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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

Throughout this Exercise Set, in computing standard divisors, standard quotas, and modified quotas, round to the nearest hundredth when necessary. A small country is comprised of four states, \(A, B, C\), and \(D\). The population of each state, in thousands, is given in the following table. Use this information to solve Exercises $1-4 . $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 138 & 266 & 534 & 662 & 1600 \\ \hline \end{array} $$ According to the country's constitution, the congress will have 80 seats, divided among the four states according to their respective populations. a. Find the standard divisor, in thousands. How many people are there for each seat in congress? b. Find each state's standard quota. c. Find each state's lower quota and upper quota.

a. A country has two states, state \(\mathrm{A}\), with a population of 9450 , and state \(B\), with a population of 90,550 . The congress has 100 seats, divided between the two states according to their respective populations. Use Hamilton's method to apportion the congressional seats to the states. b. Suppose that a third state, state \(C\), with a population of 10,400 , is added to the country. The country adds 10 new congressional seats for state C. Use Hamilton's method to show that the new-states paradox occurs when the congressional seats are reapportioned.

Describe the irrelevant alternatives criterion.

Three candidates, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\), are running for mayor. Election rules stipulate that the plurality method will determine the winner. In the event that the plurality method leads to a tie, the Borda count method will decide the winner. The election results are summarized in the following preference table. Under these rules, which candidate becomes the new mayor? $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 , 0 0 0} & \mathbf{7 5 0 0} & \mathbf{4 5 0 0} \\ \hline \text { First Choice } & \text { C } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$

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