1. Chapter 12 Electing the President
Paul Moore

2. Presentation Outline
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

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

4. How to become President
Democratic and Republican Primaries
Candidates campaign for party nomination
Party nominates candidates
National conventions
General Election
2-3 serious contenders
Electoral College

5. Math & Presidential Elections
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

6. Spatial Model: 2 Candidates
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

7. Unimodal Distribution
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

8. Unimodal Distribution
Number of voters
Number of voters
Candidate A
Candidate B
Candidate A
Candidate B M M
Voter Positions on L-R Continuum
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?

9. Unimodal Distribution
Number of voters
Candidate A
Candidate B M
“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

10. Unimodal Distribution
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

11. 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)?

12. 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.

13. 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?

14. Bimodal Distribution
Number of voters 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
Any others?

15. Bimodal Distribution 2 Cases for possible equilibriums (other than M)
Number of voters M
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.

16. Bimodal Distribution Number of voters
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

17. Bimodal Distribution
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 continuum
ni = number of voters at position i
li = location of position i on continuum
Σ is from i = 1 to k
Weighted average – location of each position is weighted by number of voters at that position
Exercise 1!

18. 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?
Median: 0.6, Mean: 0.56

19. Bimodal Distribution 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)
Position, i 1 2 3 4 5 6 7 8
Location (li) of position I
0.2
0.3
0.4
0.5
0.7
0.8
0.9
Number of voters (ni) at position i

20. Bimodal Distribution 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
0.2
0.3
0.4
0.5
0.7
0.8
0.9
Number of voters (ni) at position i

21. Bimodal Distribution M Mean = 0.5 Median = 0.45
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

22. Spatial Models: +2 Candidates
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
Is it rational for a 3rd candidate to enter the race? (are there any positions offering the candidate a chance at success?)
Number of voters
Candidate A
(red)
Candidate B
(blue) M

23. Spatial Models: +2 Candidates
Candidate A
(red)
Midway between
A/B and C
Number of voters
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
Candidate B
(blue)
Candidate C
(pink) M C
A/B
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?

24. Spatial Models: +2 Candidates
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 A B M

25. Spatial Models: +2 CandidatesB M
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

26. Spatial Models: +2 Candidates
Not exactly to scale
1/6 each M A B
1/6
2/3
1/6

27. Spatial Models: +2 Candidates
Exercise 2 C C A B A B M M

28. Election Reform 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

29. Election Reform Approval Voting 2000 Election
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

30. Electoral College
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

31. 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 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

32. Maximizing Popular Vote
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
Basically a weighted average, weighted by
probabilities

33. Electoral College
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

34. Maximizing Popular VoteCollege
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/( )]
Strategy sound familiar?
Candidates are at equilibrium
Now Exercise…what happens when departing from equilibrium?

35. Maximizing Popular Vote
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/( )] = 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*
Way to Go!!

36. Maxing Electoral Votes
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 state
Pi = probability that the Democrat wins more than 50% of popular votes in state i

37. 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

38. 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
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

39. Maxing Electoral Votes
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)

40. Conclusions & Discussion
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 state
bias
HW: Chapter 12
(45, 51)
(7th Ed)