[EM] Mean Weighted Political Distance

[Dan Bishop]

A while back, I proposed a measure called "mean minimum political 
distance" (MMPD) for evaluating multi-winner election methods.  It works 
pretty well on the uniform linear political spectrum, but rather poorly 
for a polarized electorate.

Surely, a voter's evaluation of the election result would consider ALL 
of the winners, not just the one nearest their views.  And we can do 
this with a straightforward extension of MPD.

Define the "weighted political distance" (WPD) between a voter V and the 
winning candidate set S as the dot product of D and W, where D = 
sorted(distance(V, C) for C in S), and W is an array of weights.  The 
"mean weighted political distance" (MWPD) is the mean of the WPD over 
all voters.

I choose W based on the polarized two-party model, i.e., p of the voters 
at position 0.0 of the linear spectrum, and 1-p at position 1.0, giving 
the ballots

p: A1=A2=A3=A4=... > B1=B2=B3=B4=...
1-p: B1=B2=B3=B4=... > B1=B2=B3=B4=...

If "a" denotes the number of winning A candidates, and "b" denotes the 
number of winning B candidates, then
MWPD = p*sum(W[a:]) + (1-p)*sum(W[b:])

In a 2-winner election with W= [1, x], the possibilities are:
a=0, b=2 --> MWPD = p*(1+x)
a=1, b=1 --> MWPD = p*x + (1-p)*x = x
a=2, b=0 --> MWPD = (1-p)*(1+x)

If we want MWPD minimization to be Droop-proportional, then x=1/2.

In a 3-winner election with W = [1, 1/2, x],
a=0, b=3 --> MWPD = p*(3/2+x)
a=1, b=2 --> MWPD = p*(1/2+x) + (1-p)*x
a=2, b=1 --> MWPD = p*x + (1-p)*(1/2+x)
a=3, b=0 --> MWPD = (1-p)*(3/2+x)

Using the same Droop-proportionality criterion, we get x=1/3.

In a 4-winner election with W = [1, 1/2, 1/3, x],
a=0, b=4 --> MWPD = p*(11/6+x)
a=1, b=3 --> MWPD = p*(5/6+x) + (1-p)*x
a=2, b=2 --> MWPD = p*(1/3+x) + (1-p)*(1/3+x) = 1/3+x
a=3, b=1 --> MWPD = p*x + (1-p)*(5/6+x)
a=4, b=0 --> MWPD = (1-p)*(11/6+x)

For which x=1/4.  By now, the pattern is clear: W=[1, 1/2, 1/3, 1/4, 
..., 1/N].

Now, consider the uniform linear political spectrum.  Where do the 
election winners need to be to minimize MWPD?   Based on my simulations 
so far (brute force with candidates i/60 and voters (i+0.5)/100):

S={0.5} --> MWPD=0.25
S={0.36666666666666664, 0.6166666666666667} --> MWPD=0.34383333333333338
S={0.31666666666666665, 0.5, 0.68333333333333335} --> 
MWPD=0.3949111111111111

I was expecting it to be the Droop multiples.  Looks like it's a little 
more centrist-biased than that, but not intolerably so.

Note that in the single-winner case, MWPD minimization elects the 
Condorcet winner (although this would not necessarily be true for 
asymmetric spectrums).