 dimensions, sigma = 1

Ext-Minmax                   0.950199
Resistant,Ext-Minmax         0.949857
Plurality                    0.918477
IRV                          0.945945
Smith,IRV                    0.949666
Resistant,Eurovision         0.950432

Relative resistant set compliance cost: 0.035% (Minmax)

Number of voters: 99, number of candidates: 6
Gaussian spatial distribution, 8 dimensions, sigma = 1

Ext-Minmax                   0.923995
Resistant,Ext-Minmax         0.923015
Plurality                    0.813871
IRV                          0.9066
Smith,IRV                    0.921964
Eurovision                   0.937177
Resistant,Eurovision         0.927298

Relative resistant set compliance cost: 0.1% (Minmax), 1.05% (Eurovision)

VSE:

Number of voters: 99, number of candidates: 6
Gaussian spatial distribution, 8 dimensions, sigma = 1

Ext-Minmax                   0.994717
Resistant,Ext-Minmax         0.994538
Plurality                    0.965466
IRV                          0.991495
Smith,IRV                    0.994306
Eurovision                   0.996406
Resistant,Eurovision         0.995076

Relative resistant set compliance cost: 0.02% (Minmax), 0.13% (Eurovision)

The Eurovision method is a weighted positional method that gives the 
highest ranked candidate 12 points, then 10, then 8 down to zero: (12, 
10, 8, 7, ..., 1, 0, 0, 0...)

So for methods that already pass majority, it doesn't seem that we lose 
much in terms of utility. The main costs of resistant set compliance are 
instead, in my opinion, that we lose monotonicity and summability, and 
that calculating the set is quite complex.

-km