Arrow’s Theorem on Dictatorship. Thought of the Day 49.0

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 ⇐⇒ ap_i 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.

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