<p>I can't think of any theorems which specifically focus on the <em>number</em> of social aggregation rules (of some type) satisfying axioms <em>X</em>, <em>Y</em>, and <em>Z</em>.  However, aside from impossibility theorems, most of the theorems in social choice theory fall into one of two categories:</p>
<ol>
<li>An axiomatic characterization of a <em>single</em> rule, e.g. &quot;Rule <em>F</em> is the <em>unique</em> rule satisfying axioms <em>X</em>, <em>Y</em>, and <em>Z</em>.&quot;</li>
<li>An axiomatic characterization of a well-defined <em>family</em> of rules, e.g. &quot;Rules of class <em>C</em> are the <em>only</em> rules satisfying axioms <em>X</em>, <em>Y</em>, and <em>Z</em>.&quot;</li>
</ol>
<p>In the first case, the &quot;number&quot; of rules is obviously <em>one</em>, which is not very interesting.   In the second case, the class <em>C</em> is typically defined by one or more &quot;parameters&quot;, so the number of rules in <em>C</em> is just the number of valid parameter values.  Since these parameters are usually real numbers, this means that the number of rules in <em>C</em> is usually infinite.  Indeed asking about the &quot;cardinality&quot; of <em>C</em> is generally not the right question.  The right question is the <em>dimension</em> of the parameter space.</p>
<p>For example: suppose <em>C</em> is the set of <em>scoring rules</em> on a set of <em>N</em> alternatives (e.g. the Borda rule, (anti)plurality rule, etc.).  Any scoring rule is defined by <em>N</em> numbers: the &quot;scores&quot; <span class="math-container">$s_1\leq s_2\leq \cdots \leq s_N$</span> assigned to the <em>N</em> possible ranks in any voter's preference order.  Without loss of generality, we can assume <span class="math-container">$s_1=0$</span> and <span class="math-container">$s_N=1$</span>.  Furthermore, since the scores are an increasing sequence, they are completely described by the score <em>differences</em> <span class="math-container">$\delta_1=s_2-s_1=s_2$</span>,  <span class="math-container">$\ \delta_2=s_3-s_2$</span>, <span class="math-container">$\ldots$</span>, <span class="math-container">$\delta_{N-1}=1-s_{N-1}$</span>. Observe that <span class="math-container">$\delta_1+\cdots+\delta_{N-1}=1$</span>.  In other words, the vector <span class="math-container">$(\delta_1,\ldots,\delta_{N-1})$</span> is an element of the <em>probability simplex</em> <span class="math-container">$\Delta^{N-1}$</span>, which is a convex set of dimension <span class="math-container">$N-2$</span>.</p>
<p>From this parameterization, we see that the cardinality of the set of scoring rules is <span class="math-container">$\beth_1$</span>.  But the parameterization is much more informative than the cardinality.</p>

CC BY-SA 4.0
2022-11-29T08:54:50.213Z
Thank you very much for that answer, that's exactly the type of things I am looking for. Do you have any reference to suggest?
MiKiDe
CC BY-SA 4.0
2022-11-30T09:09:14.653Z
There isn't really a reference on this exact question.  But you can find many related results in the *Handbook of Social Choice and Welfare*, volumes I and II (Editors: Kenneth Arrow, Amartya Sen, and Kotaro Suzumura, Publisher: Elsevier)
Marcus Pivato
CC BY-SA 4.0
2022-11-27T11:24:29.773Z
2022-11-27T11:24:29.773Z
Marcus Pivato
<p>I know many papers about showing the existence (or the impossibility) of a social choice rule satisfying a set of properties.</p>
<p>However, I was wondering whether there were some references trying to answer the following question: given a set of desiderata, what is the cardinal of the set of aggregation functions?</p>
<p>The desiderata in question could be anything:</p>
<ul>
<li>A certain structure of preferences at the individual or the social level</li>
<li>The usual universal domain, Pareto property, independence of irrelevant alternative...</li>
</ul>
<p>I know in the literature, there are many distinctions between different properties of the aggregation function (SWF, SWFL, Collective choice rule...). I am only interested in any result that concern an aggregation function of any type, i.e. something that takes a profile of the voters preferences and return a choice (or a rationalization of the choice, as a social preference).</p>
<p>All I am looking for is any result that says something like the following:</p>
<ul>
<li>Given some restriction on the aggregation rule (like saying it is a social welfare function) the number of SWF satisfying some set of properties is ...</li>
<li>There is strictly more CCR satisfying ... than SWF. (It is trivial when there is an impossibility for SWF and not for CCR, but maybe this is true whenever there is a possibility for SWF too?)</li>
</ul>

CC BY-SA 4.0
2022-11-26T12:56:11.900Z
Usually, "independence of irrelevant alternative" is a property imposed on social welfare functions and not social choice functions. Could you be a bit more precise?
Michael Greinecker
CC BY-SA 4.0
2022-11-26T13:15:02.340Z
@MichaelGreinecker Sorry that was my mistake, I wrote "social choice function" instead of aggregation function (that I use as an umbrella term for anything that takes a profile of preference and lead to social preference or a choice)
MiKiDe
CC BY-SA 4.0
2022-11-26T22:44:59.693Z
One of the cases you seem to be interest in is straightforward: Since there are as many disctatorial SWFs and SCFs as there are agents, that gives you the number of such functions under the assumptions of the relevant impossibility theorems.
Michael Greinecker
<reference-request><social-choice>
Possibilities in social choice: number of possible collective decision function?
MiKiDe
# Possibilities in social choice: number of possible collective decision function?

(Asked by: username_0 on 2022-11-26)

I know many papers about showing the existence (or the impossibility) of a social choice rule satisfying a set of properties.

However, I was wondering whether there were some references trying to answer the following question: given a set of desiderata, what is the cardinal of the set of aggregation functions?

The desiderata in question could be anything:

* A certain structure of preferences at the individual or the social level
* The usual universal domain, Pareto property, independence of irrelevant alternative...

I know in the literature, there are many distinctions between different properties of the aggregation function (SWF, SWFL, Collective choice rule...). I am only interested in any result that concern an aggregation function of any type, i.e. something that takes a profile of the voters preferences and return a choice (or a rationalization of the choice, as a social preference).

All I am looking for is any result that says something like the following:

* Given some restriction on the aggregation rule (like saying it is a social welfare function) the number of SWF satisfying some set of properties is ...
* There is strictly more CCR satisfying ... than SWF. (It is trivial when there is an impossibility for SWF and not for CCR, but maybe this is true whenever there is a possibility for SWF too?)

**Tags:** reference-request, social-choice
> **Comments:**
> **username_1**: Usually, "independence of irrelevant alternative" is a property imposed on social welfare functions and not social choice functions. Could you be a bit more precise?  
> **username_0**: @MichaelGreinecker Sorry that was my mistake, I wrote "social choice function" instead of aggregation function (that I use as an umbrella term for anything that takes a profile of preference and lead to social preference or a choice)  
> **username_1**: One of the cases you seem to be interest in is straightforward: Since there are as many disctatorial SWFs and SCFs as there are agents, that gives you the number of such functions under the assumptions of the relevant impossibility theorems.  

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## 1 answers

### Answer by username_2 on 2022-11-27. Score: 3 | Accepted answer

I can't think of any theorems which specifically focus on the *number* of social aggregation rules (of some type) satisfying axioms *X*, *Y*, and *Z*. However, aside from impossibility theorems, most of the theorems in social choice theory fall into one of two categories:

1. An axiomatic characterization of a *single* rule, e.g. "Rule *F* is the *unique* rule satisfying axioms *X*, *Y*, and *Z*."
2. An axiomatic characterization of a well-defined *family* of rules, e.g. "Rules of class *C* are the *only* rules satisfying axioms *X*, *Y*, and *Z*."

In the first case, the "number" of rules is obviously *one*, which is not very interesting. In the second case, the class *C* is typically defined by one or more "parameters", so the number of rules in *C* is just the number of valid parameter values. Since these parameters are usually real numbers, this means that the number of rules in *C* is usually infinite. Indeed asking about the "cardinality" of *C* is generally not the right question. The right question is the *dimension* of the parameter space.

For example: suppose *C* is the set of *scoring rules* on a set of *N* alternatives (e.g. the Borda rule, (anti)plurality rule, etc.). Any scoring rule is defined by *N* numbers: the "scores" $s\_1\leq s\_2\leq \cdots \leq s\_N$ assigned to the *N* possible ranks in any voter's preference order. Without loss of generality, we can assume $s\_1=0$ and $s\_N=1$. Furthermore, since the scores are an increasing sequence, they are completely described by the score *differences* $\delta\_1=s\_2-s\_1=s\_2$, $\ \delta\_2=s\_3-s\_2$, $\ldots$, $\delta\_{N-1}=1-s\_{N-1}$. Observe that $\delta\_1+\cdots+\delta\_{N-1}=1$. In other words, the vector $(\delta\_1,\ldots,\delta\_{N-1})$ is an element of the *probability simplex* $\Delta^{N-1}$, which is a convex set of dimension $N-2$.

From this parameterization, we see that the cardinality of the set of scoring rules is $\beth\_1$. But the parameterization is much more informative than the cardinality.

> **Comments:**
> **username_0**: Thank you very much for that answer, that's exactly the type of things I am looking for. Do you have any reference to suggest?  
> **username_2**: There isn't really a reference on this exact question.  But you can find many related results in the *Handbook of Social Choice and Welfare*, volumes I and II (Editors: Kenneth Arrow, Amartya Sen, and Kotaro Suzumura, Publisher: Elsevier)  

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