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Chapter 1: Problem 58

Explain why the plurality method satisfies the monotonicity criterion.

Short Answer

Expert verified
The plurality method satisfies the monotonicity criterion because if a candidate who wins an election under plurality voting receives more votes in a new election, their total votes increase, and since they already had the most votes, they continue being the winner. No one passes them in votes, hence monotonicity is satisfied.

Step by step solution

01

Understand the Plurality Method

In the plurality method, the candidate who receives more votes than any other candidate wins the election. Each voter is allowed to vote for only one candidate. Let's say we have a candidate X who wins an election using the plurality method.
02

Understand the Monotonicity Criterion

With the monotonicity criterion, if candidate X wins, and in a subsequent election all voters still keep their preferences with some of them changing their votes in favor of X, then candidate X should remain the winner.
03

Apply the Monotonicity Criterion to the Plurality Method

If candidate X wins an election using the plurality method, it means that they received the most number of votes. If in a new election some voters change their votes in favor of candidate X, this will increase candidate X's vote count, and not decrease it. Therefore, as candidate X already had the highest number of votes, increasing this number can't make him lose. So the plurality method satisfies the monotonicity criterion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plurality Method
The plurality method is a simple and widely used voting system. It determines the winner by the count of first-place votes each candidate receives. In this method:
  • Voters cast their vote for a single candidate.
  • The candidate with the most votes wins.
This method does not consider any further preferences of the voters beyond their first choice. It is straightforward, making it popular in many elections. However, plural voting might not reflect the voters鈥� preferences fully, especially when there are more than two candidates. This is because it solely focuses on the person with the most votes, without considering how much support might exist for other candidates.
Monotonicity Criterion
The monotonicity criterion is an important concept in voting theory, ensuring fairness and consistency in election outcomes. According to this principle:
  • If a candidate wins an election, they should not lose in a subsequent election if they gain additional support.
  • Conversely, if a candidate loses, they should not win if they lose support.
This criterion checks that gaining more votes should not disadvantage a candidate. For example, if candidate X wins and then gains more votes, they should still be the winner. Monotonicity ensures stability in the voting system and upholds the notion that more support strengthens a candidate's position, rather than weakening it.
Election Methods
Election methods are various systems used to conduct elections and determine winners. The choice of method impacts how votes are counted and how winners are announced. Some common election methods include:
  • Plurality Voting: Winner takes all based on who gets the most votes.
  • Runoff Voting: If no candidate gets a majority, the top candidates have another round.
  • Instant-runoff Voting: Voters rank candidates, and the counting simulates a series of runoffs.
Each method has its advantages and drawbacks. For example, while the plurality method is simple, it might not always reflect the most broadly approved candidate. Different contexts may call for different methods to more accurately represent voters' preferences and ensure fair outcomes.
Voting Systems
Voting systems encompass the entire framework and procedures of managing and conducting elections. They cover everything from voter registration processes to vote counting and ensuring transparency and fairness. Key aspects of voting systems include:
  • Fairness: Ensuring each vote carries equal weight.
  • Security: Protecting the integrity of the vote.
  • Accessibility: Offering all eligible voters easy ways to participate.
An ideal voting system accurately reflects the will of the people while maintaining trust through secure and fair handling of the election process. Different voting theory approaches may be applied to design systems that cater to specific societal needs and democratic principles.

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Most popular questions from this chapter

Consider a modified Borda count where a first-place vote is worth \(F\) points \((F>N\) where \(N\) denotes the number of candidates) and all other places in the ballot are the same as in the ordinary Borda count: \(N-1\) points for second place, \(N-2\) points for third place, \(\ldots, 1\) point for last place. By choosing \(F\) large enough, we can make this variation of the Borda count method satisfy the majority criterion. Find the smallest value of \(F\) (expressed in terms of \(N\) ) for which this happens.

Explain why when there are only two candidates, the four voting methods we discussed in this chapter give the same winner and the winner is determined by straight majority. (Assume that there are no ties.)

The Demublican Party is holding its annual convention. The 1500 voting delegates are choosing among three possible party platforms: \(L\) (a liberal platform), \(C\) (a conservative platform), and \(M\) (a moderate platform). Seventeen percent of the delegates prefer \(L\) to \(M\) and \(M\) to \(C\) Thirty-two percent of the delegates like \(C\) the most and \(L\) the least. The rest of the delegates like \(M\) the most and \(C\) the least. Write out the preference schedule for this election.

Imagine that in the voting for the American League Cy Young Award ( 7 points for first place, 4 points for second, 3 points for third, 2 points for fourth, and 1 point for fifth) there were five candidates \((A, B, C, D,\) and \(E)\) and 50 voters. When the points were tallied \(A\) had 152 points, \(B\) had 133 points, \(C\) had 191 points, and \(\mathrm{D}\) had 175 points. Find how many points \(E\) had and give the ranking of the candidates.

Explain why the method of pairwise comparisons satisfies the monotonicity criterion.
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