/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 Explain why the plurality method... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An election with six candidates \((A, B, C, D, E,\) and \(F)\) is decided using the method of pairwise comparisons. If \(A\) loses four pairwise comparisons, \(B\) and \(C\) both lose three, \(D\) loses one and ties one, and \(E\) loses two and ties one, (a) find how many pairwise comparisons \(F\) loses. (b) find the winner of the election.

The following simple variation of the conventional Borda count method is sometimes used: last place is worth 0 points, second to last is worth 1 point,..., first place is worth \(N-1\) points (where \(N\) is the number of candidates). Explain why this variation is equivalent to the conventional Borda count described in this chapter (i.e., it produces exactly the same winner and the same ranking of the candidates).

Explain why the Borda count method satisfies the monotonicity criterion.

The AP college football poll is a ranking of the top 25 college football teams in the country. The voters in the AP poll are a group of sportswriters and broadcasters chosen from across the country. The top 25 teams are ranked using a conventional Borda count: a first-place vote is worth 25 points, a second- place vote is worth 24 points, a third-place vote is worth 23 points, and so on. A last-place vote is worth 1 point. Table \(1-44\) shows the ranking and total points for each of the top three teams at the end of the 2006 regular season. (The remaining 22 teams are not shown here because they are irrelevant to this exercise.) $$\begin{array}{|l|c|}\hline {\text { Team }} & \text { Points } \\\\\hline 1 . \text { Ohio State } & 1625 \\\\\hline \text { 2. Florida } & 1529 \\\\\hline \text { 3. Michigan } & 1526 \\\\\hline\end{array}$$ (a) Given that Ohio State was the unanimous first-place choice of all the voters, find the number of voters that participated in the poll. (b) Given that all the voters had Florida in either second or third place, find the number of second-place and the number of third-place votes for Florida. (c) Given that all the voters had Michigan in either second or third place, find the number of second-place and the number of third-place votes for Michigan.

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.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.