The half-win set and the geometry of spatial voting games

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 vot ing games knowledge of the major i ty-preference relat ion between some given alternative, ai, and each of the remain ing alternatives aj e A tells us no th ing whatsoever abou t the direct ionali ty of major i ty 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 s tructure on alternatives would impose some constraints on the overall s t ructure of major i ty preferences. But a remarkably strong result holds. If we know the geometry of the win set of any poin t 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 major i ty preference for the space; we need not know either the n u m b e r of voters or the locat ion of voters ' ideal points. D e f i n i t i o n 1: The win set o f y , denoted Win(y), is the set of al ternatives xeX such that xPy. * The listing of authors is alphabetical. We are indebted to the staff of the Word Processing Center, School of Social Sciences, UCI, for typing earlier drafts of this manuscript, to Cheryl Larsson for preparing the figures, and to Dorothy Gormick for bibliographic assistance. This research was partially supported by NSF Grant #SES 85-06397, Program in Management Sciences, awarded to the second-named author.