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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.

Andrés E. Caicedo
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John Moser
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1 Answers1

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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 $\mathscr{A}$.

FZS
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  • Ah, so the superscript isn't an exponent notation, or it's something in some mathematical domain I don't understand. Thanks! It's hard to get my head around the representation; I understand the mathematics of social choice functions quite well—I've been demonstrating how they're broken, how to force an election without breaking any laws, and how to make elections where every vote has equal impact—but the technical notation of those relationships? It's beyond my learning. – John Moser Jul 15 '20 at 16:13
  • Also I probably should have included this: http://research.illc.uva.nl/COMSOC/estoril-2010/slides/Brandt.pdf – John Moser Jul 15 '20 at 16:14
  • In SCT, it is common to use $A^B$ to denote the set of mappings from $A$ to $B$. I am not the right person to opine on whether this is right or wrong, but some authors in that field argue this is just fine, e.g., "The notation $X^N$ is simpler than $X^{|N|}$ and is just as correct, as $X^N$ is the set of mappings from $N$ to $X$." from "Fleurbaey, Marc, and François Maniquet. A theory of fairness and social welfare. Vol. 48. Cambridge University Press, 2011". – FZS Jul 15 '20 at 16:29