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

Question

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., $P+P’$).

Answer 1

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 $C\cap C'$.