**The
Bizarre Math of Elections**

Low voter turnout can be a
healthy sign

that the electoral system
is working well

Richard A. Muller

Technology for Presidents

Technology Review Online

October 2003

An unexpected
thing happened in the California recall election: the winner got more votes
than the loser.

Few people thought that
would happen. Under California's unique law, the incumbent Gray Davis had to
get 50 percent of the vote on the recall portion of the ballot, or he was out.
He got 45 percent. In the separate gubernatorial vote, his replacement needed
only a plurality. With 135 candidates, that meant that -- in principle -- the new
governor could be elected by less than 1 percent of the voters, even though 45
percent liked the incumbent!

But it didn't
turn out that way. Arnold Schwarzenegger won with 49 percent. Even some
opponents were grateful that, at least, the results were decisive. Californians
had grown tired of the pre-election ridicule of their ridiculous law. Yet
peculiar election results aren't new. In the last presidential election, the
winner, George W. Bush, received 47.9 percent of the votes, while the loser, Al
Gore, got more: 48.4 percent. Bill Clinton won the presidency in 1992 with an
even smaller percentage: 43 percent.

Election math is
screwy. Why don't we fix it? Well, the problem may not be as simple as changing
California's law, or abolishing the Electoral College. Election math is
fundamentally unfixable. That is a celebrated result of a mathematical theorem
proven in 1952 by Kenneth Arrow, who won the Nobel Prize in Economics for this
and other work.

Consider, for
example, the 'instant runoff' system, in which every voter ranks every
candidate. This method is already
used in several municipalities, including Cambridge, Massachusetts, and it has
been proposed as a replacement for the current national electoral college
system. From Arrow's
theorem, we expect to be able to find cases in which instant runoffs are
unfair, and indeed such examples are not hard to find.

Here's how instant
runoffs work. Imagine three candidates named Left, Middle, and Right -- L, M
and R for short. After the vote, the top candidates are retained, and the
eliminated candidate has his votes allocated to them. The ultimate winner
always has a majority, although it may be a combination of first place, second,
and lower ranking votes. Isn't that the best way to decide an election?

Not necessarily.
Consider this plausible situation: L gets 34 percent, M gets 32 percent, and R
gets 34 percent of first place votes. Everyone who voted for L or R puts M in
their second slot. Yet the moderate M, who represents the center-of-mass of the
voters, is eliminated because he narrowly lost the first-round plurality. The
decision will be determined by second place votes. The country will get one of
the extremists, either L or R, despite the fact that 100 percent of the voters
put M as either their first or second choice.

Moreover, there
is good reason to think that candidates cannot, in principle, be put in an
order of preference. This fact is so counter-intuitive that it requires a
specific example. Please bear with me through the following math. It will be
worth it. And just to make the case even more convincing, let's begin not with
politics, but with chess.

Imagine three
chess teams, A, B, and C, each with three players. A match between two teams,
say A and B, consists of each player of A playing each player of B. The team
with the most victories wins the match.

Let's assume
that luck is not involved, so that the stronger chess player always beats the
weaker. I'll set up the teams in such a way that A always beats B, and B always
beats C. In math notation we say A > B > C. A seems to be the strongest.
Now here's the paradox: if A plays C, then A will lose. In symbols, C > A.
How can that be?

Here is one way
to do it. Assume that the names of
the players on team A are A2, A6, and A7.
(The numbers indicate their skill.) For team B, the names are B1, B5, and B9, and for C they are
C3, C4, and C8. First, consider
the match between teams A and B:

A2
beats B1 and
loses against B5 and B9

A6
beats B1 and B5 and
loses against B9

A7
beats B1 and B5 and
loses against B9

So team A wins 5
of the 9 games, and A > B.

When team B
plays team C, it works out as follows:

Team B wins 5 of
the 9 games. So B > C. You would expect that A >
C. But look what happens when they
play:

A6
beats C3 and C4 and
loses to C8

A7
beats C3 and C4 and
loses to C8

C wins 5 of the
9 games. We have A > B, B > C, and C > A. If all three teams compete, the winner will be decided by
the order in which they play. If A and C play in the first round, A is
eliminated. If B and C play in the first round, then C is eliminated.

Such peculiar
behavior is not actually peculiar at all. In math, we say the chess competition
is 'not transitive.' That means that A > B and B > C does not necessarily
imply that A > C. We can also say the objects (the teams) cannot be
'ordered' under the operation of round-robin competition. Non-orderable objects
abound in math. And also in the real world.

Does this
example also relate to baseball, football, tennis, and soccer? Yes, the order
of the playoffs can determine the winners, regardless of the real strengths. If
you are a sports fan, you probably already knew that. (That's why your team
lost.) Does it relate to politics? Yes, in the same way. We have playoffs in
politics too. They're called primaries.

The paradox even
works for your individual choice of candidate. Suppose you are a
middle-of-the-road voter and you rank each candidate on three issues, such as
their stands on human rights, on use of military forces, and on taxes. Your evaluation of candidate L on these
issues is (2, 7, 6). For M it is (9, 5, 1), and for R it is (4, 3, 8). When you
compare L to M, you'll prefer L on military and taxes, but not on human rights;
since he is better on two of three issues, you decide L is better than M, i.e.
that L > M. When you compare M to R, you'll find M is better on two issues,
human rights and military, so M > R.
Thus far we have L > M and M > R, so you would think L > R,
right? Wrong. Compare L directly to R.
You prefer R on human rights, L on use of military, and R on taxes. R wins on two of three issues, so R > L. Political preferences can be
intransitive.

If the playoff
system doesn't work, can we come up with a system that will? That brings us
back to Kenneth Arrow's notorious theorem. Arrow made a series of postulates
that were so reasonable that every voting system should obey them, and then he
proceeded to show that they were incompatible with each other. In other words,
there is no voting system that will always satisfy fundamental criteria of
fairness.

So any system we
might choose to replace current local, state, or national election procedures
would have its own frustrating flaws.
It wouldn't cover all possible situations. But does that mean all hope is lost?

No. Arrow's
theorem only guarantees that you can find a situation in which the election is
unfair; it doesn't guarantee bad results in all cases. Moreover, the theorem is true only when
there are three or more candidates. Let's consider the two candidate case: L
and R. Everyone votes for the one who is closest to their preference. In real
elections, the positions of the candidates on issues may not be absolutely
immutable. Both L and R realize this, so to maximize their chance of winning,
they both start shifting towards the middle. They both know that whoever best
takes possession of the middle will get the most votes

By the time the voting takes place, the
candidates positions are almost indistinguishable. Voters complain that they
have no real choice. And that is true. The center has been found -- the position
where M would have been, had he (or she) run. Both candidates, to get elected, have moved to the position
where they best represent the average of all voters.

Notice how the
existence of a primary election can interfere with this center-finding
process. A candidate who wins in a
primary is often the extreme one who represent the center of his party rather
than the center of the whole population.

If, however, people
vote in the primaries for a candidate who 'can win' rather than one who is
closest to their own preference, then the two party system works well, and
results in candidates who are close to the center. If there is sufficient time during the subsequent campaign,
the candidates can move even closer to the middle position. Democrats can co-opt Republican issues,
and Republicans can co-opt Democratic stands. This leads to a surprising irony. With both candidates moving towards the center, many people
ultimately see such little difference that they lose interest in the election.
Turnout is lowÑbut for this example, that actually reflects the fact that the
election process is working well.