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The One-Dimensional Voting Model allows for certain assumptions to hold that result in a Condorcet winner and no cycle. Voters choose among points on a line so that each voter prefers points on the line closest to his or her favorite point on the line. The point on the line that beats all of the other options, under the Majority Rule, is the point that the median voter favors the most. Although this model proves there can be a Condorcet winner under the Majority Rule of voting, it is a simplistic model, since it can only prove this result when certain assumptions are made, such as in a single regression rather than in multiple regressions.
In the One-Dimensional Model, the outcome of ...view middle of the document...

If the default budget is $0, then almost anything is better than $0. Since this would be on the farthest left option on the linear model explained earlier. In the linear model explained earlier, this default option can be expressed as D=0. Therefore, if the only other option on the ballet is a very large budget, L, this is the obvious option the majority will vote for. So in this case, the outcome will result in passing the over-priced school budget, since this outcome is better than the default outcome of no school budget. This example proves that the outcome depends on what the default outcome is and what other option is available on the ballet.
The outcome in the previous example would not have passed, however, when instead of a public referendum, the proposed budget could be amended. The diagram below represents what would happen if the voters had been allowed to propose another alternative closer to the median, through an amendment:

L a

L D a D
In the above model, L represents the proposed budget, a represents the amended proposal to change the original proposal, and D represents the default outcome. Since the amended proposal is the closest option to the median, the outcome of this example will be that a will pass.
When voters vote for candidates rather than propositions, the electorate is pushed towards the middle as well. In the case where voters vote for people rather than propositions, the candidates tend to move toward the middle rather than remain on extreme ends of the line. Since political competitors must receive the most votes to win under Majority Rule, each candidate competes by moving across the line, to the middle. To make this conclusion, however, it is assumed that there are two candidates, the voters know where each candidate is on the line, the candidates know enough to know where the median is, and the candidates are mobile so they can respond to the other’s position by moving toward the middle. This example is expressed in the following model:

C2 C2 C1 C1

≤1/2 # of voters ≤1/2 # of voters

In the model above, there are equal voters on each side of the median, the yellow triangle. C1 represents Candidate #1 and C2 represents Candidate #2. The orange and green lines represent the majority on the left and on the right respectively. In order to gain a greater majority, C2 makes an initial move toward the middle. In response, C1 makes an even greater move toward the middle, to gain a larger majority of votes. Therefore, competition between the two candidates leads to the outcome in which both of them meet in the middle.
In the previous models, the One-Dimensional Model has proved two results. The first conclusion is that the median is the Condorcet winner, since it is the stable outcome in which no other outcome can beat it. The second result is that a median must exist in the linear One-Dimensional model,...

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