# Mean minimum political distance

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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

### 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.