Let A be a set of alternatives, and J be a set of individuals.
P(A) is a set of preference relations on A. These are usually taken to be weak orders (transitive, connected and irreflexive).
P(A)J is the set of profiles or ballots, which assign a preference relation on the alternatives of each individual – a ‘vote’.
A social welfare function is a map
σ: P(A)J → P(A)
Such a map produces a single ranking on alternatives – a social choice – from a profile.
Two conditions are standardly considered on such functions:
– Independence of Irrelevant Alternatives (IIA). The social decision on the relative preference between two alternatives a, b depends only on the individual preferences between these alternatives. It is independent of their rankings with respect to other alternatives.
– The Pareto or Uniformity Principle (P). If every individual prefers a to b, then so should the social welfare function.
So, what then is Arrow’s Theorem?
If |A| > 2 and J is finite, then any social welfare function satisfying IIA and P is a dictatorship, i.e. for some individual i ∈ J ∀ profiles p ∈ P(A)J and alternatives a, b ∈ A:
a σ(p) b ⇐⇒ api b
Thus, the social choice function, under these very plausible assumptions, simply copies the choices of one fixed individual – the dictator.
A closely related result is the Gibbard-Satterthwaite theorem on voting systems.
If |A| > 2 and J is finite, then any voting system
v: P(A)J → A
which is non-manipulable is a dictatorship.
For an area of study to become a recognized eld, or even a recognized subfield, two things are required: It must be seen to have coherence, and it must be seen to have depth. The former often comes gradually, but the latter can arise in a single flash of brilliance. . . . With social choice theory, there is little doubt as to the seminal result that made it a recognized field of study: Arrow’s impossibility theorem.