An Introduction to Voting
Theory: History and Procedures
Arnold B. Urken
Professor of Political Science
Division of Humanities and Social Science
Stevens Institute of Technology
aurken@stevens.edu
DIMACS Workshop, May 10, 2004
© 2004. Arnold B. Urken All Rights Reserved
1
Outline

Top Six Voting Systems

Pre-18th Century Voting Theory

18th Century France: The Golden Age?

The Rediscovery of Voting Theory

Preference Aggregation Issues

Competence in Voting Theory
© 2004. Arnold B. Urken All Rights Reserved
2
Top Six Voting Systems

Plurality voting

Borda voting

Condorcet scoring

Copeland scoring

Approval voting

STV (IRV)
© 2004. Arnold B. Urken All Rights Reserved
3
Top Six Voting Systems
[continued]
Voting systems include rules for
Vote Endowment: number of votes used to
express preferences
Vote Allocation: Saving or trading?
Vote Aggregation: Standard for producing a
collective outcome.
Allocation => “fungible voting,” which
allows votes to be saved and traded
© 2004. Arnold B. Urken All Rights Reserved
4
Hypothetical Data Set
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
Nine voters rank
1. Bush
2. Kerry
3. Nader
© 2004. Arnold B. Urken All Rights Reserved
5
Top Six Voting Procedures
[continued]
Plurality Voting
Endowment: One vote for the most preferred choice
Allocation: Trading/saving not explicitly allowed
Aggregation: Choice with the most votes wins (plurality)
© 2004. Arnold B. Urken All Rights Reserved
6
Plurality Voting Results
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Bush
4 votes
Kerry
3 votes
Nader
2 votes
© 2004. Arnold B. Urken All Rights Reserved
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
7
Plurality vs. Majority
What’s the Difference?
 Absolute vs. relative majority
 Historically
 Sanior et major pars
 Right/healthy and greater part
 Used to overturn outcomes
 Sanior difficult to measure, so major used
© 2004. Arnold B. Urken All Rights Reserved
8
Top Six Voting Procedures
[continued]
Borda Voting
Endowment: Assign ranks to choices
Allocation: Trading/saving not explicitly allowed
Aggregation: Choice with the most votes wins
(plurality)
© 2004. Arnold B. Urken All Rights Reserved
9
Borda Voting Results
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Bush
Nader
Kerry
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
18 points
18 points
18 points
Plurality aggregation not satisfied.
© 2004. Arnold B. Urken All Rights Reserved
10
Borda and Rankings
First Place
Bush
Kerry
Nader
Second Place Third Place
x
x
Illegal in some elections
© 2004. Arnold B. Urken All Rights Reserved
11
Borda and Rankings
[continued]
First Place
Bush
Kerry
Nader
x
Second Place Third Place
x
x
Not used this way
© 2004. Arnold B. Urken All Rights Reserved
12
Top Six Voting Procedures
[continued]
Condorcet Scoring
Endowment: Ordinal rankings assigned to choices
Allocation: Trading/saving not explicitly allowed
Aggregation: Winner is the choice with the most
victories in binary comparisons
© 2004. Arnold B. Urken All Rights Reserved
13
Condorcet Scoring Results
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Joe
Nader
Kerry
Bush
Bush
Kerry
Nader
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
9 points
9 points
9 points
Plurality aggregation not satisfied.
© 2004. Arnold B. Urken All Rights Reserved
14
Top Seven Voting Procedures
[continued]
Copeland Scoring
Endowment: Ordinal rankings assigned to choices
Allocation: Trading/saving not explicitly allowed
Aggregation: Winner is the choice with greatest net
score in binary comparisons
© 2004. Arnold B. Urken All Rights Reserved
15
Copeland Scoring Results
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Bush
Nader
Kerry
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
0 points
0 points
0 points
Plurality aggregation not satisfied.
© 2004. Arnold B. Urken All Rights Reserved
16
Top Seven Voting Procedures
[continued]
Approval Voting
Endowment: N votes where N = number of choices
Allocation: One vote cast for each approved
choice; no trading/saving
Aggregation: Plurality, majority, or unanimity
© 2004. Arnold B. Urken All Rights Reserved
17
Approval Voting Results
Assuming one approval vote is cast for 1st and 2nd place choices
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Bush
Nader
Kerry
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
5 points
6 points
5 points
Nader is the plurality winner!
Based on the number of voters who approve him
© 2004. Arnold B. Urken All Rights Reserved
18
Top Seven Voting Procedures
[continued]
Observations about Approval Voting
 Empirical observation: Voters cast an approval
vote for each choice ≥ average utility
 Ties possible under plurality, majority, and
unanimous aggregation rules
 Definitions of base for aggregation
 All allocated votes
 The number of voters casting votes
© 2004. Arnold B. Urken All Rights Reserved
19
Top Seven Voting Procedures
[continued]
STV (IRV--Proportional Representation)
Endowment: Assign ranks to choices
Allocation: One choice for each rank, trading/saving: not explicitly
allowed
Aggregation: Majority of first place votes, but if no choice wins, eliminate
the most choice most frequently ranked last and count first
place preferences again until a majority winner is produced
© 2004. Arnold B. Urken All Rights Reserved
20
STV (IRV—Proportional) Scoring Results
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Bush
Kerry
Nader
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
4 points
3 points
2 points
Majority aggregation not satisfied.
© 2004. Arnold B. Urken All Rights Reserved
21
STV (IRV—Proportional) Scoring Results
One Round of Elimination
Ranks John Mary Joe
1st
Kerry Kerry Nader
2nd
Nader Nader Kerry
3rd
Bush
Kerry
Nader
Tom Ruth Steve Debby Ed
Jack
Kerry Nader Nader Kerry
Nader Kerry
Nader Kerry Kerry Nader Kerry Nader
Deleted Information
Eliminated
5 votes
4 votes
Kerry is the majority winner!
© 2004. Arnold B. Urken All Rights Reserved
22
PR with Strategic Voting
Ranks
1st
2nd
3rd
John
Bush
Kerry
Nader
Mary
Kerry
Nader
Bush
Joe
Nader
Kerry
Bush
Tom Ruth
Kerry Nader
Nader Bush
Bush Kerry
Steve
Bush
Nader
Kerry
Debby
Bush
Kerry
Nader
Ed
Bush
Nader
Kerry
Jack
Kerry
Nader
Bush
Strategic Voting
Rank John Mary Joe Tom Ruth Steve Debby Ed
Jack
1st
Bush Kerry Nader Kerry Nader Bush Bush
Bush Kerry
2nd
Nader Kerry Nader Kerry
Nader
3rd
First Round Elimination
Rank John Mary Joe Tom Ruth Steve Debby Ed
Jack
1st
Bush Kerry Kerry Kerry Kerry Bush Bush
Bush Kerry
2nd
3rd
© 2004. Arnold B. Urken All Rights Reserved
23
Summary of Results
Method
Plurality
Majority
Borda
Condorcet
Copeland
Approval (Plurality)
Approval (Majority)
PR (IRV) Sincere
PR (IRV) Strategic
© 2004. Arnold B. Urken All Rights Reserved
Winner
Bush
None
Tie
Tie
Tie
Nader
None
Kerry
Kerry
24
Pre-18th Century Voting Theory
General Observations
 Theoretical insights were derived from
practical problem solving
 Knowledge was not cumulative
 The communication of votes was an
issue “Science” was
• “pre-normal” Kuhnian framework
• early stage Popperian “metaphysical”
research program
© 2004. Arnold B. Urken All Rights Reserved
25
Pre-18th Century Voting Theory
[continued]
 Pliny the Younger
 Ramon Lull
 Nicolaus Cusanus
 The Venetian Mehod
© 2004. Arnold B. Urken All Rights Reserved
26
Pre-18th Century Voting Theory
[continued]
 Pliny the Younger

Letter to Titius Aristo, A.D. 105

Agenda manipulation in the trial of Afranius Dexter’s
slaves

Slaves accused of murdering his master

Options
 Acquittal
 Banishment
 Death
© 2004. Arnold B. Urken All Rights Reserved
27
Pre-18th Century Voting Theory
[continued]

Execution faction leader leads switch from death to
banishment

Banishment is the majority choice

Pliny’s faction favored leniency, but included less than
one-half of all votes
© 2004. Arnold B. Urken All Rights Reserved
28
Pre-18th Century Voting Theory
[continued]

Pliny calls for ternary vote (with division of the whole)

Pliny knew that the opposition had the following
preference orders:
Death > Banishment > Acquittal
Banishment > Acquittal > Death
© 2004. Arnold B. Urken All Rights Reserved
29
Pre-18th Century Voting Theory
[continued]
 Why?
 Neither Acquittal nor Death would get a majority in the
first round of voting—in binary comparisons
 In the second round of voting, the winner of the first
round of voting (Acquittal or Death) would lose to
Banishment
 Sincere and manipulated voting produce the same
outcome!
 Pliny uncomfortable: inconsistent with Senate customs?
© 2004. Arnold B. Urken All Rights Reserved
30
Pre-18th Century Voting Theory
[continued]
 Issues Raised
 Sincere voting: honest communication of
preferences
 Strategic voting: changing “sincere” votes to
manipulate the collective outcome
 Pliny anticipates Robin Farquharson,
Theory of Voting. Yale, 1969
© 2004. Arnold B. Urken All Rights Reserved
31
Pre-18th Century Voting Theory
[continued]
Ramon Lull A.D. 1232-1316
 Explored methods for honest church elections
 Two methods based on selections of pairs of
choices from a larger set of ranked choices

Blanquera (1285)

De Arte Eleccionis (1299)
© 2004. Arnold B. Urken All Rights Reserved
32
Pre-18th Century Voting Theory
[continued]
Blanquerna (1285)

Mixed method (“art”) Borda and Condorcet

Electors choose Blanquerna as bishop without
following the “art” they generate an indecisive outcome
and the decision must be appealed to the Pope to
produce a winner

Work reflects ambivalence about preference
aggregation and making the right choice.
© 2004. Arnold B. Urken All Rights Reserved
33
Pre-18th Century Voting Theory
[continued]
De Arte Eleccionis (1299)

Condorcet scoring

Uses matrix notation (next used by Dodgson in the 19th
century)

Method does not address collective intransitivity (later
discovered by Condorcet and Arrow)
© 2004. Arnold B. Urken All Rights Reserved
34
Pre-18th Century Voting Theory
[continued]
Nicolaus Cusanus (1430)

Goal: design an “honest” voting procedure to elect a Holy
Roman Emperor to end a long schism in the papacy

Proposes a Borda system


Applies it to propositions with more than two choices
Criticizes manipulation of electorsand criticizes attempts to
control the collective outcome by manipulating electors.

Implicitly suggests that voting by ballot is new
© 2004. Arnold B. Urken All Rights Reserved
35
Pre-18th Century Voting Theory
[continued]
The Venetian Method (13th Century)
 Similar to approval voting
 Simplified the process of selecting 41
electors from an initial assembly of 1500
members.
© 2004. Arnold B. Urken All Rights Reserved
36
18th Century France: The Golden Age?
 Voting in the French Academy of
Sciences
 Borda, Condorcet, and others
 Condorcet and the French Revolution
 Daunou and after
 Proportional voting
© 2004. Arnold B. Urken All Rights Reserved
37
18th Century France: The Golden Age?
[continued]
Voting in the French Academy of Sciences
 Scientists recommend top three candidates
to the King of France
 Plurality voting used since 1699, ties rare.
 1770 Borda talk about plurality voting
 Borda paper not published until 1784
© 2004. Arnold B. Urken All Rights Reserved
38
18th Century France: The Golden Age?
[continued]
Voting in the French Academy of Sciences

Borda and Condorcet were political
enemies

Borda fought in the American Revolution

Condorcet, a modernist, won a
manipulated election as Secretary
© 2004. Arnold B. Urken All Rights Reserved
39
18th Century France: The Golden Age?
[continued]
Voting in the French Academy of Sciences

No evidence of actual voting debate

Condorcet regards Borda’s work as physicaille
(petty experiments)

Condorcet’s 1785 Essai
Essai sur l’application d’analyse à probabilité des
décisions rendues à la pluralité des voix
© 2004. Arnold B. Urken All Rights Reserved
40
18th Century France: The Golden Age?
[continued]
Voting in the French Academy of Sciences
 The 1785 Essai

Goal: analyze the probability of making a
correct collective choice

Introduction: identifies collective
intransitivity

Body: 13 hypothetical situations
© 2004. Arnold B. Urken All Rights Reserved
41
Condorcet “Jury Theorem”
Question: How does majority
rule affect the group probability
of making a correct choice?
Assumptions
• 50 or more
voters
1.0
• Binary choice
0.5
• One Person,
One Vote
• Preferences a
random variable
0
0
0.5
1.0
Individual Voter Competence
© 2004. Arnold B. Urken All Rights Reserved
• Individual
competence
statistically
independent
42
18th Century France: The Golden Age?
[continued]
Condorcet and the French Revolution

Creates practical voting plan for the
Republican Constitution with binary agendas

Recommends jury design for the trial of the
King of France

Robespierre’s hit list drives him underground

Dies in prison?
© 2004. Arnold B. Urken All Rights Reserved
43
18th Century France: The Golden Age?
[continued]
Daunou and after

FAS becomes the Institute of France

New election method needed


Napoléon interested

Borda and Daunou on commission
Daunou writes critique of Borda voting
© 2004. Arnold B. Urken All Rights Reserved
44
18th Century France: The Golden Age?
[continued]
Daunou and after (continued)

Voting theory is lost in French probability
theory (Cf. Daston)

Ideas rediscovered by Dodgson (Lewis
Carroll)

Nanson (Australia) refers to Condorcet’s
ideas in designing elections for scientists
© 2004. Arnold B. Urken All Rights Reserved
45
18th Century France: The Golden Age?
[continued]
Daunou and after (continued)

Proportional voting developed for
allocating seats in legislatures

Ideas are not integrated with voting
theorists
© 2004. Arnold B. Urken All Rights Reserved
46
The Rediscovery of Voting Theory
Black
 Does archival research on Condorcet
 Coins “jury theorem” to explain
Condorcet’s interest in competence
 Develops “single-peakedness” concept
to explain collective intransitivity
© 2004. Arnold B. Urken All Rights Reserved
47
The Rediscovery of Voting Theory
[continued]
Arrow
 Relies on Black to understand Condorcet
 Invents the term “social choice”
 Axiomatizes collective intransitivity
problem in impossibility theorem
© 2004. Arnold B. Urken All Rights Reserved
48
The Rediscovery of Voting Theory
[continued]
Arrow





Unrestricted domain or universality
Non-imposition or citizen sovereignty
Non-dictatorship
Monotonicity
Independence of irrelevant
alternatives
Impossible to satisfy all conditions simultaneously
© 2004. Arnold B. Urken All Rights Reserved
49
The Rediscovery of Voting Theory
[continued]
Brams and Fishburn
 Develop formal proposal for approval
voting
 Scientific societies adopt approval voting
 Articulate theoretical and empirical
arguments
© 2004. Arnold B. Urken All Rights Reserved
50
The Rediscovery of Voting Theory
[continued]
Saari
 Develops a geometric framework for
comparing voting methods for three
choices
 Does not address



Ties
Truncated preferences
Competence
© 2004. Arnold B. Urken All Rights Reserved
51
The Rediscovery of Voting Theory
[continued]
Preference Aggregation Issues

Vote trading and fungible voting

Manipulation: potential vs. actual

Voter use of voting methods
 Ranking choices (STV)
 Identifying approved set of choices
© 2004. Arnold B. Urken All Rights Reserved
52
The Rediscovery of Voting Theory
[continued]
Competence in Social Choice

Young: Maximum likelihood
interpretation of Condorcet’s rule

Grofman (Owen, Feld)
 Explore models of competence
 Show that Condorcet solved Rousseau’s
problem of reconciling “general will” and
the “will of all”
© 2004. Arnold B. Urken All Rights Reserved
53
The Rediscovery of Voting Theory
[continued]
Grofman-Shapley Theorem

How to weight votes in interdependent collective decisions

Don’t weight votes by using the ratio of
p/1-p (ratio of competence to incompetence)

Instead use
ln p/1-p
Experimental Confirmation
© 2004. Arnold B. Urken All Rights Reserved
54
Average individual
competence equals
group competence.
Average
individual does
better than the
group.
1.0
Group does
better than the
average
individual
0.5
00
0.5
1.0
Individual Voter Competence
Non-monotonic pattern in approval voting
© 2004. Arnold B. Urken All Rights Reserved
55
Reconciling Competence and Preferences
1.0
Minimum group
competence
Optimal
group
competence
0.5
Better than
minimum
performance
0
Suboptimal
performance
0.5
1.0
Average Voter Competence
Low
© 2004. Arnold B. Urken All Rights Reserved
High
56
Perspective
 History not of purely antiquarian interest
 Draws our attention to models and
problems of integrating ideas

Unresolved dualism

Preference aggregation

Competence
© 2004. Arnold B. Urken All Rights Reserved
57