SPATIAL MODELS OF MAJORITY RULE

Chapter 5 of *Analyzing Politics *

K.A. Shepsle

A Summary

**Spatial Formulation**

- If preferences order themselves in particular ways that reflect a sort of consensus then group decisions by majority rule can occur. When preferences are mapped on a graph with utility on the
*y*axis and a scale of what is being decided on the*x*axis (e.g. a left wing right wing scale), and each individual has a most preferred point and preferences that decline as points further away in each direction are taken up, then preferences are said to be*single peaked*.

- The above graph shows single peaked preferences.
- So if a policy scale of Left to Right is used, an individual
*i*has a most preferred policy*x**. This is the*bliss point*the most preferred point on the scale, the point where he gets the maximum utility. As the policy moves away from*x**in either direction his utility decreases. So if the policy is at a point*x*which is away from his preferred point, we can describe a set of alternatives (*winset*) which he would strictly prefer to*x*and naturally this set will include the point*x**. If*x>x**then he prefers all polices*x-x**and also all policies*x*-(x-x*)*i.e. the set is symmetrical around his preferred point*x**. This is the case for all individuals, and the sets will overlap to a certain degree meaning there are policy points in common that certain individuals prefer to the point*x*. - We need to know which set of points a majority prefers to
*x*. This is the majority win set. If there is some alternative to*x,*say*Y*that has an empty winset (written W(*Y*) = ∅) then there is no possible coalition that any majority of choosers prefers to this point. This would be a good candidate for the group choice. If the winset is not empty, then there is a point preferred to*Y*that a majority prefers so it would be hard to argue that the group choice should be*Y*. - However, we know because of Arrow that it is possible for majority preferences to cycle meaning there is no point at which the winset is empty. But, if preferences are restricted by being single peaked it is easy to show that the ideal point of the median voter has an empty winset.
- Imagine a group of 5 legislators with single peaked preferences, as in the above graph. If policy is at any point that to the right of the 3 legislator’s bliss point, then a majority of legislators 1,2,3 will prefer the bliss point of the legislator 3 to that policy. For any policy to the right of the bliss point of legislator 3, then a coalition of 3,4,5 will prefer the bliss point of legislator 3 to that policy. However, when the policy is at the bliss of legislator 3 exactly, 4 and 5 may prefer points to the right, and 1 and 2 may prefer points the left, but no majority will prefer any point to either the left or the right. Hence the
*x**of legislator 3 has an empty winset and is the majority choice. - The assumptions behind this are that there are an odd number of legislators (otherwise the preferred point will be midway between the bliss points of the middle two legislators), that everyone participates in the vote, and that those voting do so sincerely.

**Multidimensional**

- Things become more complex when more than one dimension is being decided upon. See the chapter for a clear discussion.
- The upshot is that in a case with 3 legislators deciding upon 2 issues simultaneously, if legislators 1 and 2’s bliss points are distributed symmetrically around the bliss point of the median legislator, then the majority choice is legislator 2’s bliss point.
- With more legislators, Plott’s theorem states that if the ideal points are distributed in a radically symmetrical fashion then the winset of the preferred point of the median legislator has an empty winset. This describes the centripetal tendency observed in the one dimensional setting studied by Black.
- However, this result is highly sensitive to movements in the ideal points of the legislators. In reality it would be very odd if ideal points were distributed radically symmetrically around that of the median.
- Indeed McKelvey’s Chaos Theorem shows that in the absence of such a symmetrical distribution there is no empty winset point Instead there is chaos, no Condorcet winner, and anything can happen with the result that the final outcome can be determined by whoever controls the order of voting. As some point is always preferred by a majority it is difficult to justify any particular group choice.

**Spatial Elections**

- Political competition is a contest between those seeking to be in power. They appeal to voters using different policy platforms. For Antony Downs this meant that politicians seek to maximize votes.
- Politicians are represented at some point on the left-right scale. Each voter has single peaked preferences and a preferred point on this ideological scale. Each voter chooses the politician closest to their ideal point.
- If a leftist candidate is positioned somewhere to the left of the median voter and a right wing candidate is deciding what platform to offer, in order to maximize votes he should go a tiny way to the right of the Leftist candidate, and since L is already to the left of the midpoint of voter distribution R will get more than half the votes and win the election.
- L’s location divides the electorate in 2, those with ideals further right, and those with ideals further left. R must then position himself just next to L on the side of the larger group. L knows he will then get the smaller group, so in order to maximize his votes he needs to make that smaller group as large as possible. She does this by selecting ideal point of the median voter. R does the same and they now win the election with 50% probability each.
- If they announce their platforms simultaneously (rather than sequentially as above), the only point on the scale that is not vulnerable is the ideal point of the median voter.
- Perhaps they can change their platforms once in office. If the both announce policies on the same side of the median, there will be leapfrogging toward the median voter. If the announce on opposite sides they will both home in on the median from each side. In either case there is convergence toward the ideal point of the median voter.
- If both L and R are at the median, a third party could snuggle up close on one side and claim close to 50% of the vote, leaving L and R to share the remaining 50%. If L and R are sufficiently widely dispersed then she can enter in between and win all votes between her position and midway to the L position, and midway to the R position. If this part of the electorate is sufficient, she will win the election.
- There must in that case be entry-deterrence
- Other instances will occur when politicians have their own preferences which are known to the electorate. In this case they may run on policy of the median voter but voters will not believe them. They will suspect that once in office the politician will implement his preferred policies. Voters will vote on which politician’s true preferences are closer to their ideal point. The politician is unable to credibly commit. Thus there will not be convergence.
- Even supposing that candidates can make credible promises and that they care deeply about what policies are implemented, they will still have incentive to converge on the median. If the position of L and R are equidistant from the median, then they have a 50/50 chance of winning. But, L could move just a tiny shade inwards at little cost to her and win the election outright. But R has now lost the election, and L’s policies are very costly for him. He could avoid this outcome if he too moved just a shade toward the centre. And so on, until the median is reached.
- We do indeed observe that leaders try to tone down their extremist policies in order to win elections.
- Convergence is not always complete. Politicians may fear alienating their base supporters and being punish in the polls, or they fear a new entrant.

**Spatial Legislatures**

- As above it is the position of the median legislator that is important for policy decisions. If changes to the status quo can be offered and voted upon repeatedly, policy will converge upon his ideal point.

Things change if there is committee system in place. There are three possible decision making regimes:

- Pure majority rule – there is a status quo and any legislator can offer a motion to change it
- Closed-rule committee system – committee decides whether to propose a motion to change status quo and the legislature can either accept or reject the proposal but they cannot amend it.
- Open-rule committee system – committee decides whether to offer motion and the legislature can amend the proposal.

Pure majority rule: policy converges on the median policy and it will not move from there. This is the case no matter what the status quo is. All we need to know is the SQ and M_{L} point.

Closed rule committee system: the M_{C }(median committee member) point is also relevant. The set of points around the median legislator bounded by the SQ and its reflection on the opposite side of the median legislator is the opportunity set for the committee. They can propose policies within this set and get a yes vote. If outside this set they get a no vote, and hence will not propose any motion. There are three cases:

- SQ<M
_{L}<M_{C}: In this case*x**is the right bound of the preferred to SQ set of the median legislator. If M_{C}<*x**then the committee proposes its preferred point and gets a yes vote as its ideal point is within the set that will be approved by legislative majority. If M_{C}>*x**then the best the committee can do is to propose*x**which will move the policy closer to its ideal point. It cannot propose anything closer to its ideal point and get a yes vote in the legislature. - SQ<M
_{C}<M_{L}: In this case the ideal committee point is within the preferred to SQ set of the legislature, so the committee can propose its preferred point and get a yes vote in the legislature. - M
_{L}<SQ<M_{C}: The committee and the house want to move in opposite directions. Thus the committee keeps the gates closed as it would be impossible to get a yes vote for any policy closer to its ideal point.

- The upshot of this is that agenda power can prevent centripetal forces applying as outcomes are tugged in the direction of the agenda setting group.

Open Rule committee system: When the gates remain closed, the SQ prevails. If the gates are opened the M_{L} point is reached by amendments from the house. Nothing else is possible. Thus if the M_{C} prefers M_{L} to SQ then it will make a motion, if not it will keep the gates closed.

- If policy space is multidimensional and the ideal points are not distributed symmetrically around the median then the Chaos Theorem holds. However, if an agenda setter has the power to sequence the decisions, the outcome is the multidimensional median. The median ideal point on each dimension prevails.