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,
77% of respondents said cutting spending was the best way to handle the federal budget deficit
9% preferred raising taxes
(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:
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:
- raise taxes
- cut defense
- cut Medicare
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:
- should we cut spending or raise taxes?
- the answer we will get is “cut spending” (67/33)
But if we ask:
- should we cut the defense budget?
- the answer might be “no” (33/67)
- should we cut Medicare?
- the answer might be “no” (33/67)
Notice... we have gotten to an impasse on cutting the deficit...
- and... people are not completely irrational...
- each individual voter has a perfectly rational position...
- the problem comes with how we “aggregate” the positions.
In this example, we are using made-up numbers, but the reality is pretty close to this...
- Americans don’t want to “keep every program” – there are plenty of non-worthwhile programs that could be cut.
- The problem is, there is no consensus on which programs are the worthless ones
The information we get from polls doesn’t give us a clear winner
This is what we are hearing:
“If the program benefits me, it is worthwhile – it needs to be saved at all costs.”
This is a selfish position, but it isn’t stupid!
“Majority rules” works well for making decisions when there are only two options.
- It doesn’t work so well for complex cases -- contradictions start to appear!
Another poll example
Here is another opinion poll case... about Obamacare:
CNN/ORC poll in May 2013:
- 43% favored Obamacare
- 35% said it was too liberal
- 16% said it wasn’t liberal enough
There are three possible policies here... and each of them is opposed by a majority of Americans!
- Fox News might report: “Majority of Americans oppose Obamacare!” (51/43)
- MSNBC might say: “Majority of Americans want to preserve or strengthen Obamacare!” (59/35)
- And both of them are right!!
So, when we are looking at public opinion, can polls help?
Here is our first lesson:
- For complicated public policy, it is difficult to ask a “fair” poll question
- (We want a poll question that doesn’t have some kind of contradiction or paradox lurking in the data)
Let’s look at elections instead of polls...
Example – the 1992 presidential election:
- Bill Clinton: 43% of the popular vote
- George H. W. Bush: 38%
- Ross Perot: 19%
- A majority of voters (57%) didn’t want Bill Clinton
- A majority of voters (62%) didn’t want George H. W. Bush
- And a really big majority of voters (81%) didn’t want Ross Perot
This isn’t a major problem in elections:
- for U.S. Presidential elections, it’s the Electoral College that decides
- and in other elections, whoever has the largest number of votes wins (even if it isn’t an absolute majority)
But, suppose the 19% of Perot voters were split this way:
- 13% who preferred Bush (as their “second choice”)
- 6% who preferred Clinton
(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:
- Bill Clinton: 49%
- George H. W. Bush: 51%
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):
- Instant Runoff Voting (IRV)
Instead of voting for one candidate, a voter can put “numbers” next to each candidate:
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)
- first count only the “1” votes -- if one candidate has 50% plus 1, he/she is the winner
- if no one has 50%, eliminate the candidate with the lowest total of “1” votes
- and recount the votes... and count the lowest number of remaining candidates
 - Bill Clinton
 - George H. W. Bush
 - Ross Perot
Is this a fairer vote??
Maybe... This sounds like magic:
- it takes more effort for voters to fill out the ballot,
- but somehow it seems to feel fairer in a multi-way race.
However... there are still paradoxes in the IRV system.
A real example
Here is a case from a mayoral election in Burlington VT
- Three candidates: Conservative, Centrist, Progressive (incumbent)
Second round of vote counting
Second round – use the “2” votes for ballots for Centrist
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.
- 4067 voters preferred Centrist to Progressive
- 3477 voters preferred Progressive to Centrist
- 4597 voters preferred Centrist to Conservative
- 3688 voters preferred Conservative to Centrist
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!
Advanced topic: What makes a fair election?
Ellenberg’s book goes further:
Ellenberg explains parts of a mathematical theory of elections created by the French mathematician Condorcet in the late 1700s.
Condorcet started with an axiom:
- If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.
Condorcet then explored how to create a voting system that would satisfy this axiom...
- He tried but failed to find a voting system that would work...
- and in 1951, the mathematician Kenneth Arrow proved that even with a weaker set of axioms, there would be paradoxes.
Conclusion: There Is No Such Thing as Public Opinion
This work is licensed under a Creative Commons Attribution 4.0 International License.
Last modified: Oct. 26, 2018
Dennis Mancl - http://manclswx.com