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

Supplement: one more way to count “ranked choice” votes

If we really want second-place votes to be important – for example, if ranked choice is used in party primary elections – there is another way to add up the votes that is somewhat simpler than the “immediate runoff” algorithm.  This is the system that is used for sports polls in the US: for example, for deciding on the top-ranked college basketball teams.  It is also similar to the “match points” scheme used to score bridge tournaments.

In this method, each voter still puts numbers next to each candidate, and each candidate is assigned a number of points based on the ordering on each ballot.  The least-favorite candidate gets zero points, the next-least-favorite gets 1 point, and so on.  In a three-candidate race, the favorite candidate will get 2 points.

If we look at the numbers in the Burlington VT mayoral race, we see that Progressive has 2982 first-place votes and 1827 second-place votes -- which gives him 7791 points.  Centrist has 2554 first-place votes and 3556 second-place votes for a total of 8664 points, and Conservative has 3297 first-place votes and 1138 second-place votes for a total of 7732 points.  If we count the votes this way, Centrist is the big winner because so many voters were willing to have him as their second choice.

[Vote example table]

This scheme sounds more fair (except maybe to Progressive, who was actually reelected using the IRV algorithm), but it doesn’t solve all of the problems.  There are three issues to consider:

First, what should we do with the ballots that only name a first choice (no second or third choice).  If these people really don’t care which of the other two candidates are elected if their first choice fails, maybe any candidate not mentioned on a ballot should get one-half point! (the average of what they would have received if every voter were forced to give an ordering for all candidates).

Second, maybe there should be a way for a voter to indicate that their top two choices are “equally good” – and each of the two candidates should get 1.5 points.  In a 3-person race, the “match point” voting forces the voter to say that one person is exactly twice as good as another.

Third, it might create some interesting side-effects in election campaigns.  A candidate who is worried about a close competitor might send a message to his followers to rank that competitor last, so that the close competitor will collect fewer second-place votes and therefore fewer points.  In the Burlington VT example, Progressive might tell all of his supporters, “Please put Conservative as your second choice, because I’m worried that Centrist might get too many second-place points.”  Other kinds of tactical voting may also occur: several hopeless candidates might be included in the ballot (the equivalent of the Monty Python “Silly Party”), and voters may tactically choose one or more of these candidates as their second or third choices in order to reduce the number of points for the other serious candidates.

The lesson is that multiway elections are really difficult to design, and there will inevitably be some kinds of paradoxes in any complex voting scheme.

Another recent election - Alaska special Congressional election, August 2022

Alaska now uses Ranked Choice Voting for statewide elections, and there were some interesting results in August 2022 in Alaska’s special election for their Congressional seat.

full data:

In Alaska’s election system, all candidates run together in a primary, and the top 4 vote-getters go to the general election – with Ranked Choice Voting.


(48 candidates had run in the primary. One of the top four finishers (Al Gross) who qualified for the general election ballot decided to drop out after the primary.)

In the general election, the “first round” of vote counting only counted the first-choice for each voter

Begich is dropped, because he came in 3rd

In the second round, Begich’s votes were redistribued, based on each voter’s second choice. Here is a summary of the transfers-

Second round votes

Peltola is elected

In order for Palin to win, she needed a higher percentage of second-place votes from Begich voters. Assuming that all Begich voters choose a second-place candidate, Palin would have needed 35319 second-place votes = 66.9% of Begich’s vote total.

Of course, that threshold is lower if some voters decide not to choose a second-place candidate. With the actual 11243 votes cast for Begich and no one else, Palin would have required 29197 second-place votes = 55.3% of Begich’s vote total.

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Last modified: Sept. 5, 2022
Previously modified: Mar. 3, 2020
Previously modified: Oct. 26, 2018
Dennis Mancl -