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2023版美赛投票模型
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此模块的上下文是投票系统。本模块以简短的初读“选举美国众议院议长”开始,重点关注 2023 年凯文·麦卡锡的选举。讨论多数和多数选举决定。初读之后是简短的初步活动,多数和多数选举结果,学生分析与选举美国众议院议长有关的情况。 在活动1“多元模型”中,学生分析使用复数模型决定的选举结果,这是美国决定选举最常用的方法。使用偏好图,学生发现模型中的缺陷。 在活动 2“径流模型”中,学生分析决选数据。人们期望使用决选可以避免多数缺陷。但是,学生发现使用径流模型也存在问题。 在活动3“排名选择投票模型”中,向学生介绍了排名选择投票(RCV)(也称为即时决选投票(IRV))模型,该模型目前正在一些州和城市使用。但是排名选择投票模型并不总是完美运作。活动 3 以建模警报结束!专注于肯尼斯·阿罗(Kenneth Arrow),他证明了没有合理的选举模式可以完全没有缺陷。 活动4是关于批准投票模型的项目。学生将批准投票的选举结果与本抽出中涵盖的其他投票模型的结果进行比较。该项目以学生使用互联网研究批准投票的利弊结束。
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MarshaDavis
ACOMAPMODELINGMODULE
Voting Models
Voting Models
A COMAP Modeling Module
Copyright © 2023 by COMAP, Inc.
The Consortium for Mathematics and Its Applications (COMAP)
175 Middlesex Turnpike, Suite 3B
Bedford, MA 01730
Published and distributed by
All rights reserved. The text of this publication, or any part thereof, may not
be reproduced or transmi
ed in any form or by any means, electronic or
mechanical, including photocopying, recording, storage in an information
retrieval system, or otherwise, without wri
en permission of the publisher.
ISBN 978
‑0‑9971490‑7‑4
www.comap.com
A COMAP Modeling Module
—1—
Voting Models
T
he context for this Module is voting systems. This Mod‑
ule begins with a short Preliminary Reading, Electing a
Speaker of the U.S. House of Representatives, which focuses on
the 2023 election of Kevin McCarthy. Majority and plurality
election decisions are discussed. The Preliminary Reading is
followed by a brief Preliminary Activity, Majority and Plurality
Election Results, in which students analyze situations related
to electing a Speaker of the U.S. House of Representatives.
In Activity 1, Plurality Models, students analyze results of
elections that are decided using a plurality model, which is
the most used method for deciding elections in the United
States. Using preference diagrams, students discover a flaw
in the model.
In Activity 2, Runoff Models, students analyze data from run‑
off elections. The expectation is that using runoff elections
can avoid the plurality flaw. However, students discover
that there are problems with using runoff models as well.
In Activity 3, Rank‑Choice Voting Models, students are intro‑
duced to rank‑choice voting (RCV) (also called instant runoff
voting (IRV)) models, which are currently in use in some
states and cities. But rank‑choice voting models don’t always
work perfectly. Activity 3 concludes with a Modeling Alert!
focusing on Kenneth Arrow, who proved that no reasonable
election model can be entirely free from flaws.
TEACHER MATERIAL
Activity 4 is a project on approval voting models. Students
compare the election outcome from approval voting to the
outcomes from other voting models covered in this Module.
The project ends with students using the Internet to
research the pros and cons of approval voting.
TEACHER NOTES
Prerequisites
Students should be familiar with calculating percentages.
Mathematical Content
• Percentages
• Basic arithmetic calculations
• Number sense
• Interpreting preference diagrams
Materials Needed
Students should have access to calculators
(a basic calculator is fine).
LESSON NOTES
Preliminary Reading: Electing a Speaker
of the U.S. House of Representatives
Students should read the Preliminary Reading before they
complete the brief Preliminary Activity that follows. The
purpose of this reading is to introduce the majority model
for deciding elections, which is the model used to elect the
Speaker of the U.S. House of Representatives. The election
is held by a roll call vote in which each Representative when
called states a person’s name or simply says “Present.” A
Speaker is elected once a majority of those stating a person’s
name agree on the same name. If no person gets a majority,
then the process is repeated. In 2023, it took 15 rounds and
four days before Kevin McCarthy was elected Speaker.
Preliminary Activity: Majority and Plurality
Election Results
This brief activity is a follow‑up to the Preliminary Reading.
Students apply the definitions of majority and plurality to
analyze results from the 2023 election of the Speaker of the
U.S. House of Representatives.
—2—
A COMAP Modeling Module
Voting Model
LESSON NOTES
Activity 1: Plurality Models
Activity 1 opens with a general description of the basic ele‑
ments in a voting model.
Question 1 shows the popular vote totals for the 2016 pres‑
idential election. The plurality winner was Hillary Clinton.
However, Donald Trump won the election based on a
majority of the Electoral College votes. Presidential elections
are run by the states. Most states use the plurality method
to decide the winner for their state – and except in Maine
and Nebraska, the winner gets all of that state’s Electoral
College votes. There are 538 Electoral College votes up for
grabs and 270 are needed to win. For more information,
have students check out this site:
www.archives.gov/electoral‑college/about
Question 2 turns to the 2022 race for Alaska’s State House
seat in District 11. You can revisit this race in Activity 3 (see
Lesson Notes), because Alaska is one of two states that uses
rank‑choice voting to decide many of its races. However, for
question 2 the winner is determined by the plurality model
and students discover a flaw in that model. Given there were
two Republican candidates in this race, Republican voters
split their votes between them and, using the plurality
model, the winner would be the Independent candidate.
Question 3 focuses on a 2019 recall election for the mayor
of Fall River, Massachuses. Jasiel Correia was recalled by
61% of the voters and then re‑elected as mayor by a plurality
due to the presence of four opposition candidates.
Question 4 presents results from a 1970 New York Senate
race. There were three candidates in this race, which was
decided by a plurality. However, a majority of the voters
rated the winner last. Analysis of this race is continued in
question 5 where voter preferences for the candidates are
represented with preference diagrams. Students will work
with preference diagrams again in Activity 3.
Question 6 deals with the 1992 Presidential race between
Bill Clinton, George H.W. Bush, and Ross Perot. In analyzing
this election, students find that the plurality model flaw does
not occur even though there are more than two candidates.
Consider using question 7 as a discussion or reflection ques‑
tion. Given that the plurality model is the method used to
determine the winners of most elections in the United States,
make sure that students understand one of its major flaws:
In a race with more than two candidates, the plurality
winner could be the candidate that is disliked by a majority
of the voters. The plurality model may not be the best way
to decide winners of an election. When a flaw is discovered
in a model, it’s time to look for a beer model. So this ques‑
tion is a good lead in to Activity 2, where students are intro‑
duced to the runoff voting model.
Activity 2: Runoff Models
The runoff model is explained at the start of Activity 2.
Questions 1 – 4 deal with the 2020 and 2022 elections for
Senator in Georgia. In Georgia, if no candidate in a race gets
a majority, then the top two candidates compete in a runoff
election. One flaw in runoff models is that fewer voters tend
to participate in the runoff election compared to the general
election. They are also expensive to run, which is noted in
the In the News excerpt following question 4.
Question 5 is based on the runoff model used in North Car‑
olina for primary elections. The threshold percentage that
triggers a runoff election is reduced from 50% (as used in
Georgia) to 30%. You might ask students if the North Caro‑
lina runoff model would have changed the results of the Fall
River mayoral election from Activity 1, question 3.
Question 6 summarizes the flaws in runoff models. Discuss
this with the class and allow them to make suggestions for
a beer model.
Activity 3: Rank‑Choice Voting Models
Activity 3 begins by examining Maine’s 2018 election for the
U.S. House of Representatives. Question 1 walks students
through the steps of rank‑choice voting that led to Jared
Golden being elected to the U.S. House of Representatives
from District 2 over the incumbent Bruce Poliquin, even
though Poliquin was first‑choice among the most voters.
Figure 3.1 shows a sample ballot for this race. Maine voters
are allowed to rank up to five candidates (the 5
th
candidate
is a write‑in candidate) according to their preferences. Each
voter can select as few or as many choices as they want, but
each choice has to be a different candidate.
As of 2022, Alaska has adopted rank‑choice voting. Before
students begin question 2, revisit Activity 1, question 2,
which gives the results in the race for Alaska House from
District 11, which are shown below.
In Activity 1, question 2 students determined that the plu‑
rality winner was the Independent candidate, Featherly. The
two other candidates, both Republican, split the Republican
vote and ended up with fewer votes than the Independent
candidate. As noted in the solutions to this question, a
Featherly win would probably upset most of the 4,187 voters
(54.58% of the voters) who selected a Republican candidate.
However, that’s not what happened in this race. Due to
rank‑choice voting, Bieling was eliminated in Round 1. For
Candidate Party Votes Percentage
Featherly, Walter T. Independent 3,484 45.42%
Bieling, Ross P. Republican 1,186 15.46%
Coulombe, Julie Republican 3,001 39.12%
—3—
A COMAP Modeling Module Voting Model
Round 2, Bieling’s
fi
rst‑
choice votes were transferred to their
second
‑
choice candidate. It turns out that 429 of these voters
did not list a second choice, 81 listed Featherly, and 670
listed Coulombe as their second choice. The Round 2 results
appear below, and Coulombe is the winner.
Activity 3,
q
uestion 2
focuses on the
2022 Alaskan race for
the U.S. Senate. For this race, the plurality winner turns out
to be the rank
‑
choice winner. While four candidates are
listed in Table 3.2, Buzz Kelly dropped out of the race throw
‑
ing his support to Kelly Tshibaka. However, his withdrawal
came too late to remove his name from the ballot.
For
questions 1 and 2, raw data from rank‑
choice voting
ballots were organized into tables. That took a computer
program. (Take a look at Table 3.5, an excerpt from the
Maine election discussed in
question 1
, to see why a com
‑
puter program was needed to analyze the data.) If you
would like students to set up a rank
‑
choice election of their
own, a good resource is RankedVote
(
www.rankedvote.co
)
which will allow you to set up a rank
‑choice election and
return the results. To do this, you have to create an account,
which allows you to run
one election free
.
Questions 3 – 6 are based on voting data that has been or
‑
ganized into preference diagrams. This allows students to
compare the winners selected by rank‑
choice models with
winners determined by plurality or runo
ff
models. After stu
‑
dents have completed
questions 3 and 4
, discuss their
answers in class. Students can then apply what they have
learned to
questions 5
and
6
.
In the preference diagrams used for
questions 3
–
6
, it was
assumed that voters rated all the candidates from
fi
rst to
last. However, for the Maine and Alaska ballots, voters did
not have to rank all of the candidates. You might ask stu
‑
dents what preference diagrams might look like in the case
of Maine or Alaska. The set of preference diagrams below is
based on the data from Table 3.5. (Vote counts are listed
below each preference diagram.)
LESSON NOTES
Activity Answers
Activity 3 ends with a Modeling Alert! that should be dis‑
cussed because it demonstrates the mathematical mod‑
eling process and its broad reach (into the political
sciences, in this case). Sound modeling requires that mod‑
elers establish reasonable criteria to guide their work. Of
course, they must be able to defend those criteria as good
ones. Arrow’s search for a model that met his criteria
proved frustrating, and he subsequently proved that no
reasonable election model can be entirely free from flaws.
However, some models are beer than others, and accord‑
ing to Arrow, we should continue to work to improve vot‑
ing models. For an elementary discussion of Arrow’s
criteria, see Lesson 1.4 of Discrete Mathematics: Modeling
Our World, which is available from COMAP.
Activity 4: Project – Approval Voting Models
Questions 1‑3 involve the approval voting model, a model
that is used in St. Louis, Missouri and Fargo, North Dakota
but otherwise not widely used in U.S. elections. This
model is simple to explain, and you could determine the
winner by summing columns in a spreadsheet, no special
program is needed. In fact, you could ask students to enter
voting data from Table 4.1 into an Excel spreadsheet and
use the sum command to tabulate the votes.
Question 4 asks students to conduct an Internet search on
the pros and cons of approval voting. Here are some pos‑
sible sites that students can search.
• hps://givingcompass.org/article/alternative‑voting‑
methods‑pros‑cons
• hps://electionscience.org/voting‑methods/
ten‑critiques‑and‑defenses‑on‑approval‑voting/
• hps://electionbuddy.com/blog/2022/02/16/
significant‑approval‑voting‑pros‑and‑cons/
Consider asking teams of students to take sides, one side
presenting the pros for approval voting and the other pre‑
senting the cons for approval voting.
ACTIVITY ANSWERS
Preliminary Activity: Majority and
Plurality Election Results
1. If the minority‑party Representatives all vote for a single
person within their party, requiring a majority vote for
Speaker ensures that the majority party can choose the
Speaker from their party even if initially members of the
majority party are divided.
2. To determine the number of Representatives voting, we
can go back to the 15
th
ballot. There were 216 (McCarthy)
+ 212 (Jeffries) + 6 (Present) = 434 Representatives voting.
Half of 434 is 217. So, McCarthy would need 218 votes
to have a majority.
Candidate Party
Votes + Transfer
Votes from Bieling
Percentage
Featherly,
Walter T.
Independent 3,484 + 81 = 3,565 49.27%
Coulombe, Julie Republican 3,001 + 670 = 3,671 50.73%
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