The spatial model of politics initially focused on the analysis of two agents, j and k, competing in a policy space X for electoral votes. The two agents (whether candidates or party leaders) are assumed to pick policy positions zj, zk, both in X, which they present as manifestos to a large electorate. Suppose that each member of the electorate votes for the agent that the voter truly prefers. When X involves two or more dimensions, then under conditions developed by Plott (1967), Kramer (1973), McKelvey (1976, 1979), Schofield (1978, 1983, 1985), and many others, there will generically exist no Condorcet or core point unbeaten under majority rule. That is to say, whatever position, zj, is picked by j there always exists a point zk that will give agent k a majority over agent j.
However, the existence of a Condorcet point has been established in those situations where the policy space is one-dimensional. In this case, the agents can be expected to converge to the position of the median voter (Downs, 1957). When X has two or more dimensions, it is known that a Condorcet point exists when electoral preferences are represented by a spherically symmetric distribution of voter ideal points. Even when the distribution is not spherically symmetric, a Condorcet point can be guaranteed as long as the decision rule requires a sufficiently large majority (Caplin and Nalebuff, 1988). Although a PNE generically fails to exist in competition between two agents under majority rule, there will exist mixed strategy equilibria whose support lies within a central electoral domain called the uncovered set.