Difference between social choice functions and social decision functions? A social decision function (SDF) à la Sen (1970) is defined as a collective choice rule whose range is restricted to social preference relations which generate a choice function. From Gaertner (2009), a preference relation $R$ generates a choice function over a set $X$ if and only if $R$ is reflexive, complete and acyclical over $X$. I thus struggle to understand where exactly the difference between a SDF and a social choice function (SCF) à la Gibbard-Satterthwaite lies. A SCF is itself a choice function, so the preference relation $R$ generating it must satisfy the same conditions as the one generating a SDF.
Let $X$ be the set of alternatives.
A social decision function maps profiles of preference orderings to relations on $X$ such that every nonempty subset of $X$ has at least one maximum under this relation.
A social choice function maps profiles of preference orderings to elements of $X$.
Now let $P$ be a profile of preferences, $f$ a social decision function, and $g$ a social choice function.
There might be more than one $f(P)$-maximum in $X$, so $f$, in contrast to $g$, will not always pin down a single choice in $X$.
On the other hand, suppose the alternative $g(P)$ is not available for some reason (say, the winning candidate died). Then $g$ is of no help in finding an alternative from the remaining set of alternatives $X\setminus\{g(P)\}$. But $f(P)$ will allow us to rank the alternatives in this remaining set (provided $g(P)$ is not the only element of $X$), though, again, there might be more than on $f(P)$-maximum in this set.