Necessary and Sufficient Conditions for a Majority Winner in n-Dimensional Spatial Voting Games: An Intuitive Geometric Approach

Abstract

Necessary and sufficient conditions for there to be a majority winner in n-dimensional spatial voting games are well known, but they are customarily stated in symbolic terms in a fashion which is virtually incomprehensible to those without some reasonable degree of mathematical training, and the proofs of the basic results are even less accessible to the nonmathematically sophisticated reader. We offer proofs of the key results in this area restricted to the important case where voter preferences are a simple function of distance, that is, where, in a choice between any two alternatives, voters prefer the alternative that is closer to their ideal outcome. Our proofs, unlike those customary in the literature, can be understood by anyone who can remember high school geometry. The nature of our proofs is such as to show how the basic n-dimensional results, including the Plott (1967) conditions, the Kramer (1973) sequential voting theorem, the McKelvey (1976, 1979) agenda manipulation result, the Shepsle (1979) germaneness restriction result, and the McCubbins and Schwartz (1985) budget constraint result, can all be derived as reasonably straightforward extensions of Duncan Black's famous median voter result in the one-dimensional case.