MGF 1107/ Classroom examples/ Chapter 9

 

1. If there are 49 voters, how many are needed to constitute a majority?

2. If there are 50 voters, how many are needed to constitute a majority?

3. Suppose we poll the class to see whether you prefer the pizza at Domino’s, Pizza Hut, Papa John’s, Godfather’s or Little Caesars.  Suppose there are 55 students who vote as follows:

                        Domino’s         18

                        Papa John’s      12

                        Pizza Hut          10

                        Little Caesars     9

                        Godfather’s        6

Who wins the election?

4. In problem 3, we ask each of the 55 voters to rank all 5 chains in order of preference.  After gathering all 55 rankings we see that many of the voters turned in the same ranking.  In fact, there are only six distinct rankings:

   18 votes               12 votes                10 votes           9 votes                4 votes                2 votes

1st

Domino’s

Papa John’s

Pizza Hut

Little Caesars

Godfather’s

Godfather’s

2nd

Little Caesars

Godfather’s

Papa John’s

Pizza Hut

Papa John’s

Pizza Hut

3rd

Godfather’s

Little Caesars

Godfather’s

Godfather’s

Little Caesars

Little Caesars

4th

Pizza Hut

Pizza Hut

Little Caesars

Papa John’s

Pizza Hut

Papa John’s

5th

Papa John’s

Domino’s

Domino’s

Domino’s

Domino’s

Domino’s

Who wins the election?

5. Who wins the election in problem 4 using sequential pairwise voting and the agenda

PJ, G, D, PH, LC?

6. Sequential pairwise voting arises naturally in the legislative process.  When the U.S. House of Representatives considers a bill, an amendment to the bill can be offered.  Two votes are then taken.  The first vote is whether to accept the amendment.  If accepted, the second vote is between the amended bill and no bill at all.  If the amendment is defeated, the second vote is between the original bill and no bill at all.  In 1956 the House was considering a bill that would provide federal funding for the construction of schools.  An amendment was offered that would only provide the federal funding to states with integrated school systems.  The House could more or less be divided into three groups: Republicans, northern Democrats, and southern Democrats.  The Republicans generally opposed federal aid, but favored integration.  So their first choice was no bill, but they preferred the amended bill to the original bill.  Northern Democrats favored the federal aid and integration so their first choice was the amended bill, but preferred the original bill to no bill.  Southern Democrats, who came from states with segregated school systems, wanted the aid, but abhorred the amendment.  The three preference lists looked like this (with the breakdown of Democrats an estimate):

Republicans (203)        Northern Democrats (116)       Southern Democrats (116)

1. No bill                      1. Amended bill                        1. Original bill

2. Amended bill            2. Original bill                           2. No bill

3. Original bill               3. No bill                                  3. Amended bill

a) What was the outcome of the two votes?

b) If you were a northern Democrat, should you have introduced the amendment?

c) If you were a Republican, should you have introduced the amendment?

d) Is there a Condorcet winner in this election?

7. The three members of FIU’s Student Entertainment Committee vote on which of four movies will be shown in the Graham Center that week.  The alternatives are Star Wars (SW), Dances with Wolves (DW), Pearl Harbor (PH), and Jurassic Park (JP).  The preference lists are as follows:

                                    Juan                 Maria               Sally

                                    1. SW              1. JP                1. DW

                                    2. DW              2. SW              2. PH

                                    3. PH               3. DW              3. JP

                                    4. JP                4. PH               4. SW

Use sequential pairwise voting with the following agendas:

a) PH, JP, DW, SW

b) DW, SW, PH, JP

c) JP, SW, DW, PH

d) SW, DW, JP, PH

8. A college has just received a donation of $1,000,000 with the only condition being that it is all used for the same purpose. At a meeting of the Board of Trustees, four possible uses are proposed: increasing financial aid (IFA), construction (C), increasing salaries (IS), and hiring new faculty (HNF).  The 26 trustees have the following preferences:

                        8 votes             5 votes             6 votes             7 votes

                        IFA                  C                     IS                     HNF

                        C                     IS                     C                     C        

                        IS                     HNF                HNF                IS

                        HNF                IFA                  IFA                  IFA

Which alternative prevails under the plurality method?

9. Three candidates run for president of the local chapter of their labor union.  There are 100 voters with the following preference lists:

                        38 votes           30 votes           25 votes           7 votes

                            A                       C                     B                    B

                            B                        A                     C                   A

                            C                       B                      A                   C

The voting method is plurality with runoff.  Who wins?

10. Referring to problem 9, suppose in her acceptance speech, A announces a change in one of her positions.  This causes the 7 voters in the last column to now prefer A to B.  The first and last columns being the same, we combine them.

45 votes           30 votes           25 votes          

                            A                       C                     B                   

                            B                        A                     C                  

                            C                       B                      A

Who would win now with the same voting method?

11. The five members of a city commission have to determine where to build a new baseball stadium.  There are three proposed sites: next to the park (P), along the river (R), and at the intersection (I) of the two main thoroughfares.  The preference lists are:

                        3 votes             2 votes

                          P                      R

                          R                      I

                          I                       P

A Borda count is used.  Who wins?

12. The Hare system was used by the International Olympic Committee to choose the site of the 2000 Summer Olympic games.  Eighty-eight voters took part in the election that was won by Sydney, Australia.  The site of the 2012 Summer Olympics has not yet been chosen.  Suppose 88 voters have the following preference lists:

                        44 votes                       22 votes                       22 votes

                        Beijing                          Istanbul                        Manchester

                        Istanbul                        Manchester                  Istanbul

                        Manchester                  Beijing                          Beijing

Who wins under the Hare system?

13. Referring to problem 12, suppose Manchester and Istanbul are switched on the last preference list.  Who wins now under the Hare system?

14. Consider the following preference lists:

                        12 votes           10 votes           8 votes             x votes

                             A                      B                    C                     C

                             D                      D                   B                      D

                             C                      A                   D                     B

                             B                       C                   A                     A

Find all values of x that make D the Borda count winner but not the Condorcet winner.

15. In an election with 5 alternatives and 20 voters, what is the sum of the Borda points?           

16. In a few sentences, explain why plurality with runoff satisfies the Pareto condition.

17. In a few sentences, explain why the Borda count satisfies monotonicity.

18. Suppose 100 voters are asked to rank 3 football teams: Miami (UM), Florida State (FSU), and Nebraska (N).  Here are the preference lists:

                        52 votes           38 votes           10 votes

                           UM                  FSU                    N

                           FSU                  UM                  FSU

                             N                     N                    UM

Who wins using a Borda count?

19. Referring to problem 17, suppose the 38 voters in the middle decide to vote against their true beliefs and flip UM and N.  Now who wins using the Borda count?

20. Six candidates (A, B, C, D, E, and F) are competing for a spot on the board of directors of a company.  The current nine members each vote for as many candidates as they wish using approval voting.  An X indicates a vote of approval.

A

X

X

 

X

 

X

X

X

 

B

 

 

X

 

X

 

X

 

 

C

X

 

 

X

 

 

 

 

 

D

 

 

 

X

 

X

 

 

 

E

X

X

X

X

X

 

X

X

X

F

 

 

 

X

 

 

 

 

X

Which candidate gets the vacant board seat?

21. The Baseball Writers Association of America polls its membership each year to elect new members to the baseball Hall of Fame.  In 1999, 497 writers cast a ballot.  Writers can vote for as many players as they feel are deserving of enshrinement.  To be elected, a player must be listed on at least 75% of the ballots.  Here are the top six finishers in the 1999 balloting along with the number of votes each received. 

Nolan Ryan           491   

George Brett         488   

Robin Yount         385   

Carlton Fisk          330   

Tony Perez            302   

Gary Carter           168   

a) Which ones were elected to the Hall?

b) What percent of voters who did not vote for Carter would have had to change and vote for him in order for Carter to be elected?