1. Chapter 12: Electing the President Lesson Plan
For All Practical Purposes
Electing the President
Spatial Models for Two-Candidate Elections
Spatial Models for Multicandidate Elections
Narrowing the Field
What Drives Candidates Out?
Election Reform: Approval Voting
The Electoral College
Is There a Better Way to Elect a President?
Mathematical Literacy in Today’s World, 8th ed. 1
© 2009, W.H. Freeman and Company

2. Chapter 12: Electing the President Electing the President
Several Phases of the Presidential Election Process:
First phase – Democratic and Republican candidates seek their party’s nomination for president by running in state primaries.
Winning the state primaries almost always assures a candidate the party’s nomination in its national convention.
Final phase – General election in the fall, which typically involves only two serious contenders—Republican and Democratic nominees—but sometimes includes significant other candidates.
Two prominent reform proposals are analyzed for electing President: ) approval voting and 2) popular vote, instead of electoral college 2

3. Chapter 12: Electing the President Spatial Models for Two-Candidate Elections
Most common contests in the general election are between just two contenders.
Assumptions for the model of two-candidate elections:
We assume that voters primarily respond to the positions that the
Spatial Models – The representation of candidate positions along a left-right continuum in order to determine the equilibrium or optimal positions of the candidates.
candidates take on issues. Also, we assume other factors, such as personality, ethnicity, religion, race, etc., have no effect on outcomes.
We assume there is a single overriding issue on which the candidate must take a definite stand.
Voter Distribution – A graph that shows the attitudes of the individual voters being represented along a horizontal line (ranging from left-liberal to right-conservative). The number (percentage) of voters who share the attitude is represented by the vertical height. 3

4. Chapter 12: Electing the President Spatial Models for Two-Candidate Elections
Unimodal Distribution – A voter distribution that has one peak (mode) and is symmetric.
Median, M, is the point on the horizontal axis where half of the voters’ attitudes lie left and half lie right.
Maximin position for a candidate is one where no other position can guarantee a better outcome.
Equilibrium occurs if neither candidate has an incentive to depart from either one of his or her positions.
Median-voter theorem: In a two-candidate election with an odd number of voters, M is the unique equilibrium position. (True for any distribution.) 4

5. Skew to the left, Mean < Median Ī < M
Chapter 12: Electing the President Spatial Models for Two-Candidate Elections
Bimodal Distribution – Mean and median are different.
A voter distribution that has two peaks and is not symmetric.
Median, M, is once again the maximin and equilibrium position of the two candidates; the bulk of voters is at the two modes.
Mean, Ī, of a voter distribution is:
Ī = ∑ ni li
Where:
k = number of different positions i that voters take on the continuum
ni = number of voters at position i
Ii = location of position i on the continuum
n = ∑ ni = n1 + n2 + … + nk = total number of voters
Skew to the left, Mean < Median Ī < M k
i = 1 1 n
When skewed, candidates may not want to take the position of the mean. k
i = 1 5

8. Chapter 12: Electing the President Spatial Models for Two-Candidate Elections
Unimodal Distribution – Mean and median are the same.
When the mean Ī and median M are the same, you could possibly have a bimodal (two-peak) distribution and symmetry (mirror image) about M.
However, M and Ī can coincide with asymmetric (not symmetric) distribution.
Discrete distribution of voters is where voters are located at only certain positions along the left-right continuum.
Mean, Ī, can be calculated using the same formula and a table of values.
Median, M, for an even number of voters may result in an extended median—in which there may be no median position. 8

9. Chapter 12: Electing the President Spatial Models for Multi-Candidate Elections
Entry of a Third Candidate in a Two-Candidate Race
Suppose that two candidates have already entered a primary and they both take position at M and therefore split the vote.
Now a third candidate, C, enters the race and takes a position on either side of the median M in this unimodal distribution.
Candidate C can take a position with less than 1/3 of the voters in the distribution to his or her right.
C can still win the election if C’s area (yellow) is greater than half of A/B’s area (blue). Why? Because C will attract voters not only to the right of his or her position but also some voters to the left. 9

10. Chapter 12: Electing the President Spatial Models for Multi-Candidate Elections
1/3 Separation Obstacle
If A and B are equidistant from the median and separated by no more than 1/3 of the area under the curve, there will be no position C can take to win.
2/3 Separation Opportunity
If A and B are equidistant from the median and separated by at least 2/3 of the area under the curve, then C can take the median M position and win.
Optimal entry of two candidates, anticipating a third entrant:
Candidates A and B would be best to enter at ¼ and ¾ . Then, when third candidate C enters, at most he can win 25% of the vote. 10

11. Chapter 12: Electing the President Narrowing the Field
Voters Viewpoint of the Spatial Game of Elections:
When a candidate drops out, who will the voters choose in his/her place? To whom will the dropout’s supporters shift their votes?
Example: Three candidates take positions, from left to right, A-B-C.
If A or C drops out, his/her supporters most likely will switch to B.
What if B is the first to drop out? Who will get the votes?
Example: Four candidates take positions, left to right, A-B-C-D.
If one of the extremists A or D drops out, his/her supporters most likely will switch to one of the two centrists, B or C.
What if C drops out? The voters can go to B or D.
Spoiler Problem – When a candidate cannot win but “spoils” the election for a candidate who otherwise would win.
Bandwagon Effect – When a candidate gains momentum (beats expectations), resulting in a presumption that a candidate will win and inducing voters to vote for the presumed winner, independent of merit. 11

12. Chapter 12: Electing the President What Drives Candidates Out?
Why Candidates Drop Out of an Election:
Candidates drop out of the race because of poor performance at the polls or early primaries. As expectations are not met, the candidate may lose momentum and drop further in the polls.
Poll Assumption
Voters adjust, if necessary, their sincere voting strategies to differentiate between the top two candidates as revealed in the poll, voting for the one they prefer.
Condorcet Winner Unsuccessful
Given the poll assumption, a Condorcet winner will always lose if he or she is not one of the top two candidates identified by the poll.
Condorcet Winner Successful
Given the poll assumption, a Condorcet winner will always win if he or she is one of the top two candidates identified by the poll.
Remember, a Cordorcet winner is a candidate who can defeat each of the other candidates in pairwise contests. 12

15. Chapter 12: Electing the President Election Reform: Approval Voting
Problem with Plurality Voting and Electoral College
When there are multicandidate elections, the candidate who wins under plurality voting may not be a Condorcet winner.
A Simple Election Reform: Approval Voting
Under approval voting, voters can vote for as many candidates as they like or find acceptable. Each candidate approved of receives one vote, and the candidate with the most approval votes wins.
Different Voter Predictions When Using Approval Voting:
Voting Only for a Second Choice – In a three-candidate election under approval voting, it is never rational for a voter to vote only for a second choice. A voter would most likely vote for both second and first choice.
Dichotomous Preferences – A voter has dichotomous preferences if he or she divides the set of candidates into two subsets—a preferred and nonpreferred subset—and is indifferent among all candidates in each.
Effect of Dichotomous Preferences – A Condorcet winner will always be elected under approval voting if all voters have dichotomous preferences and choose their dominant strategies. 15

18. Chapter 12: Electing the President The Electoral College
Electoral College – How We Elect Our U.S. President
The number of electors allotted to a state is equal to the size of its congressional delegation (representatives + 2 senators).
There are 538 electoral votes: A candidate needs 270 to win.
2000 presidential election: George W. Bush won with 271 electoral votes, even though he lost the popular vote. The real battle was fought in the so-called battleground states, or toss-up states, where a close race was expected.
Expected Popular Vote (EPV) of the Democratic candidate in a toss-up state.
Where: EPVD = ∑ ni pi
t = number of toss-up states
ni = number of voters in a toss-up state i
pi = probability that a voter in a toss-up state i votes for the Democratic candidate is pi = di / (di + ri )
di , ri = the resources each party (Dem. & Rep.) spent in state i. t
i=1 18

19. Chapter 12: Electing the President The Electoral College
Proportional Rule
The strategy of the Democrat who maximizes his or her EPVD (indicated by the asterisk), given that the Republican also chooses a maximizing strategy, is:
di * = (ni / N ) D
Where:
N = ∑ ni the total number of voters in the toss-up states, and
D = ∑ di the sum of the Democrat’s expenditures across all states
Expected Electoral Vote
The expected electoral vote (EEV) of the Democratic candidate in a toss-up state, EEVD, is:
EEVD = ∑ vi Pi
vi is the number of electoral votes of toss-up state i , and Pi is the probability that the Democrat wins more than 50% of the popular votes in this state, which would give the Democrat all that state’s electoral votes, vi .
Republicans would have similar equations to both Proportion Rule and Expected Electoral Vote. t
i = 1 t
i = 1 t
i = 1 19

21. Chapter 12: Electing the President The Electoral College
The 3/2’s Rule
The strategies of the Democratic and the Republican candidates that maximize their EEVs are:
di * = (vi √ ni ) D ri * = (vi √ ni ) R
S and S
Where:
S = ∑ vi √ ni
Local Maximum
A maximizing strategy from which small deviations are nonoptimal but large deviations may be optimal.
Global Maximum
A maximizing strategy from which all deviations (small or large) are nonoptimal. The proportional rule is a global maximum (and equilibrium) for candidates whose goal is to maximize their expected popular vote, whereas the 3/2’s rule is only a local maximum for candidates whose goal is to maximize their expected electoral vote. t
i = 1 21

22. Chapter 12: Electing the President Is There a Better Way to Elect a President?
Taking the choice of a president out of the hands of voters and putting it in the hands of members of the electoral college may no longer be justified.
Some people feel that using the electoral college may create a large-state bias due to the winner-take-all feature. This is due to the fact that all the electoral votes of one state go to only one candidate—no matter how close the race was in that state.
Example: In the 2000 election, only a few hundred voters in the large toss-up state of Florida determined the outcome of all of Florida’s electoral votes going to Bush, resulting in his winning the presidency.
Some people think that approval voting would better enable voters, especially in the early presidential primaries where candidates express their preferences. It may also reduce the role of spoilers.
Others may think direct popular-vote election of the president would be most fair. Then all voters, wherever they reside, would count equally. 22