Finite Math A Chapter 2, Weighted Voting Systems 1

ONE PERSON – ONE VOTE is an democratic idea of equality But what if the voters are not PEOPLE but are governments? countries? states? If the institutions are not equal, then the number of votes they control should not be equal.

This situation where each voter is not equal in the number of votes they control is called:

2.1 An Introduction to weighted voting

Important terms: _________________________________________: A voting situation where voters are not necessarily equal in the number of votes they control. _________________: A vote with only two choices. (usually yes/no) _________________: The voters (symbolized by P1, P2, P3, etc.) _________________: The number of votes a player controls. _________________: The smallest number of votes required to “pass” a motion.

Discrete Mathematics Notes

Chapter 2: Weighted Voting Systems – The Power Game

Academic Standards: PS.ED.2: Use election theory techniques to analyze election data. Use weighted voting techniques to decide voting power within a group.

Stock Holders/Shareholders: The more stock you own, the more say you have in decision making for the company.

The United Nations Security Council – 15 voting nations: 5 permanent members (Britain, China, France, Russia, United States), 10 nonpermanent members appointed for a 2-year rotation. Permanent members have more “votes” than non permanent members.

The Electoral College—Each state gets a number of “electors” (votes) equal to the number of Senators plus the number of Representatives in Congress. California has 55 votes but North Dakota only has 3 votes. Each state is a voter but states with heavy concentration of population receive a bigger “vote”.

Finite Math A Chapter 2, Weighted Voting Systems 2

Notation

[q: w1, w2, w3, . . ., wn] q = w’s = Example 1. [14: 8, 6, 5, 1]

quota = Player 1 (P1) = controls ___ votes /“has a weight of ____”

total votes = Player 2 (P2) = controls ___ votes

Player 3 (P3) = controls ___ votes

Player 4 (P4) = controls ___ vote

Example 2. Given the weighted voting system [16: 8, 6, 4, 4, 3, 1], state the following: The number of players: ____ The total number of votes: ____ The weight of P5: ____ The minimum % of the quota to nearest whole %: ____

Common Types of Quotas: Simple majority/Strict majority

Two-thirds majority

Unanimity

Weighted Voting Issues Example 3: Four partners decide to start a business. P1 buys 8 shares, P2 buys 7 shares, P3 buys 3 shares and P4 buys 2 shares. One share = one vote.

a. The quota is set at two-thirds of the total number of votes. Describe as a weighted voting system. b. The partnership above decides the quota is too high and changes the quota to 10 votes. c. The partnership above decides to make the quota equal to 21 votes.

U.S. Senate: Simple Majority to pass an ordinary law (51 votes) 60 votes to stop a filibuster 2/3 of the votes (67) to override a presidential veto.

Finite Math A Chapter 2, Weighted Voting Systems 3

For a weighted voting system to be legal: the quota must be at least a __________________________________ and no more than ___________________________________________

Symbolically: If 1 2 3 ... nV w w w w , then

2

Vq V

Example 3d. What if our partnership changed the quota to 19? 4. [q: 7, 2, 1, 1, 1]

What is the smallest legal quota? _____ What is the largest legal quota? _____ What is the value of the quota if at least two-thirds of the votes are required to pass a motion? ______ What is the value of the quota if more than three-fourths of the votes are required to pass? ______

5. A committee has 4 members (P1, P2, P3, P4). P1 has twice as many votes as P2. P2 has twice as many votes

as P3. P3 and P4 have the same number of votes. The quota is 49. Describe the weighted voting system using the notation [q: w1, w2, w3, w4] given the definitions of quota below. (Hint: write the weighted voting system as [49: 4x, 2x, x, x] and then solve for x.

a) The quota is a simple majority b) The quota is more than three-fourths

Finite Math A Chapter 2, Weighted Voting Systems 4

Dictators, Dummies, and Veto Power Example 6: [11: 12, 5, 4] What do you notice about P1 ?

P1 has all the power

P2 and P3 have no power

Example 7: [30: 10, 10, 10, 9] Example 8: [12: 9, 5, 4, 2] Is there a dictator ?

If P1 chooses to vote against the motion, can the other players combine weight to meet the quota?

Example 9. Determine which players, if any, are: dictators, veto power, dummies a) [15: 16, 8, 4, 1] b) [18: 16, 8, 4, 1] c) [24: 16, 8, 4, 1] Example 10. Consider [q: 8, 4, 2]. Find the smallest value of q for which

a) all three players have veto power b) P2 has veto power, but P3 does not c) P3 is the only dummy

Note: If any player is a dictator, then EVERY OTHER PLAYER is a dummy. Even if there is no dictator, there may still be dummies.

If a player is not a dictator, but the other players cannot meet the quota without his votes, we say he has veto power. Sometimes, more than one player will have veto power.

Finite Math A Chapter 2, Weighted Voting Systems 5

2.2/2.3 The Banzhaf Power Index Who is the most POWERFUL player?

_________________: A group of players who choose to vote together

_______________________________: The set of all voters. This represents a unanimous vote.

Weight of the coalition:

Winning coalitions—

Losing coalitions— ________________________ : Any player who MUST BE PRESENT in a winning coalition in order for it to remain a winning coalition.

Example 1: Find the critical player or critical players in each of the following coalitions.

[15: 13, 9, 5, 2] a) {P1, P4} b) {P2, P3, P4} c) {P3, P4} d) {P1, P2, P3}

[51: 30, 25, 25, 20] a) {P1, P3} b) {P1, P2, P3}

c) {P2, P3, P4} d) {P2, P3} The Banzhaf Power Index: A player’s power is proportional to the number of coalitions for which that player

is critical. The more often a player is critical, the more power he holds.

Note: If you subtract the critical player’s votes from the coalition, the number of votes drops below the quota.

Finite Math A Chapter 2, Weighted Voting Systems 6

Calculate the Banzhaf Power Index: Idea: Step 1: Make a list of all WINNING coalitions.

Step 2: Determine which players are critical in each coalition. (circle, underline, highlight)

Step 3: Count the total number of times each player is critical Step 4: Add the total number of times each player is critical to find the grand total number of critical players.

The Banzhaf Power INDEX number for each player = step 3 ÷ step 4 The Banzhaf Power DISTRIBUTION for the weighted voting system is the % of power each player holds.

Example 2: Find the Banzhaf Power index for the weighted voting system:

[101: 99, 98, 3]

Example 3: Find the Banzhaf Power Distribution for

[4: 3, 2, 1]

Finite Math A Chapter 2, Weighted Voting Systems 7

How do you know you have all the possible coalitions written down?

Be systematic or use the formula!

How many coalitions if 4 players? How many coalitions if 5 players? Example 4: Find the Banzhaf Power Distribution for

[6: 4, 3, 2, 1]

Example 5: Consider the weighted voting system [q: 8, 4, 2, 1] . Find the Banzhaf Power Distribution of this weighted voting system when:

a) q = 8 b) q = 10 c) q = 14

Banzhaf Coalitions: 4 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P1, P2, P4} {P3} {P1,P4} {P1, P3, P4} {P4} {P2,P3} {P2, P3, P4}

{P2,P4} {P1, P2, P3, P4} {P3,P4}

If n = number of players in a weighted voting system,

Then the number of possible coalitions is: 2n – 1

Finite Math A Chapter 2, Weighted Voting Systems 8

What I expect to see for “work” on your homework: 1. Write down all possible coalitions and cross off losers OR just the winning coalitions. 2. Critical Players should be circled or underlined. 3. Show fraction of BPI for each player AND calculate the % for BPD.

Possible Coalitions:

Use these to help you: Where weighted voting systems/Banzhaf are used: Banzhaf is used to QUANTIFY the amount of power each player holds.

1. Nassau County Board of Supervisors (see p. 55): Votes were given to districts according to population and quota was simple majority. [58: 31, 31, 28, 21, 2, 2] Banzhaf showed that two of the six counties actually had no voting power—that they were actually dummy voters. Final result: 1993 court decision abolishing weighted voting in New York States. “Districts” were created of roughly the same population and each given one voted.

2. United Nations Security Council: Banzhaf shows that a permanent member of the council holds more than 10 times the amount of power as one of the non-permanent members. There are 5 permanent members (Britain, China, France, Russia, US) and 10 non-permanent members. This voting arrangement may change as others are being considered for permanent membership.

3. European Union Banzhaf quantifies the amount of power each nation has and shows that smaller nations such as Luxembourg and Malta still hold some power.

Banzhaf Coalitions: 3 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P3} {P2,P3}

Banzhaf Coalitions: 4 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P1, P2, P4} {P3} {P1,P4} {P1, P3, P4} {P4} {P2,P3} {P2, P3, P4}

{P2,P4} {P1, P2, P3, P4} {P3,P4}

Finite Math A Chapter 2, Weighted Voting Systems 9

2.4/2.5 The Shapley Shubik Power Index:

The Shapley-Shubik Power Index:

A player’s power is proportional to the number of sequential-coalitions for which that player is pivotal. The more times a player is pivotal, the more power he holds.

Sequential coalition:

Banzhaf: { P1, P2, P3} Shapley-Shubik: ⟨𝑃1 , 𝑃3, 𝑃2⟩ These 3 players decide to vote together. These 3 players decide to vote together. They form a coalition. P1 votes 1st, P3 votes 2nd , P2 votes 3rd. Order listed in the { } doesn’t matter. They form a sequential coalition. Order listed in the ⟨ ⟩ is important.

Pivotal player:

Example: Find the Pivotal Player 1. Given the weighted voting system [5: 3,2,1,1} find the pivotal player for the given sequential coalition.

a) [P1,P4,P3,P2] b) [P3,P1,P2,P4] c) [P4,P3,P2,P1]

Finite Math A Chapter 2, Weighted Voting Systems 10

Counting Sequential Coalitions: List the possible sequence for 3 players. How many are there?

How many sequential coalitions are there for 4 players? For 5 players?

Shapley-Shubik Power Distribution Step 1: Make a list of all sequential coalitions

Step 2: For each sequential coalition, determine the pivotal player.

Step 3: For each player, count the number of times they are pivotal and divide by the number of sequential coalitions.

Calculate the %.

Example 2: Find the Shapely Shubik Power Distribution for [4: 3, 2, 1]

Multiplication Rule: If there are m ways to do task 1 and n ways to do task 2, then there are mxn ways to do both tasks together.

Factorials: If N= the number of players, then the number of sequential coalitions is N! N! = N x (N-1) x . . . x 3 x 2 x 1

Sequential Coalitions: 3 Players

[P1,P2,P3]

[P1,P3,P2]

[P2,P1,P3]

[P2,P3,P1]

[P3,P1,P2]

[P3,P2,P1]

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Example 3: Find the Shapley-Shubik Power Distribution for [6: 4, 3, 2, 1]

Example 4: Find the Shapley-Shubik Power Distribution for [10: 8, 4, 2, 1]

Sequential Coalitions: 4 Players

[P1,P2,P3,P4] [P2,P1,P3,P4] [P3,P1,P2,P4] [P4,P1,P2,P3] [P1,P2,P4,P3] [P2,P1,P4,P3] [P3,P1,P4,P2] [P4,P1,P3,P2] [P1,P3,P2,P4] [P2,P3,P1,P4] [P3,P2,P1,P4] [P4,P2,P1,P3] [P1,P3,P4,P2] [P2,P3,P4,P1] [P3,P2,P4,P1] [P4,P2,P3,P1] [P1,P4,P2,P3] [P2,P4,P1,P3] [P3,P4,P1,P2] [P4,P3,P1,P2] [P1,P4,P3,P2] [P2,P4,P3,P1] [P3,P4,P2,P1] [P4,P3,P2,P1]

Sequential Coalitions: 4 Players

[P1,P2,P3,P4] [P2,P1,P3,P4] [P3,P1,P2,P4] [P4,P1,P2,P3] [P1,P2,P4,P3] [P2,P1,P4,P3] [P3,P1,P4,P2] [P4,P1,P3,P2] [P1,P3,P2,P4] [P2,P3,P1,P4] [P3,P2,P1,P4] [P4,P2,P1,P3] [P1,P3,P4,P2] [P2,P3,P4,P1] [P3,P2,P4,P1] [P4,P2,P3,P1] [P1,P4,P2,P3] [P2,P4,P1,P3] [P3,P4,P1,P2] [P4,P3,P1,P2] [P1,P4,P3,P2] [P2,P4,P3,P1] [P3,P4,P2,P1] [P4,P3,P2,P1]