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.