<p>Let <span class="math-container">$X$</span> be the set of alternatives.</p>
<p>A <em>social decision function</em> maps profiles of preference orderings to relations on <span class="math-container">$X$</span> such that every nonempty subset of <span class="math-container">$X$</span> has <em>at least one</em> maximum under this relation.</p>
<p>A <em>social choice function</em> maps profiles of preference orderings to elements of <span class="math-container">$X$</span>.</p>
<p>Now let <span class="math-container">$P$</span> be a profile of preferences, <span class="math-container">$f$</span> a social decision function, and <span class="math-container">$g$</span> a social choice function.</p>
<p>There might be more than one <span class="math-container">$f(P)$</span>-maximum in <span class="math-container">$X$</span>, so <span class="math-container">$f$</span>, in contrast to <span class="math-container">$g$</span>, will not always pin down a single choice in <span class="math-container">$X$</span>.</p>
<p>On the other hand, suppose the alternative <span class="math-container">$g(P)$</span> is not available for some reason (say, the winning candidate died). Then <span class="math-container">$g$</span> is of no help in finding an alternative from the remaining set of alternatives <span class="math-container">$X\setminus\{g(P)\}$</span>. But <span class="math-container">$f(P)$</span> will allow us to rank the alternatives in this remaining set (provided <span class="math-container">$g(P)$</span> is not the only element of <span class="math-container">$X$</span>), though, again, there might be more than on <span class="math-container">$f(P)$</span>-maximum in this set.</p>

CC BY-SA 4.0
2020-09-05T08:59:21.663Z
Thank you! So in other words, the $R$ generating a social decision function is complete over all of $X$, while the one generating a social choice function must only be complete for the best alternative? What, then, is the difference between a SDF and a social welfare function? Can't we construct a social welfare function from the information encoded in a SDF (assuming acyclicality?)
decisionsdecisions
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2020-09-05T11:55:11.310Z
Edit to my comment above: is it correct that the only difference between SWFs and SDFs is that the former require transitivity ($xPy$ and $yIz \Rightarrow xPz$) while the latter only quasi-transitivity (e.g. it could be that $xPy$ and $yIz$ and $xIz$, such as in Sen's (1969) Pareto extension rule)?
decisionsdecisions
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2020-09-05T12:47:43.797Z
In Sen's 1970 book, the values of an (Arrovian) SWF have to be "orderings" (reflexive, complete, and transitive relations) on $X$, while the values of an SDF need only induce a choice function for all nonempty subsets. I don't have a characterization of that property at hand, but for infinite sets, an SWF need not be an SDF.
Michael Greinecker
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2020-09-04T20:53:05.140Z
2020-09-04T20:53:05.140Z
Michael Greinecker
<p>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 <span class="math-container">$R$</span> generates a choice function over a set <span class="math-container">$X$</span> if and only if <span class="math-container">$R$</span> is reflexive, complete and acyclical over <span class="math-container">$X$</span>. 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 <span class="math-container">$R$</span> generating it must satisfy the same conditions as the one generating a SDF.</p>

<microeconomics><social-choice><choice-theory>
Difference between social choice functions and social decision functions?
decisionsdecisions
# Difference between social choice functions and social decision functions?

(Asked by: username_0 on 2020-09-04)

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.

**Tags:** microeconomics, social-choice, choice-theory

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## 1 answers

### Answer by username_1 on 2020-09-04. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you! So in other words, the $R$ generating a social decision function is complete over all of $X$, while the one generating a social choice function must only be complete for the best alternative? What, then, is the difference between a SDF and a social welfare function? Can't we construct a social welfare function from the information encoded in a SDF (assuming acyclicality?)  
> **username_0**: Edit to my comment above: is it correct that the only difference between SWFs and SDFs is that the former require transitivity ($xPy$ and $yIz \Rightarrow xPz$) while the latter only quasi-transitivity (e.g. it could be that $xPy$ and $yIz$ and $xIz$, such as in Sen's (1969) Pareto extension rule)?  
> **username_1**: In Sen's 1970 book, the values of an (Arrovian) SWF have to be "orderings" (reflexive, complete, and transitive relations) on $X$, while the values of an SDF need only induce a choice function for all nonempty subsets. I don't have a characterization of that property at hand, but for infinite sets, an SWF need not be an SDF.  

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