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 .
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