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Is There a Discrete Analog of the Median Voter Theorem?. James Bradley Calvin College Public Choice 2003. One dimensional median voter theorem. The one-dimensional spatial model: Policy space: the real line, Ideal points: voter i has a preferred position x i in
Public Choice 2003
The most widely appreciated benefit of the spatial model has been the explanatory power of the one-dimensional median voter theorem. If the discrete model is to be as helpful, it needs to provide comparable explanatory power.
In this case the model is very simple. All we can say is that in majority voting with two candidates, the one that gets a majority of the votes will win.
This is trivial, but it tells us the model makes sense in an elementary base case. Maybe the multidimensional model can offer more ...
In the n-dimensional case, n > 1, the natural analog to the median in all directions is the corner of the hypercube that corresponds to the majority’s preference on each issue separately. Call this the majority point.
Note that the model addresses simultaneous voting on composite outcomes. The majority point is formed by taking every voter’s first choice, then counting the votes on each issue separately.
Under what circumstances would the majority position necessarily win?
What factors are operative when it loses?
1. All voters cast their ballots lexicographically and rank the n dimensions in the same order of importance.
2. Voters attach a numerical weight to each dimension and assign points to alternatives additively. Furthermore, each voter attaches precisely the same weight to each dimension.
3. If there are k dimensions, the fraction of voters supporting the majority position is greater than (2k-1)/2k on every dimension.
Suppose all voters have separable preferences and suppose x wins in pairwise voting against all other alternatives. Then x is the majority point.
Corollary If all voters have separable preferences and x is not the majority point, x cannot defeat all other alternatives in pairwise voting.
If preferences are separable, an alternative positioned at the majority point may lose, but if it does, no other position is safe either.
What are the factors that could bring about a loss by the majority position?
At this time, the discrete spatial model has not yielded a result with the intuitive appeal and explanatory power of the single dimensional median voter theorem. Nevertheless, it has the potential to be more helpful in the multidimensional case than the continuous spatial model.