Editing Arrow's impossibility theorem (section)

=== Formal statement ===
Let <math>A</math> be a set of ''alternatives''. A voter's [[preference (economics)|preferences]] over <math>A</math> are a [[Connected relation|complete]] and [[Transitive relation|transitive]] [[binary relation]] on <math>A</math> (sometimes called a [[total preorder]]), that is, a subset <math>R</math> of <math>A \times A</math> satisfying:
# (Transitivity) If <math>(\mathbf{a}, \mathbf{b})</math> is in <math>R</math> and <math>(\mathbf{b}, \mathbf{c})</math> is in <math>R</math>, then <math>(\mathbf{a}, \mathbf{c})</math> is in <math>R</math>,
# (Completeness) At least one of <math>(\mathbf{a}, \mathbf{b})</math> or <math>(\mathbf{b}, \mathbf{a})</math> must be in <math>R</math>.
The element <math>(\mathbf{a}, \mathbf{b})</math> being in <math>R</math> is interpreted to mean that alternative <math>\mathbf{a}</math> is preferred to alternative <math>\mathbf{b}</math>. This situation is often denoted <math>\mathbf{a} \succ \mathbf{b}</math> or <math>\mathbf{a}R\mathbf{b}</math>. Denote the set of all preferences on <math>A</math> by <math>\Pi(A)</math>. Let <math>N</math> be a positive integer. An [[Ranked voting|''ordinal (ranked)'']] ''social welfare function'' is a function<ref name="Arrow1950"/>
: <math> \mathrm{F} : \Pi(A)^N \to \Pi(A) </math>
which aggregates voters' preferences into a single preference on <math>A</math>. An <math>N</math>-[[tuple]] <math>(R_1, \ldots, R_N) \in \Pi(A)^N</math> of voters' preferences is called a ''preference profile''.

'''Arrow's impossibility theorem''': If there are at least three alternatives, then there is no social welfare function satisfying all three of the conditions listed below:<ref name="Gean">{{cite journal |last=Geanakoplos |first=John |year=2005 |title=Three Brief Proofs of Arrow's Impossibility Theorem |url=https://cowles.yale.edu/sites/default/files/files/pub/d11/d1123-r4.pdf |url-status=live |journal=[[Economic Theory (journal)|Economic Theory]] |volume=26 |issue=1 |pages=211–215 |citeseerx=10.1.1.193.6817 |doi=10.1007/s00199-004-0556-7 |jstor=25055941 |s2cid=17101545 |archive-url=https://ghostarchive.org/archive/20221009/https://cowles.yale.edu/sites/default/files/files/pub/d11/d1123-r4.pdf |archive-date=2022-10-09}}</ref>
; [[Pareto efficiency]]
: If alternative <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> for all orderings <math>R_1, \ldots, R_N</math>, then <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> by <math>F(R_1, R_2, \ldots, R_N)</math>.<ref name="Arrow1950" />
; [[Dictatorship mechanism|Non-dictatorship]]
: There is no individual <math>i</math> whose preferences always prevail. That is, there is no <math>i \in \{1, \ldots, N\}</math> such that for all <math>(R_1, \ldots, R_N) \in \Pi(A)^N</math> and all <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, when <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> by <math>R_i</math> then <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> by <math>F(R_1, R_2, \ldots, R_N)</math>.<ref name="Arrow1950" />
; [[Independence of irrelevant alternatives]]
: For two preference profiles <math>(R_1, \ldots, R_N)</math> and <math>(S_1, \ldots, S_N)</math> such that for all individuals <math>i</math>, alternatives <math>\mathbf{a}</math> and <math>\mathbf{b}</math> have the same order in <math>R_i</math> as in <math>S_i</math>, alternatives <math>\mathbf{a}</math> and <math>\mathbf{b}</math> have the same order in <math>F(R_1, \ldots, R_N)</math> as in <math>F(S_1, \ldots, S_N)</math>.<ref name="Arrow1950" />