Please think about the following questions by the Tuesday, June 9 class:

• • Look up the history of the electoral college and why it was introduced. Do you think the current Electoral College system is fine as it is now? We talked about various alternatives to the Electoral College -- abolish it, add electoral votes, etc. (see class notes for a more complete list). What do you think the best alternative method might be and why?

  • Try to find up some instances of elections with exactly two candidates where the majority rule does not decide the winner.

  • Look up some ways that ties are broken in elections in the U.S. You’ll run across some fun things like poker hands and coin flips.

  • Can you think of some arguments in favor of plurality voting?

Please think about the following questions by the Thursday, June 11 class:

• • What do you think a “fair” social choice function (like voting) should mean? Try to look up some definition of fairness? Is there a definite, mathematical way to define it?

  • Individual preferences are usually transitive, i.e. if a person prefers A over B and B over C, then they probably prefer A over C. Give some examples where a society’s preferences aren’t necessarily transitive.

  • Find some places and situations (governments, institutions, organizations) where instant runoff, Borda, or Condorcet methods are used. What methods are used in your town, local, and state elections?

Please turn in these problems by Friday, June 12 at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, June 16 class:

• • What voting method(s) are used in your town, local, and state elections?

  • Suppose you and your friends are planning to go out to eat. You love Indian food, but this is the second choice for most of your other friends and it does not have many first-place votes. A few others are advocating for Thai, and a few others for Mexican. You have a good idea of your friends’ preferences, so which voting method should you push for to influence the group to decide on Indian? Why?

  • One of the things we might do when we break up into groups on Tuesday is figure out how the voting is done for some of the following:

    • • U.N. Security Council

      • Academy awards

      • TV shows Survivor or American Idol

      • Song contest Eurovision (if you haven’t heard of Eurovision, you have to see this Colbert clip)

      • Choosing the host city for the Olympics

      • ATP tennis rankings

      • MPV in baseball’s National League

      • World figure skating championship

Please think about the following questions by the Thursday, June 18 class:

• • No new questions, we'll break up into groups and investigate how the voting is done for some of the organizations/contests listed above.

Please turn in these problems by Friday, June 12 at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, June 23 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 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. Are any of these methods used in your town, local, and state elections?

Please think about the following questions by the Thursday, June 25 class:

• • No questions to think about; in class we'll divide into groups and do an example of the computation of the Banzhaf power index.

Please turn in these problems by Friday, June 26 at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, June 30 class:

• • 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 think about the following questions by the Thursday, July 2 class:

• • 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?

  • 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, July 3 at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, July 7 class:

• 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?

Please think about the following questions by the Thursday, July 9 class:

• • No questions for today since you'll be presenting your final projects.

Please turn in these problems by Friday, July 10 at 5 pm. And here are the solutions.