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

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&le;i&le;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&le;i&lt;n} is elected.