Formal Statement of The Theorem
Let be a set of outcomes, a number of voters or decision criteria. We shall denote the set of all full linear orderings of by .
A (strict) social welfare function (preference aggregation rule) is a function which aggregates voters' preferences into a single preference order on . The -tuple of voters' preferences is called a preference profile. In its strongest and simplest form, Arrow's impossibility theorem states that whenever the set of possible alternatives has more than 2 elements, then the following three conditions become incompatible:
- unanimity, or Pareto efficiency
- If alternative a is ranked above b for all orderings, then a is ranked higher than b by . (Note that unanimity implies non-imposition).
- non-dictatorship
- There is no individual i whose preferences always prevail. That is, there is no such that .
- independence of irrelevant alternatives
- For two preference profiles and such that for all individuals i, alternatives a and b have the same order in as in, alternatives a and b have the same order in as in .
Read more about this topic: Arrow's Impossibility Theorem
Famous quotes containing the words formal, statement and/or theorem:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)
“A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
—J.L. (John Langshaw)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)