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

Dan Bishop danbishop04 at gmail.com
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).

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 ****

Results for 10,000 simulations:

Mean = 0.29600265035811468
St. Dev. = 0.049225275160608858
95% C.I. for mean = (0.29503785269367599, 0.29696744802255337)
Simplest fraction in C.I. = 8/27

**** 5 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.30229572580110786
St. Dev. = 0.051063084679029438
95% C.I. for mean = (0.30129490773200368, 0.30329654387021204)
Simplest fraction in C.I. = 10/33

**** 6 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.30668385765348882
St. Dev. = 0.052246238355619305
95% C.I. for mean = (0.30565985019844172, 0.30770786510853593)
Simplest fraction in C.I. = 4/13

**** 7 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.31194122641028588
St. Dev. = 0.054292402651190418
95% C.I. for mean = (0.31087711487198105, 0.3130053379485907)
Simplest fraction in C.I. = 5/16

**** 8 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.31583421713519955
St. Dev. = 0.054691865511392539
95% C.I. for mean = (0.31476227626870318, 0.31690615800169591)
Simplest fraction in C.I. = 6/19

**** 9 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.31839339216709406
St. Dev. = 0.05584311495999409
95% C.I. for mean = (0.31729888722603289, 0.31948789710815523)
Simplest fraction in C.I. = 7/22

**** 10 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.32199377022074216
St. Dev. = 0.056722075258496291
95% C.I. for mean = (0.32088203797439191, 0.32310550246709241)
Simplest fraction in C.I. = 9/28

**** 11 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.3250827093110063
St. Dev. = 0.058751979685123995
95% C.I. for mean = (0.32393119166897361, 0.326234226953039)
Simplest fraction in C.I. = 12/37

**** 12 CANDIDATES ****

Results for 10,000 simulations:

Mean = 0.3256730946327252
St. Dev. = 0.059259775440600918
95% C.I. for mean = (0.32451162437677011, 0.32683456488868029)
Simplest fraction in C.I. = 13/40

**** GENERAL OBSERVATION ****

I haven't come up with a formula yet, but I have noticed that the lowest
(i.e., best) expected MMPD occurs when there are 2 candidates.  As more
candidates are added, the expected MMPD gets greater (i.e., worse) and
appears to approach a limit of 1/3: An election with an extremely large
number of candidates is equivalent to a dictatorship.

```