# What is the “Reinforcement Axiom” for a social choice function?

I recently came across this Computational Social Choice lecture note from Stanford, 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., $$P+P’$$).

• The axiom is mentioned and discussed at 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. Commented Aug 3 at 22:59
• 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. Commented Aug 4 at 7:17

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 $$C\cap C'$$.