12.6 12.5 Spatial Models and the Electoral College
The Electoral College had a decisive effect in the 2000 presidential election, in which Al Gore received 537,000 more popular votes (0.5%) than did George W. Bush, but Bush won the electoral-vote tally by 4 votes. Thirty-six days after the election, a 5–4 Supreme Court decision blocked further vote recounts in Florida and determined that Bush won the state by a razor-thin margin of 537 votes (less than 0.01% of those cast). By capturing all 25 of Florida’s electoral votes, Bush received a majority in the Electoral College and became the 43rd president of the United States.
What is the justification for the Electoral College? Its 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 that it could make more deliberative choices. As for its composition, each state gets 2 electoral votes for its two senators (total for all states: 100). In addition, a state receives 1 electoral vote for each of its representatives in the House of Representatives, whose numbers are based on population (see Chapter 14) and range from 1 representative for the seven smallest states to 53 representatives for the largest state, California. The House has a of 435 representatives. The District of Columbia, like the smallest states, is given electoral votes. Altogether, there are 538 electoral votes, and a candidate needs 270 to win. In 2000, George W. Bush got 271 electoral votes.
Although there is nothing in the U.S. Constitution mandating that the popular- vote winner in a state receive all its electoral votes, this has been the tradition almost from the founding of the republic. Only in Maine and Nebraska can the electoral votes be split among candidates, depending on who wins each of the two congressional districts in Maine and the three congressional districts in Nebraska. Because the statewide winner receives the two senatorial electoral votes, the closest split possible in these two states is 3-1 in Maine and 3-2 in Nebraska. In the actual election, Gore won all of Maine’s 4 electoral votes, and Bush won all of Nebraska’s 5 electoral votes, so winner-take-all prevailed in all 50 states and the District of Columbia.
The spatial voting models developed so far can be extended to examine some of the complexities introduced by the winner-take-all nature of the Electoral College. The models also allow for a comparison with two proposals. The first is to abolish the Electoral College and replace it by a direct popular-vote election, and the second is the National Popular Vote law. As of July 2015, ten states and the District of Columbia—which account for 165 electoral votes—have passed laws to give all their electoral votes to the national popular-vote winner; the law goes into effect when states with a majority of electoral votes (that is, 270) pass such a law. If enough states pass such a law, then the popular-vote winner would be guaranteed a victory in the Electoral College.
EXAMPLE 9 Spatial Modeling of an Election with Winner-Take-All Districts
Let’s return to the mayoral election between Ann and Bob described in Example 3. Suppose that the tiny town is divided into three districts. One district consists of 5 voters with ideal points 2, 3, 3, 3, and 6; we denote this district by {2, 3, 3, 3, 6}. The other two districts have 7 and 9 voters, respectively; their ideal points are represented as {0, 2, 2, 3, 3, 6, 6} and {3, 4, 4, 4, 6, 8, 8, 10, 10}. Assume that the candidate who wins a majority of the votes from a district with voters wins district votes—akin to electoral votes. The candidate with the majority of the district votes is elected mayor.
525
Bob is aware of the median of the ideal points of the 21 voters and announces as his policy position. Ann is shrewder. She realizes that she needs to win two of the three districts and therefore calculates the medians of each district. The median of districts {2, 3, 3, 3, 6}, {0, 2, 2, 3, 3, 6, 6}, and {3, 4, 4, 4, 6, 8, 8, 10, 10} are 3, 3, and 6, respectively. By announcing a policy position of 3, she wins a majority of the votes in two of the three districts, becoming mayor by a tally of 12 to 9 district votes. Indeed, 3 is the unique equilibrium policy position and cannot be defeated by any other policy position.
Self Check 8
Assume that a population of 9 voters has the following ideal points: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Find the median of the ideal points. Assume the population is broken into the three districts: {1, 3, 5}, {2, 6, 8}, and {4, 7, 9}. Find the median for each district. If each district awards one district vote to the candidate that receives a majority of the votes in the district, then what is the equilibrium policy position?
- The median of the 9 voters’ ideal points is 5. The medians of the districts {1, 3, 5}, {2, 6, 8}, and {4, 7, 9} are 3, 6, and 7, respectively. The equilibrium position is 6.
Ann shrewdly calculated the median for each district. And it so happened that the median of the medians was the unique equilibrium strategy. Indeed, this occurred in Self Check 8, too. However, this may not happen in general. In the two previous examples, it was the case because winning any two of the three districts would ensure enough of the district votes to win the election. Because electoral votes from the Electoral College are based on the population of each state (according to the most recent census), a state with a greater population has at least as many electoral votes as a state with a smaller population. It is the size disparity that prevents the median of the medians from yielding an equilibrium strategy in general.
EXAMPLE 10 The Median of the Medians May Not Be an Equilibrium Strategy
Student Council is divided into groups , , and , so that and each consist of 3 students and consists of 8 students. Each student votes for a candidate based on the candidates’ announced policy positions on the left-right continuum from 1 to 9. Assume that the median positions of the distribution of the voters’ ideal points for groups , , and are 5, 6, and 8, respectively. Assume that a candidate must win a majority of the “group votes” and that groups , , and award 2, 2, and 5 group votes, respectively, to the candidate who receives a majority of the votes from the students in the group.
The median of the medians is 6 because 6 is the middle value in 5, 6, and 8. However, if a candidate announces 6 as his or her policy position, then an opponent who announces 8 would defeat the candidate. This follows because group is so large in comparison with the other groups and its group votes alone are a majority of all group votes. (In Chapter 11, group is referred to as a dictator, and groups and are referred to as dummies in the corresponding simple weighted-voting game that models this voting scenario.)
The median of each group is weighted by the number of group votes each group has to award. Groups , , and have weights 2, 2, and 5, respectively. The weighted median is the median of 5, 5, 6, 6, 8, 8, 8, 8, 8. The data string 5, 5, 6, 6, 8, 8, 8, 8, 8 includes the median of each group multiple times, according to the group’s weight. Thus, 5 is listed twice because group has weight 2, 6 is listed twice because group has weight 2, and 8 is listed five times because group has weight 5. The median of this new data is 8, which is an equilibrium strategy.
526
Weighted Median DEFINITION
The weighted median of with positive integer weights , respectively, is the median of the data , where appears in the list times for each .
When the sum is odd, the list consists of an odd number of terms and the weighted median is a single value. When the sum is even, the weighted median may be an interval, as it is the extended median of an even number of terms.
Self Check 9
Find the weighted median of 1, 2, 3 with corresponding weights 1, 2, and 3.
- The weighted median is the extended median of 1, 2, 2, 3, 3, 3; it is the interval [2, 3].
We can now extend the median-voter theorem to model two-candidate elections under the Electoral College.
Weighted Median Voter Theorem THEOREM
In a two-candidate election with states where, for all , the median distribution of voters in state is and state has electoral votes, the weighted median of with positive integer weights is an equilibrium position.
The furor caused by the divided outcome in the 2000 presidential election, in which George W. Bush won the electoral vote and Al Gore won the popular vote, spurred efforts for reform of the election system. That the candidate with the most electoral votes may not be the same as the popular-vote winner is only one criticism of the Electoral College. Another criticism is that states have unequal power under the Electoral College, as larger states have been shown to wield more power in the winner-take-all awarding of electoral votes in the Electoral College. Under the Electoral College, candidates are less likely to court voters in states for which the election is de facto determined, just as voters are less likely to turn out to vote if one political party dominates the state. These behaviors create swing or toss- up states that garner the attention of the candidates. Indeed, Florida was the key swing state in the 2000 presidential election, deciding the election between Bush and Gore. Exercise 45 (page 534) highlights how a state with few electoral votes may have a limited impact on an election.
Besides the call to abolish the Electoral College, the immediate discussion of election reform after the 2000 election centered on making balloting more accurate and reliable and eliminating election irregularities—especially those practices that discriminate among different types of voters, such as people who do not have a state driver’s license. Post-2000, there have been other initiatives, including the aforementioned National Popular Vote law that was proposed in 2006. But, for the most part, such initiatives have been focused on the state level because the U.S. Constitution grants each state the power to allocate its electoral votes.
As mentioned previously, all but two states use a winner-take-all system to allocate their electoral votes. Historically, states have used other methods, such as dividing the state into electoral districts. This is similar to how states are divided into districts for a primary, where the popular-vote winner of the district would receive all electoral votes for the district. A major criticism for this method worked its way into the English language: Gerrymandering, in which electoral districts are constructed to gain political advantage, is named for the 1812 salamander-shaped district that was created to favor the party of then Massachusetts Governor Gerry. By 1832, no states were using electoral districts, until Michigan re-introduced winner-take-all districts for the 1892 election.
527
After the 2008 and 2012 presidential elections, a number of states proposed changes to the method used to allocate their electoral votes. In 2010, Republicans in Pennsylvania proposed using the state’s congressional districts as electoral districts. This seemed to be in response to the outcome of the 2008 presidential election, in which Barack Obama was the popular-vote winner yet won only a minority of the districts. If the change had been in effect for the 2008 election, Obama would have received 11 of Pennsylvania’s 21 electoral votes, including the 2 electoral votes to the candidate who received the most statewide votes.
Republicans in Virginia, Wisconsin, Ohio, and Michigan have also considered electoral district proposals to change how the electoral votes are awarded in their states. Perhaps not surprisingly, the states that are considering these proposals are those for which electoral districts would have the biggest impact. Republicans ironically control the states in which Democrats would benefit the most from the use of electoral districts; these include Arizona, Georgia, North Carolina, and Texas.
The motivation the founding fathers had for placing the decision of who will be president in the hands of Electoral College members, rather than in the hands of voters, may no longer be justified. A simpler method of determining the president without the Rube Goldberg-esque complications of the Electoral College would be to count equally the votes of all voters, wherever they reside. A direct popular-vote election of a president would best accomplish this goal; but because this would require the abolishment of the Electoral College and a change to the U.S. Constitution, it seems unlikely to occur. However, enacting the National Popular Vote law would not only ensure that the electoral-vote winner is the popular-vote winner if states with a majority of electoral votes signed on, but it would also not require changing the Constitution.
Mathematics illuminates the effect of different ways of awarding delegates and electoral votes and also highlights the strategic decisions candidates make during a campaign. Furthermore, it provides a method to analyze possible changes to the Electoral College, whether proposed now or in the future.