Ever since Condorcet described the famous "paradox of voting" some 200 years ago, 1 political scientists have been aware that the process of direct majority-rule decision-making wilt not in general produce a stable outcome when the choice to be made is between more than two alternatives. Since majority-rule voting is a fundamental part of our decision-making apparatus, it is natural to ask what conditions are needed in Order to guarantee the existence of a stable decision. This question has intrigued both political scientists and mathematicians for some time. Some of the most interesting stability conditions have been derived using the so-called "spatial" model, in which alternative social states are viewed as points in a convex policy space, such as E n. Black and Newing [3] present a very complete and general analysis in geometrical terms for the 3-person case, where alternatives can be represented as points in E2; they extend some of their results to the N-person case. Plott [9] has found necessary and sufficient conditions for local stability in the finite-population case, where every individual's preferences can be represented by a differentiable