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 xi in

Preferences: y >i z <=> |y-xi| < |z-xi|

Decision rule: simple majority

Conclusion: If y is the median position for the electorate, then y cannot lose the election.

Multidimensional median voter theorem

The multidimensional spatial model:

Policy space: Euclidean space, n

Ideal points: voter i has a preferred position xi in n

Preferences: y >i z <=> WED (y, xi )< WED(z, xi)

Decision rule: simple majority

Conclusion: If y is the median position in all directions for the electorate, then y cannot lose the election.

Median in all directions

Instability

Contract curve

More sophisticated versions of the basic spatial model

• Regard the previous model as applicable to committee voting where voters have direct access to policy space

• Treat candidates as having a fixed position in a one or two dimensional predictive space and voters as selecting candidates who are mapped into a multidimensional policy space. Voters now have only indirect access to policy space and are represented by matrices.

• Combine discrete non-policy issues with real-valued policy issues using utility measures.

Some reflections on the basic model

• Euclidean space

Reasonable approximation for monetary & proportionality issues

However, many issues are intrinsically discrete - e.g., death penalty, Arctic oil drilling, war with Iraq.

• Voter ideal point

Abstracts away the process of selecting it

• Preferences based on Euclidean distance

There are other metrics.

Discrete model

The discrete spatial model:

Policy space: Corners of a hypercube, {0,1}n

Ideal points: voter i has a preferred corner xi

Preferences: given by a preference matrix

Decision rule: simple majority

Conclusion: ...

Preference matrices

111

110

101

011

001

000

100

010

Why preference matrices?

• They don’t depend on the arbitrary assumption of preferences being representable by a Euclidean metric.

• They can represent any preferences on {0,1}n.

• They are tractable as long as n isn’t too large.

What would we lose by employing a discrete rather than

a continuous model?

• Geometric intuition

•The capacity to deal with continuous policy issues.

• The contract curve and the subsequent analysis of cyclicity it yields.

• The capacity to minimize distances from the ideal point by using calculus.

• Most importantly, the median voter theorem.

What would we gain?

• Another tool that can enhance our understanding of voting.

• Freedom from the need to artificially force situations into a real-valued framework.

• Some fresh insights into voting.

•A united treatment of both the policy space and non-policy issues in a single approach.

The goal

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.

One dimensional discrete model

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 ...

The majority point

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.

Cyclicity

A candidate (or a proposal in committee voting) positioned at the majority point may not win an election. For example, consider three voters and two issues.

V1 V2 V311 01 0010 11 1001 00 0100 10 11

Note that all three voters have separable preferences. 01 is the majority position. Nevertheless, in majority voting

00 > 10 > 01 > 11 > 00.

The central questions

Under what circumstances would the majority position necessarily win?

What factors are operative when it loses?

Sufficient conditions for a winning majority point

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.

A necessary condition involving the majority point

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.

What insights can we glean from this?

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?

The key factors seem to be:

• Inseparability

• Differing perspectives on the importance of issues

• The presence of single issue voters in the electorate

Perceptions of the importance of issues

A candidate in an election who perceives herself as being in the minority on an issue has three plausible campaign strategies:

• Try to fuzzy both her position and her opponent’s.

• Try to reduce the importance voters give to the issue.

• Try to convince her supporters to be single-issue voters on this issue while going after her opponent’s voters on other issues.

Perceptions of the importance of issues

The majority candidate on an issue also has strategies:

• Try to sharpen voters’ perceptions of both his and his opponent’s position.

• Raise the importance of the issue to voters.

Single issue voters

• A candidate whose positions do not match the majority on any issue can win an election by assembling a coalition of single issue voters on various issues.

• Single issue voting can be an effective form of strategic voting.

• This model helps explain why special interest groups like the Sierra Club and Right to Life urge their supporters to be single issue voters.

Conclusion

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.