Chapter 12Electing the

PresidentPaul Moore

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)

Election Reform◦ Approval Voting

Electoral College◦ Strategies to maximize:

Popular Votes Electoral Votes

Presentation Outline

Elections every 4 years 35 years old Native born citizens US residents for 14 years No third term

Running for President

Democratic and Republican Primaries◦ Candidates campaign for party nomination

Party nominates candidates ◦ National conventions

General Election◦ 2-3 serious contenders◦ Electoral College

How to become President

Campaign strategies◦ Choosing states to campaign based on electoral

college weight

Effects of reform on these strategies◦ Approval voting◦ Popular voting (without Electoral College)

Candidates getting a leg up in the primaries to help them win their party’s nomination◦ Spatial Models

Math & Presidential Elections

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) Unimodal vs Bimodal

◦ Voter distribution represented by curve, giving number of voters with attitudes at different points on L-R continuum

Spatial Model: 2 Candidates

Unimodal Distribution

•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

Num

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f vot

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Voter Positions on L-R Continuum

Candidate A Candidate BM

Unimodal DistributionNu

mbe

r of v

oter

s

Candidate A Candidate BM

Voter Positions on L-R Continuum

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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?

MCandidate A 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

Unimodal Distribution

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Candidate A Candidate BM

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

Unimodal Distribution

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

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)?

Unimodal Distribution

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.

Unimodal Distribution

…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?

Unimodal Distribution

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

Any others?

Bimodal Distribution

M

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Bimodal Distribution

M

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2 Cases for possible equilibriums (other than M) 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.

Bimodal Distribution

M

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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

How does M compare with the mean of the distribution

Mean of Voter Distribution

where :n = total number of voters = n1+n2+n3+…+nk

k = number of different positions i that voters take on continuumni = number of voters at position ili = location of position i on continuumΣ is from i = 1 to k

Weighted average – location of each position is weighted by number of voters at that position

Exercise 1!

Bimodal Distribution

Exercise 1:

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

From Median Voter Theorem: ◦ if the distribution is skewed, then it may not be rational for candidates to

choose the mean of the distribution

Even number of voters?◦ Equilibrium positions?

Bimodal DistributionMedian: 0.6, Mean: 0.56

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)

Consider example:◦ n = 26◦ k = 8 different positions over interval [0, 1]

◦ Mean = 0.5◦ Median = 0.45 (average of 0.4 and 0.5)

Bimodal Distribution

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 DistributionPosition, 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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9012345678

Bimodal Distribution

Both candidates at M, they’re in equilibrium◦ Is this equilibrium position still unique?

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

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

Bimodal Distribution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9012345678

Bimodal Distribution

Mean = 0.5Median = 0.45

M

Primary elections often have more than 2 candidates

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

Spatial Models: +2 Candidates

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Candidate A(red)

Candidate B(blue)M

Is it rational for a 3rd candidate to enter the race? (are there any positions offering the candidate a chance at success?)

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

Spatial Models: +2 Candidates

Num

ber o

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ers Candidate A

(red)Candidate B

(blue) Candidate C(pink)

MA/B

C

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

C can also enter on the left side of M, still winning by the same logic

Can a 4th candidate, D, enter the race and win?

Midway betweenA/B and C

Median M no longer appealing to candidates◦ Vulnerable

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

Spatial Models: +2 Candidates

MA B

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

Spatial Models: +2 Candidates

MA B

Spatial Models: +2 Candidates

A BM

2/3 1/61/6

1/6 each

Not exactly to scale

Exercise 2

Spatial Models: +2 Candidates

MA B

C

MA B

C

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

Approval Voting

Election Reform

Approval Voting◦ Voters can vote for as many candidates as they like or find

acceptable. Candidate with the most approval votes wins.

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

Election Reform

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

Electoral College

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

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

Electoral College

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

The probability of a voter voting Republican is 1 – pi

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

Maximizing Popular Vote

Basically a weighted average, weighted by probabilities

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

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)

Exercise 3

Electoral College

The optimal spending strategy for each state (from di* and ri

*) is ◦ ($14M : $21M : $28M) on states 2, 3, and 4 respectively

Meaning the probability that either candidate will win any state i is 50%◦ pi = 50% for all i toss up states

So at optimal strategy, the EPV is the same for both candidates (D = R)◦ EPVD or R = 2[14/(14+14)] + 3[21/(21+21)] + 4[28/(28 + 28)]

Strategy sound familiar?◦ Candidates are at equilibrium

Now Exercise…what happens when departing from equilibrium?

Maximizing Popular VoteCollege

Exercise 3 Calculating p for the Republican candidate in 3 states

◦ p1 = 14/14◦ p2 = 21/(21+27)◦ p3 = 28/(28+36)

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

Can the Republican candidate do even better?◦ Change his spending to ($2M : $26M : $35M) to achieve an ◦ EPVR = 5.44 votes

*Departure from popular-vote maximizing strategy lowers candidates expected popular vote*

Maximizing Popular Vote

Way to Go!!

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

Expected Electoral Votes (EEV)

Where vi = number of electoral votes of toss-up statePi = probability that the Democrat wins more than 50% of popular votes in state i

Maxing Electoral Votes

Expected Electoral Votes (EEV)

Calculating Pi ◦ Must determine all probabilities that majority of voters in i will vote

Democratic

3 states: A, B, C with 2, 3, 4 electoral votes 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

Maxing Electoral Votes

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

Maxing Electoral Votes

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

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) = 27d2

* = 163/2 = 16 √(16) = 16(4) = 64d3

* = 253/2 = 25√(25) = 25(5) = 125

Optimal allocation of resources:(27 : 64 : 125)

Maxing Electoral Votes

Mathematics is certainly used, though not obviously, in strategic aspects of campaigning and voting in presidential elections

Is there a better way to elect president?

Many believe in approval voting◦ Believe it would better enable voters to

express their preferences◦ What do you think?

Electoral College creates a large statebias

Conclusions & Discussion

HW: Chapter 12(45, 51)(7th Ed)