9.3 Other Voting Systems for Three or More Candidates

Size: px
Start display at page:

Download "9.3 Other Voting Systems for Three or More Candidates"

Transcription

1 9.3 Other Voting Systems for Three or More Candidates With three or more candidates, there are several additional procedures that seem to give reasonable ways to choose a winner. If we look closely at each of these, however, we find that each one has certain problems. The first of these other voting systems is called Plurality Voting, in which only first place votes are considered. The shortcoming of this method has already been mentioned, as in the case of the 2000 presidential election. The candidate with the most votes wins, even though they may not get a majority of the votes. As Example 2 (p ) illustrates, by using the plurality voting method, George Bush was the winner of the election, even though Al Gore was the Condorcet winner, meaning that he would have won if Condorcet's method had been used. A voting system is said to satisfy the Condorcet winner criterion (CWC) provided that, for every possible sequence of preference list ballots, either (1) there is no Condorcet winner (as is often the case) or (2) the voting system produces exactly the same winner for the election as does Condorcet's method. The Florida vote in the 2000 presidential election shows that plurality voting fails to satisfy the CWC. Use the preference list ballot above to determine the: a) plurality winner b) Condorcet winner 1

2 Another drawback of plurality voting is that it doesn't give voters a chance to express any preferences except for naming his or her top choice. Another shortcoming of plurality voting is that it is sometimes to a voter's advantage to submit a ballot that misrepresents his or her true preferences, which is referred to as manipulability. A voting system is subject to manipulability (or is manipulable) if there are elections in which it is to a voter's advantage to submit a ballot that misrepresents his or her true preferences. For example, suppose you really like and respect the Green party candidate in a particular race, but you don't want to "throw away" your vote by voting for him or her, so you vote for one of the major (Democratic or Republican) party candidates instead. It turns out that Condorcet's method is not manipulable, which is another reason that it is good. 2

3 Let's create another preference list ballot involving favorite restaurants and use it to determine the: a) plurality winner b) Condorcet winner 3

4 HW: p. 351: 9-11, 14, 15, 18 4

5 In many elections that use preference list ballots, the goal is to arrive at a final group rank ordering of all the candidates that best expresses the wishes of the voters. The purpose is not only to determine the winner, but also to find a second place, third place, etc., or for honorable mention, or also ran status. One common mechanism for achieving this objective is to assign points to each voter's ranking and then to sum these for all voters to obtain the total points for each candidate. If there are 4 candidates, for example, we could assign 3 points to the 1st choice, 2 points to the 2nd choice, 1 point to the 3rd choice, and 0 points to the 4th choice. Description of Rank Methods and the Borda Count: A rank method of voting assigns points in a non increasing manner to the ordered candidates on each voter's preference list ballot and then sums these points to arrive at a group's final ranking. The special case in which there are n candidates with each first place vote worth n 1 points, each second place vote worth n 2 points, and so on, down to each last place vote worth 0 points, is known as the Borda count. The actual point totals are referred to as a candidate's Borda score. Example: Calculate the winner below using the Borda count

6

7 7

8 The Borda count is named after Jean Charles de Borda, who was a contemporary of Condorcet. It seems to be a reasonable way to choose a winner from among several candidates, but it too has shortcomings, one of which is the failure of a property known as independence of irrelevant alternatives. A voting system is said to satisfy independence of irrelevant alternatives (IIA) if it is impossible for a candidate X to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had X and the winning candidate ranked. For example, suppose an election yields A as a winner and B as a nonwinner. Suppose a new election is held and some people change their ranking, but they don't move B above A. If B were to win the new election, this would not satisfy the IIA. Condorcet's method satisfies IIA, but Borda count does not, as seen in the following example. The Borda count shows that A is the winner, while B and C are nonwinners. Now, consider the following vote. 8

9 HW: p : 14b, 15b, 16a&b, 17a&b, 18b, 19a&b, 20a&b 9

10 The next voting method we will discuss is a type that uses an agenda, which is a listing of the candidates in some order. To avoid confusion, we will present agendas as horizontal lists and continue to present preference list ballots as vertical lists. Sequential pairwise voting starts with an agenda and pits the first candidate against the second in a one on one contest. The winner then moves on to confront the third candidate in the list one on one. Losers are deleted. This process continues throughout the entire agenda, and the one remaining at the end wins. We will see that the particular agenda can change the outcome of the vote. Also, as long as the number of voters is odd, sequential pairwise voting never ends in a tie. Example: Assume we have four candidates and that the agenda is A, B, C, D. Consider the following preference list ballot. 10

11 There is something very troubling about the outcome of that example, since everyone prefers B to D, but D ends up winning! This example shows that sequential pairwise voting fails to satisfy what is called the Pareto condition, named after the Italian economist Vilfredo Pareto. A voting system is said to satisfy the Pareto condition provided that in every election in which every voter prefers candidate X to candidate Y, the latter candidate Y is not among the winners. Note that the agenda order can affect the outcome of the election. Consider the agenda C, A, D, B with the same preference list ballot from the previous example. 11

12 Use the preference list ballots below with the given agenda to find the pairwise sequential winner. 1) Agenda: C, A, B 2) Agenda: B, A, D, C 3) Agenda: A, C, D, B 12

13 The voting system known as the Hare system, which was introduced by Thomas Hare in 1861, is used to elect public officials in Australia, Malta, and Ireland. The Hare system arrives at a winner by repeatedly deleting candidates that are "least preferred" in the sense of being at the top of the fewest ballots. If a single candidate remains after all others have been eliminated, he or she alone is the winner. If two or more candidates remain and all of these remaining candidates would be eliminated in the next round (because they all have the same number of first place votes), then these candidates are declared to be tied for the win. Example: Suppose we have the following preference list ballot. Since C only has one first place vote (compared to two for A and B), C is eliminated in the first round. The second round ballot deletes all of C's votes and moves the candidates up to look as follows: Since B only has two first place votes (compared to three for A), B is eliminated in the second round. Since A is the only candidate remaining, A is the winner. 13

14 Example: Calculate the winner in the ballot below using the Hare system. Now, consider the same ballot with one change: The last voter moved A above B, which should only help A, right? Calculate the winner of this ballot using the Hare system. This example shows that the Hare system fails to satisfy what is called monotonicity. A voting system for three or more candidates is said to satisfy monotonicity provided that, for every election, if some candidate X is a winner and a new election is held in which the only ballot change made is for some voter to move this winning candidate X higher on his or her ballot (and to make no other changes), then X will remain a winner. The fact that the Hare system does not satisfy monotonicity is considered by many to be a glaring defect. However, it is still used in important ways today, such as to determine that Rio de Janeiro would be the site of the 2016 Summer Olympics. 14

15 Example: Calculate the winner in each ballot below using the Hare system. 15

16 HW: pp : 14c&d, 15c&d, 16c&d, 17c&d, 18c&d, 19c&d, 20c&d and read

17 The last voting system in this section is the Plurality runoff method, in which there is a runoff (that is, a new election using the same ballots) between the two candidates receiving the most first place votes. If there are ties, then the runoff is among either those tied for the most 1st place votes, or the lone candidate with the most 1st place votes along with those tied for 2nd place (and plurality voting is used.) This is the system used in most elections in the U.S. (other than presidential elections). It seems similar to the Hare system, but the following example illustrates the difference. 17

18 Find the winner of each ballot below using the Plurality runoff method. 18

Chapter 9: Social Choice: The Impossible Dream Lesson Plan

Chapter 9: Social Choice: The Impossible Dream Lesson Plan Lesson Plan For All Practical Purposes An Introduction to Social Choice Majority Rule and Condorcet s Method Mathematical Literacy in Today s World, 9th ed. Other Voting Systems for Three or More Candidates

More information

Elections with Only 2 Alternatives

Elections with Only 2 Alternatives Math 203: Chapter 12: Voting Systems and Drawbacks: How do we decide the best voting system? Elections with Only 2 Alternatives What is an individual preference list? Majority Rules: Pick 1 of 2 candidates

More information

Social Choice: The Impossible Dream. Check off these skills when you feel that you have mastered them.

Social Choice: The Impossible Dream. Check off these skills when you feel that you have mastered them. Chapter Objectives Check off these skills when you feel that you have mastered them. Analyze and interpret preference list ballots. Explain three desired properties of Majority Rule. Explain May s theorem.

More information

Chapter 9: Social Choice: The Impossible Dream Lesson Plan

Chapter 9: Social Choice: The Impossible Dream Lesson Plan Lesson Plan For ll Practical Purposes Voting and Social hoice Majority Rule and ondorcet s Method Mathematical Literacy in Today s World, 7th ed. Other Voting Systems for Three or More andidates Plurality

More information

Voting: Issues, Problems, and Systems, Continued

Voting: Issues, Problems, and Systems, Continued Voting: Issues, Problems, and Systems, Continued 7 March 2014 Voting III 7 March 2014 1/27 Last Time We ve discussed several voting systems and conditions which may or may not be satisfied by a system.

More information

SOCIAL CHOICES (Voting Methods) THE PROBLEM. Social Choice and Voting. Terminologies

SOCIAL CHOICES (Voting Methods) THE PROBLEM. Social Choice and Voting. Terminologies SOCIAL CHOICES (Voting Methods) THE PROBLEM In a society, decisions are made by its members in order to come up with a situation that benefits the most. What is the best voting method of arriving at a

More information

Mathematical Thinking. Chapter 9 Voting Systems

Mathematical Thinking. Chapter 9 Voting Systems Mathematical Thinking Chapter 9 Voting Systems Voting Systems A voting system is a rule for transforming a set of individual preferences into a single group decision. What are the desirable properties

More information

Chapter 10. The Manipulability of Voting Systems. For All Practical Purposes: Effective Teaching. Chapter Briefing

Chapter 10. The Manipulability of Voting Systems. For All Practical Purposes: Effective Teaching. Chapter Briefing Chapter 10 The Manipulability of Voting Systems For All Practical Purposes: Effective Teaching As a teaching assistant, you most likely will administer and proctor many exams. Although it is tempting to

More information

Math for Liberal Arts MAT 110: Chapter 12 Notes

Math for Liberal Arts MAT 110: Chapter 12 Notes Math for Liberal Arts MAT 110: Chapter 12 Notes Voting Methods David J. Gisch Voting: Does the Majority Always Rule? Choosing a Winner In elections with more then 2 candidates, there are several acceptable

More information

Exercises For DATA AND DECISIONS. Part I Voting

Exercises For DATA AND DECISIONS. Part I Voting Exercises For DATA AND DECISIONS Part I Voting September 13, 2016 Exercise 1 Suppose that an election has candidates A, B, C, D and E. There are 7 voters, who submit the following ranked ballots: 2 1 1

More information

Voting: Issues, Problems, and Systems, Continued. Voting II 1/27

Voting: Issues, Problems, and Systems, Continued. Voting II 1/27 Voting: Issues, Problems, and Systems, Continued Voting II 1/27 Last Time Last time we discussed some elections and some issues with plurality voting. We started to discuss another voting system, the Borda

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Arrow s Impossibility Theorem Some announcements Final reflections due on Monday. You now have all of the methods and so you can begin analyzing the results of your election. Today s Goals We will discuss

More information

The Manipulability of Voting Systems. Check off these skills when you feel that you have mastered them.

The Manipulability of Voting Systems. Check off these skills when you feel that you have mastered them. Chapter 10 The Manipulability of Voting Systems Chapter Objectives Check off these skills when you feel that you have mastered them. Explain what is meant by voting manipulation. Determine if a voter,

More information

1.6 Arrow s Impossibility Theorem

1.6 Arrow s Impossibility Theorem 1.6 Arrow s Impossibility Theorem Some announcements Homework #2: Text (pages 33-35) 51, 56-60, 61, 65, 71-75 (this is posted on Sakai) For Monday, read Chapter 2 (pages 36-57) Today s Goals We will discuss

More information

Fairness Criteria. Majority Criterion: If a candidate receives a majority of the first place votes, that candidate should win the election.

Fairness Criteria. Majority Criterion: If a candidate receives a majority of the first place votes, that candidate should win the election. Fairness Criteria Majority Criterion: If a candidate receives a majority of the first place votes, that candidate should win the election. The plurality, plurality-with-elimination, and pairwise comparisons

More information

Fairness Criteria. Review: Election Methods

Fairness Criteria. Review: Election Methods Review: Election Methods Plurality method: the candidate with a plurality of votes wins. Plurality-with-elimination method (Instant runoff): Eliminate the candidate with the fewest first place votes. Keep

More information

Math for Liberal Studies

Math for Liberal Studies Math for Liberal Studies There are many more methods for determining the winner of an election with more than two candidates We will only discuss a few more: sequential pairwise voting contingency voting

More information

Chapter 1 Practice Test Questions

Chapter 1 Practice Test Questions 0728 Finite Math Chapter 1 Practice Test Questions VOCABULARY. On the exam, be prepared to match the correct definition to the following terms: 1) Voting Elements: Single-choice ballot, preference ballot,

More information

Voting Criteria: Majority Criterion Condorcet Criterion Monotonicity Criterion Independence of Irrelevant Alternatives Criterion

Voting Criteria: Majority Criterion Condorcet Criterion Monotonicity Criterion Independence of Irrelevant Alternatives Criterion We have discussed: Voting Theory Arrow s Impossibility Theorem Voting Methods: Plurality Borda Count Plurality with Elimination Pairwise Comparisons Voting Criteria: Majority Criterion Condorcet Criterion

More information

Desirable properties of social choice procedures. We now outline a number of properties that are desirable for these social choice procedures:

Desirable properties of social choice procedures. We now outline a number of properties that are desirable for these social choice procedures: Desirable properties of social choice procedures We now outline a number of properties that are desirable for these social choice procedures: 1. Pareto [named for noted economist Vilfredo Pareto (1848-1923)]

More information

Today s plan: Section : Plurality with Elimination Method and a second Fairness Criterion: The Monotocity Criterion.

Today s plan: Section : Plurality with Elimination Method and a second Fairness Criterion: The Monotocity Criterion. 1 Today s plan: Section 1.2.4. : Plurality with Elimination Method and a second Fairness Criterion: The Monotocity Criterion. 2 Plurality with Elimination is a third voting method. It is more complicated

More information

The Mathematics of Voting. The Mathematics of Voting

The Mathematics of Voting. The Mathematics of Voting 1.3 The Borda Count Method 1 In the Borda Count Method each place on a ballot is assigned points. In an election with N candidates we give 1 point for last place, 2 points for second from last place, and

More information

The Impossibilities of Voting

The Impossibilities of Voting The Impossibilities of Voting Introduction Majority Criterion Condorcet Criterion Monotonicity Criterion Irrelevant Alternatives Criterion Arrow s Impossibility Theorem 2012 Pearson Education, Inc. Slide

More information

12.2 Defects in Voting Methods

12.2 Defects in Voting Methods 12.2 Defects in Voting Methods Recall the different Voting Methods: 1. Plurality - one vote to one candidate, the others get nothing The remaining three use a preference ballot, where all candidates are

More information

Math116Chap1VotingPart2.notebook January 12, Part II. Other Methods of Voting and Other "Fairness Criteria"

Math116Chap1VotingPart2.notebook January 12, Part II. Other Methods of Voting and Other Fairness Criteria Part II Other Methods of Voting and Other "Fairness Criteria" Plurality with Elimination Method Round 1. Count the first place votes for each candidate, just as you would in the plurality method. If a

More information

Write all responses on separate paper. Use complete sentences, charts and diagrams, as appropriate.

Write all responses on separate paper. Use complete sentences, charts and diagrams, as appropriate. Math 13 HW 5 Chapter 9 Write all responses on separate paper. Use complete sentences, charts and diagrams, as appropriate. 1. Explain why majority rule is not a good way to choose between four alternatives.

More information

Voting Criteria April

Voting Criteria April Voting Criteria 21-301 2018 30 April 1 Evaluating voting methods In the last session, we learned about different voting methods. In this session, we will focus on the criteria we use to evaluate whether

More information

Chapter 9: Social Choice: The Impossible Dream

Chapter 9: Social Choice: The Impossible Dream Chapter 9: Social Choice: The Impossible Dream The application of mathematics to the study of human beings their behavior, values, interactions, conflicts, and methods of making decisions is generally

More information

Economics 470 Some Notes on Simple Alternatives to Majority Rule

Economics 470 Some Notes on Simple Alternatives to Majority Rule Economics 470 Some Notes on Simple Alternatives to Majority Rule Some of the voting procedures considered here are not considered as a means of revealing preferences on a public good issue, but as a means

More information

Computational Social Choice: Spring 2007

Computational Social Choice: Spring 2007 Computational Social Choice: Spring 2007 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today This lecture will be an introduction to voting

More information

Head-to-Head Winner. To decide if a Head-to-Head winner exists: Every candidate is matched on a one-on-one basis with every other candidate.

Head-to-Head Winner. To decide if a Head-to-Head winner exists: Every candidate is matched on a one-on-one basis with every other candidate. Head-to-Head Winner A candidate is a Head-to-Head winner if he or she beats all other candidates by majority rule when they meet head-to-head (one-on-one). To decide if a Head-to-Head winner exists: Every

More information

answers to some of the sample exercises : Public Choice

answers to some of the sample exercises : Public Choice answers to some of the sample exercises : Public Choice Ques 1 The following table lists the way that 5 different voters rank five different alternatives. Is there a Condorcet winner under pairwise majority

More information

MATH4999 Capstone Projects in Mathematics and Economics Topic 3 Voting methods and social choice theory

MATH4999 Capstone Projects in Mathematics and Economics Topic 3 Voting methods and social choice theory MATH4999 Capstone Projects in Mathematics and Economics Topic 3 Voting methods and social choice theory 3.1 Social choice procedures Plurality voting Borda count Elimination procedures Sequential pairwise

More information

The Mathematics of Voting

The Mathematics of Voting The Mathematics of Voting Voting Methods Summary Last time, we considered elections for Math Club President from among four candidates: Alisha (A), Boris (B), Carmen (C), and Dave (D). All 37 voters submitted

More information

Measuring Fairness. Paul Koester () MA 111, Voting Theory September 7, / 25

Measuring Fairness. Paul Koester () MA 111, Voting Theory September 7, / 25 Measuring Fairness We ve seen FOUR methods for tallying votes: Plurality Borda Count Pairwise Comparisons Plurality with Elimination Are these methods reasonable? Are these methods fair? Today we study

More information

Mathematics and Social Choice Theory. Topic 4 Voting methods with more than 2 alternatives. 4.1 Social choice procedures

Mathematics and Social Choice Theory. Topic 4 Voting methods with more than 2 alternatives. 4.1 Social choice procedures Mathematics and Social Choice Theory Topic 4 Voting methods with more than 2 alternatives 4.1 Social choice procedures 4.2 Analysis of voting methods 4.3 Arrow s Impossibility Theorem 4.4 Cumulative voting

More information

Voting rules: (Dixit and Skeath, ch 14) Recall parkland provision decision:

Voting rules: (Dixit and Skeath, ch 14) Recall parkland provision decision: rules: (Dixit and Skeath, ch 14) Recall parkland provision decision: Assume - n=10; - total cost of proposed parkland=38; - if provided, each pays equal share = 3.8 - there are two groups of individuals

More information

Intro to Contemporary Math

Intro to Contemporary Math Intro to Contemporary Math Independence of Irrelevant Alternatives Criteria Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Independence of Irrelevant Alternatives Criteria

More information

Voting: Issues, Problems, and Systems. Voting I 1/36

Voting: Issues, Problems, and Systems. Voting I 1/36 Voting: Issues, Problems, and Systems Voting I 1/36 Each even year every member of the house is up for election and about a third of the senate seats are up for grabs. Most people do not realize that there

More information

(67686) Mathematical Foundations of AI June 18, Lecture 6

(67686) Mathematical Foundations of AI June 18, Lecture 6 (67686) Mathematical Foundations of AI June 18, 2008 Lecturer: Ariel D. Procaccia Lecture 6 Scribe: Ezra Resnick & Ariel Imber 1 Introduction: Social choice theory Thus far in the course, we have dealt

More information

The Mathematics of Voting

The Mathematics of Voting Math 165 Winston Salem, NC 28 October 2010 Voting for 2 candidates Today, we talk about voting, which may not seem mathematical. President of the Math TA s Let s say there s an election which has just

More information

Recall: Properties of ranking rules. Recall: Properties of ranking rules. Kenneth Arrow. Recall: Properties of ranking rules. Strategically vulnerable

Recall: Properties of ranking rules. Recall: Properties of ranking rules. Kenneth Arrow. Recall: Properties of ranking rules. Strategically vulnerable Outline for today Stat155 Game Theory Lecture 26: More Voting. Peter Bartlett December 1, 2016 1 / 31 2 / 31 Recall: Voting and Ranking Recall: Properties of ranking rules Assumptions There is a set Γ

More information

Josh Engwer (TTU) Voting Methods 15 July / 49

Josh Engwer (TTU) Voting Methods 15 July / 49 Voting Methods Contemporary Math Josh Engwer TTU 15 July 2015 Josh Engwer (TTU) Voting Methods 15 July 2015 1 / 49 Introduction In free societies, citizens vote for politicians whose values & opinions

More information

Math Circle Voting Methods Practice. March 31, 2013

Math Circle Voting Methods Practice. March 31, 2013 Voting Methods Practice 1) Three students are running for class vice president: Chad, Courtney and Gwyn. Each student ranked the candidates in order of preference. The chart below shows the results of

More information

The search for a perfect voting system. MATH 105: Contemporary Mathematics. University of Louisville. October 31, 2017

The search for a perfect voting system. MATH 105: Contemporary Mathematics. University of Louisville. October 31, 2017 The search for a perfect voting system MATH 105: Contemporary Mathematics University of Louisville October 31, 2017 Review of Fairness Criteria Fairness Criteria 2 / 14 We ve seen three fairness criteria

More information

Voting: Issues, Problems, and Systems. Voting I 1/31

Voting: Issues, Problems, and Systems. Voting I 1/31 Voting: Issues, Problems, and Systems Voting I 1/31 In 2014 every member of the house is up for election and about a third of the senate seats will be up for grabs. Most people do not realize that there

More information

Notes for Session 7 Basic Voting Theory and Arrow s Theorem

Notes for Session 7 Basic Voting Theory and Arrow s Theorem Notes for Session 7 Basic Voting Theory and Arrow s Theorem We follow up the Impossibility (Session 6) of pooling expert probabilities, while preserving unanimities in both unconditional and conditional

More information

Name Date I. Consider the preference schedule in an election with 5 candidates.

Name Date I. Consider the preference schedule in an election with 5 candidates. Name Date I. Consider the preference schedule in an election with 5 candidates. 1. How many voters voted in this election? 2. How many votes are needed for a majority (more than 50% of the vote)? 3. How

More information

Voting Methods

Voting Methods 1.3-1.5 Voting Methods Some announcements Homework #1: Text (pages 28-33) 1, 4, 7, 10, 12, 19, 22, 29, 32, 38, 42, 50, 51, 56-60, 61, 65 (this is posted on Sakai) Math Center study sessions with Katie

More information

Section Voting Methods. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section Voting Methods. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 15.1 Voting Methods What You Will Learn Plurality Method Borda Count Method Plurality with Elimination Pairwise Comparison Method Tie Breaking 15.1-2 Example 2: Voting for the Honor Society President

More information

Social welfare functions

Social welfare functions Social welfare functions We have defined a social choice function as a procedure that determines for each possible profile (set of preference ballots) of the voters the winner or set of winners for the

More information

Many Social Choice Rules

Many Social Choice Rules Many Social Choice Rules 1 Introduction So far, I have mentioned several of the most commonly used social choice rules : pairwise majority rule, plurality, plurality with a single run off, the Borda count.

More information

MATH 1340 Mathematics & Politics

MATH 1340 Mathematics & Politics MATH 1340 Mathematics & Politics Lecture 6 June 29, 2015 Slides prepared by Iian Smythe for MATH 1340, Summer 2015, at Cornell University 1 Basic criteria A social choice function is anonymous if voters

More information

Voting: Issues, Problems, and Systems

Voting: Issues, Problems, and Systems Voting: Issues, Problems, and Systems 3 March 2014 Voting I 3 March 2014 1/27 In 2014 every member of the house is up for election and about a third of the senate seats will be up for grabs. Most people

More information

Section Voting Methods. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section Voting Methods. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 15.1 Voting Methods INB Table of Contents Date Topic Page # February 24, 2014 Test #3 Practice Test 38 February 24, 2014 Test #3 Practice Test Workspace 39 March 10, 2014 Test #3 40 March 10, 2014

More information

Homework 7 Answers PS 30 November 2013

Homework 7 Answers PS 30 November 2013 Homework 7 Answers PS 30 November 2013 1. Say that there are three people and five candidates {a, b, c, d, e}. Say person 1 s order of preference (from best to worst) is c, b, e, d, a. Person 2 s order

More information

In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible.

In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible. Voting Theory 1 Voting Theory In many decision making situations, it is necessary to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company decides

More information

Section 3: The Borda Count Method. Example 4: Using the preference schedule from Example 3, identify the Borda candidate.

Section 3: The Borda Count Method. Example 4: Using the preference schedule from Example 3, identify the Borda candidate. Chapter 1: The Mathematics of Voting Section 3: The Borda Count Method Thursday, January 19, 2012 The Borda Count Method In an election using the Borda Count Method, the candidate with the most points

More information

CS 886: Multiagent Systems. Fall 2016 Kate Larson

CS 886: Multiagent Systems. Fall 2016 Kate Larson CS 886: Multiagent Systems Fall 2016 Kate Larson Multiagent Systems We will study the mathematical and computational foundations of multiagent systems, with a focus on the analysis of systems where agents

More information

Voting Systems. High School Circle I. June 4, 2017

Voting Systems. High School Circle I. June 4, 2017 Voting Systems High School Circle I June 4, 2017 Today we are going to start our study of voting systems. Put loosely, a voting system takes the preferences of many people, and converted them into a group

More information

How should we count the votes?

How should we count the votes? How should we count the votes? Bruce P. Conrad January 16, 2008 Were the Iowa caucuses undemocratic? Many politicians, pundits, and reporters thought so in the weeks leading up to the January 3, 2008 event.

More information

Presidential Election Democrat Grover Cleveland versus Benjamin Harrison. ************************************ Difference of 100,456

Presidential Election Democrat Grover Cleveland versus Benjamin Harrison. ************************************ Difference of 100,456 Presidential Election 1886 Democrat Grover Cleveland versus Benjamin Harrison Cleveland 5,540,309 Harrison 5,439,853 ************************************ Difference of 100,456 Electoral College Cleveland

More information

Explaining the Impossible: Kenneth Arrow s Nobel Prize Winning Theorem on Elections

Explaining the Impossible: Kenneth Arrow s Nobel Prize Winning Theorem on Elections Explaining the Impossible: Kenneth Arrow s Nobel Prize Winning Theorem on Elections Dr. Rick Klima Appalachian State University Boone, North Carolina U.S. Presidential Vote Totals, 2000 Candidate Bush

More information

that changes needed to be made when electing their Presidential nominee. Iowa, at the time had a

that changes needed to be made when electing their Presidential nominee. Iowa, at the time had a Part I The Iowa caucuses are perhaps the most important yet mysterious contest in American politics. It all began after the 1968 Democratic National Convention protest, the party decided that changes needed

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Chapter 1 Review SHORT ANSWER. Answer each question. Circle your final answer. Show all work. Determine whether any of the listed candidates has a majority. 1) Four candidates running for congress receive

More information

Section Voting Methods. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section Voting Methods. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 15.1 Voting Methods What You Will Learn Plurality Method Borda Count Method Plurality with Elimination Pairwise Comparison Method Tie Breaking 15.1-2 Example 2: Voting for the Honor Society President

More information

Introduction: The Mathematics of Voting

Introduction: The Mathematics of Voting VOTING METHODS 1 Introduction: The Mathematics of Voting Content: Preference Ballots and Preference Schedules Voting methods including, 1). The Plurality Method 2). The Borda Count Method 3). The Plurality-with-Elimination

More information

In this lecture we will cover the following voting methods and fairness criterion.

In this lecture we will cover the following voting methods and fairness criterion. In this lecture we will cover the following voting methods and fairness criterion. Borda Count Method Plurality-with-Elimination Method Monotonicity Criterion 1 Borda Count Method In the Borda Count Method

More information

Math for Liberal Studies

Math for Liberal Studies Math for Liberal Studies As we have discussed, when there are only two candidates in an election, deciding the winner is easy May s Theorem states that majority rule is the best system However, the situation

More information

Possible voting reforms in the United States

Possible voting reforms in the United States Possible voting reforms in the United States Since the disputed 2000 Presidential election, there have numerous proposals to improve how elections are conducted. While most proposals have attempted to

More information

PROBLEM SET #2: VOTING RULES

PROBLEM SET #2: VOTING RULES POLI 309 Fall 2006 due 10/13/06 PROBLEM SET #2: VOTING RULES Write your answers directly on this page. Unless otherwise specified, assume all voters vote sincerely, i.e., in accordance with their preferences.

More information

Make the Math Club Great Again! The Mathematics of Democratic Voting

Make the Math Club Great Again! The Mathematics of Democratic Voting Make the Math Club Great Again! The Mathematics of Democratic Voting Darci L. Kracht Kent State University Undergraduate Mathematics Club April 14, 2016 How do you become Math Club King, I mean, President?

More information

Main idea: Voting systems matter.

Main idea: Voting systems matter. Voting Systems Main idea: Voting systems matter. Electoral College Winner takes all in most states (48/50) (plurality in states) 270/538 electoral votes needed to win (majority) If 270 isn t obtained -

More information

Warm-up Day 3 Given these preference schedules, identify the Plurality, Borda, Runoff, Sequential Runoff, and Condorcet winners.

Warm-up Day 3 Given these preference schedules, identify the Plurality, Borda, Runoff, Sequential Runoff, and Condorcet winners. Warm-up Day 3 Given these preference schedules, identify the Plurality, Borda, Runoff, Sequential Runoff, and Condorcet winners. Plurality: Borda: Runoff: Seq. Runoff: Condorcet: Warm-Up Continues -> Warm-up

More information

Social Choice & Mechanism Design

Social Choice & Mechanism Design Decision Making in Robots and Autonomous Agents Social Choice & Mechanism Design Subramanian Ramamoorthy School of Informatics 2 April, 2013 Introduction Social Choice Our setting: a set of outcomes agents

More information

What is the Best Election Method?

What is the Best Election Method? What is the Best Election Method? E. Maskin Harvard University Gorman Lectures University College, London February 2016 Today and tomorrow will explore 2 Today and tomorrow will explore election methods

More information

: It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria was proven in 1949.

: It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria was proven in 1949. Chapter 1 Notes from Voting Theory: the mathematics of the intricacies and subtleties of how voting is done and the votes are counted. In the early 20 th century, social scientists and mathematicians working

More information

Sect 13.2 Flaws of Voting Methods

Sect 13.2 Flaws of Voting Methods 218 Sect 13.2 Flaws of Voting Methods From an example the previous section, we had 48 sports writers rank the top four Spurs players of all time. Below is the preference table. Number of votes 20 14 10

More information

Introduction to Theory of Voting. Chapter 2 of Computational Social Choice by William Zwicker

Introduction to Theory of Voting. Chapter 2 of Computational Social Choice by William Zwicker Introduction to Theory of Voting Chapter 2 of Computational Social Choice by William Zwicker If we assume Introduction 1. every two voters play equivalent roles in our voting rule 2. every two alternatives

More information

Lecture 11. Voting. Outline

Lecture 11. Voting. Outline Lecture 11 Voting Outline Hanging Chads Again Did Ralph Nader cause the Bush presidency? A Paradox Left Middle Right 40 25 35 Robespierre Danton Lafarge D L R L R D A Paradox Consider Robespierre versus

More information

Simple methods for single winner elections

Simple methods for single winner elections Simple methods for single winner elections Christoph Börgers Mathematics Department Tufts University Medford, MA April 14, 2018 http://emerald.tufts.edu/~cborgers/ I have posted these slides there. 1 /

More information

The mathematics of voting, power, and sharing Part 1

The mathematics of voting, power, and sharing Part 1 The mathematics of voting, power, and sharing Part 1 Voting systems A voting system or a voting scheme is a way for a group of people to select one from among several possibilities. If there are only two

More information

n(n 1) 2 C = total population total number of seats amount of increase original amount

n(n 1) 2 C = total population total number of seats amount of increase original amount MTH 110 Quiz 2 Review Spring 2018 Quiz 2 will cover Chapter 13 and Section 11.1. Justify all answers with neat and organized work. Clearly indicate your answers. The following formulas may or may not be

More information

Social Choice. CSC304 Lecture 21 November 28, Allan Borodin Adapted from Craig Boutilier s slides

Social Choice. CSC304 Lecture 21 November 28, Allan Borodin Adapted from Craig Boutilier s slides Social Choice CSC304 Lecture 21 November 28, 2016 Allan Borodin Adapted from Craig Boutilier s slides 1 Todays agenda and announcements Today: Review of popular voting rules. Axioms, Manipulation, Impossibility

More information

Public Choice. Slide 1

Public Choice. Slide 1 Public Choice We investigate how people can come up with a group decision mechanism. Several aspects of our economy can not be handled by the competitive market. Whenever there is market failure, there

More information

The Math of Rational Choice - Math 100 Spring 2015

The Math of Rational Choice - Math 100 Spring 2015 The Math of Rational Choice - Math 100 Spring 2015 Mathematics can be used to understand many aspects of decision-making in everyday life, such as: 1. Voting (a) Choosing a restaurant (b) Electing a leader

More information

In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible.

In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible. Voting Theory 35 Voting Theory In many decision making situations, it is necessary to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company decides

More information

The Mathematics of Voting Transcript

The Mathematics of Voting Transcript The Mathematics of Voting Transcript Hello, my name is Andy Felt. I'm a professor of Mathematics at the University of Wisconsin- Stevens Point. This is Chris Natzke. Chris is a student at the University

More information

Voting Methods for Municipal Elections: Propaganda, Field Experiments and what USA voters want from an Election Algorithm

Voting Methods for Municipal Elections: Propaganda, Field Experiments and what USA voters want from an Election Algorithm Voting Methods for Municipal Elections: Propaganda, Field Experiments and what USA voters want from an Election Algorithm Kathryn Lenz, Mathematics and Statistics Department, University of Minnesota Duluth

More information

Is Majority Rule the Best Voting Method? Partha Dasgupta and Eric Maskin

Is Majority Rule the Best Voting Method? Partha Dasgupta and Eric Maskin Is Majority Rule the Best Voting Method? by Partha Dasgupta and Eric Maskin June 2003 The authors are, respectively, the Frank Ramsey Professor of Economics at the University of Cambridge, UK, and the

More information

The Mathematics of Voting and Elections: A Hands-On Approach. Instructor s Manual. Jonathan K. Hodge Grand Valley State University

The Mathematics of Voting and Elections: A Hands-On Approach. Instructor s Manual. Jonathan K. Hodge Grand Valley State University The Mathematics of Voting and Elections: A Hands-On Approach Instructor s Manual Jonathan K. Hodge Grand Valley State University January 6, 2011 Contents Preface ix 1 What s So Good about Majority Rule?

More information

Section 7.1: Voting Systems. Plurality Method The candidate who receives the greatest number of votes is the winner.

Section 7.1: Voting Systems. Plurality Method The candidate who receives the greatest number of votes is the winner. Section 7.1: Voting Systems Plurality Method The candidate who receives the greatest number of votes is the winner. Borda Count Method Each voter s last choice receives one point, each voter s second-to-last

More information

1.1 The Basic Elements of an Election 1.2 The Plurality Method

1.1 The Basic Elements of an Election 1.2 The Plurality Method 1.1 The Basic Elements of an Election 1.2 The Plurality Method Some announcements Math Center study sessions with Katie Greene (TA). Tuesday and Wednesday 7pm-9pm in Kirby 120. First Math colloquium this

More information

(c) 2013 Janice L. Epstein Voting Methods 1

(c) 2013 Janice L. Epstein Voting Methods 1 (c) 2013 Janice L. Epstein Voting Methods 1 Majority Rule: Each voter votes for one candidate. The candidate with the majority of the votes wins. n n + 1 When there are n votes, the majority is + 1 [n

More information

Rationality of Voting and Voting Systems: Lecture II

Rationality of Voting and Voting Systems: Lecture II Rationality of Voting and Voting Systems: Lecture II Rationality of Voting Systems Hannu Nurmi Department of Political Science University of Turku Three Lectures at National Research University Higher

More information

Introduction to the Theory of Voting

Introduction to the Theory of Voting November 11, 2015 1 Introduction What is Voting? Motivation 2 Axioms I Anonymity, Neutrality and Pareto Property Issues 3 Voting Rules I Condorcet Extensions and Scoring Rules 4 Axioms II Reinforcement

More information

Voting Definitions and Theorems Spring Dr. Martin Montgomery Office: POT 761

Voting Definitions and Theorems Spring Dr. Martin Montgomery Office: POT 761 Voting Definitions and Theorems Spring 2014 Dr. Martin Montgomery Office: POT 761 http://www.ms.uky.edu/~martinm/m111 Voting Method: Plurality Definition (The Plurality Method of Voting) For each ballot,

More information

Algorithms, Games, and Networks February 7, Lecture 8

Algorithms, Games, and Networks February 7, Lecture 8 Algorithms, Games, and Networks February 7, 2013 Lecturer: Ariel Procaccia Lecture 8 Scribe: Dong Bae Jun 1 Overview In this lecture, we discuss the topic of social choice by exploring voting rules, axioms,

More information

Warm-up Day 3. Phones OFF and in pockets! 1) Given these preference schedules, identify the Condorcet, Runoff, and Sequential Runoff winners.

Warm-up Day 3. Phones OFF and in pockets! 1) Given these preference schedules, identify the Condorcet, Runoff, and Sequential Runoff winners. Warm-up Day 3 1) Given these preference schedules, identify the Condorcet, Runoff, and Sequential Runoff winners. Phones OFF and in pockets! Condorcet: Runoff: Seq. Runoff: 2) If each voter approves of

More information

Grade 6 Math Circles Winter February 27/28 The Mathematics of Voting - Solutions

Grade 6 Math Circles Winter February 27/28 The Mathematics of Voting - Solutions Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles Winter 2018 - February 27/28 The Mathematics of Voting - Solutions Warm-up: Time

More information