Mean minimum political distance

Mean minimum political distance (MMPD) is a political spectrum statistic defined as the mean distance between a voter and the nearest elected candidate.

Example

Assume a one-dimensional political spectrum with the voter distribution

• 15% at position 0
• 20% at position 0.25
• 30% at position 0.5
• 20% at position 0.75
• 15% at position 1

If the candidate set {0.25, 0.75} is elected, then

voters	position	nearest winner	distance	voters × distance
0.15	0.00	0.25	0.25	0.0375
0.20	0.25	0.25	0.00	0.0000
0.30	0.55	either	0.25	0.0750
0.20	0.75	0.75	0.00	0.0000
0.15	1.00	0.75	0.25	0.0375
sum of voters × distance				0.1500

The MMPD of this example is 0.15.

Special cases

On a uniform linear political spectrum:

Random Ballots

The mathematically expected MMPD for n winners randomly selected from uniform(0,1) is (n+3)/(2(n+1)(n+2)), which is 1/3 for a single winner, and asympotically 1/(2n) as the number of seats approaches infinity.

Droop Multiples

Electing the candidates {i/(n+1): 1≤i≤n} gives an MMPD of (n+3)/(4(n+1)²). As n approaches infinity, this is asymptotically equal to the optimal value of 1/(4n).

Optimal Winners

The minimum possible MMPD in a uniform linear spectrum is 1/(4n), which occurs when the candidate set {(2i+1)/(2n): 0≤i<n} is elected.