Mathematics and Politics HW # 1, Due 2/15/2017 Do any 8 out of the following 10 problems! 1. Consider the profile A A B A A A B B B B B B B B B A B B Who wins under: (a) The simple majority method (b) The 2/3 majority method (c) Weighted voting where the first 6 voters get 3 votes , the next 6 voters get 2 and the rest get 1. 2. (a) There are 435 seats in the US House of Representatives, and bills are passed using the simple majority method. How many votes are needed for the house to pass a bill? Assuming every member votes, are ties possible? (b) In the senate there 100 seats, and again a bill can pass the senate with a simple majority. How many votes are needed in the Senate? Explain why ties are possible. (c) Because ties are possible in the Senate, the Constitution provides for the Vice President to cast a tie breaking vote. Describe this as a voting method with 101 voters. Is this system anonymous? Hint: Would it make any difference if the Vice President was always allowed to vote? (d) The US Federal System is the voting method described in the Constitution for passing a bill. It involves the Senate, the House, the Vice President and the President, for a total of 537 voters. For a bill to pass it must have (i) a majority in the House, (ii) a majority in the Senate, with the possibility of the Vice President breaking a tie, (iii) the signature of the President, or (iv) if the President does not sign the bill (called a veto), a 2/3 majority in both the House and Senate. Is the US Federal system anonymous? neutral? monotone? minimal ties? no ties? Explain briefly 3. Consider the profile below, in which the voters are numbered 1, 2, ... , 9 corresponding to the columns C C B A A A A B B B B A B B B B C C A A C C C C C A A Write down the corresponding tabulated profile. Determine who wins under the: (a) Plurality method, (b) Borda count method, (c) Hare method, (d) Copeland method, and (e) Dictator method, when voter 7 is the dictator. Does any candidate have a majority of first place votes? 4. A counterexample is a single example that demonstrates that in at least one instance a certain method fails to satisfy a given criterion. A counterexample may require one or two profiles. (A) The 2/3 (super-) majority method does not satisfy the minimal ties criterion (one profile). (B) A status quo voting method is not neutral (two profiles). (C) The monarchy method is not neutral. (D) The dictatorship method is not anonymous. (E) Weighted voting is not anonymous. In case E), you will first need to set up a weighted voting system, by specifying some number of voters and assigning them some weights. I would recommend four voters to keep things simple. If it helps, you can think of them as partners in a small business. You will then need to give a counterexample with two profiles to see that your weighted voting system is not anonymous. In 5 and 6 are some easy questions of the type \"Does method X satisfy criterion Y ? Remember that a yes answer requires a \"proof\" (a short explanation of why the method always satisfies the conditions specified in the criterion); A no always requires a counterexample. See if you can do these without looking at your notes or the chapters (or if you have to, just look for the definitions of the methods and criteria) 5.. (a) Does Copeland's method satisfy the Condorcet criterion? Does it satisfy the Anti Condorcet criterion? (b) Does the plurality method satisfy the Pareto criterion? (c) Does the Borda count method satisfy Condorcet Criterion? 6.. (a) Explain why (i.e., prove that) the Dictator method satisfies the Pareto and independence of irrelevant alternatives (IIA) criteria. (b) Does the dictator method satisfy monotonicity? (c) Does the dictator method satisfy Condorcet Criterion? 7. Some examples: (a) Find an example showing that the Hare method and the Coombs method do not always produce the same winner. (b) Find an example showing that the plurality winner can come in last place in the Borda count. (c) Find an example showing that the Borda count winner need not have any first- place votes. 8.. A counterexample (a) In each of the profiles shown below, who wins by the Hare method 8 5 4 7 5 4 1 A C B A C B B B A C B A C A C B A C B A C (b) How was the profile on the right obtained from the profile on the left? That is, what happened? (c) Conclusion: The ________ method (does / does not) satisfy the _______ criterion. The next two problems involve the antimajority criterion which says that a candidate with the majority of last place votes should not win (call this the antimajority candidate). 9. Find a profile where there is no antimajority candidate. Prove that if there is an antimajority candidate, that candidate must be the Condorcet loser too. Conclude that any method that satisfies the Condorcet loser criterion must satisfy the antimajority criterion (i.e, explain why this is the case). 10. Does the antipluarality method (i.e., the winner is the candidate with the fewest last place votes) satisfy the antimajority criterion? Does the plurality method satisfy the antimajority criterion? Does the Borda count method? Does the Copeland method