Gerrymandering and the Political Process week 7

Investigation 7: Simple Voting Games and the Banzhaf Index

 

In the lecture, we discussed simple voting games:  voting structures used by different organizations such as the United Nations Security Council and the Electoral College when decisions need to be made. In this investigation, you will explore some of these simple voting games and learn about how to measure the “power” of a voting player in these games.

 

  1. (To be gone over as a class)

One of the early proposals to amend the Canadian constitution was the 1971 Victoria Charter.  Canada has 10 provinces. In this system, a coalition was winning if:

(i) it contained the 2 largest provinces, Ontario and Quebec

(ii) it contained at least 2 of the Atlantic provinces (these are: Nova Scotia, New Brunswick, PEI, and Newfoundland)

(iii) it contained at least 2 of the western provinces representing at least half of the population of the western provinces (these are: Manitoba 5%, Saskatchewan 5%, Alberta  7% and British Columbia 9%).

  1. a) Identify 2 different winning coalitions. Ontario/Quebec/PEI/Nova Scotia/Alberta/British Columbia or Ontario/Quebec/New Brunswick/Newfoundland/Manitoba/British Columbia
  2. b) Find a winning coalition in which not all members are critical.  (A member is critical to a winning coalition if the coalition becomes losing if they leave it.)  Identify the critical members in the winning coalition. Ontario/Quebec/Nova Scotia/Brunswick/Manitoba/Saskatchewan/Alberta/British Columbia. The critical members are Ontario/Quebec/Nova Scotia/Brunswick/ Alberta/British Columbia. If we subtracted manitoba/ saskatchewan the coalition could still win which makes them not critical.

 

  1. In her book, The Tyranny of the Majority: Fundamental Fairness in Representative Democracy,

Lani Guinier discussed how 51% of the voters can wield 100% of the power. Let’s explore that in a simple voting game. Suppose a town council contains 10 members.  Six of them belong to a “majority” group, 3 of them belong to a “minority group,” and 1 belongs to a second “minority group.” Assume that the members in each group tend to vote in a block.  Thus the town council acts like a weighted voting game of 3 blocks with weights equal to 6, 3 and 1. Let’s call these blocks A, B and C.

  1. a) Assume the council makes decisions using simple majority rule (a quota of 6). List all the winning coalitions. A, A/B, A/B/C, A/C
  2. b) List all the winning coalitions in which A is critical. A, A/B, A/B/C A/C
  3. c) List all the winning coalitions in which B is critical. Repeat for C. B/C are not critical for winning
  4. d) A member of a voting game is said to be a “dictator” if they are the only member who is critical in any winning coalition. Is there a member in this voting game that is a dictator?  Yes, A is critical because they can meet the quota of 6, without A none of the other blocks can reach the 6 quota.
  5. e) Why might this situation be likened to a “tyranny of the majority”?

Because A is the majority and will always win despite the fact that there are two minority votes that make up 40% of the votes. So even though A technically has a majority it ignores a large minority and dictates all of the decisions like a tyranny would.

 

  1. One of the proposals Lani Guinier makes is to consider other voting mechanisms in town councils and other bodies that pass legislation.  An adaptation of one of her ideas is a Minority Veto System.

Assume A, B and C are members of a “majority group,” and D and E are members of a “minority group.”  A coalition is winning if it contains at least three of the five voters and also approval of at least one of the two minority voters.  

  1. a)  Identify one winning coalition that A is a member of.  Is A critical to this coalition? A/B/D no A is critical to this coalition because A helps make up the ⅗ voter win.
  2. b)  Identify one winning coalition that A is a member of in which A is not A/B/C/E
  3. c) How many winning coalitions is A a critical member of? 4 – A/B/D A/B/E A/C/D A/C/E
  4. d) Identify one winning coalition that D is a member of in which D is critical. A/D/E
  5. e) Identify one winning coalition that D is a member of in which D is not A/B/D/E
  6. f) How many winning coalitions is D a critical member of? 4- A/B/D A/C/D B/C/D A/B/C/D

 

  1. During the lecture, we talked about the European Economic Community.  Its original structure consisted of 6 members countries with the following weights: France 4, Italy 4, Germany 4, Belgium 2, Netherlands 2, Luxembourg. 1.  A coalition is winning if its total weight is at least 12.
  2. a) Identify a winning coalition containing both France and Luxembourg.  Is France critical in this coalition? Yes France is Critical because without it the coalition would not be able to reach the winning weight. What about Luxembourg? Luxembourg is not because if Luxembourg was removed then the weight would still be 12 and that would still win . France/Luxembourg/ Italy/ Germany
  3. b) List all the winning coalitions in which France is critical.  How many are there? 9– F/I/G/B F/I/G/N F/I/B/N F/G/B/N F/G/B/N/L F/I/B/N/L F/I/G/L F/I/G/B/L F/I/G/N/L
  4. c) Repeat (b) for Luxembourg. None

 

  1.  Suppose there are 3 members in a weighted voting game, A, B and C.  They have weights 2, 2 and 1 respectively. A coalition is winning it its total weight is at least 3.
  2. a) List all the winning coalitions. A/B A/C B/C
  3. b) How many winning coalitions is A critical in?  2 What about B?  2 What about C? 2

 

  1. In 1964, the Nassau County Board of Supervisors (on Long Island) had 115 seats, distributed among the towns in the county as follows: Hemstead1 = 31;   Hemstead2 = 31;   N. Hemstead = 21;  Oyster Bay= 28; Glen Cove = 2;  Long Beach = 2

A coalition was winning it its total weight was at least 68.

  1. a) Identify a winning coalition that Glen Cove is a member of.  Is Glen Cove critical in this coalition? Hemstead, N. Hemstead, Oyster Bay, Glen Cove, No it is not critical.
  2. b) Can you find any winning coalition in which Glen Cove is critical?

No

  1. One way to analyze the roles each member plays in a voting game is to compare their Banzhaf indices.  The Banzhaf index is named after the lawyer John Banzhaf who was hired by the town of Glen Cove (from problem #6). Glen Cove sued the  Nassau County Board of Supervisors, claiming they were not given fair representation.

To understand the Banzhaf index, let’s start by looking at the Minority Veto game in problem 3.

  1. a) Look back at your answers in problem 3.  How many winning coalitions is A a critical member of? 5
  2. b) How many winning coalitions is B a critical member of?  (Hint– it’s probably a lot like A). 5
  3. c) How many winning coalitions is C in?  What about D and E? 5,7,7
  4. d) Add up all your answers (the total number of winning coalitions that each member is critical in). 29
  5. e) Take your answer in (a) and divide by the total in (d).  Express this as a percentage. This is called the Banzhaf index for A. .1%
  6. f) Find the Banzhaf index for B, C, D and E. 17%, 17%, 24% 24%

How do the Banzhaf indices for “majority” members compare to those of “minority” members?

The percents are larger for those who are majority and smaller for minorities because A,B,C make up a majority and D,E are a minority group.

  1. a) What is the Banzhaf index of Glen Cove in problem 6? 0%
  2. b)  Do you think Glen Cove was correct that it was unfairly represented the Nassau County Board of Supervisors? Why? Yes because when looking at winning coalitions Glen Cove is not critical to any winning coalitions.
  3. c) How is the situation of Glen Cove in the Nassau County Board of Supervisors similar to the situation of Luxembourg in the EEC? It’s similar because their weights are both so small that it doesn’t really make a difference which way they vote. They will never be the deciding factor in a vote for something and are not critical members.
  4. d) In your own words, describe what the Banzhaf index measures. It measures the importance of each member in comparison to the whole and helps determine who has how much power.

 

  1. a) Find the Banzhaf index of the 3 players in problem 5.  
  2. b) What do you notice about your answer?  Is it surprising in any way?

They are all 33.3% this is surprising because C only has a weight of one but is still just as important as A and B which have 2.

  1. In many large companies, decisions are made by shareholders.  Each shareholder has a “weight” equal to the number of shares they own.  Suppose a board consists of 4 members, A, B, C and D. They have weights equal to  50, 20, 15, and 15 respectively. A coalition is winning if its total weight is at least 60.
  2. a) List all the winning coalitions. A/B A/C A/D
  3. b) How many winning coalitions is A critical in? All of them
  4. c) How many winning coalitions are B, C and D critical in? 1 each
  5. d) A member of a voting game is said to have veto power if they are critical in every winning coalition. Is there a member in this voting game veto power? A Has the veto power because A is the critical member in coalition. Without A nothing will get approved.
  6. e) Find the Banzhaf power of each member. A 50% B 16.% C16.6% D 16.6%
  7. f) What do you notice about your answers? A has 50% of the power which is a lot considering a majority is 51% the other 3 members share 50% giving them only a fraction of the power.

 

  1. a) Find the Bhanzhaf index for each block in problem 2.  

A=100% B/C=0%

  1. b) What do you notice about your answer? A has 100% of the power and the other members have no power because without A the other members can not get anything down.
  2. c) How would your answers in (a) change if the quota was raised to 7?

A= 60% B=20% C=20%

A/3 b/1 c/1

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