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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
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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 $a<b<c$.

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If I understand your problem correctly your setting is as follows:

  • 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 $A$.

Beware that when social welfare functions associate ordering profiles with a (social) ordering of the alternatives, people tend to call these functions social ordering functions. It may be clearer to limit the use of the term social welfare function to function associating social welfare levels to vectors of utilities. These two may be viewed as equivalent only when the ordering can be represented by a utility function (e.g. when the ordering is continuous).

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