Restricted Domain
Suppose that there is an order $>$ on $X$ such that for each $n$ there is an $x^*_n$ such that if
$x\lt y \lt x^*_n$, then $x\prec_n y\prec_n x^*_n$ and if $x^*_n\lt x\lt y$, then $y\succ_n x\succ
x^*_n$. These are single-peaked preferences. Define majority rule:
\begin{equation*}
x\succeq y\text{ iff }\#\{n:x\succ_ny\}\geq \#\{n:y\succ_nx\}.
\end{equation*}
On this set of preferences, majority rule satisfies all the Arrow axioms.
Relaxing the range restriction
Suppose that the output of the social welfare function need only be transitive, but not negatively
transitive.
Theorem. Arrow's axioms imply that if negative transitivity is
relaxed to transitivity of strict preference, then the SWF must be the Pareto order.
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