Discrete Math A Chapter 2, Weighted Voting Systems 1
ONE PERSON – ONE VOTEis 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:
Weighted Voting System:
Motion:
Players: (symbolized by P1, P2, P3, etc.)
Weight:
Quota:
Notation[q: w1, w2, w3, . . ., wn]q =
w’s =
Example:[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
Common Types of Quotas:
Simple majority/Strict majority
Two-thirds majority
Unanimity
Weighted Voting Issues
Example 2.1:
Venture CapitalismFour 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. The quota is set at two-thirds of the total number of votes. Describe as a weighted voting system.
Example 2.2
AnarchyThe partnership above decides the quota is too high and changes the quota to 10 votes.
Example 2.3
GridlockThe partnership above decides to make the quota equal to 21 votes.
For a weighted voting system to be legal: the quota must be at least a ______
and no more than ______
Symbolically: If , then
Example 2.4
One Person – One VoteWhat if our partnership changed the quota to 19?
Dictators, Dummies, and Veto Power
Example 2.5:[11: 12, 5, 4]
Dictators
What do you notice about P1 ?
P1 has all the power
P2 and P3 have no power
Example 2.6:[30: 10, 10, 10, 9]
Unsuspecting Dummies
Example 2.7: [12: 9, 5, 4, 2]
Veto Power
Is there a dictator ?
If P1 chooses to vote against the motion, can the other
players combine weight to meet the quota?
Practice Problems
1.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 P4: ____The minimum % of the quota to nearest whole %: ____
2.[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? ______
3.A committee has 4 members (P1, P2, P3, P4). P1 has twice as many votes as P2. P2has 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 two-thirds
c) The quota is more than three-fourths
4.Determine which players, if any, are: (i) dictators, (ii) veto power, (iii) dummies
a) [15: 16, 8, 4, 1] b) [18: 16, 8, 4, 1] c) [24: 16, 8, 4, 1]
5.Consider [q: 8, 4, 2]. Find the smallest value of q for which
a) all three players have veto powerb) P2 has veto power, but P3 does not
c) P3 is the only dummy
2.2/2.3 The Banzhaf Power Index
Who is the most POWERFUL player?
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.
Coalition:
Grand coalition:
Weight of the coalition:
Winning coalitions—
Losing coalitions—
Critical player:
Example 2.8:[101: 99, 98, 3]
Weirdness
Step 1:Make a list of all WINNING coalitions coalitionC.P.’s
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 BanzhafPower 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 2.9: Find the Banzhaf Power Distribution for [4: 3, 2, 1]
Example 2.10 Find the Banzhaf Power Distribution for [6: 4, 3, 2, 1]
How can you be sure you have all the winning coalitions written down?
One Method Brute Force: Write down ALL POSSIBLE coalitions. Check each one.
How do you know you have all the possible coalitions written down?
Be systematic or use the formula!
How many coalitions if 5 players?How many coalitions if 6 players?
Where weighted voting systems/Banzhaf are used:
Banzhaf is used to QUANTIFY the amount of power each player holds.
- 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.
- 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.
- European Union (see chart p. 56): Banzhaf quantifies the amount of power each nation has and shows that smaller nations such as Luxembourg and Malta still hold some power.
What I expect to see for “work” on your homework:
Exercise 17e:[q: 8, 4, 2, 1] when q=14, Find BPD
1. Write down all 4-player 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:
Copy/Paste these as needed
to a word document for
use on homework
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:
Pivotal player:
Banzhaf: { P1, P2, P3}Shapley-Shubik:
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.
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]
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?
Example 2.17:Find the Shapley-Shubik Power Distribution for
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 numberof times they are pivotal
and divide by the number of sequential coalitions. Calculate the %.
==
Example 2.18:Find the Shapley-Shubik Power Distribution for
4 players = ____ sequential coalitions
==
Other Interesting Weighted Voting Scenarios
Example: No Weights, Just Description
Example: Mergers
Example: Decisive Voting
Example: Antagonists