In the spatial context, when preferences can be characterized by circular indifference curves, we show that we can derive all the information about the majority preference relationship in a space from the win-set of any single point. Furthermore, the size of win sets increases for points along any ray outward from a central point in the space, the point that is the center of the yolk. To prove these results we employ a useful new geometric construction, the half-win set. The implication of these results is that embedding choice in a continuous n-dimensional space imposes great constraints on the nature of the majority-preference relationship.
In finite voting games knowledge of the majority-preference relation between some given alternative, ai, and each of the remaining alternatives aj e A tells us nothing whatsoever about the directionality of majority preference between pairs in which a i is not included, for example, between a e and a k. It might seem that imposing a spatial structure on alternatives would impose some constraints on the overall structure of majority preferences. But a remarkably strong result holds. If we know the geometry of the win set of any point x, then, when preferences are characterized by circular indifference curves, we can reconstruct the win-set of any other point in the space; that is, in the spatial context, if we know a single win-set, we can specify the complete structure of majority preference for the space; we need not know either the number of voters or the location of voters' ideal points.
Definition 1: The win set of y, denoted Win(y), is the set of alternatives xeX such that xPy.