<p>In the formalism being used here, a profile <span class="math-container">$P$</span> is a <em>vector</em>, indexed by the set of all linear orders on the set <span class="math-container">$A$</span>.  Each component of this vector is a non-negative integer, indicating the number of voters who report a certain preference order.  Thus, the vector sum <span class="math-container">$P+P'$</span> represents the profile obtained by merging two disjoint populations of voters, where one population is described by the profile <span class="math-container">$P$</span> and the other population is described by the profile <span class="math-container">$P'$</span>.</p>
<p>The &quot;reinforcing&quot; property essentially says:  if the population described by profile <span class="math-container">$P$</span> selects some subset <span class="math-container">$C\subseteq A$</span>, and the population described by profile <span class="math-container">$P'$</span> selects some subset <span class="math-container">$C'\subseteq A$</span>, and <span class="math-container">$C\cap C'\neq \emptyset$</span>, then the &quot;merged&quot; population (described by the profile <span class="math-container">$P+P'$</span>) should select some subset of the (nonempty) intersection <span class="math-container">$C\cap C'$</span>.</p>

CC BY-SA 4.0
2024-08-04T07:16:57.433Z
Thank you for your answer—it was not obvious to me at all that the document I found was dealing with populations of variable size.
EoDmnFOr3q
CC BY-SA 4.0
2024-08-04T03:34:47.493Z
2024-08-04T03:34:47.493Z
Marcus Pivato
<p>I recently came across <a href="https://web.stanford.edu/%7Eashishg/msande336/aut2021/win2020/msande336_handouts/lecture1.pdf" rel="nofollow noreferrer">this Computational Social Choice lecture note from Stanford</a>, where an axiom for social choice functions called “Reinforcement” is introduced as a desirable axiom. I am trying to make sense of this axiom, but I can’t.</p>
<p>Let <span class="math-container">$N$</span> be a <em>finite voter set</em>, let <span class="math-container">$A$</span> be a <em>finite alternative set</em>, let <span class="math-container">$\mathcal{P}$</span> be the <em>set of all linear order profiles on <span class="math-container">$A$</span></em>, and let <span class="math-container">$f:\mathcal{P}\to 2^A$</span> be a <em>social choice function</em>.</p>
<p>A social choice function <span class="math-container">$f$</span> is <strong>reinforcing</strong> if for any two profiles <span class="math-container">$P,P’\in\mathcal{P}$</span> such that <span class="math-container">$f(P)\cap f(P’)\neq\emptyset$</span>, <span class="math-container">$f(P+P’)\subseteq f(P)\cap f(P’)$</span>.</p>
<p>I don’t understand the axiom because I can’t quite figure out the meaning of adding two preference profiles (i.e., <span class="math-container">$P+P’$</span>).</p>

CC BY-SA 4.0
2024-08-03T22:59:36.663Z
The axiom is mentioned and discussed at https://staff.fnwi.uva.nl/u.endriss/pubs/files/BrandtEtAlMAS2012.pdf It seems to refer to slightly different settings where the group of voters can vary.
Michael Greinecker
CC BY-SA 4.0
2024-08-04T07:17:00.370Z
Thank you for the comment and the very useful document you shared—it was not obvious to me at all that the document I found was dealing with populations of variable size.
EoDmnFOr3q
<theory><social-choice>
What is the “Reinforcement Axiom” for a social choice function?
EoDmnFOr3q
# What is the “Reinforcement Axiom” for a social choice function?

(Asked by: username_0 on 2024-08-03)

I recently came across [this Computational Social Choice lecture note from Stanford](https://web.stanford.edu/%7Eashishg/msande336/aut2021/win2020/msande336_handouts/lecture1.pdf), where an axiom for social choice functions called “Reinforcement” is introduced as a desirable axiom. I am trying to make sense of this axiom, but I can’t.

Let $N$ be a *finite voter set*, let $A$ be a *finite alternative set*, let $\mathcal{P}$ be the *set of all linear order profiles on $A$*, and let $f:\mathcal{P}\to 2^A$ be a *social choice function*.

A social choice function $f$ is **reinforcing** if for any two profiles $P,P’\in\mathcal{P}$ such that $f(P)\cap f(P’)\neq\emptyset$, $f(P+P’)\subseteq f(P)\cap f(P’)$.

I don’t understand the axiom because I can’t quite figure out the meaning of adding two preference profiles (i.e., $P+P’$).

**Tags:** theory, social-choice

> **Comments:**
> **username_1**: The axiom is mentioned and discussed at https://staff.fnwi.uva.nl/u.endriss/pubs/files/BrandtEtAlMAS2012.pdf It seems to refer to slightly different settings where the group of voters can vary.  
> **username_0**: Thank you for the comment and the very useful document you shared—it was not obvious to me at all that the document I found was dealing with populations of variable size.  

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## 1 answers

### Answer by username_2 on 2024-08-04. Score: 4 | Accepted answer

In the formalism being used here, a profile $P$ is a *vector*, indexed by the set of all linear orders on the set $A$. Each component of this vector is a non-negative integer, indicating the number of voters who report a certain preference order. Thus, the vector sum $P+P'$ represents the profile obtained by merging two disjoint populations of voters, where one population is described by the profile $P$ and the other population is described by the profile $P'$.

The "reinforcing" property essentially says: if the population described by profile $P$ selects some subset $C\subseteq A$, and the population described by profile $P'$ selects some subset $C'\subseteq A$, and $C\cap C'\neq \emptyset$, then the "merged" population (described by the profile $P+P'$) should select some subset of the (nonempty) intersection $C\cap C'$.

> **Comments:**
> **username_0**: Thank you for your answer—it was not obvious to me at all that the document I found was dealing with populations of variable size.  

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