in case no candidate has an electoral majority, but the 1824 election re-
vealed that the system had not been fixed. The leading candidate, the son
of a former president of the United States, failed to win the popular vote.
The election was eventually decided not by the voters but by a relative
handful of highly placed government officials. It may sound like a de-
scription of the 2000 presidential election, but history does have a ten-
dency to repeat itself.
The 1824 election featured four major candidates: the charismatic gen-
eral Andrew Jackson, who had helped defeat the British in the War of
1 812; John Quincy Adams, the son of a former president who was the
secretary of state at the time of the election; William Crawford, the secre-
tary of the treasury; and Henry Clay, the Speaker of the House. After the
votes were cast, Jackson had received a plurality of both the popular vote
and the Electoral College vote, but he had not obtained the needed Elec-
toral College majority. As provided by the Twelfth Amendment, the elec-
tion went to the House of Representatives (a specter that brief ly reappeared
during the 2000 election as well); but the Twelfth Amendment stipu-
lated that only the top three vote getters could be considered. This elimi-
nated Clay, who encouraged his electors to vote for Adams, a man he
disliked personally but with whom he shared some important political
views. As a result, Adams won, despite the fact that Jackson had received
not only the most popular votes, but also the most votes in the Electoral
College. When Adams later appointed Clay as secretary of state, it seemed
to many that this was the payoff for Clay’s votes.
Even today, the system still has not been fixed. The Electoral College
places a different weight on popular votes cast in different states, and try-
ing to assess the relative weights of those votes is not an easy task. If one
defines the value of an individual’s vote as the fraction of an electoral vote
that it represents, then votes of individuals in low-population states with
three electoral votes are often considerably more valuable than the votes
of individuals in populous states such as California or New York. Using
this method of evaluation, a Wyoming voter has almost four times the
Electoral College clout as a California voter.^1
There is an alternative—and more mathematically interesting—way to
measure the weight of a vote. The Banzhaf Power Index (BPI) counts how
many coalitions (a coalition is a collection of votes) a voting entity can join
such that its joining that coalition changes the coalition from a losing one
to a winning one.
Although the BPI can be computed for both individual voters and blocs
of voters, it is easier to understand how it is computed in the context of an
electoral college. To see how the BPI is computed, suppose there are three
206 How Math Explains the World