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Talk is cheap, voting is free; take it to the polls.
 Nanette L. Avery
People want spending cut, but are opposed to cuts in anything but foreign aid...The conclusion is inescapable: Republicans have a mandate to repeal the laws of arithmetic.
 Paul Krugman
Once upon a time, in a parallel universe, three famous French timetraveler philosopherpoliticians were relaxing at a coffee shop in the Pacific Northwest.
"The local politics in this alternate universe are refreshingly simple," said Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet.
"Yes," affirmed his mentor, Alexis Claude Clairaut. "There are three main issues: planting trees, equipping forest firefighters, and improving driving safety."
"All three issues are currently being funded," added Jean le Rond d'Alembert. "But the state is low on money. One program must be defunded."
The three timetravelers talked more, and realized they disagreed about which of the three programs to defund. Their opinions are summarized in a table:
Person  Wants to Keep Funding Two Programs  Wants to Defund  

Marquis of Condorcet  
Alexis Claude Clairaut  
Jean le Rond d'Alembert 
Each individual had a strong preference. But there was no overall preference. Each program got two "keep it" votes. Each program got one "defund it" vote.
Does it seem strange that every voter can have a strong preference but overall no outcome is preferred?
This is named the Condorcet Paradox, after that first famous philosopherpoliticianmathematician.
When can the Condorcet Paradox appear? Maybe it only happens when there were three voters and three options?
Draw another table showing the paradox appearing with two voters and two options.
Draw another table showing the paradox appearing with four voters and four options.
What if the number of voters is less than the number of options? Can you still make the paradox happen?
What if the number of voters is more than the number of options? Can you still make the paradox happen?
If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people's choice.
 from the writings of Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet, shortly before he discovered his paradox
Another day once upon a time, in that parallel universe, it was time to name the first interstellar battleship for the United States Space Force. The excited American populace was allowed to vote. In the final round three options remained. Voters were asked to rank these by their first, second, and third choice.
The most popular opinion, picked by almost half the voters, was to name the ship the U.S.S. Kirk, and as a second choice option the U.S.S. Solo. Most people had never heard of Buckaroo Banzai and did not want his name on the battleship.
Almost half the voters had a first choice of U.S.S. Solo. But these were almost evenly split between whether their second choice was for James T. Kirk or Buckaroo Banzai.
Only three percent of the voters had a first choice of the U.S.S. Banzai, and all of these preferred Han Solo over James T. Kirk for their second choice.
These opinions are summarized in a table:
Voting Group  First Choice  Second Choice  Third Choice  Percentage of Votes 

One  49  
Two  25  
Three  23  
Four  3 
When the votes were counted, the people who preferred the name U.S.S. Solo were outraged!
"Buckaroo ruined the election!" they complained. "The only reason that Kirk won by 1% was because the stupid Buckaroo Banzai fans had their idol as an option. If the choice had only been between Kirk and Solo then our name would have won 51to49. Not far!
Does it seem strange that adding an insignificant option can change which significant option the voters prefer?
This is named the Chernoff Paradox, after another famous mathematician.
Can you somehow get around the Chernoff Paradox by taking into consideration second choice preferences?
Many people have proposed ways around the Chernoff Paradox. The most famous proposal says "Give two points for every firstchoice vote, and only one point for every secondchoice vote. (Or something like that.)"
That proposal is always named after JeanCharles de Borda, although it was actually invented more than three hundred of years earlier by the much more famous Nicholas of Cusa. Life isn't fair.
What happens if we do give two points for every firstchoice vote, and one point for every secondchoice vote, to the votes in our Space Force example?
The name U.S.S. Kirk gets 49 × 2 + 25 = 123 points.
The name U.S.S. Solo gets 48 × 2 + 52 = 148 points (the winner!).
The name U.S.S. Banzai gets 3 × 2 + 23 = 29 points.
Now the people who preferred the name U.S.S. Kirk were outraged!
"Mathematicians ruined the election!" they shouted. "We won fair and square before. The only reason that you say Han Solo wins is because you invented biased formulas. Kirk got a ton of third place votes which your stupid formulas ignore. If third place votes also gave points then Kirk would have won. Not fair!"
Remember how the famous proposal concluded with "Or something like that"? Perhaps it is too simple to just give first place votes two points, second place votes one point, and ignore all other preferences?
Can you invent a better point scheme to make the voting with preferences more fair?
Is considering third choice preferences always enough? If there were four choices, is it safe to ignore each voter's least favorite?
You can't buy happiness, but you can buy ice cream and that's pretty much the same thing.
 internet meme
A hypothetical college is planning for its annual ice cream social. This year the supplier has a special offer. Some very fancy coffee and pistachio ice creams are available, and because they are less popular flavors they are available at the same price as the "standard" flavors of vanilla, chocolate, and strawberry.
Discussions happen. The pricing and budget issues require ordering three flavors. The college decides that vanilla should be an option because it is by far the most popular second choice. Enough people like coffee and pistachio ice cream to get some of those also while they are a special deal.
How much of each flavor should the college order? A volunteer asks 67 faculty, staff, and students to rank their preferences. These opinions are summarized in a table:
Voting Group  First Choice  Second Choice  Third Choice  Number of Votes 

One  11  
Two  10  
Three  20  
Four  4  
Five  17  
Six  5 
The number of first choice votes are 21 for vanilla, 24 for coffee, and 22 for pistachio.
That seems wrong. Clearly the fans of coffee and pistachio are excited to try the special deals. But vanilla is indeed the favored second choice option, and many of the people who ask for a second serving pick a different flavor from their first serving. It seems wrong to order about the same amount of each flavor, with the least vanilla.
The college could use the simple de Borda scheme with two points for every firstchoice vote and one point for every secondchoice vote. The results become:
Vanilla gets 21 × 2 + 37 = 79 points (the winner!).
Coffee gets 24 × 2 + 16 = 64 points.
Pistachio gets 22 × 2 + 14 = 58 points.
This scheme puts vanilla farther ahead, which seems reasonable. But we have seen that the simple de Borda system can also lead to bias.
Some mathematicians distrust de Borda systems for political reasons. Maybe it is too easy for a corrupt regime to use whatever point scheme favors their own candidate! Surely the use of point schemes will cause important elections to become courtroom battles. Is there a better way to vote?
Another idea is throw out the option with the fewest first place votes and then make comparisons. Perhaps all we really need to do is toss out the insignificant Buckaroo Banzais of the world that interfere with the significant options? We could make a "second pass" over the votes once the option with the fewest first choice votes is gone. That kind of runoff voting should make the result clear, right?
But in this case vanilla ice cream received the fewest first choice votes, despite seeming very significant because of receiving the most combined first and second choice votes. Hm. Tossing out the option with the fewest first place votes seems to be a plan that might also be ripe for bias and abuse.
How would you interpret the survey? What percentage of the total ice cream ordered do you think should be vanilla, coffee, and pistachio? Can you defend your decision from claims of unfairness or bias?
Do you have a preference among these methods of voting (first choices, de Borda, runoff)? If so, why?
Would a combination of these methods of voting (first choices, de Borda, runoff) make you happiest? If so, why?
Most systems are not going to work badly all of the time. All I proved is that all can work badly at times.
 Kenneth Arrow
The quest for a perfect voting system has no end. Arrow's Impossibility Theorem is a mathematical proof that any voting system will be unfair or biased in certain situations.
If you are interested, read more, or search for videos about that theorem. Here are two videos to get you started.
Our conclusion is immensely practical.
Knowing that no system is perfect, we must look at when and how different voting systems are problematic.
Then we can act wisely, and pick the voting system whose flaws interfere least with our goals when we try to make a decision using voting.