Problems with the Median Voter Hypothesis
The existence problem is exacerbated if there is more than one issue in the election. In Fig. 12.5, we have depicted three voters’ preferences for two issues, G and H, in the left-hand panel and ideal points for voters A, B, and C in the right-hand panel. Each agent has single-peaked preferences. Segment AB is the set of common tangency points for voters A and B, segment BC is the set of common tangency points for voters B and C, and so on.
There is no point that intersects all three segments AB, BC, and AC; thus, there is no majority rule equilibrium even though each voter has single-peaked preferences. Consider a point on segment AC, say e, and suppose it is an equilibrium. Voter B can propose point f to voter A and voter A will support it over point e since it is closer to her or his bliss point. However, voter C can respond by offering voter B a point that she or he prefers to point f, and so on.
However, Plott (1967) showed that a “median voter” hypothesis may be salvaged if there is an ideal point for at least one voter and all the other voters come in pairs that are diametrically opposed to one another. In Fig. 12.6, we have depicted the ideal points for a number of voters.
First, consider the left-hand panel. The set of points on the line segment AB is a set of common tangency points between voters A and B, the set AC is a set of tangency points between voters A and C, and so on. Notice that point C, voter C's ideal point, is a majority voting equilibrium for the three voters. A point strictly in
Fig. 12.5 Multi-dimensional voting
Fig. 12.6 Generalizing the median voter hypothesis
between points A and C makes voter A better off but makes B and C worse off and would be voted against by B and C. The same applies for points strictly between B and C. Point C is the only point in the intersection of the sets AB, AC, and CB.
Now suppose we add a pair of voters, D and E, whose preferences are diametrically opposed to one another relative to voter C. Point C is still a majority voting equilibrium. Consider a point on the segment CE. Voter E is better off; however, A, B, C, and D are worse off and vote against it. The same applies for any other move away from point C. Voter C is a median voter in all directions. This assumption is quite restrictive and Plott concluded that it was unlikely to occur. Other analysts also provided necessary and sufficient conditions for existence. However, the required conditions tend to be highly restrictive.
The question then becomes: How prevalent is the problem of cycles? The conditions supporting the existence of a majority rule equilibrium are very restrictive and probably do not hold in the real world. If so, why is there so much apparent stability in the politics of the real world? If we seek more realistic information about voter preferences, the existence of a cycle may not be a serious problem. See Batina and Ihori (2005) for more discussions on this topic.