Uncovered set: Difference between revisions
→Example
m (Clarify that the "k-step beatpath" is exclusive (i.e. 1-step beatpath from A is A>X). Remove Banks statement (see talk)) |
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Notice that the Smith set includes all candidates (this can be seen by observing that there is a [[beatpath]] of x>y>z>v>x, or alternatively by observing that no matter how many candidates you look at from top to bottom, there is still some candidate outside of the group being looked at that one of the candidates in the group lose or tie to). But the uncovered set is all candidates except z; this is because y>z and all candidates who beat y (just x) also beat z. <ref>https://economics.stackexchange.com/a/27691</ref> (Notice that the Copeland set is even smaller; it is just v and x).
An alternative way of understanding the uncovered set in this example is to show the size of the smallest-size beatpath from each candidate to another, if one exists (if x>y is 1 here, this means x pairwise beats y. If it's 2, it means x pairwise beats someone who pairwise beats y, etc.). Any candidate with a smallest-size beatpath of 3 or more to another candidate is not in the uncovered set:
{| class="wikitable"
! colspan="5" |Size of smallest-size beatpath
between each pair of candidates
|-
!
!x
!y
!v
!z
|-
|x
| ---
|1
|Lose
|1
|-
|y
| 2
| ---
|1
|1
|-
|v
|1
|2
| ---
| 2
|-
|z
|2
|'''<big><u>3</u></big>'''
|1
| ---
|}
Notice that all candidates except z have beatpaths of size 1 or 2, whereas z>y is (z has a smallest beatpath to y of) 3 steps (z>v>x>y), therefore z is not in the uncovered set.
==Notes==
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