Integer Programming and Arrovian Social Welfare Functions
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
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Given $m \geq 3$ alternatives and $n \geq 2$ voters, let $\sigma(m, n)$ 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 $\sigma(3, 2) = 6$ and that for each $m$, $\sigma(m, n)/4^n$ approaches 0 monotonically as $n$ gets large. In addition, for each $n$ and $\epsilon 0$, $\sigma(m, n)/(\log_2 m)^{2+\epsilon}$ 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.