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Chapter 13: Problem 21

Describe the head-to-head criterion.

Short Answer

Expert verified
The head-to-head criterion is a concept in voting theory, stating that if an option would win every head-to-head competition against all other options, it should be the overall winner. This criterion, though often associated with fairness, is not always respected in all voting systems.

Step by step solution

01

Understand the Head-to-Head Criterion

Firstly, understand what the head-to-head criterion is. It's a concept used in decision-making processes, most notably elections and voting. Suppose there are various candidates or options to choose from. The head-to-head criterion states that if option A would beat every other option in a one-on-one competition, option A should be the winner even when all options are considered simultaneously.
02

Explain the usage of the Criterion

It's also important to know when and how to use this criterion. The head-to-head criterion is used in different voting systems to determine a winner. Usually, you would list out all possible one-on-one matchups between each pair of candidates, and the candidate who wins in all head-to-head matchups is considered the 'overall' win according to this criterion.
03

Discuss its significance

Lastly, highlight the implication of the head-to-head criterion. This criterion is often associated with the concept of 'fairness' within voting systems. If a candidate or option can beat every other option in a head-to-head contest, most would agree that it makes sense for that option to be the winner. However, not all voting systems respect the head-to-head criterion, which can lead to contentious results

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Most popular questions from this chapter

Is it possible to have election results using a particular voting method that satisfy all four fairness criteria? If so, does this contradict Arrow's Impossibility Theorem?

Three people pool their money to buy 30 shares of stock. The amount that each person contributes is shown in the following table. Use Adams's method to apportion the shares of stock. (Hint: Find the standard divisor. A modified divisor that is greater than this standard divisor will work.) $$ \begin{array}{|l|c|c|c|} \hline \text { Person } & \text { A } & \text { B } & \text { C } \\ \hline \text { Amount } & \$ 795 & \$ 705 & \$ 525 \\ \hline \end{array} $$

Four professors are running for chair of the Natural Science Division: Professors Darwin (D), Einstein (E), Freud (F), and Hawking (H). The votes of the professors in the natural science division are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & 30 & 22 & 20 & 12 & 2 \\ \hline \text { First Choice } & \text { D } & \text { E } & \text { F } & \text { H } & \text { H } \\ \hline \text { Second Choice } & \text { H } & \text { F } & \text { E } & \text { E } & \text { F } \\ \hline \text { Third Choice } & \text { F } & \text { H } & \text { H } & \text { F } & \text { D } \\ \hline \text { Fourth Choice } & \text { E } & \text { D } & \text { D } & \text { D } & \text { E } \\ \hline \end{array} $$ Who is declared the new division chair using the plurality method?

Voters in a small town are considering four proposals, A, B, \(C\), and D, for the design of affordable housing. The winning design is to be determined by the Borda count method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & 300 & 120 & 90 & 60 \\ \hline \text { First Choice } & \text { D } & \text { C } & \text { C } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { A } & \text { A } & \text { D } \\ \hline \text { Third Choice } & \text { B } & \text { B } & \text { D } & \text { B } \\ \hline \text { Fourth Choice } & \text { C } & \text { D } & \text { B } & \text { C } \\ \hline \end{array} $$ a. Which design has a majority of first-place votes? b. Using the Borda count method, which design will be used for the affordable housing? c. Is the majority criterion satisfied? Explain your answer.

What is the plurality-with-elimination method? Why is it advantageous to rank the candidates when using this method?
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