No Such Thing As Public Opinion

An explanation of some mathematical paradoxes with polls and elections.

(From the book How Not to be Wrong by Jordan Ellenberg)

In theory, our government officials should respect the “will of the people.” But it isn’t easy to determine what people want. 

In a January 2011 CBS News poll,


(Note: This is a general trend in polls -- people prefer cutting government programs to paying more taxes... but... which government programs should be cut?)

What do people want?

Should we spend more or spend less on certain programs? Here are the results of a Pew Research poll in February 2011:

[Poll results tablel]


Are we rational?

Of course, we might say that what we all want is a free lunch... But let’s look at the decision-making process in more detail.

Suppose we just consider three ways to reduce the deficit:

and suppose that 1/3 of the people prefer A, 1/3 prefer B, and 1/3 prefer C

The answers we get depend on the questions we ask.

If we ask:

But if we ask:

Notice... we have gotten to an impasse on cutting the deficit...

In this example, we are using made-up numbers, but the reality is pretty close to this...

The information we get from polls doesn’t give us a clear winner

This is what we are hearing:

“Majority rules” works well for making decisions when there are only two options.

Another poll example

Here is another opinion poll case... about Obamacare:

CNN/ORC poll in May 2013:

There are three possible policies here... and each of them is opposed by a majority of Americans!

So, when we are looking at public opinion, can polls help?

Here is our first lesson:

Let’s look at elections instead of polls...

Election paradoxes

Example – the 1992 presidential election:

Let’s analyze:

This isn’t a major problem in elections:

But, suppose the 19% of Perot voters were split this way:

(We don’t actually have any real data about second choices of Perot voters... but for this example, let’s pretend we do.)

If you ran the election with just two candidates (Clinton and Bush), this might have been the vote totals:

Question: Should we run elections differently – to take second choices into account?

More complex voting scheme (IRV)

Here is an interesting system of voting (in Australia and Ireland):

Instead of voting for one candidate, a voter can put “numbers” next to each candidate:

[Rank voting example]

The system for counting the votes is complex (in the case where no one gets an absolute majority)

In order to tally the results of the election:

So if Perot was thrown out in the first round, the ballot below would be counted for Bush (as the voter’s second choice)

Is this a fairer vote??

Maybe... This sounds like magic:

However... there are still paradoxes in the IRV system.

A real example

Here is a case from a mayoral election in Burlington VT

[Vote example table]

Second round of vote counting

Second round – use the “2” votes for ballots for Centrist

[Vote example table]

Arguments over the vote results

Centrist is unlucky... there are a lot of people who like him, but he is a “second choice” for most people.

Centrist might argue that maybe the IRV vote count algorithm isn’t so good...

Another paradox (with slightly different data)

Suppose we “add votes” to Progressive, it shouldn’t change the result!

[Vote example table]

Advanced topic: What makes a fair election?

Ellenberg’s book goes further:

Condorcet started with an axiom:

Condorcet then explored how to create a voting system that would satisfy this axiom...

Conclusion: There Is No Such Thing as Public Opinion

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Last modified: Oct. 26, 2018
Dennis Mancl -