# What is the difference between social choice function and social welfare function?

I am trying to understand a shard proof on Arrow's Impossibility Theorem and Gibbard-Satterthwaite Theorem.

I stumbled upon these 2 different functions, and I cannot understand the difference between them:

f:L^n→A - social choice function

F:L^n->L* - social welfare function


Assume that finite set of alternatives are a,b and c.

A social choice function can have one single output which can be a or b or c.

A social welfare function can have any ranking as output such as $a<b<c$.

• List item A set of alternatives $A$, with $\mathcal{L}$ the set of orderings over $A$.
• List item A finite set $N$ of individuals with cardinality $n$.
• List item Each individual has a preference relation $L\in \mathcal{L}$.
• List item A preference profile is a list of preferences relation for each individuals in the population $L^n \in \mathcal{L}^n$.
Then a social choice function associates every profile in your domain with an alternative, that is it picks a chosen alternative in $A$ for every profile of preferences.
A social welfare function on the other hand associates every profile of preferences with a complete ranking of the alternatives -- not only with the best alternative in $A$.