2. Does the median voter always determine the outcome of any majority vote?
Let’s note some of the conditions which led to our outcome. The most important
concerns the ordering of each individual’s voting preferences. In the figure below,
we rank the preferences for each voter.
With individual 1 (I1), Bill C is preferred to B, and B is preferred to A. The blue line connects these (ordered) points. Similarly for individuals 2 and 3. Note that, for each individual, moving away from the 1st choice pick results in steadily lower choice picks. For example, none of these individuals has a preference ordering of C to A to B. In such a case, as we move from the top choice of C we arrive at the third choice (B) and then the second choice (A), rather than steadily decreasing as we do above.
What would happen if individual 3 adopts the preference ordering C to A to B. Now the 3 individuals have these preferences:
The figure, from above, changes to become:
Using the majority voting order from above, we get the following result.
| Bill A vs. Bill B | A wins |
| Bill A vs. Bill C | C wins |
However, suppose we revise the order (A vs. C, winner vs. B)? If so, we get:
| Bill A vs. Bill C | C wins |
| Bill B vs. Bill C | B wins |
Similarly, we can make the order B vs. C, winner vs. A. That yields A as the eventual
winner. Unlike our earlier example, the outcome of this majority voting situation depends
upon the order of the vote. Whoever controls the order controls the outcome. This makes
room for something called agenda setting.
The reason for the different outcome, where the median voter doesn’t determine the
outcome, is that the preferences are different. In our earlier example, preferences are
"single peaked". That is, by moving away from the top choice, the preferences
decrease in value accordingly (unlike what we see with C to A to B).
This leads to something called the median voter theorem, which states "If preferences
are single-peaked, then the median voter determines the outcome in a majority vote."