POLI 325                                                                                                                                                                 due 10/13/04

EXERCISES PERTAINING TO VOTING RULES

1.                                                          Preference Profile 1 

                                    # of voters                   4          4          2          9

                                     1st pref.                       A         B         B         C

                                     2nd pref.                     B         A         D         D

                                     3rd pref.                      D         D         A         A

                                     4th pref.                      C         C         C         B

                        Voting Rule                                                     Winning Candidate

                        Simple Plurality Voting                                              ____

                        Approval Voting (when each voter votes

                                    for two candidates)                                         ____

                        Plurality Runoff (or IRV variant 1)                            ____

                        Alternative Vote (or IRV variant 2)                           ____

                        Borda Point Voting                                                    ____

                        Condorcet Voting                                                       ____

            Here are two more voting systems:

Coombs Voting: proceed in the manner of the Alternative Vote (see p. 3 in Voting to Elect Several Candidates) but, instead of eliminating the remaining candidate with the fewest first preferences, eliminate the candidate with the most last preferences.

Sequential Voting: pair two candidates in a straight fight and eliminate the loser; pair the winner with a third candidate and eliminate the loser; proceed until every candidate but one has been eliminated; the last candidate standing is the winner.

                        Voting Rule                                                     Winning Candidate

                        Coombs Voting                                                          ____

                        Sequential Voting (when the candidates are

                                    paired in alphabetical order)                           ____

Does the Sequential Voting winner change when the candidates are paired in other orders?

OVER =>

2.         You should have found that, under Preference Profile 1, both variants of IRV (equivalent to Plurality Runoff and Alternative Vote, respectively — see footnote 1 in Voting to Elect Several Candidates) give the same winner. Can you construct a preference profile in which the two variants produce different winners?

3.                                                          Preference Profile 2

                        # of voters                   4          2          2          2          4          1

                         1st pref.                       A         A         B         B         C         C

                         2nd pref.                     B         C         A         C         B         A

                         3rd pref.                      C         B         C         A         A         B

                        Voting Rule                                                     Winning Candidate

                        Simple Plurality Voting                                              ____

                        Approval Voting (when each voter votes

                                    for two candidates)                                         ____

                        Plurality Runoff (or IRV variant 1)                            ____

                        Alternative Vote (or IRV variant 2)                           ____

                        Borda Point Voting                                                    ____

                        Condorcet Voting                                                       ____

                        Coombs Voting                                                          ____

                        Sequential Voting (when the candidates are

                                    paired in alphabetical order)                           ____

Does the Sequential Voting winner change when the candidates are paired in other orders?

4.         The 15 members of a Congressional committee are choosing among five different proposed funding levels for a government program. Here are the proposals and the first preferences of all committee members.

                        Proposal 1       $10 million which is the first preference of 5 members

                        Proposal 2       $12 million which is the first preference of 2 members

                        Proposal 3       $18 million which is the first preference of 1 member

                        Proposal 4       $19 million which is the first preference of 4 members

                        Proposal 5       $21 million which is the first preference of 3 members

With respect to second and lower preferences, each committee ranks proposals according to their proximity (“closeness”) to his or her first preference. For example, the preference ordering of the member who most prefers Proposal 3 is this:

                                                                   Proposal    Proximity to first pref.

                                    first pref.                     $18                  $0

                                    second pref.                $19                  $1

                                    third pref.                    $21                  $3

                                    fourth pref.                 $12                  $6

                                    fifth pref.                    $10                  $8

The Committee will adopt one proposal on the basis of Sequential Voting (which is more or less how legislative committees actually take votes). Can you determine which proposal will win in committee? Does the winning proposal depend on the order in which poposals are paired for votes?

Hint: Is there a Condorcet winner among the five proposals? If so, is there some way by which we can quickly determine which proposal it is?

5.         For “Extra Credit”: Show by example that — as claimed on p. 5 of Voting to Elect Several Candidates — no PR apportionment formula can satisfy both Intra-Election Monotonicity. and Inter-Election Monotonicity, Hint: consider what any PR formula must do in the special case in which m = 1 (e.g., when applied in a single-member district).