Since the seminal contribution of Downs (1957), spatial models have been used to analyze the electoral process. However, their utility has been severely limited by (at least) four stringent assumptions. 1 First, typical spatial models, henceforth to be called Euclidean models, require that the messages candidates transmit to voters be the points of a Euclidean issue space. A point message indicates a candidate's promised issue outcome. Perfect candidate mobility and a perfect flow of information from candidates to voters are two aspects of this assumption. Secondly, in the basic spatial models all promises are believed -the issue outcome that a voter believes will occur if a candidate is elected is assumed to be identical to the candidate's point message. Thirdly, every individual's preferences are required to be complete over the entire issue space and often to decline with distance from an ideal point. Finally, candidates are usually assumed to perceive the preferences of all voters over all points in the issue space.
These requirements of Euclidean spatial models have been questioned by political scientists -Page (1975) is particularly critical. In this paper, a weakening of each of the above assumptions will be shown to lead naturally to a model employing a non-Euclidean outcome space which can be viewed as the set of points on the surface of a hypersphere. Under the primary interpretations to be offered in Section 1, this space is composed of the directions in which a status quo point in a Euclidean issue space can shift.
In Section 2 the basic model is described as a two-person plurality game in which the candidates adopt shift directions as strategies. Equilibrium directions in this game, however, are shown to be in the core of a corresponding n-person absolute majority rule game. Necessary and sufficient conditions are then easily established for the existence of an equi~-brium direction.
In Section 3 optimal strategies for a candidate competing against a rigid