The Politics of Numbers: Is Majority Rule Always Fair?
| DOI | 10.1111/j.2041-9066.2010.00039.x |
| Date | 01 December 2010 |
| Published date | 01 December 2010 |
| Author | George Szpiro |
| Subject Matter | Feature |
in Florida, Nader probably denied Al
Gore the few hundred votes that he
would have required in order to win
the state and become president.
Knowing full well that their can-
didate stood no chance, rational
Greens should have voted for their
second most preferred candidate.
By giving their vote to a spoiler, the
Greens managed to get a candidate
elected whom the majority clearly
did not want. Did the democratic
tradition let the American people
down?
The Paradox of Majority Rule
The sad truth is that majority rule
may not reflect the true will of the
electorate. This has been known for
centuries. In 1785, the nobleman
Jean-Marie-Antoine Nicolas de Cari-
tat, Marquis de Condorcet published
a 200-page pamphlet titled Essay on
the Application of Probability Analysis
to Majority Decisions. He presented an
astonishing paradox that can be il-
lustrated by a simple example. Let us
say Peter, Paul and Mary must decide
what to buy for their after-dinner
drinks. Their preferences are:
• Peter:
Amaretto > Grappa > Limoncello
• Paul:
Grappa > Limoncello > Amaretto
• Mary:
Limoncello > Amaretto > Grappa
(‘>’ indicates ‘preferred to’)
Committed as they are to demo-
cratic values, the three decide to go
by the majority opinion. A majority
prefers Amaretto to Grappa (Peter
The Politics of Numbers:
Is Majority Rule Always Fair?
Imagine the following discussion
that allegedly took place many
years ago in a New York restau-
rant. A waiter informs the diner,
‘for dessert, we have apple pie and
brownies’. The customer decides on
apple pie. A few moments later the
flustered waiter returns to inform the
patron that he had forgotten to men-
tion that the restaurant also offers
ice cream. ‘In that case I will have a
brownie’, the guest announces after
some short reflection.
This anecdote, ascribed to the phi-
losopher Sidney Morgenbesser from
Columbia University, encapsulates
a problem that has forever plagued
decision-making procedures. Obvi-
ously, the diner did not care one way
or another about ice cream, since he
did not choose it, even when it was
offered. But its sudden availability
did reverse his choice between the
first two alternatives. Something
like this just should not happen. But
things like that do happen, and more
often than one thinks. Indeed they
can have much graver consequences
than the simple choice of a dessert.
In the 2000 US presidential elec-
tions, a clear, if slim, majority of
the electorate preferred Al Gore to
George Bush (50,999,897 votes for
Gore, 50,456,002 for Bush). Bush
nevertheless became president be-
cause − after intervention by the
US Supreme Court − he garnered
Florida’s 25 electoral votes. It was
the presence of a complete outsider
on the ballot, the Green Party can-
didate Ralph Nader, that decided the
election result. With only 2,882,995
votes across the United States, Nader
never stood a chance of winning. But
by attracting a measly 97,421 votes
The electorate is always right, isn’t it? Not necessarily. As George Szpiro shows, majority
decisions can result in unintended cycles, a dunce may upset a sure thing, and dishonest
electors are able to manipulate elections.
and Mary) and a majority prefers
Grappa to Limoncello (Peter and
Paul). Based on these two rounds
they can make their decision: pur-
chase a crate of Amaretto. But
surprise, surprise: Paul and Mary
protest. What happened? The most
reasonable selection method was
used – one person, one vote – and
they are still not happy? Paul and
Mary have a legitimate grumble.
They point out that they would
prefer even Limoncello, the lowest
ranked option, over Amaretto. How
come? Well, had the three campers
had a third round of voting, between
Limoncello and Amaretto, a major-
ity would have preferred Limoncello
(Paul and Mary). But if they bought
Limoncello, Peter and Paul would
protest just as vigorously. They pre-
fer Grappa to Limoncello. So here we
have it, a paradox. Try as you might,
the final result is that Amaretto is
preferred to Grappa, Grappa to
Limoncello, Limoncello to Amaretto,
Amaretto to Grappa…
Condorcet’s Paradox, as the conun-
drum was henceforth called, can be
the source of much abuse (see Box).
For example, a person setting the
agenda at a board meeting can subtly
influence the outcome of decisions by
manipulating the order in which votes
are taken. So what is the solution?
The depressing answer is that there
is none. With majority rule there is
no way out of Condorcet’s Paradox.
What a let-down for democracy.
The Borda Count
Enter Condorcet’s contemporary and
intellectual sparring partner, Jean-
We have
known for
centuries
that majority
rule does not
always reect
the electorate’s
true will
90 Political Insight
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