Paul Moore
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What math can tell us about elections and strategy
behind them
Spatial Models (Candidate positions on issues)
◦ 2 candidates (Unimodal, Bimodal)
◦ 2+ candidates (1/3 separation, 2/3 opportunity)
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Election Reform
◦ Approval Voting
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Electoral College
◦ Strategies to maximize:
 Popular Votes
 Electoral Votes
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Elections every 4 years
35 years old
Native born citizens
US residents for 14 years
No third term
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Democratic and Republican Primaries
◦ Candidates campaign for party nomination
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Party nominates candidates
◦ National conventions
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General Election
◦ 2-3 serious contenders
◦ Electoral College
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Campaign strategies
◦ Choosing states to campaign based on electoral
college weight
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Effects of reform on these strategies
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Candidates getting a leg up in the primaries
to help them win their party’s nomination
◦ Approval voting
◦ Popular voting (without Electoral College)
◦ Spatial Models
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Model Assumptions:
◦ Voters respond to positions on issues
◦ Single overriding issue, candidates must chose side
◦ Voter attitudes represented as “left-right
continuum” (very liberal to very conservative)
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Unimodal vs Bimodal
◦ Voter distribution represented by curve, giving
number of voters with attitudes at different points
on L-R continuum
Number of voters
Candidate A
M
Candidate B
Voter Positions on L-R Continuum
•Unimodal – one peak, or mode
•Pictured as continuous for simplicity
•Median, M – of a voter distribution is the point on the horizontal axis
where half the voters have attitudes that lie to the left, and half to right
Candidate B
Voter Positions on L-R Continuum
Number of voters
Number of voters
Candidate A M
Candidate A M
Candidate B
Voter Positions on L-R Continuum
•Attitudes are a fixed quantity, decisions of voters depend on position
of candidates
•Candidate positions: Candidate A (yellow line), Candidate B (blue line)
•Assume voters vote for candidate with attitudes closest to their own
(and that all voters vote)
•What happens in models above?
Number of voters
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Candidate A
M
Candidate B
“A” attracts all voters to the left of M, while “B” attracts all voters to
the right of M
Any voters on the horizontal distribution between “A” and “B” (when
they are not side by side) are split down the middle
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Maximin – the position for a candidate at which there is no
other position that can guarantee a better outcome (more
voters), no matter what the other candidate does
At what position is a candidate in maximin?
Is there more than one maximin position?
Taking position at M guarantees a candidate 50% of the votes,
no matter what the other candidate does
Is there any other position that can guarantee a candidate
more?
◦ No, there is no other position guaranteeing more votes
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Further more M is stable, meaning that once a candidate
chooses this position, the other candidate has no incentive to
choose any other position except M.
◦ M is a maximin for both candidates, and they are in equilibrium
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Equilibrium – when a pair of positions, once chosen by
candidates, does not offer any incentive to either candidate to
depart from it unilaterally
Is there another equilibrium position(s)?
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Equilibrium Positions…unique?
◦ 2 cases:
 Common point – both candidates take the same position
 Distinct positions – one taken by each
◦ Case 1: Common Point
 If candidates are in at a common point, to the left of M for example,
then one candidate can always do better by moving right but staying on
the left of M. The same idea can be applied to common points to the
right of M. So common position other than M cannot be equilibrium
◦ Case 2: Distinct Positions
 If candidates are in two different positions then one candidate may
always do better by moving alongside of the other candidate, gathering
more voters. So distinct positions cannot be in equilibrium.
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…From this we get the Median Voter theorem
Median Voter Theorem – in 2 candidate elections with an odd
number of voters, M is the unique equilibrium position
Bimodal distribution?
Number of voters
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M
Use same logic as unimodal distribution to examine unique
equilibriums
Again at M, a candidate is guaranteed at least 50% of the
votes no matter what the other candidate does
◦ It is a maximin for both candidates, and an equilibrium
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Any others?
Number of voters
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M
2 Cases for possible equilibriums (other than M)
 Common point – both candidates take the same position
 Distinct positions – one taken by each
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Case 1: Common Point
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Case 2: Distinct Positions
◦ If candidates are in at a common point, to the left of M for example, then one
candidate can always do better by moving right but staying on the left of M.
The same idea can be applied to common points to the right of M. So common
position other than M cannot be equilibrium
◦ If candidates are in two different positions then one candidate may always do better
by moving alongside of the other candidate, gathering more voters. So distinct
positions cannot be in equilibrium.
Number of voters
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M
So using the same logic, we can see that M is the unique equilibrium
for bimodal distributions (Median voter theorem)
Extension:
◦ Median Voter Theorem can be applied to any distribution of electorate’s attitudes
◦ This is because the logic for the proof does not rely on any modal characteristics.
Only the idea that to the left and right of the median lies an equal distribution of
voter attitudes
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How does M compare with the mean of the distribution
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Mean of Voter Distribution
where :
n = total number of voters = n1+n2+n3+…+nk
k = number of different positions i that voters take on continuum
ni = number of voters at position i
li = location of position i on continuum
Σ is from i = 1 to k
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Weighted average – location of each position is weighted by number
of voters at that position
Exercise 1!
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Exercise 1:
Median: 0.6, Mean: 0.56
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Mean need not coincide with median M
In exercise, distribution is skewed to the left
◦ Area under the curve is less concentrated to the left of M than to the right
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From Median Voter Theorem:
◦ if the distribution is skewed, then it may not be rational for candidates to
choose the mean of the distribution
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Even number of voters?
◦ Equilibrium positions?
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Even # of Voters, Discrete Distribution
◦ Discrete Distribution of voters - where voters are located at only certain positions
along the left-right continuum (like in the exercise)
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Consider example:
◦ n = 26
◦ k = 8 different positions over interval [0, 1]
Position, i
1
2
3
4
5
6
7
8
Location (li) of position I
0
0.2
0.3
0.4
0.5
0.7
0.8
0.9
Number of voters (ni) at position i
2
3
4
4
2
3
7
1
◦ Mean = 0.5
◦ Median = 0.45 (average of 0.4 and 0.5)
Position, i
1
2
3
4
5
6
7
8
Location (li) of position I
0
0.2
0.3
0.4
0.5
0.7
0.8
0.9
Number of voters (ni) at position i
2
3
4
4
2
3
7
1
◦ Mean = 0.5
◦ Median = 0.45 (average of 0.4 and 0.5)
Bimodal Distribution
8
7
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Bimodal Distribution
M
8
7
Mean = 0.5
Median = 0.45
6
5
4
3
2
1
0
0
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Both candidates at M, they’re in equilibrium
◦ Is this equilibrium position still unique?
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Any pair of positions between 0.4 and 0.5 is in equilibrium
◦ Following that, distinct positions 0.4 and 0.5 are also in equilibrium
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In general
◦ With even number of voters and 2 middle voters have different positions then the
candidates can choose those 2 positions, or any in between, and be in equilibrium
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Primary elections often have more than 2 candidates
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Under what conditions is a multicandidate race “attractive”?
◦ Using a similar model, will examine the different positions of an entering
3rd candidate
Consider the unimodal 2 candidate race with both at M
Number of voters
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Is it rational for a 3rd candidate
to enter the race? (are there any
positions offering the candidate
a chance at success?)
Candidate A
M
(red)
Candidate B
(blue)
Number of voters
Candidate A
(red)
Candidate B
(blue)
Midway between
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A/B and C
Candidate C
(pink)
Candidate C enters race at
position C on graph
◦ C’s area of voters is yellow
◦ A and B have to split the light blue
voters
◦ C wins plurality of votes
M C
A/B
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Upon entry C gains support of voters to the right, and some to left
Blue votes are split between A and B, and C is left with the majority
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C can also enter on the left side of M, still winning by the same logic
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Can a 4th candidate, D, enter the race and win?
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Median M no longer appealing to candidates
◦ Vulnerable
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However, a 3rd candidate C will not necessarily win against both A
and B
1/3-Separation Obstacle
◦ If A and B are distinct positions equidistant from M of symmetric distribution, and
separated from each other by at most 1/3 of total area, then C can take no position
that will displace A and B and enable C to win
A
M
B
A
B
M
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2/3 Separation Opportunity
◦ If A and B are distinct positions equidistant from M on a symmetric
distribution and separated by at least 2/3 of the area, then C can defeat
both candidates by taking position at M
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Not exactly to scale
1/6
each
A
1/6
M
2/3
B
1/6
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Exercise 2
C
C
A
B
M
A
B
M
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Abolition of the Electoral College
More accurate and reliable ballots
Eliminating election irregularities
Most reforms ignore problem with
multicandidate elections
◦ Candidate who wins is not always a Condorcet
winner
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Approval Voting
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Approval Voting
◦ Voters can vote for as many candidates as they like or find
acceptable. Candidate with the most approval votes wins.
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2000 Election
◦ Came down to the “toss-up” state of Florida, where Bush
won the electoral votes by beating Gore by a little over 300
popular votes
◦ According to polls, Gore was the second choice of most
Nader voters
 In an approval voting system Gore would have almost certainly
won the election since Nader supporters could have also given a
vote of approval to Al Gore
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Original purpose was to place the selection of a president in
the hands of a body that, while its members would be chosen
by the people, would be sufficiently removed from them so
that it could make more deliberative choices.
How it works
◦ Each state gets 2 electoral votes (for the 2 senators)
◦ Also receives 1 additional electoral vote for each of its representatives in
the House of Representatives (number of members for each state based on
population)
 Ranges from 1 in the smallest states to 53 in California
◦ Altogether there are 538 electoral votes, so candidate needs 270 to win
 In 2000, Bush received 271
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Advantages in small states?
◦ Technically California voters are about three times more powerful as
individuals than those in the smallest states
◦ Though in smallest states with 1 representative and 2 senators (3 electoral
votes), the population receives a 200% (2/1) boost from having 2
senatorial electoral votes automatically.
◦ California receives less than 4% (2/53) boost from senatorial electoral
votes
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Now that we know how it works, let’s examine it’s role in the
2000 election
◦ Look at it as a game between 2 major party candidates
◦ Develop 2 models
 Candidates seek to max their expected popular vote
 Candidates seek to max their expected electoral vote
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Both models use the assumption that the probability of a
voter in a “toss up state” i votes for the Democratic candidate
is:
 where di and ri represent the proportion of campaign resources spent in state i by
the Democratic and Republican candidates, respectively
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The probability of a voter voting Republican is 1 – pi
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Expected Popular Vote (EPV)
◦ Is toss up states, the EPV of the Democratic candidate (EPVD) is the number
of voters, ni, in toss up state i, multiplied by the probability, pi, that a
voter in this toss up state votes Democrat, summed up across all toss up
states is
Basically a weighted average, weighted by
probabilities
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Candidates allocate resources across toss up states and
attempt to do so in an optimal fashion
Democratic candidate seeks strategy di to maximize EPVD
◦ Much like profit maximization among feasibility regions in Chapter 4
◦ Here some of the constraints are amount of campaign resources, time
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Proportional Rule
◦ Strategy of Democratic candidate to maximize EPVD, given Republican
candidate also chooses maximizing strategy is:
 Summed up across toss up states
◦ Candidate allocates resources in proportion to the size of each state (ni/N)
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Exercise 3
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The optimal spending strategy for each state (from di* and ri*)
is
◦ ($14M : $21M : $28M) on states 2, 3, and 4 respectively
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Meaning the probability that either candidate will win any
state i is 50%
◦ pi = 50% for all i toss up states
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So at optimal strategy, the EPV is the same for both
candidates (D = R)
◦ EPVD
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or R
= 2[14/(14+14)] + 3[21/(21+21)] + 4[28/(28 + 28)]
Strategy sound familiar?
◦ Candidates are at equilibrium
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Now Exercise…what happens when departing from equilibrium?
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Exercise 3
Calculating p for the Republican candidate in 3 states
◦ p1 = 14/14
◦ p2 = 21/(21+27)
◦ p3 = 28/(28+36)
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EPV R
◦ EPV R= 2[14/14] + 3[21/(21+27)] + 4[28/(28 + 36)] = 5.06 votes
◦ or 56% of the 9 votes in those 3 states
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Can the Republican candidate do even better?
◦ Change his spending to ($2M : $26M : $35M) to achieve an
◦ EPVR = 5.44 votes
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*Departure from popular-vote maximizing strategy lowers
candidates expected popular vote*
Way to Go!!
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Assume now goal is to max electoral votes
Candidate may think of throwing all resources into
11 largest states
◦ 11 largest states have majority of electoral voters (271)
◦ However, opponent may simply spend enough in 1 big
state to defeat and use rest to spend small amounts in
other 39, winning them
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Expected Electoral Votes (EEV)
 Where vi = number of electoral votes of toss-up state
Pi = probability that the Democrat wins more than 50% of
popular votes in state i
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Expected Electoral Votes (EEV)
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Calculating Pi
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3 states: A, B, C with 2, 3, 4 electoral votes
◦ Must determine all probabilities that majority of voters in i will
vote Democratic
 Again, assume number of pop votes = number of electoral votes
◦ State A: both voters
 PA = (pA)(pA)) = (pA)2
◦ State B: 2 of 3 (3 ways) or all
 PB = 3[(pB)2(1 – pB)] + (pB )3
◦ State C: 3 of 4 (4 ways) or all 4
 PC = 4[(pC)3(1 – pC)] + (pC)4
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Strategies to maximize ( 3/2’s Rule )
◦ Candidates should allocate resources in proportion to
number of electoral votes of each state (vi ) multiplied by
the square root of its size (ni).
◦ Can also be used to approximate maximizing strategies
 Number of electoral voters is roughly proportional to number of
voters in each state
So, if the candidates allocate same amount to each toss up,
3/2’s rule says they should spend approximately in
proportion to 3/2’s power of the # of electoral votes in
order to maximize EEV
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Applying 3/2’s Rule
◦ 3 States: A, B, C with 9, 16, 25 electoral voters respectively
◦ Candidates want to know how much to use in each state
◦ Use approximation
◦ If all states are toss ups, then 3/2’s rule says candidates should allocate
resources accordingly, spending, in total, the approximate value of S
(S = D)
d1* = [ 93/2 / S ]*D
= 93/2 = 9 √(9) = 9(3) = 27
d2* = 163/2 = 16 √(16) = 16(4) = 64
d3* = 253/2 = 25√(25) = 25(5) = 125
Optimal allocation of resources:
(27 : 64 : 125)
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Mathematics is certainly used, though not
obviously, in strategic aspects of campaigning and
voting in presidential elections
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Is there a better way to elect president?
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Many believe in approval voting
◦ Believe it would better enable voters to
express their preferences
◦ What do you think?
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Electoral College creates a large state
bias
HW: Chapter 12
(45, 51)
(7th Ed)