1. There Is No Such Thing as Public Opinion
(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
This is a general trend in polls -- people prefer cutting government programs to paying more taxes
But... Which government programs should be cut?

2. What do people think? A Pew Research poll in February 2011
Americans were asked about increasing or decreasing government spending for 13 categories
Spend more
Spend less
Education 62 11
Veterans benefits and services 51 6
Health care 41 24
Medicare 40 12
Combating crime 39 18
Energy 36 23
Scientific research
Environmental protection 26
U.S. anti-terrorism defenses 33 21
Agriculture 32
Military defense 31 30
Unemployment assistance 27 28
Global poverty assistance 45
People want to decrease government spending in total, but when asked about specific areas of spending, they want to keep spending more on most of the areas
Spend less here!

3. 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
If we ask:
should we cut spending or raise taxes?
the answer we will get is “cut spending” (67/33)
If we ask:
should we cut the defense budget?
the answer might be “no” (33/67)
should we cut Medicare?

4. Rational but complicated
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

5. Conclusions about polls
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!

6. Another 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!!

7. Can we learn about public opinion?
Here is the 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)
Maybe it would be easier to just look at elections instead of polls…

8. Election paradoxes
Here is an interesting example – the 1992 presidential election:
Bill Clinton: 43% of the popular vote
George H. W. Bush: 38%
Ross Perot: 19%
Let’s analyze:
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)

9. Second choices
But, suppose the 19% of Perot voters were split this way:
13% who preferred Bush (as their “second choice”)
6% who preferred Clinton
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%
1992 presidential election results:
Bill Clinton: 43% of the popular vote
George H. W. Bush: 38%
Ross Perot: 19%
Question: Should we run elections differently -to take second choices into account?

10. 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)
Bill Clinton
George H. W. Bush
Ross Perot
“Ross Perot is my #1 choice, and I prefer Bush to Clinton”

11. Counting votes In order to tally the results of the election:
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
So if Perot was thrown out in the first round, the ballot below would be counted for Bush (as the voter’s second choice)
Bill Clinton
George H. W. Bush
Ross Perot

12. 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.

13. Example Here is a case from a mayoral election in Burlington VT
Three candidates: Conservative, Centrist, Progressive (incumbent)
There were 8833 voters, so to have an absolute majority in the first round, a candidate must have 4417
Progressive, Centrist, Conservative 2043
Conservative, Centrist, Progressive 1513
Centrist, Progressive, Conservative 1332
Conservative
Centrist, Conservative, Progressive 767
Progressive
Conservative, Progressive, Centrist
Centrist
Progressive, Conservative, Centrist
First round totals
Conservative: 3297
Progressive: 2982
Centrist: 2554 x
Need a second round: eliminate Centrist, who is the low candidate

14. Example – second round
Second round – use the “2” votes for ballots for Centrist
Progressive, Centrist, Conservative 2043
Conservative, Centrist, Progressive 1513
Centrist, Progressive, Conservative 1332
Conservative
Centrist, Conservative, Progressive 767
Progressive
Conservative, Progressive, Centrist
Centrist
Progressive, Conservative, Centrist
First round totals
Conservative: 3297
Progressive: 2982
Centrist: 2554 x
Second round totals
Progressive: 4314
Conservative: 4064
And... Progressive keeps his job, thanks to IPV

15. Centrist should have won?? (Maybe)
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
Maybe Centrist should be mayor!!
4597 voters preferred Centrist to Conservative
3688 voters preferred Conservative to Centrist
Progressive, Centrist, Conservative 2043
Conservative, Centrist, Progressive 1513
Centrist, Progressive, Conservative 1332
Conservative
Centrist, Conservative, Progressive 767
Progressive
Conservative, Progressive, Centrist
Centrist
Progressive, Conservative, Centrist
Maybe the IRV vote count algorithm isn’t so good…

16. Another paradox
Suppose we “add votes” to Progressive, it shouldn’t change the result!
Progressive, Centrist, Conservative 2043
Conservative, Centrist, Progressive 1513
Centrist, Progressive, Conservative 1332
Conservative > change to Conservative 989, Progressive 300
Centrist, Conservative, Progressive 767
Progressive
Conservative, Progressive, Centrist > change to Progressive 495
Centrist
Progressive, Conservative, Centrist
First round totals
Progressive: 3777
Centrist: 2554
Conservative: 2502
Second round totals
Centrist:
Progressive: 3777 x
Centrist wins! And we only added votes to Progressive! How did Progressive lose??

17. 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

18. Credits
This talk is based on Chapter 17 of Jordan Ellenberg’s book How Not to be Wrong
Talk materials were prepared by Dennis Mancl,
MSWX ◊ Mancl ◊
Software ◊ Experts ◊
This work is licensed under a Creative Commons Attribution 4.0 International License