Median voter theorem:-
Show that when the policy space is one dimensional and the players' preferences are single-peaked the unique Condorcet winner is the median of the players' favorite positions. (This result is known as the median voter theorem.) A one-dimensional space captures some policy choices, but in other situations a higher dimensional space is needed.
For example, a government has to choose the amounts to spend on health care and defense, and not all citizens' preferences are aligned on these issues. Unfortunately, for most configurations of the players' preferences, a Condorcet winner does not exist in a policy space of two or more dimensions, so that the core is empty. To see why this claim is plausible, suppose the policy space is two-dimensional and there are three players. Place the players' favorite positions at three arbitrary points, like x∗1, x∗2, and x∗3 in Figure 1.
Assume that each player i's distaste for a position x different from her favorite position x∗i is exactly the distance between x and x∗i, so that for any value of r she is indifferent between all policies on the circle with radius r centered at x∗i.
