Please turn in these problems by Friday, September 22, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday, September 21 class:

• • How are elections in your state run? Try to look up the process for electing the governor, senators, Congress representatives, state legislature representatives, and local elections (city/town government, school council, etc.). If you are not from the U.S., look this up for your home country.

  • Try to find 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.

Please turn in these problems by Friday, September 29, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday, September 28 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.  Think about 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?

  • We talked about the 2000 presidential elections in class. Do you think there was a Condorcet winner in that election?

Please turn in these problems by Friday, October 13, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday, October 12 class:

• • Arrow's Theorem says that a ranked voting system cannot simultaneously satisfy monotonicity (or Pareto), 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 monotonicity (or Pareto) and IIA, so we have to give up one of these. Which do you think would be more important 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?

Please turn in these problems by October 20, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday, October 19 class:

• • 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 turn in these problems by Friday,October 27, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday. October 26 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? What are some arguments for preserving it?

  • 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? 

  • If you were a presidential candidate, how would you campaign differently if popular vote was used instead of the Electoral College? Would using popular vote change the fact that campaigning is currently focused on a small number of swing states while most of the rest of the country is ignored?

Your assignment for this week is to think about your final project topic and email me about it. The final deadline for this is 5 pm on Monday, October 30.

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

Please think about the following questions by the Thursday, November 2 class:

• • What does "power" mean to you? When we say someone or something has "power," what does that represent to you? Does it have positive or negative connotation? Do you think of it in some strong sense, as in force, authority, influence, or it something "softer," like intelligence, education, or skill?

  • 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, November 10, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday, November 9 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 2016 presidential election in particular.  Namely, research if anyone has figured out whether the outcome of that election 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?

Study for the exam taking place in class on Thursday, November 16. See resources for studying on the Exam page.

Please turn in these problems by December 1, at 5 pm. And here are the solutions.

Please think about the following questions by the Thursday, November 30 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 turn in these problems by Friday, December 8, at 5 pm. And here are the solutions.

There are no discussion questions for this material.