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Notes
The actual mechanism of forming the aggregating preference need not necessarily be by voting.
Here \(V=|{{{\textbf {V}}}}|\) is the size of the population, generally assumed finite.
Weren’t they always?
One might wonder why one would need this pretentious language; it is mostly to point out one of the “resolutions” of Arrow’s paradoxical result along the lines of [15]: in an infinite population of voters, one can have (nonprincipal) ultrafilters, thus no dictator, or rather, no personalized dictator, but instead some transcendent cabal, invisible yet all-powerful .... But let’s not get carried away.
Helmer knew Arrow, who, as a Columbia undergrad, had helped him with a translation of a book by Alfred Tarski [13].
Although one suspects that given what Arrow was trying to model, he could just as well have intuited the outcome.
And so it still remains: Google Scholar lists 23,466 citations, 1650 of them from the first half of this year; MathSciNet shows 468 citations as of this writing.
One typically invokes some political spectrum: liberal vs. conservative or market vs. étatist.
I should mention a recent rich research thread that studies, roughly speaking, the statistics of paradoxes on ensembles of preferences, shifting the desideratum from satisfying some axioms on all inputs to satisfaction on almost all inputs, the idea already present in Black’s work. The recent survey [18] covers most of this story; we won’t go into it here.
Another famed economist; see Smale’s comment on Debreu’s “Nobel Prize” in this journal [21].
We note that for the computations that follow to succeed, one does not need all possible permutations to preserve the output; it is enough to have invariance with respect to a group of permutations acting transitively on the set of voters.
The degree is the number of times the circle domain is wrapped around the circle codomain; it is positive or negative depending on the (mis)match of the orientations.
Corresponding to one of the voters varying her preferences while others keep theirs fixed.
This means that for every group element a, the equation \(nx=a\) has a unique solution for all \(n\in {\mathbb {Z}}\).
Stating that trivial homotopy groups in dimensions below k imply that the kth homotopy and homology groups are isomorphic.
Weinberger refers to this property as Solomonism—from the parable of cutting a child in two to satisfy two women each claiming to be its mother—a solution pretty remote from what either woman (allegedly) wanted.
Political scientists are unlikely to lose any sleep after learning this.
Good covers of topological spaces are those for which all intersections of the covering subspaces are either empty or contractible.
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Baryshnikov, Y. Around Arrow. Math Intelligencer 45, 224–231 (2023). https://doi.org/10.1007/s00283-023-10290-6
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DOI: https://doi.org/10.1007/s00283-023-10290-6