Arrow’s Theorem
Learning Objectives
Students will be able to...
➵ Identify deirable and undesirable characteristics of a fair election.
➵ Analyze ranked choice voting scenarios and identify the Pros and Cons of a ranked choice voting systems
➵ Describe the limitiations of ranked choice voting illuminated by Arrow's Theorem
➵ Identify the different characteristics of ranked choice voting systems
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Discussion Questions
➵ What are desirable characteristics in a voting system?
➵ What does a fair voting system look like? Do all votes count the same? Should one person cast one vote for one candidates, or should they rank the candidates?
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What are some bad voting systems?
Some Examples
Monarchy
A monarchy is a voting system where one candidate always wins, regardless of votes.
Dictatorship
A dictatorship is a voting system where one voter gets to pick the winner–even if everyone else picks a different candidate, that voter's choice wins.
What are features of a good voting system?
Anonymity
A voting system is anonymous if it treats every voter equally.
Ex. A dictatorship, where one candidate picks the winner, isn't anonymous because one voter has more power.
Neutrality
A voting system is neutral if it treats every candidate equally–a monarchy isn't neutral because one candidate is guaranteed to win
Monotonicity
A voting system is monotone if more people voting for the winner or fewer people voting for the loser doesn't change the outcome: if doing better in the election causes you to lose, the system isn't monotone!
Pareto Criterion
A voting system satisfies Pareto Criterion if, when everyone prefers Candidate A to Candidate B, Candidate B should not win.
When wouldn't this happen?
Consider a voting system where we first compare candidates A and B, then compare the winner to candidate C, and compare that winner to candidate D. Now, consider that the voters vote as below. Note that everyone likes A better than D.
Candidate B beats candidate A (a candidate wins a match up by being ranked higher than its opponent by more voters), then candidate C beats candidate B, and candidate D beats candidate C. So, even though everyone liked A better, Candidate D wins! This is an example of Pareto Criterion gone wrong.
Video Demonstration of a Voting System where Pareto Criterion fails
Independence of Irrelevant Alternatives
A voting system satisfies Independence of Irrelevant Alternatives (IIA) when only voters' preferences between A and B determines which of A and B is ranked higher.
Example: Michelle Kwan at the 1995 Figure Skating World Championship
Before Michelle Kwan skated, rankings were:
➵First Place: Chen Lu (China)
➵Second Place: Nicole Bobek (USA)
➵Third Place: Surya Bonaly (France)
Rankings after Michelle Kwan skated:
➵First place: Chen Lu (China)
➵Second Place: Surya Bonaly (France)
➵Third Place: Nicole Bobek (USA)
➵Fourth Place: Michelle Kwan (USA)
Note that second and third places switched after Kwan skated despite her placing fourth!
This is an example of IIA failure--in other words, the judge's preferences of Bonaly and Bobek changed from an irrelevant alternative (Kwan's skating).
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Kenneth Arrow
➵American Economist who lived from 1921-2017
➵Professor at Stanford and Harvard
➵Won the Nobel Prize in Economics in 1972 alongside John Hicks
Arrow's Impossibility Theorem:
A ranked voting systems where voters pick one candidate cannot satisfy Pareto's Criterion and Independence of Irrelevant Alternatives without being a dictatorship.
What does this mean?
No traditional ranked voting system is perfect–all of them are vulnerable to at least one of these flaws.
“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
Example: Voting Systems Simulation
In this example, we will see how voting systems meet or fail to meet the Pareto Criterion and Independence of Irrelevant Alternatives.
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Closing Questions/Wrap Up
Arrow proves that for ranked choice systems to satisfy both the IIA and Pareto, a system must be a dictatorship.
Assuming we don't want a dictatorship, which criterion would you give up? (Which is more important, IIA or Pareto?)
Is democracy/non-dictatorship a workable voting system if it cannot satisfy IIA and Pareto?
How has learning about Arrow's Theorem changed your views on what makes a good election?
Are there voting systems that avoid the issues from Arrow's Theorem? (Hint: Cardinal Voting Systems)
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Further Exploration
Kenneth Arrow discusses his early education and the appeal of probability theory, his efforts during World War II to improve flight routes by calculating wind patterns (20:26), how the Impossibility Theorem was born (36:15), and why his theories on health “changed the whole texture of how people thought about the problem” (43:15).