# 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 Ã— 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 Ã— distance 0.1500

The MMPD of this example is 0.15.

## Special cases

### 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)Â²). 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.