The U.S. presidential election of 2000 will leave lasting impressions on those who attempted to follow the five weeks of controversy following Election Day. Many of us swam through the legal and political issues with deficient vocabularies (what is a butterfly ballot or a per curiam opinion?) and limited knowledge (what does 3 U.S.C. Sec 5 really say?). But beneath the headlines and the partisan wrangling were some interesting mathematical issues. Without trying to weigh in on the final outcome of the election, it might be worth looking at these questions; some will likely surface again before the elections of 2002 and 2004.
First some background for those who may have dozed off. On December 18, 2000, the members of the Electoral College cast their votes and by a margin of 271 to 266 (with one abstention) officially elected George W. Bush the 43rd President of the United States. The 25 electoral votes of Florida made the difference and could have swung the election either way. According to the certified vote, Bush's winning margin in Florida was 537 votes out of over 6 million votes cast; that margin represents about 0.009% of the total votes cast. It's also about 0.02% of the 2.5 million registered Florida voters who did not vote for President - another testimony to low voter turnout that was 51.2% (of all eligible voters) nationwide in this election.
It was an unusual election in many respects. For only the second time in American history, the winner of the electoral vote was not the winner of the popular vote (joining Harrison in 1888). For only the third time in history, the winner did not take the two largest states, California and New York (along with Wilson in 1916 and Truman in 1948). It was also the third closest electoral vote in history (behind Jefferson in 1800 and Hayes in 1876).
Perhaps the first lesson lies in the popular vote count. In the final tally of popular votes, Al Gore received 50,996,064 votes to 50,456,167 for Bush, 2,864,810 for Ralph Nader, and 1,068,501 for a dozen other candidates. These popular vote totals are reported to the nearest vote, suggesting that every vote cast was counted and counted accurately. If we didn't suspect it earlier, the 2000 election told us convincingly that such precision is illusory. The current system is a bit like timing a 100-meter dash with a sundial and reporting the winning time to the nearest tenth of a second. Such deceptive precision is not unusual in this country. You can be sure that when Census 2000 results are released, the population of the nation and states will be reported to the nearest person. Similarly, figures in the federal budget (both projected and final) are given to the nearest dollar, as if uncertainties on the order of billions of dollars did not exist!
Because of mechanical, electronic, and human errors, the actual precision of vote counting is several orders of magnitude worse than what is needed to determine an election as close as Florida's. (The next four closest state elections with margins of victory less than 1% all went to Gore: New Mexico, 0.06%, Wisconsin, 0.20%, Iowa, 0.32%, and Oregon, 0.45%.). Nationwide, an estimated 2 to 3 million votes were not counted. Assuming that some of these votes were genuine "no-votes," it still leaves an uncertainty of about 1-2%. (Error rates of for the punch card systems appear to be as high as 3-4%, while optical scanning systems reduce errors to less than 1%). In many elections, such a margin of error is acceptable, particularly if the election is not close and the errors more or less compensate for each other. However, if there is bias in these so-called undervotes (for example, antiquated voting machines in poor precincts), then such errors can easily cast doubt on the outcome of an election.
Two changes might be considered for future elections. Currently, anything less than a 0.1% margin of victory in a state election triggers a recount. This cutoff margin should be increased to reflect the precision of the vote counting system. The problem with this solution, demonstrated perfectly in Florida, is that any recount with punch card ballots invites the hopeless question of interpreting voter intent. A more expensive, but inevitable, solution is to replace the antiquated systems, used in approximately half of all precincts in this country, by more reliable equipment.
Another piece of election reform that is already attracting plenty of attention is the Electoral College. Written directly into the U.S. Constitution (Article II, Section 1, modified by the 12th Amendment), the Electoral College was invented by the framers as a way to protect states with smaller populations, to prevent the "tyranny of the majority," and to cure "the mischiefs of faction." Whether the device is serving that purpose or serving it too well is the subject of considerable political and, surprisingly, mathematical debate.
Recall that each state's share of electors equals the number of Representatives and Senators the state has in the Congress. That number runs from 3 electors in several states (and the District of Columbia) to 54 in California. There are 538 electoral votes and a candidate needs a majority of the votes (at least 270) to win the Presidency.
If we compute the number of electors per million people in each state, we get a rough estimate of how well states are represented in the Electoral College. The figure below is a plot of each state's population and its number of electors (one dot for each state) . It suggests that the Electoral College is serving its purpose of protecting the smaller states: 10 of the smallest states have more than 3 electors (per million people), with Wyoming having roughly four times the representation of California.
It's interesting to notice that a candidate can win the electoral vote (in a two-candidate race) by winning only the 11 largest states. Assuming that it takes 50% of the votes to win a state, this amounts to slightly more that 25% of the popular vote nationwide. Conversely, a candidate can also win the electoral vote by winning 40 of the smallest states. Again assuming that 50% of the vote is required to win a state, this amounts to about 22% of the popular vote nationwide. So a winner can take a majority of the electoral votes and still have the approval of only 25% of the voting population.
There is another peculiarity about the Electoral College: it is brittle or unstable. The lesson in Florida in 2000 was that an incredibly small number of voters (0.009% of all voters in the state) can command a large number of electoral votes and swing the election either way. Without going into what if scenarios, it's easy to imagine how the outcome of the entire election could have changed if only a few people had done something differently.
This instability could be changed if the electoral votes in a state were divided according congressional districts, as is currently done in Maine and Nebraska: if a candidate wins half of the congressional districts, she wins half of the electoral votes. A projection done by USA Today concludes that Bush would have won the 2000 election with these rules. Alternatively, if each state's electoral votes were precisely divided according to the popular vote in the state (a candidate winning 39.3% of the popular vote receives exactly 39.3% of the electoral votes, using fractions of votes), then Gore would have won the 2000 election. Of course, the way to remove almost all of this instability is to use the popular vote to determine the winner - a method the founding fathers deliberately avoided.
But that's not the end of the story. If the debate on the Electoral College goes mathematical, you can be sure to hear about something called the Banzhaf power index. Invented by Georgetown University law professor John Banzhaf III in the 1950s, it relies on an ingenious definition of voter power. The power of a voter or block of voters is the number of ways in which it can cast a critical vote that changes the outcome of the election. By leaving a winning coalition, a critical voter changes it to a losing coalition. And by joining a losing coalition, a critical voter changes it to a winning coalition. Needless to say, these ideas apply directly to a wide range of political and corporate situations.
A couple of quick examples show how it works. Consider a small company in which there are three stockholders who hold 45%, 45%, and 10% of the shares. Company rules say that a majority vote is required to approve any measure. Clearly, it takes two stockholders to approve any measure. Although one stockholder holds far fewer stocks, they all have the same effective voting power.
Now consider a company consisting of stockholders A, B, and C, who each hold 26% of the shares, and D who holds 22% of the shares. Again a majority is needed to make a decision. Now any two of A, B, and C command a majority. However, D, who holds only slightly fewer shares than the others, cannot form a winning coalition with any of them. In this case, A, B, and C are have equal voting power and D is powerless.
When these practical ideas are applied to the far more complex electoral system, the results are perhaps unexpected. In a large state, such as California, each individual voter naturally has less power than in a smaller state. However, because California wields more electoral votes, the actual voting power of individuals in large states is greater than in smaller states. The figure below shows the Banzhaf power index for the voters in each state compared to the state with the smallest power (Montana). For example, California voters, the rightmost dot, have about 3.3 times the power of Montana voters, according to Banzaf's definition of voting power. Said differently, a California voter has about 3.3 times the chance of changing an election than a Montana voter.
The Banzhaf power index is widely respected and it agrees with other independent measures of voting power (at least for the Electoral College). For those who defend the Electoral College in terms of its original purpose (protecting the smaller states), the Banzhaf presents troublesome evidence. On the other hand, those who believe the first figure above (voting power defined as electors per voter) can vigorously argue that the Electoral College already gives too much power to the smaller states. Caught in this crossfire, it is likely that the Electoral College will receive close scrutiny in the days ahead. And you can be sure that the debate will be political, legal, and ... mathematical.