Suppose seven people are trying to decide whether to get a pizza with pepperoni, a pizza with sausage and pepperoni, or a pizza with everything on it. Four people want everything, one wants pepperoni and sausage, and two want pepperoni only. Assume that each person prefers a pizza closer to his or her first choice to a pizza that is unlike the first choice. What is the preference of the median voter? Which pizza will be selected if the majority rules?
In my answer to this question, I will assume that there is going to be some sort campaigning going on with changes in people’s positions. Otherwise, there is no real issue here – the seven people will have a pizza with everything no matter what.
If we go with majority rule here, the seven will clearly have a pizza with everything. There are four people who want that sort of a pizza and only three who do not want it. Therefore, they will certainly buy a pizza with everything.
If we just look at what the median voter wants, the people will still get a pizza with everything. The median voter is the one who has an equal number of people on either side of him or her. The median voter in this case would be the fourth voter who wants a pizza with everything. There would be three who want as much or more on the pizza and three who want less.
But this does not seem like a satisfactory answer to this question because it tells us nothing about the meaning of the median voter theorem. So let us think about what would happen if voters somehow had to settle for their second choices. If this were to happen, the people would get a pizza with pepperoni and sausage. This is because that would be the second choice of 6 of the 7 people. All of the people who want everything on their pizza would rather have pepperoni and sausage than just pepperoni. The two people who want only pepperoni would rather have pepperoni and sausage than a pizza with everything. Thus, the seven would end up having a pizza with just pepperoni and sausage. This would mean that fewer people got their first choice, but it would also mean that everyone would be reasonably happy with the outcome.
I agree with your theory on the median-voter middle, as the median voter in an electorate often determines public choices, which is normally why political candidates try to appeal to the median voter, in order to get elected.