# elect-2 - Colorado Mesa University

```The Mathematics of Elections
Part II: Voting
Mark Rogers
Last time…
• We discussed systems of apportionment.
– how we determine how many representatives
there should be in an elective body and how
those representatives are to be distributed
among various subgroups of the population
as a whole
• Several methods can be used.
– Founding Fathers: Hamilton, Jefferson,
– Huntington-Hill method: used since WWII
Last time…
• Problems can result from any type of apportionment.
– Alabama paradox: in an even larger group of representatives,
one group could end up losing one, even with the same
percentage of the population
– Population paradox: one group can lose a representative to
another group, even though they are growing at a faster rate
– New states paradox: one group can lose a representative to
another group if a new group is added, with their own delegation
considered separately
– Violation of the apportionment criterion: each representative
should represent (approximately) the same number of voters
– Violation of the quota rule: using some methods, a group can
end up with a number of representatives very different than their
“deserved” proportion of the entire body (too many or too few)
Last time…
• Gerrymandering: redrawing district lines to
favor or hinder one particular group
– Can be done to “pack” similar voters into the same
district, or to “crack” or split similar voters among
several districts, thus diluting them
– Done by state legislatures to maximize their party’s
chances of winning the Congressional seats
– This works because most of our elections follow a
simple “majority wins” method, in which candidates
with a minority of votes (and their supporters) will
have no say in that district.
1992 & 2000 Presidential Elections
1992 Candidate
Popular Vote
Electoral Vote
Bill Clinton
44,908,254 (43.0%)
370 (68.8%)
George W. Bush
39,102,343 (37.7%)
168 (31.2%)
2000 Candidate
Popular Vote
Electoral Vote
George W. Bush
50,456,002 (47.9%)
271 (50.4%)
Al Gore
50,999,897 (48.4%)
267 (49.6%)
• The lopsided 1992 results and the upside-down 2000
results are due to the fact that our Electoral College
system is based on a subdivided measure of the national
popular vote, and one which does not necessarily
require a majority of it.
Majorities and supermajorities
• “majority”: more than 50% of the votes (“50% plus 1”)
• “supermajority”: a higher-than-majority percentage of
– The specific definition can vary according to the situation.
– In the U.S. Senate, a 60% supermajority is needed to end a
filibuster.
– In both houses of Congress, a two-thirds supermajority is
needed to override a Presidential veto.
• The majority criterion: if a majority of voters rank
candidate X as their first choice, then X deserves to win
the election.
– This seems like common sense, but often a majority cannot be
attained.
What if a majority is not achieved?
• Many elections are run using the plurality method: the candidate
chosen by the most voters as their first choice wins, regardless of
whether they attain a majority or not.
• Many others use the plurality-with-elimination method: if no
candidate attains a majority, then a second round of voting (or
possibly several) takes place, with some candidates (possibly as
many as all but the top two from the first round) eliminated from the
later ballot(s).
• The plurality-with-elimination method can often result in the secondplace candidate winning in the second round of voting, as
supporters of the dropped candidates flock to his or her side.
– Often, a coalition of voters forms to defeat the candidate who was in first
place after the first round.
2007 French presidential elections
• 12 candidates, including four major ones in an extremely tight race:
–
–
–
–
Conservative Nicolas Sarkozy, favored in early polls by 26% of voters
Socialist Ségolène Royal, favored by 25%
Centrist François Bayrou, favored by 24%
Far-right nationalist Jean-Marie Le Pen, favored by 16%
• The top two candidates would advance to a runoff.
• However, Bayrou’s support was more widespread; not being as
despised by members of the other extreme as either Sarkozy or
Royal, he would have been able to attract far more voters in a runoff
than either of them…if he could make the runoff.
• Unfortunately for him, he did indeed finish in 3rd place, leaving him
unable to surpass the two frontrunners in this manner.
• In the end, Sarkozy won the runoff against Royal 53% to 47%.
• While this is not surprising or even unexpected, curious paradoxes
can occur in certain other situations.
The problem with a “revote”
• The consistency criterion: if the electorate is split into two or more
divisions, and candidate X is the winner in each portion, then X
deserves to win the overall election. (This idea is somewhat
analogous to Simpson’s paradox in statistics.)
• The independence-of-irrelevant-alternatives criterion: if
candidate X wins an election, and a recount is taken after some nonwinner(s) are eliminated from the ballot, then candidate X should still
win the election.
• The monotonicity criterion: if candidate X wins an election, and if
in a revote the only voters who change their preference do so in
favor of X, then X should still win the revote.
– Again, these both sound like the expected result, but in certain
circumstances candidate X will actually lose the second round despite
winning the first round and being the only person to gain supporters!
1992 & 2000 Presidential Elections
• Stronger-than-usual showings by third-party candidates are thought
to have tilted the balance in recent elections.
1992 Candidates
National Vote
2000 Candidates
Florida Vote
Bill Clinton
44,908,254
George W. Bush
2,912,790
George H.W. Bush
39,102,343
Al Gore
2,912,253
Ross Perot
8,085,294
97,488
• Experts theorize that if even “most” (say, 2/3) of the third-party
voters had stayed with the main candidate of their (former) party,
both elections would have swung the other way.
Electoral
Vote
With “2/3”
support
Electoral
Vote
With “2/3”
support
Clinton
370
267
Bush Jr.
271
246
Bush Sr.
168
271
Gore
267
292
• Many voters see this sort of “splitting” of the vote of one
end of the political spectrum to be self-defeating.
– By opting for their favorite candidate instead of a slightly lessfavored alternative whose higher support levels might have given
them a more realistic chance of winning, they “hand the election”
to the close-running candidate of the opposite persuasion.
• Cumulative voting: Voters can choose to split their
vote into fractional portions divided among several
candidates, or concentrate it for a single candidate.
– Voters could give small-party candidates enough fractional votes
to help make them viable, but reserve the bulk of their vote for
mainstream candidates who might otherwise lose a close race.
An unexpected election result
• First-round results:
Preference 12 voters
9 voters
3 voters
8 voters
1st
Charles
Bonnie
2nd
Charles
Charles
Bonnie
3rd
Bonnie
Bonnie
Charles
• Since Bonnie was the first choice of the fewest number of voters,
she is eliminated and a runoff held.
• Given each voter’s runner-up preferences, the following results are
likely in the second round:
Preference 12 voters
9 voters
3 voters
8 voters
1st
Charles
Charles
2nd
Charles
Charles
• Thus, Charles wins the second-round runoff.
• However, suppose that due to irregularities, the election is ordered
to be started from scratch. Charles should still win, right?
An unexpected election result
•
•
•
•
This time, the group of three voters in the third column decide that they were
disappointed by Adam’s behavior in the runoff, and decide to switch their first-round
vote from him to Charles (whom they just watched “win” anyway).
New first-round results:
Preference 12 voters
9 voters
3 voters
8 voters
1st
Charles
Bonnie
Charles
2nd
Charles
Bonnie
3rd
Bonnie
Bonnie
Charles
This time, Adam was the first choice of the fewest number of voters, so he is
eliminated from the runoff. The revised likely runoff preferences are:
Preference 12 voters
9 voters
3 voters
8 voters
1st
Charles
Bonnie
Charles
Bonnie
2nd
Bonnie
Charles
Bonnie
Charles
Thus, Bonnie wins the second-round runoff, despite no changes to her votes.
–
She wasn’t even in the original runoff!
1991 Louisiana Gubernatorial Race
• Former KKK member David Duke ran a close second to Edwin
Edwards, a candidate long suspected of corruption.
• In Louisiana’s open-party, multiple-candidate, plurality-withelimination system, the top two candidates advanced to a runoff.
Candidate
Edwin Edwards
523,096
1,057,031
David Duke
491,342
671,009
Buddy Roemer
410,690
Others
124,127
• In a case of “better the devil you know,” most voters flocked to
Edwards (a candidate they distrusted) in the runoff in order to block
the more embarrassing Duke.
1991 Louisiana Gubernatorial Race
Candidate
1st-round
Edwin Edwards
523,096
Buddy Roemer
410,690
David Duke
410,342
Others
124,127
•
•
•
•
Similarly, many Duke supporters distrusted
Edwards, but could not gather enough
support to defeat him directly. Suppose
away from the polls.
With Duke out of the runoff, voters would
have felt comfortable voting for the safer
Roemer; polls indicated that this would
have occurred by a wide margin.
Thus, David Duke’s supporters could have
done more to defeat their chief opponent
by not voting!
A similar result occurred in France in 2002.
– Far-right nationalist candidate Jean-Marie
Le Pen upset the Socialist challenger to the
conservative president Jacques Chirac to
force a runoff, in which the 3rd-place
Socialists were compelled to vote for their
archrival Chirac.
– Le Pen voters could have defeated Chirac
by being willing to finish in third place.
A reusable ballot
• One way to avoid the time and expense of a runoff round
of voting is to use ballots that allow voters to express
their levels of preference for each candidate, by ranking
them in order.
– These “preference ballots” can then be reused as needed,
since they are presumed to express the widest possible intention
of the voters in a variety of scenarios.
– The ballots could be used as a preemptive runoff ballot, with
voters getting either their original 1st choice (if still in the running)
or their highest-ranked runner-up still in the running. This is also
known as instant-runoff voting.
Winner take all?
• Unfortunately, any plurality-based voting system rewards
only the winner, with the runners-up receiving no credit
for being a “close second.”
– It also makes little distinction between a runner-up in 2nd place
and a runner-up in 10th place.
• Both were simply “less favored” than the 1st choice, who typically
was the only one who got that person’s vote.
• A weighted system that awarded each candidate points
based on how high they placed in the preference ballot
would acknowledge how close the 2nd-place candidate
was to the 1st-place candidate (and how not close either
of them was to the 10th-place candidate).
The Borda count method
• Jean-Charles de Borda (1733-1799), French
mathematician and mariner
• If there are n candidates in an election, each voter
ranks their preferences in order from 1st place
through nth place.
–
–
–
–
The candidate in 1st place receives n points.
The candidate in 2nd place receives (n – 1) points.
The candidate in 3rd place receives (n – 2) points.
And so on, until the candidate in nth (last) place
• Then all such points are totaled for each
candidate, and the one with the highest point total
is the winner.
Bigger than an election…
• Most sports polls use the Borda count method.
– The “top 25” teams are ranked based on ballots from
sportswriters, coaches, etc., ranking their personal
top 25 choices.
– Each 1st-place team receives 25 points.
– Each 2nd-place team receives 24 points.
– (And so on…) Each 25th-place team receives 1 point.
– The weighted averages are then used to rank all
teams in order.
– Typically, as many as 40 or 50 teams may receive
votes (points) from at least a few voters.
NCAA Women’s Preseason Poll
ESPN/USA Today (31 writers)
Team/Rank
Points
Team/Rank
Points
1. Connecticut
one 3rd-place)
18. Arizona State
215
2. Stanford
676
19. Ohio State
188
3. Rutgers
639
20. Old Dominion
163
4. North Carolina
629
21. Auburn
154
5. Maryland
624
22. LSU
133
6. Tennessee
602 (one 1st-place vote)
23. Pittsburgh
104
7. Oklahoma
600
24. Florida State
101
8. Duke
508
25. Georgia
94
9. California
495
26. Xavier
87
10. Texas A&M
493
27. Purdue
85
11. Louisville
481
28. Kansas State
72
12. Vanderbilt
419
29. Iowa State
63
13. Texas
315
30. Marist
48
14. Notre Dame
294
…
…
15. Virginia
279
41. (3 teams)
3
15. Oklahoma State
279
44. (4 teams)
2
17. Baylor
272
48. (5 teams)
1
A method from very close to home
• The Bucklin method: a multi-stage variation on
the Borda count method
– Voters rank their choices on a preference ballot.
– The preferences are considered one level at a time.
• Initially, only the 1st choices are considered. If one
candidate receives a majority of them, that candidate is the
winner.
• If not, the 2nd choices are added to each candidate’s total.
– There are now twice as many “votes,” cast by the same
number of voters.
– If any candidate now has the support of a majority of voters,
that candidate is the winner.
• If multiple candidates surpass a majority in the same round,
the one with the highest level of support is the winner.
• This process repeats through as many stages as needed.
– Invented by James W. Bucklin (1856-1919), co-founder
of Grand Junction.
The pairwise-comparison method
• Also known as Copeland’s method
• Voters first rank all candidates with a preference ballot.
• Then, for each combination of two candidates, we consider their
• If candidate A was ranked higher than candidate B on a majority of
the ballots, regardless of by how much, then A receives 1 point.
– If each one was ranked higher than the other by the same number of
voters, then each one receives ½ point.
• After all possible combinations are compared, the total points
awarded to each candidate are summed up.
– The total of all points awarded will be the total number of matchups
(which will be nC2, a fact we can use to make sure we haven’t missed
any matchups).
• This is also the round-robin method used in many competitions, in
Another pairwise-comparison method
• One alternative version is the Kemeny-Young method.
• Voters again rank all candidates with a preference ballot and tally
• This time, we keep track of how many voters ranked candidate A
higher than candidate B on the ballots, not just “who was higher.”
• For each of the possible n! permutations of the rankings of the n
candidates, we tally the percent of voters who preferred each of the
n candidates to each of the ones below them.
– Example: For ranking ACB (1 of 6 A/B/C ones), find how many
preferred A over C, A over B, and C over B. (no other comparisons)
• After all of those matchups are compared, the sum of all the
percentages calculated gives the Kemeny sequence score.
– Whichever specific sequence of the n candidates has the highest
Kemeny sequence score is decreed to be the “winning sequence.”
• Its rankings are considered the aggregate voters’ 1st choice, 2nd choice, etc.
•
•
In the 2007 French elections, François Bayrou
would have likely defeated all of his opponents
using either the Borda count or pairwisecomparison methods.
Condorcet’s criterion: if candidate X can defeat
X deserves to win the election.
– Marie Jean Antoine Nicolas Caritat, the Marquis de
Condorcet (1743-1794), French mathematician
•
Clearly, anyone who is ranked first on all ballots
will win the election.
matchup, for an unsurpassable total of (n - 1) points.
•
However, it is possible for a candidate who would
win using the plurality method to lose using the
Borda count method, if they are ranked low on
many other ballots.
Using the plurality method
Preference 15 voters
7 voters
13 voters
5 voters
2 voters
1st
Helen
Eddie
Grover
Donna
Eddie
2nd
Donna
Donna
Flora
Grover
Flora
3rd
Eddie
Grover
Eddie
Flora
Grover
4th
Flora
Flora
Donna
Eddie
Donna
5th
Grover
Helen
Helen
Helen
Helen
• For each column above, the heading tells us how many voters
submitted an identical ballot of that type.
• Using this method, Helen would be declared the winner.
Using the plurality-with-elimination method
Preference 15 voters
7 voters
13 voters
5 voters
2 voters
1st
Helen
Eddie
Grover
Donna
Eddie
2nd
Donna
Donna
Flora
Grover
Flora
3rd
Eddie
Grover
Eddie
Flora
Grover
4th
Flora
Flora
Donna
Eddie
Donna
5th
Grover
Helen
Helen
Helen
Helen
• Suppose that only Helen and Grover, the two candidates with the
– Barring any unforeseen change of heart, the likely result for the runoff would be:
Preference 15 voters
7 voters
13 voters
5 voters
2 voters
1st
Helen
Grover
Grover
Grover
Grover
2nd
Grover
Helen
Helen
Helen
Helen
• Helen is “hated” by most voters, but her supporters are numerous
enough to get her into the runoff, where all other voters flock to
Grover, who would be declared the winner using this method.
Using the Borda count method
Preference 15 voters
7 voters
13 voters
1st (5 pts.) Helen (75)
Eddie (35)
Grover (65) Donna (25)
2nd (4 pts.)
Donna (60)
Donna (28)
Flora (52)
3rd (3 pts.)
Eddie (45)
Grover (21) Eddie (39)
Flora (15)
Grover (6)
4th (2 pts.)
Flora (30)
Flora (14)
Donna (26)
Eddie (10)
Donna (4)
5th (1 pt.)
Grover (15) Helen (7)
Helen (13)
Helen (5)
Helen (2)
Candidate
Total Points
Donna
143
Eddie
139
Flora
119
Grover
127
Helen
82
5 voters
2 voters
Eddie (10)
Grover (20) Flora (8)
• Donna would be declared the winner
using this method.
– She was the top choice for very few voters, but
was an acceptable alternative for many others,
whose combined support increased her total.
• Helen suffers the most, a victim of the
low “weight” she is given by most voters.
Using the pairwise-comparison method
Preference 15 voters
7 voters
13 voters
5 voters
2 voters
1st
Helen
Eddie
Grover
Donna
Eddie
2nd
Donna
Donna
Flora
Grover
Flora
3rd
Eddie
Grover
Eddie
Flora
Grover
4th
Flora
Flora
Donna
Eddie
Donna
5th
Grover
Helen
Helen
Helen
Helen
Cand.
Cand.
Cand.
Cand.
Donna
20
Eddie
22
Eddie
24
Grover
18
Donna
27
Flora
15
Eddie
27
Helen
15
Donna
27
Grover
15
Flora
17
Grover
25
Donna
27
Helen
15
Flora
27
Helen
15
Eddie
24
Flora
18
Grover
27
Helen
15
Using the pairwise-comparison method
Candidate
Total
Points
Donna
3
Eddie
4
Flora
1
Grover
2
Helen
0
• Eddie would be declared the
winner using this method.
– He was perfectly balanced, finishing
ahead of Donna and Helen on
roughly half of the ballots and ahead
of Flora and Grover on the other
half.
– With the “margin of victory”
matchups.
• Helen again suffered from being
the bottom choice for most
voters, insuring she would lose
all possible matchups.
Recap of the “Four-Method Faceoff”
•
•
•
•
•
•
majority method: no immediate winner
plurality method: Helen wins
plurality-with-elimination method: Grover wins
Borda count method: Donna wins
pairwise-comparison method: Eddie wins
Only Flora, a mediocre candidate in all voters’
minds, fails to win under any of these methods.
Recap of the “Four-Method Faceoff”
• Each method yielded different results because each was
violating a different “fairness criterion.”
Violates
majority
criterion?
Violates
Concorcet’s
criterion?
Violates
indep.-ofirrelevant-alt.
criterion?
Violates
monotonicity
criterion?
Plurality method
No
Yes
Yes
No
Plurality-withelimination
No
Yes
Yes
Yes
Borda count
method
Yes
Yes
Yes
No
Pairwise-comp./
Kemeny-Young
No
No
Yes
No
• So...which voting method should we use?
• Once again, we ask the question, “Is
there a best way to do this?”
• Once again, we receive the unfortunate
• Arrow’s Impossibility Theorem: In
any election involving more than two
candidates, there is no voting system
that will satisfy all four of the “fairness
criteria.”
– Kenneth Arrow (1921- ), RAND Corporation
– youngest-ever winner of the Nobel Prize in
Economic Sciences, at age 51
• won for his work in equilibrium theory
More Equal Than Others
• “All animals are created equal…but some animals are
more equal than others.”
– George Orwell, Animal Farm
• Under weighted voting systems, some voters receive
“supervotes” that count more than others.
– U.N. Security Council: 15 members, 5 of which have a veto
• All 5 founding members must vote yes (or abstain) for any measure to pass.
– Juries: in criminal cases, a single holdout can prevent a decision
• Each of the jurors thus has veto power, able to prevent a verdict either way.
– Stockholder meetings: preferred-stock shares, which grant
heavier voting power…for a premium share price
– Dictators: their single “vote” outweighs all others combined
• Roman Empire: the tribune’s cry of “Veto!” (“I forbid it!”) could overturn any
decision of the Senate
Building a Coalition
• quota: the number of votes necessary to pass a
measure
• Whether the voting system is equally weighted
or not, the voters can attempt to form coalitions
to consolidate their block of voting power.
– If their combined voting power is greater than or equal
to the quota, then it is a winning coalition.
– In systems where a supermajority quota is needed,
minority blocs can still form a blocking coalition
capable of preventing a measure from passing.
• U.S. Senate: 41 members can sustain a filibuster
Keeping a Coalition
• A voter or group whose participation in a coalition
is necessary for it to remain a winning coalition is
called a critical voter.
• A voter or group could thus proclaim its
importance by calculating how often it is the
critical component of a coalition.
• John F. Banzhaf III (1940- ): GWU law professor
Banzhaf pow erindex =
# of times a voter/group is
criticalin w inningcoalitions
total # of times any voter/group is
is criticalin w inningcoalitions
Party
MPs
Conservative
143
Liberal
77
Québécois
49
NDP
37
Others
2
Total
308
Party
BPI
Conservative
24
Liberal
8
48 = 16.6%
Québécois
8
48 = 16.6%
NDP
8
48 = 16.6%
48 = 50%
• With just short of a majority, the
Conservatives can theoretically choose
any of the other three major parties to
ally with for a ruling coalition.
• They are thus critical in every winning
coalition except the triad coalitions of the
other three (163 MPs total).
• However, so are any of the three when
only one works with the Conservatives,
or when the three band together.
• Winning coalitions: {CL}, {CQ}, {CN},
{CLQ}, {CLN}, {CQN}, {CLQN}, {LQN},
and each with either/both of the other 2
– (i.e., 4 versions of each of these 7)
• Theoretically, there are 251 possible Electoral
College outcomes if we only include two parties
as potential winners.
– Within these, there are more than 9 trillion possible
winning coalitions to consider.
– With computer assistance, we can calculate the
Banzhaf power index of each state and D.C.
•
•
•
•
California: critical in 11% of winning electoral-vote totals
Texas and New York: critical in 6% of winning combinations
Colorado: critical in 1.5% of winning combinations
7 least-populated states and D.C., each with only 3 electoral
votes: critical in 0.55% of winning combinations
A new Electoral College?
• The Constitution mandates the use of the Electoral College, so
amending it would require a ⅔ vote in the House and Senate, plus
the consent of ¾ of the state legislatures (38 of 50!).
– However, the manner in which each state names its electors is up to the
legislatures.
• Many proposals and referendums have been made for states to
award their electoral votes by Congressional district, with the winner
of the state as a whole receiving the 2 additional electoral votes.
– Maine and Nebraska are currently the only states to do so.
– This would mean battleground states were split almost evenly.
• Would candidates visit if they knew they would receive half of the votes either way?
– It would also give minority-party voters in lopsided states a voice.
• Candidates with a minority of support would still have a reason to campaign there.
• California wouldn’t be “quite so blue” of a state, Texas not “quite so red.”
• Thus, for now, same-party state legislatures in “very red” or “very blue” states have little
incentive to end their winner-take-all advantage.
– Under this system, Bush would have beaten Gore 289 to 249.
Other voting dilemmas
• Presidential primary season
– Statewide party votes staggered throughout the winter and
spring of election years
– Intense focus on early primaries (NH, IA, SC), with very little on
states late in the primary season (winners chosen before then)
– 2008: historical exception, with 10 candidates from each party
• Democratic candidate Obama: did not achieve majority of
convention delegates until final primaries in early June
• Republican candidate McCain: achieved majority of delegates
relatively early
• Candidates dropped out quickly if they found little early support
– Lower turnout in Republican primaries once the contest was “over”
– Later ballots would include only 2 or 3 of the 10 original candidates
– Republican voters in late-primary states felt robbed of their chance to
vote for their preferred candidate
• Approval voting: candidates are not ranked; instead,
voters can cast a vote for as many of the candidates as
they like.
– The winner is the candidate receiving the most votes of
approval.
– Encourages the election of consensus-building candidates
acceptable to all, rather than the ones who can build the largest
core group of supporters
– Can reduce infighting, since candidates do not have to take
support away from others to gain it for themselves
– Used to select leaders in many “collegial” organizations
•
•
•
•
•
Mathematical Association of America
Many universities
U.N. Secretary-General
Boy Scouts of America
What have we learned?
• Much like allocating a group of representatives, deciding how they
will be elected is not easy.
unhappy candidates, no method of voting is perfect.
• Depending on the voting system used, a candidate can end up with
a narrow victory, a crushing landslide, or a stunning defeat, all with
the same vote tallies.
• Even in our “two-party” system, smaller third-party candidates can
have a major mathematical impact.
• Under our current Electoral College system, large states get more
attention because there are more ways they can end up on the
“winning side.”
– If their electoral votes were split up, the “balance of power” might
change. (Of course, that’s why it probably won’t be changed.)
• And don’t even get me started about Parliamentary systems.
References
•
Most liberal-arts college-mathematics course (ex.: MATH 110) textbooks
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•
•
The Alfred Nobel Foundation: www.nobelprize.org
Banzhaf Power Index analysis of the Electoral College (2 studies):
–
–
•
•
www.thirty-thousand.org/pages/Neubauer-Zeitlin.htm
Interactive electoral maps, both historic and modern: www.270towin.com
The story of GJ’s own James W. Bucklin: www.gjhistory.org/cat/main.htm
–
•
•
Analysis of 2000 Presidential election given different House sizes:
–
•
•
www.everything2.com/node/1678149
www.cs.uiowa.edu/~dsidran/2000%20Power%20Indices.htm
Banzhaf’s own website: www.banzhaf.net
National Archive 2000 Election Returns:
–
•
Including ours, Thomas L. Pirnot’s Mathematics All Around, 3rd edition
And look for his Grand Junction charter and voting system debut at books.google.com!
And yes, of course, Google and Wikipedia.
My Mesa State homepage, at www.mesastate.edu/~mcrogers, will have this
presentation plus the previous one.
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