Editing Social welfare function (section)

== {{Anchor|Construction|Construct|Constructing}}Terminology and equivalence ==
Some authors maintain a distinction between three closely-related concepts: 

# A social ''choice'' function selects a single best outcome (a single candidate who wins, or multiple if there happens to be a tie).
# A social ''ordering'' function lists the candidates, from best to worst.
# A social ''scoring'' function maps each candidate to a number representing their quality. For example, the standard social scoring function for [[first-preference plurality]] is the total number of voters who rank a candidate first.

Every social ordering can be made into a choice function by considering only the highest-ranked outcome. Less obviously, though, every social choice function is also an ordering function. Deleting the best outcome, then finding the new winner, results in a runner-up who is assigned second place. Repeating this process gives a full ranking of all candidates.<ref>{{Cite journal |last=Quesada |first=Antonio |date=2002 |title=From social choice functions to dictatorial social welfare functions |url=https://ideas.repec.org//a/ebl/ecbull/eb-02d70006.html |journal=Economics Bulletin |volume=4 |issue=16 |pages=1–7}}</ref>

Because of this close relationship, the three kinds of functions are often conflated by [[abuse of terminology]].

=== Example ===
Consider an [[Instant-runoff voting|instant-runoff]] election between Top, Center, and Bottom. Top has the most first-preference votes; Bottom has the second-most; and Center (positioned [[Political spectrum|between the two]]) has the fewest first preferences. 
{| class="wikitable"
|+
!
!Round 1
!Round 2
|-
!Top
|40
|'''53'''
|-
!Center
|<s>26</s>
|''Eliminated''
|-
!Bottom
|34
|47
|}
Under instant-runoff voting, Top is the winner. Center is eliminated in the first round, and their second-preferences are evenly split between Top and Bottom, allowing Top to win.

To find the second-place finisher, we find the winner if Top had not run. In this case, the election is between Center and Bottom.
{| class="wikitable"
!Runner-up
!Round 1
|-
!—
|''Excluded''
|-
!Center
|'''66'''
|-
!Bottom
|34
|}
(Note that the finishing order is not the same as the elimination order for [[Sequential elimination method|sequential elimination methods]]: despite being eliminated first, Center is the runner-up in this election.)