The uncovered set and the core: Cox's result revisited

In this work first it is shown, in contradiction to the well-known claim in Cox (1987), that the uncovered set in a multidimensional spatial voting situation (under the usual regularity conditions) does not necessarily coincide with the core even when the core is singleton: in particular, the posited coincidence result, while true for an odd number of voters, may cease to be true when the number of voters is even. Second we provide a characterisation result for the case with an even number of voters: a singleton core is the uncovered set in this case if and only if the unique element in the core is the Condorcet winner.