G180 Module 05 Assignment 1: A Group Of Students Were 600564
G180 Module 05 Assignment1 A Group Of Students Were Asked To Vote On
A group of students were asked to vote on their favorite horror films. The candidate films are: Abraham Lincoln Vampire Hunter, The Babadook, Cabin Fever, and Dead Snow (A, B, C, D for short). The following table gives the preference schedule for the election. Use the pairwise-comparisons method to find the winner of the election.
Three players (April, Brandy and Cindy) are sharing a cake. Suppose that the cake is divided into three slices (s1, s2, s3). The following table gives the value of each slice in the eyes of each of the players. (A fair share would be 1/3 = .333 = 33.3% or greater.)
| s1 | s2 | s3 | |
|---|---|---|---|
| April | $4.50 | $5.50 | $5.00 |
| Brandy | $5.50 | $5.25 | $5.00 |
| Cindy | $5.60 | $5.00 | $5.00 |
a) Which of the three slices are fair shares to April?
b) Which of the three slices are fair shares to Brandy?
c) Which of the three slices are fair shares to Cindy?
d) Find a fair division of cake using s1, s2, and s3 as fair shares. If this is not possible, explain why not.
Three players (Adam, Bob and Chad) are sharing a cake. Suppose that the cake is divided into three slices (s1, s2, s3). The percentages represent the value of the slice as a percent of the value of the entire cake. (A fair share would be 1/3 = .333 = 33.3% or greater.)
| s1 | s2 | s3 | |
|---|---|---|---|
| Adam | 30% | 50% | 20% |
| Bob | 32% | 36% | 32% |
| Chad | 30% | 35% | 35% |
a) Which of the three slices are fair shares to Adam?
b) Which of the three slices are fair shares to Bob?
c) Which of the three slices are fair shares to Chad?
d) Find the fair division of the cake, using s1, s2 and s3 as fair shares. If this is not possible, explain why not.
Paper For Above instruction
The analysis of voting preferences using pairwise comparison methods provides insights into the overall preference hierarchy among candidates. In this context, the candidates are four horror films: Abraham Lincoln Vampire Hunter, The Babadook, Cabin Fever, and Dead Snow. The pairwise comparison approach involves comparing each candidate directly against every other candidate based on voters’ preferences, which reveals the strongest overall contender without the need for a voting quota or plurality method. This method is particularly useful in understanding the collective choice in elections where preferences are ranked, ensuring that the winner has a consistent majority advantage over each alternative when compared pairwise. For the given preference schedule, constructing a pairwise comparison matrix allows us to tally the number of voters favoring each candidate over the others and identify the candidate with the most wins in these comparisons.
Regarding the allocation of cake slices among players, equality and fairness are the central principles. When considering April’s valuations, the total value of the slices should be assessed to determine which slices meet or exceed the fair share threshold of one-third of the total cake value. For April, calculating the proportion of total valuation that each slice represents helps identify slices that meet or surpass this threshold. Similarly, for Brandy and Cindy, their valuation of each slice highlights the slices they consider fair shares—slices whose valuations reach or exceed their own one-third proportional share. The goal in achieving a fair division involves balancing these individual preferences to distribute slices such that each participant receives at least their fair share or an equitable division that respects their valuations.
In the scenario involving Adam, Bob, and Chad, their percentage valuations of each slice offer a clear view of what constitutes a fair share for each. A fair share is defined as at least 33.3% of the total cake’s value, which in percentage terms is approximately one-third or greater. By examining each individual’s valuation, it is possible to identify which slices they consider fair, satisfying the criterion of a third or more of the total value. Creating a division that considers these individual valuations involves matching slices or fragments thereof where each participant's minimum fair share condition is satisfied. Challenges may arise if no such division exists that satisfies all preferences simultaneously, necessitating an explanation of the impossibility of a fair share allocation based on the given valuations.
Overall, these problems illustrate essential concepts in collective decision-making and fair division, emphasizing methods such as pairwise comparison for election outcomes and proportional valuation for resource allocation. These approaches ensure that collective choices and resource sharing are conducted equitably, reflecting each participant’s valuation as closely as possible and adhering to principles of fairness and majority preference when applicable.