[Election-Methods] Linear Spectrum MMPD analysis: Single-Winner Plurality

Sun Dec 30 02:14:01 PST 2007

In the coming weeks/months/whenever-I-feel-like-it, I will be performing 
simulations to evaluate the performance of multi-winner methods.  In 
order to do this, I will make the assumptions that:

* There is a uniform linear political spectrum.  (Other models of voter 
behavior will be considered later.)
* Candidates are uniformly-distributed random variables in.
* All votes are sincere.  (i.e., a voter at position V votes A>B iff 
abs(A-V) < abs(B-V))

The last two assumptions, that no strategy is involved in either 
nominations or voting, is admittedly unrealistic.  But, as I see it, in 
order to know the best strategy to use with a method, you must first 
know how it would behave without strategy, so that's a useful thing to 
analyze.

The measure used to evaluate each method is the expected mean minimum 
political distance 
(http://wiki.electorama.com/wiki/Mean_minimum_political_distance).

I shall start with the familiar single-vote plurality method.  And, for 
now, I shall limit it to single-winner elections.

**** 1 CANDIDATE ****

Not much of an election, but quite simple: the one candidate wins.

Results for 1,000,000 simulations:

Mean = 0.33346117339223746
St. Dev. = 0.074536492271239416
95% C.I. for mean = (0.33331508455185188, 0.33360726223262305)
Simplest fraction in C.I. = 1/3

**** 2 CANDIDATES ****

def winner():
    cand1 = random.uniform(0, 1)
    cand2 = random.uniform(0, 1)
    if abs(cand1 - 0.5) < abs(cand2 - 0.5):
        return cand1
    else:
        return cand2

Results for 1,000,000 simulations:

Mean = 0.29158801343581486
St. Dev. = 0.049207169932997265
95% C.I. for mean = (0.29149156915496505, 0.29168445771666468)
Simplest fraction in C.I. = 7/24

**** 3 CANDIDATES ****

Results for 10,000 simulations (scaled down because now I'm creating 
files of linear ballots and running them through an election calculator):

Mean = 0.29272619476621009
St. Dev. = 0.04956380959184073
95% C.I. for mean = (0.29175476194884398, 0.2936976275835762)
Simplest fraction in C.I. = 12/41

**** 4 CANDIDATES ****