Mathematics and Politics, Spring '20

Assignments

Assignment for the week of February 3 - Electoral College

Please write short responses to the following questions by Monday, February 3 at 5 pm:

Please turn in these problems by Tuesday, February 4 at 5 pm. And here are the solutions.

Assignment for the week of February 10 - Majority and plurality voting

Please write short responses to at least two of the following questions by Monday, February 10 at 5 pm:

Please turn in these problems by Tuesday, February 11 at 5 pm. And here are the solutions.

Assignment for the week of February 24 - Ranked choice voting

Please write short responses to at least two of the top five questions. Everyone also has to address the last question. Responses are due by Monday, February 24 at 5 pm:

Please turn in these problems by Tuesday, February 25 at 5 pm. And here are the solutions.

Assignment for the week of March 2 - Arrow's Theorem and cardinal voting

Please write short responses to at least two of the following five questions by Monday, March 2 at 5 pm. Also look at the last question and bring your answer to class:
- Arrow's Theorem says that a ranked voting system cannot simultaneously satisfy the Pareto Condition, Independence of Irrelevant Alternatives (IIA), and non-dictatorship. There is an agreement that dictatorship is bad, so we would like a ranked voting method that is not a dictatorship. But Arrow then says that this method cannot satisfy both the Pareto Condition and IIA, so we have to give up one of these. Which do you think a more important criterion would be to try to preserve and why?
- Does Arrow's Theorem mean that we should never use any of the ranked voting methods we have seen? Why or why not?
- We saw several ordinal (ranked choice) and cardinal (approval, range, and cumulative) voting methods. Which of these is the best in your opinion and why? Consider both mathematical and practical issues in your answer. Does you answer depend on the type of election and the size of the electorate?
- Find some places and situations (governments, institutions, organizations) where approval voting, range voting, or cumulative voting is used for single-winner elections. Which methods are used in your town, local, and state elections?
- Find some places and situations (governments, institutions, organizations) where discrete cumulative voting or single transferable vote is used for multiple-winner elections. Which methods are used in your town, local, and state elections?
- Tuesday, March 3 is the 2020 Super Tuesday! So we'll run our own little primary election in class. To prepare for this, you'll vote on the Democratic candidates in several ways: ranked choice, approval vote, and cumulative vote. Then we will compile all the ballots in class and figure out who the winner is (or winners are) using various methods we have learned about. The remaining candidates are
  - Here is a website that might help you think about how to vote.

Please turn in these problems by Tuesday, March 3 at 5 pm. And here are the solutions.

Assignment for the week of March 9 - Final project assignments

Your only assignment this week is to think about the topic for your final project. Here is more information about the project. In the next couple of weeks, you should talk to me about your potential topic choice. Everyone should have a topic by Tuesday, March 17.

Assignment for the week of March 31 - Coronavirus

It is a strange and difficult time, but we might as well turn it into an opportunity to learn some math! So your assignment this week is to try to look up the mathematics behind the coronavirus. For example, what is the math behind the models use to predict how it spreads and why is it so hard to make these predictions? What is the math behind social distancing and the "flatten the curve" mantra? What is R0 and how is it computed? I'd love to hear about anything else that you can find that has to do with the mathematics surrounding the coronavirus, global response to it, or its predicted economic consequences. You do not have to write a response for this, we can just chat about it on Zoom when we meet on Tuesday, March 31.

Assignment for the week of April 6 - Apportionment

(postponed from March 17 due to extended spring break and transition to online instruction)

Please write short responses to at least two the following questions by Monday, April 6 at 5 pm:
- Find some places and situations (governments, institutions, organizations) in the U.S. or in the world where some of the apportionment methods we have seen are used. Which method is used in your state's House of Representatives?
- Try to look up the current figures for the populations of the U.S. and of your home state. According to these numbers, do you think your home state is overrepresented, underrepresented, or appropriately represented in the U.S. House of Representatives?
- Try to look up some instances when, had a different apportionment method been used prior to some presidential election, the outcome of the election would have been different because the Electoral College numbers would have been different.
- Same question, but for the last election in particular. Namely, research if anyone has figured out whether the outcome of the 2016 presidential elections would have been different if, say, the Hamilton or the Jefferson method were used for apportionment after the 2010 census. (If you don't find the answer and you're ambitious, figure it out for youself!)
- Do you think the Webster method or the Huntington-Hill method of apportionment is better? What does using the geometric mean instead of the arithmetic mean do? What does it achieve?

Please turn in these problems by Friday, April 10 at 5 pm (upload the assignment to the Google folder I created for you). And here are the solutions.

Assignment for the week of April 13 - Quantification of power

Please write short responses to at least two the following questions by Monday, April 13 at 5 pm:
- Which do you think is "better" -- the Banzhaf index or the Shapley-Shubik index? In other words, which do you think more accurately represents the distribution of power in a weighted voting system? Why?
- Banzhaf and Shapley-Shubik indices come out to be very different when the power of the President of the U.S. is calculated (4% and 16%). Which of these do you think is closer to reality? Can you find some evidence to support your thinking? How would you even start to test these percentages against actual experience?
- John Banzhaf was a lawyer, not a mathematician. Try to find out what led him to the (re)discovery of the power index that bears his name. What was the situation he applied it to?
- We have seen some interesting examples of the computation of the Banzhaf and Shapley-Shubik indices (Electoral College, European Economic Community, U.S. President, UN Security Council). Can you find some other interesting examples?

Please turn in these problems by Friday, April 17 at 5 pm (upload the assignment to the Google folder I created for you). And here are the solutions.

Assignment for the week of April 20 - Gerrymandering

Please write short responses to at least two the following questions by Monday, April 20 at 5 pm:
- Look into how gerrymandered your state is. Does your state have an independent redistricting commission?
- What was the efficiency gap in your state in the last elections?
- Try to find out something about the compactness scores for the districts in your state.
- Look up a more general version of the Isoperimetric Theorem. What does it say?
- How would you extend efficiency gap to the situation when the election has more than two parties?
- Can you find examples of when efficiency gap falsely flagged a district as overly gerrymandered?
- Can you find some examples of stacking, hijacking, or kidnapping?

There is no math assignment for this topic.

Assignment for the week of April 27 - Cryptography

Please write short responses to at least two the following questions by Monday, April 27 at 5 pm:
- Find some instances of where you (or your device) are using cryptography in your daily interactions.
- Look up the First Crypto Wars. What were they about? How did the issues get resolved?
- Can you find some current news about issues surrounding cryptography and its regulation?
- What is quantum cryptography and why will its advent obliterate most commonly used cryptosystems?
- Try to find more about what exactly the NSA did to the elliptic curve random number generator.
- Anything else you might find interesting about cryptography.

There is no math assignment for this topic.

Assignment for the week of May 4 - Statistics

This is an optional assignment. If you decide to submit the usual written response, then please do so by Monday, May 4 at 5 pm. Or you could just talk about it in class on Tuesday, May 5:
- Find some examples of misuses of statistics in politics along the lines of the examples given in the video lecture.

There is no math assignment for this topic.

Assignment for the rest of the semester - Final project

For the remaining couple of weeks, you should just focus on working on your final papers. They are due Friday, May 15, by 5 pm (end of the finals period). You should upload them into the Google folder I created for you.