METHOD HANDOUT

ARROW'S IMPOSSIBILITY THEOREM, VOTING AND POWER

Lesson: In any system of decision-making based on voting, the structure of the system and the order in which votes are taken ("controlling the agenda") -- which do not necessarily reflect "society's" preferences -- often determine the outcome.

Arrow's Impossibility Theorem: No general rule can rank preferences for a society based only on how the preferences are ranked by individual members of the society. Consider three choices (A, B and C) and three individuals (1, 2 and 3):

Ranking       Individual 1          Individual 2             Individual 3

          First choice         A                            B                              C

          Second choice      B                           C                               A

                    Third choice        C                            A                               B

If the first vote is between A and B -- with the "winner" matched against C -- C finishes on top. If the first vote is between B and C, A ultimately wins. If the first vote is between A and C, B ultimately wins.

Bill Clinton (C)

Paul Tsongas (T)

Tom Harkin (H)

Bob Kerrey (K)

Jerry Brown (B)

                          Number of Delegates with Preference Ranking

Ranking                18      12     10      9     4      2

First choice           T        C       B      K     H     H

Second choice       K       H      C       B     C      B

Third choice          H       K      H     H      K      K

Fourth choice        B       B       K      C      B      C

Solutions to Paulos Problem

Tsongas' argument: only first-place votes count, with a plurality controlling (18 to 12)

Clinton's argument: majority needed, so run-off between top two first-place vote-getters (37 to 18)

Brown's argument: majority needed, so successively eliminate lowest finisher and redistribute votes (final tally 37 to 18)

Kerrey's argument: consider overall rankings by awarding 5 points for first, 4 for second, etc. (191 to Harkin's 189, Brown's 162, Clinton's 156 and Tsongas' 127)

Harkin's argument: consider only head-to-head contests (28-27 over closest rival).

Voting Power: Consider a corporation with majority voting among shareholders.

Case I: Three shareholders of 47%, 44% and 9% have equal voting power.

Case II: Three shareholders of 27%, 26% and 25% have equal voting power, while a fourth shareholder of 22% has no voting power (a "dummy"). (Distributions of 45%, 44%, 7% and 4% would have the same result.)

Banzhaf Power Index = the number of ways she can change a losing coalition into a winning coalition or vice versa. For example, if the respective holdings for A, B, C and D are 40%, 35%, 15% and 10%, the Banzhaf indices are 10, 6, 6 and 2.

Cumulative Voting: One way to empower minorities is to give each voter a number of votes equal to the contested positions in a geographical district or to her shareholdings multiplied by the number of directors to be elected in a corporation. Votes can be "cumulated" for one candidate or distributed among candidates, with the candidates receiving the most votes in their favor being elected. In Case II above, a 20% shareholder cumulating votes for a five-director board could elect one director.

At-Large Voting: In contrast to cumulative voting, at-large voting dilutes the effectiveness of smaller voting blocs by electing multiple representatives from a pooled district by majority vote.

Approval Voting: Instead of "one person, one vote," approval voting is based on a concept of "one candidate, one vote," allowing voters to "approve" as many candidates as they like. This tends to avoid majorities "splitting" votes, but favors lesser-known, "middle-of-the-road" candidates.

Borda count: Any process of ranking candidates where higher rankings are accorded more weight than lower rankings (e.g., Kerrey's system in the previous example).