[In a well known paper, Plott has given a sufficient and a necessary condition onthe set of gradients of individual preferences at a point in a multidimensional space, for the point to be an equilibrium under simple majority voting. This paper defines a class of α-majority voting rules under which, given some α, 0 extless α extless 1, an alternative x is socially at least as good as y iff the number of individuals who prefer x to y is at least α/(1 - α) times the number who prefer y to x. Simple majority rule is α = 1/2 while setting α near 0 and 1 gives two types of unanimity rule. For all elements in this class, this paper generalizes Plott by giving necessary and sufficient conditions on the set of gradients for a point in a multidimensional space to be a voting equilibrium.]