How sensitive are the results in voting theory when just one other voter joins in? Some instances with spatial majority voting

In this paper we consider situations of (multidimensional) spatial majority voting. We explore some possibilities such that under some regularity assumptions usual in this literature, if the number of voters changes from being odd to even then some results may change somewhat drastically. For example, we show that with an even number of voters if the core of the voting situation is singleton (and the core element is in the interior of the policy space) then the core is never externally stable (i.e., the situation has no Condorcet winner). This is sharply opposite to what happens with an odd number of voters: in that case, under identical assumptions on the primitives, it is well known that if the core of the voting situation is non-empty then the singleton core is always externally stable: i.e., the core element is the Condorcet winner majority-dominating every other policy vector. We find similar strikingly contrasting results with respect to the coincidence of the core and the (Gillies) uncovered set and the size and geometry of the (Gillies) uncovered set. These results rectify some erroneous statements found in this literature.