quality:corruption
Formal statement

Let A {\displaystyle A} be a set of alternatives. A preference on A {\displaystyle A} is a complete and transitive binary relation on A {\displaystyle A} (sometimes called a total preorder), that is, a subset R {\displaystyle R} of A × A {\displaystyle A\times A} satisfying:

The element (a, b) {\displaystyle (\mathbf {a},\mathbf {b})} being in R {\displaystyle R} is interpreted to mean that alternative a {\displaystyle \mathbf {a} } is preferred to alternative b {\displaystyle \mathbf {b} }. This situation is often denoted a ≻ b {\displaystyle \mathbf {a} \succ \mathbf {b} } or a R b {\displaystyle \mathbf {a} R\mathbf {b} }. Denote the set of all preferences on A {\displaystyle A} by Π (A) {\displaystyle \Pi (A)}.Let N {\displaystyle N} be a positive integer. An ordinal (ranked) social welfare function is a function

which aggregates voters' preferences into a single preference on A {\displaystyle A}. An N {\displaystyle N} - tuple (R 1, …, R N) ∈ Π (A) N {\displaystyle (R_{1},\ldots,R_{N})\in \Pi (A)^{N}} 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: