Given m ≥ 3 alternatives and n ≥ 2 voters, let σ(m,n) be the least integer k for which there is a set of k strict preference profiles for the voters on the alternatives with the following property: Arrow's impossibility theorem holds for this profile set and for each of its strict preference profile supersets. We show that σ(3, 2) = 6 and that for each m, σ(m,n)/4n approaches 0 monotonically as n gets large. In addition, for each n and ε > 0, σ(m,n)/(log2 m)2+ε approaches 0 as m gets large. Hence for many alternatives or many voters, a robust version of Arrow's theorem is induced by a very small fraction of the set of all (m!)n strict preference profiles.
- Arrow's impossibility theorem
- Minimum profile sets
- Voter preference profiles
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