# 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.