Please turn in these problems by Tuesday, February 23, at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, February 23 class:

• • 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.

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

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

Please think about the following questions by the Friday, February 26 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?

  • 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?

  • Investigate the 1991 Louisiana gubernatorial elections. Do you think there would have been a Condorcet winner in that race? Which candidate do you think represented the will of the people best?

  • Look up how the voting is done for some of the following (a few that look the most interesting to you). We may set aside time in class to talk about some of them:

    • • Wellesley College faculty elections for Committees of Academic Council

      • 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

      • Heisman Trophy in NCAA football

      • World figure skating championship

Please turn in these problems by Tuesday, March 2 at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, March 2 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, March 5 at 5 pm. And here are the solutions.

Please think about the following questions by the Friday, March 5 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?

  • 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?

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

Please think about the following questions by the Friday, March 12 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 Tuesday, March 16 at 5 pm. And here are the solutions.

Please think about the following questions by the Tuesday, March 16 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?

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

Here is the gerrymandering activity we did in class as a group assignment.

Please think about the following questions by the Monday, March 29 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?