Editing Arrow's impossibility theorem (section)

=== Esoteric solutions ===
In addition to the above practical resolutions, there exist unusual (less-than-practical) situations where Arrow's requirement of IIA can be satisfied.

==== Supermajority rules ====
[[Supermajority]] rules can avoid Arrow's theorem at the cost of being poorly-decisive (i.e. frequently failing to return a result). In this case, a threshold that requires a <math>2/3</math> majority for ordering 3 outcomes, <math>3/4</math> for 4, etc. does not produce [[voting paradox]]es.<ref>{{Cite journal |last=Moulin |first=Hervé |date=1985-02-01 |title=From social welfare ordering to acyclic aggregation of preferences |url=https://dx.doi.org/10.1016/0165-4896%2885%2990002-2 |journal=Mathematical Social Sciences |volume=9 |issue=1 |pages=1–17 |doi=10.1016/0165-4896(85)90002-2 |issn=0165-4896}}</ref>

In [[Spatial model of voting|spatial (n-dimensional ideology) models of voting]], this can be relaxed to require only <math>1-e^{-1}</math> (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved ([[quasiconcave]]).<ref name="Caplin-1988" /> These results provide some justification for the common requirement of a two-thirds majority for constitutional amendments, which is sufficient to prevent cyclic preferences in most situations.<ref name="Caplin-1988">{{Cite journal |last1=Caplin |first1=Andrew |last2=Nalebuff |first2=Barry |date=1988 |title=On 64%-Majority Rule |url=https://www.jstor.org/stable/1912699 |journal=Econometrica |volume=56 |issue=4 |pages=787–814 |doi=10.2307/1912699 |issn=0012-9682 |jstor=1912699}}</ref>

==== Infinite populations ====
[[Peter C. Fishburn|Fishburn]] shows all of Arrow's conditions can be satisfied for [[Uncountable set|uncountably infinite sets]] of voters given the [[axiom of choice]];<ref name="Fishburn197022">{{Cite journal |last=Fishburn |first=Peter Clingerman |year=1970 |title=Arrow's impossibility theorem: concise proof and infinite voters |journal=Journal of Economic Theory |volume=2 |issue=1 |pages=103–106 |doi=10.1016/0022-0531(70)90015-3}}</ref> however, Kirman and Sondermann demonstrated this requires disenfranchising [[Almost surely|almost all]] members of a society (eligible voters form a set of [[Measure (mathematics)|measure]] 0), leading them to refer to such societies as "invisible dictatorships".<ref>See Chapter 6 of {{cite book |last=Taylor |first=Alan D. |title=Social choice and the mathematics of manipulation |publisher=Cambridge University Press |year=2005 |isbn=978-0-521-00883-9 |location=New York |postscript=none}} for a concise discussion of social choice for infinite societies.</ref>