Mean minimum political distance: Difference between revisions
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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
<table class="wikitable" border="">
<tr>
<th>voters</th>
<th>position</th>
<th>nearest winner</th>
<th>distance</th>
<th>voters × distance</th>
</tr>
<tr align="right">
<td>0.15</td>
<td>0.00</td>
<td>0.25</td>
<td>0.25</td>
<td>0.0375</td>
</tr>
<tr align="right">
<td>0.20</td>
<td>0.25</td>
<td>0.25</td>
<td>0.00</td>
<td>0.0000</td>
</tr>
<tr align="right">
<td>0.30</td>
<td>0.55</td>
<td>either</td>
<td>0.25</td>
<td>0.0750</td>
</tr>
<tr align="right">
<td>0.20</td>
<td>0.75</td>
<td>0.75</td>
<td>0.00</td>
<td>0.0000</td>
</tr>
<tr align="right">
<td>0.15</td>
<td>1.00</td>
<td>0.75</td>
<td>0.25</td>
<td>0.0375</td>
</tr>
<tr>
<th colspan="4">sum of voters × distance</th>
<td align="right">0.1500</td>
</tr>
</table>
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.
[[Category:Voting theory metrics]]
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