Arrovian independence and the aggregation of choice functions

We reappraise the Arrow problem by studying the aggregation of choice functions. We do so in the general framework of judgment aggregation, in which choice functions are naturally representable by specifying, for each menu A and each alternative x in A, whether x is choosable from A, or not. Our framework suggests a natural strengthening of Arrow's independence condition positing that the collective choosability of an alternative from a menu depends on the individual views on that issue, and that issue alone. Our analysis reveals that Arrovian impossibility results crucially hinge on what internal consistency requirements we impose on choice functions. While the aggregation of 'binary' choice functions, i.e. those satisfying both contraction (α) and expansion (γ) consistency, is necessarily dictatorial, possibilities in the form of oligarchic rules emerge for path-independent choice functions, that is, when the expansion property γ is replaced by the so-called Aizerman condition. Remarkably, the Arrovian aggregation of choice functions is shown to be almost dictatorial already under property γ alone. When giving up expansion consistency, specific quota rules become possible.