[EM] Mean Weighted Political Distance, Part 2

[Dan Bishop]

For a more efficient method of computing MWPD on a uniform political 
spectrum, observe that over any interval in which all voters have the 
same order of preference (i.e., between two points that are midpoints 
between candidates), the WPD is the sum of absolute value functions 
centered at each candidate.  If we also include the candidate positions 
themselves as boundary points, then the WPD is a straight line segment 
over each interval, so integrating it requires simply finding the area 
of a trapezoid.  Thus, the MWPD can be computed as:

def linear_mwpd(candidates):
    """
    Mean Weighted Politcal Distance on a uniform linear spectrum.
    candidates = coordinates of the candidates (between 0 and 1)
    """
    critical_points = set([0, 1])
    for cand1 in candidates:
        for cand2 in candidates:
            critical_points.add((cand1 + cand2) / 2)
    sorted_cp = sorted(critical_points)
    result = 0
    for low, high in zip(sorted_cp[:-1], sorted_cp[1:]):
        mid = (low + high) / 2
        width = high - low
        distances = sorted(abs(mid - candidate) for candidate in candidates)
        for index, dist in enumerate(distances):
            result += dist / (index + 1) * width
    return result

The results of my simulations to find the optimal candidate set are:

{0.5} --> 0.25
{0.37499999988319355, 0.62500000011680656} --> 0.34375
{0.30952378373473316, 0.5, 0.6904762162652669} --> 0.39484126984127094
{0.2676758158861115, 0.42676788016347578, 0.57323211983652422, 
0.73232418411388855} --> 0.42760942761107096
{0.23806066570959628, 0.37686611982478441, 0.5, 0.62313388017521554, 
0.76193933429040372} --> 0.45066542288719574
{0.21573576423391647, 0.34004090761866429, 0.44800610652086625, 
0.55199389347913375, 0.65995909238133565, 0.78426423576608362} --> 
0.46789281516205333

Or, as fractions:

{1/2} --> 1/4
{3/8, 5/8} --> 11/32
{13/42, 1/2, 29/42} --> 199/504
{53/198, 169/396, 227/396, 145/198} --> 127/297
{319/1340, 101/268, 1/2, 167/268, 1021/1340} --> 72467/160800
{85/394, 169/497, 517/1154, 637/1154, 328/497, 309/394} --> 
716777674392488813/1531927080643703520