Existence Of Condorcet Equilibrium

We study the necessary and sufficient conditions for the existence of equilibrium for majority rule. A new and an alternative set of conditions for the existence of equilibrium for majority rule are proposed, which increases the three Plott conditions by three. They are an angle, a convex cone, and a cardinality condition, which, incidentally, seem to be well-suited to put into some sort of order the confusion concerning the celebrated Plott conditions (the assumptions of strict pair-wise symmetry, an odd number of voters, and the existence of a centrally located voter) and the situations when some or all of Plott's conditions are not satisfied. It turns out that all these six conditions have to be checked in order to ascertain the necessary and sufficient conditions that have to be satisfied for all possible preference profiles and all possible numbers of voters. Since there are a number of relationships between these conditions, those relationships also need to be explored, which of course indeed are explored. We also study the question whether or not the expansion and/or contraction of the membership of the decision-making body has a stabilizing or a destabilizing influence on the equilibrium of the resulting decision-making body. This latter stability aspect has some highly interesting applications, notably to customs union theory of international trade and other alliances.Presented at 15th International Conference, Istanbul, Turkey, May 2005.