Chapter 2: Weighted Voting System
Introduction
When it comes to voting rights, the democratic ideal of equality translates into the principle of ______. In a diverse society, voters are ______equal, and sometimes it is actually desirable to recognize their differences by giving them different amounts of say over the outcome of an election. This is an example of weighted voting.
Definition: Weighted Voting:
Examples:
One of the best known and most controversial examples of weighted voting is the ______, used in presidential elections. Other examples include:
- Shareholder elections
- Business partnerships
- The U.N. Security Council
- The European Union Council of Ministers
Section 2-1: Weighted Voting System
Important definitions and terminology:
- Weighted Voting System: ______
______
- Motion: ______
- Every weighted voting system is characterized by three elements:
- Players:
- Weight:
- Quota:
- Notation of Weighted Voting System
Example:
1)In the weighted voting system [31: 12, 8, 6, 5 ,5 ,5 , 2], find:
a)total number of players
b)total number of votes
c)The weight of P3
d)The minimum percentage of votes needed to pass a motion rounded to the next whole percent
Weighted Voting Issues:
Venture Capitalism
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. The quota is set at two-thirds of the total number of votes. Describe as a weighted voting system using proper notation.
Anarchy
The partnership decides the quota is too high and changes the quota to 10 votes
Gridlock: The partnership above decides to make the quota equal to 21 votes.
Quota
Common Types of Quotas:
- Simple majority/strict majority
- Two-thirds majority
- Unanimity
For a weighted voting system to be legal: the quota must be at least a ______
and no more a ______.
Symbolically:
Example
Consider the weighted voting system [q: 10, 6, 5, 4, 2]
a)What is the smallest value that the quota q can take?
b)What is the largest value that the quota can take?
More important vocabulary and terminology:
1)Dictator:
Consider the weighted voting system: [11: 12, 5, 4]
What do you notice about P1?
2)Unsuspecting dummies:
Consider the weighted voting system: [30:10, 10, 10, 9]
3)Veto Power
Consider the weighted voting system [12: 9, 5, 4, 2]
Section 2-2: Banzhaf Power Index:
Motions (yes or no votes) can be passed or blocked by the players by joining forces. A group of players that might join forces to vote together is called a ______. The total number of votes controlled by a coalition is called the ______.
There are three type of coalitions:
1)Winning coalition:
2)Losing coalition:
3)Grand coalition:
When players join forces, there may be key player(s) who can influence if a motion is passed. This person is called a ______. The player must be present for a coalition to ______.
Note: If you subtract the critical player’s votes from the coalition, the number of votes ______the quota.
Example: Determine the coalitions for this weighted voting system: [101: 99, 98, 3]
The subsets of sets
Set / {P1, P2} / {P1, P2,, P3} / {P1, P2,, P3, P4} / {P1, P2,, P3, P4, P5}Number of Subsets / 4 / 8 / 16 / 32
Subsets / { }
{P1}
{P2}
{P1, P2} / / / The 16 subsets from the previous column along with each of these with thrown in.
Since each time we add a new player we are doubling the number of subsets, we will find it convenient to think in terms of powers of 2.
Players / Number of Subsets / Number of CoalitionsPower Index
We can determine how powerful a player by calculating how often the player is critical in the coalitions. John Banzhaf introduced his mathematical interpretation of power in 1965.
The Banzhaf Power Index: The player’s power is ______for which that player is critical so that the more often the player is critical, the more power he or she holds.
Steps to Calculating the Banzhaf Power Index:
1)Make a list of all possible coalitions
2)Determine the winning coalitions
3)In each winning coalition, determine the critical player
4)Count the total number of times player P is critical (we’ll call this number B)
5)Count the number of times all players are critical (we’ll call this number T)
6)Determine the Banzhaf power index of player P by dividing the B by the T
Example
1)Find the Banzhaf Power Distribution for [4: 3, 2, 1]
2)NBA Draft and Banzhaf Power Index: When the NBA teams prepare for the annual draft of college players, the decision on which college basketball players to draft may involve many people, including the management, the coaches, and the scouting staff. Typically, not all these people have equal voice in the process – the head coach’s opinion is worth more than that of an assistant coach, and the general manager’s opinion more than that of a scout. Let’s use a fictional team – the Akron Flyers
Example: In the Akron Flyers draft system, the head coach (P1) has 4 votes, the general manager (P2) has 3 votes, the director of scouting operations (P3) has 2 votes, and the team psychiatrist (P4) has 1 vote. Of the 10 votes cast, a simple majority is required for a yes vote on the team. Determine the Banzhaf power distribution.