How do I interpret the social choice function noted by Felix Brandt?

Question

How do I read this:

• Let $\mathscr{U}$ be a universe of alternatives
• Let $\mathcal{F}(\mathscr{U}):\forall\mathscr{A}\in\mathcal{F}(\mathscr{U}),\mathscr{A}\subseteq\mathscr{P}_{\geq1}(\mathscr{U})$
• A choice function is a function $\mathcal{S}:\mathcal{F}(\mathscr{U})\rightarrow\mathcal{F}(\mathscr{U})$ such that $\mathcal{S}(\mathscr{A})\subseteq\mathscr{A}$ where $\mathscr{A}\in\mathcal{F}(\mathscr{U})$
• $\mathcal{R}$ is a relation on $\mathscr{U}$
• $\mathcal{S}$ is rationalizable if $\mathcal{R}$ is transitive and complete
• Let $\mathscr{N}$ be a finite set of voters and $\mathcal{R}(\mathscr{U})$ the set of all transitive and complete relations over $\mathscr{U}$
• A social choice function (SCF) is a function $f:\mathcal{R}(\mathscr{U})^\mathscr{N}\times\mathcal{F}(\mathscr{U})\rightarrow\mathcal{F}(\mathscr{U})$ such that $f(\mathcal{R},\mathscr{A})\subseteq\mathscr{A}$

I guess $\mathcal{F}(\mathscr{U})$ is a function? If not, why is it not the set $\mathscr{F_U}$ containing all sets $\mathscr{A}$ being non-empty feasible subsets of $\mathscr{U}$?

Similarly, why is $\mathcal{R}(\mathscr{U})$ denoted as a function and not the set $\mathscr{R_U}$?

Finally, I'm not sure how to interpret $\mathcal{R}(\mathscr{U})^\mathscr{N}$, or to interpret the $\times$ in this context (what's the concept called so I can read up on this?).

Thanks in advance.

Answer 1

I am not sure what $\mathscr{P}_{\geq1}(\mathscr{U})$ means here, so I cannot be 100% sure what the whole model means either. But let me venture a few guesses and try to offer some useful guidance.

I guess $\mathcal{F}(\mathscr{U})$ is a function? If not, why is it not the set $\mathscr{F_U}$ containing all sets $\mathscr{A}$ being non-empty feasible subsets of $\mathscr{U}$?

I'll assume $\mathcal{F}(\mathscr{U}):\forall\mathscr{A}\in\mathcal{F}(\mathscr{U}),\mathscr{A}\subseteq\mathscr{P}_{\geq1}(\mathscr{U})$ means that $\mathcal{F}(\mathscr{U})$ is a set of subsets of $\mathscr{U}$ satisfying some property (though again, I can't be 100% sure because I don't know what $\mathscr{P}_{\geq1}(\mathscr{U})$ means).

Similarly, why is $\mathcal{R}(\mathscr{U})$ denoted as a function and not the set $\mathscr{R_U}$?

For whatever reason, the authors seem keen to spell out a model where the universe of alternatives $\mathscr{U}$ is the only "fundamental parameter". So they need to make everything a function of $\mathscr{U}$. In particular, for every $\mathscr{U}$, they want to generate the set of all transitive and complete relations over $\mathscr{U}$. With that goal in mind, it is somewhat natural to encode the generated set of relations through a function $\mathcal{R}(\mathscr{U})$.

Finally, I'm not sure how to interpret $\mathcal{R}(\mathscr{U})^\mathscr{N}$, or to interpret the $\times$ in this context (what's the concept called so I can read up on this?).

This is perhaps the most important and interesting part. It helps the interpretation to understand that relations in $\mathcal{R}(\mathscr{U})$ are typically meant to represent to the voters preferences over the alternatives. So $\mathcal{R}(\mathscr{U})^\mathscr{N}$ is just the set of collections of preferences over alternatives, one for each voter. A typical element of $\mathcal{R}(\mathscr{U})^\mathscr{N}$ would be a list $(R_1, R_2, \dots, R_{\# \mathscr{N}})$ representing one possible configuration of the preferences of each of the $\# \mathscr{N}$ voters.

So for every pair $[(R_1, \dots, R_{\# \mathscr{N}}), \mathscr{A}] \in \mathcal{R}(\mathscr{U})^\mathscr{N}\times\mathcal{F}(\mathscr{U})$ specifying

• the preferences over alternatives of each of the voters $(R_1, \dots, R_{\# \mathscr{N}})$, and
• the set of alternatives $\mathscr{A}$ that we are allowed to choose from
  • (somehow we ask for the voters' preferences on all alternatives in the universe but allow "feasibility constraints" on the set of alternatives the SCF can actually chose from)

the SCF $f:\mathcal{R}(\mathscr{U})^\mathscr{N}\times\mathcal{F}(\mathscr{U})\rightarrow\mathcal{F}(\mathscr{U})$ returns a "social choice" which consists of a subset of alternatives contained in $\mathscr{A}$.