# CHAPTER 2 Weighted Voting Systems

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```					                                 CHAPTER 2: Weighted Voting Systems
Section 2.1 Weighted Voting Systems
 Weighted Voting System:

One Voter with _____________________ Number of Votes
** not all voters have the same number of votes**

   Motion:

   Key Elements and Notation of a Weighted Voting System
o Players:

For N players: ______________________

o Weights:

For N Players: ______________________

o Quota, q:

o Summary of Weighted Voting System: [__________; ________________________]

Example 1: [13: 7, 4, 3, 3, 2, 1]                   Example 2: [31: 12, 8, 6, 5, 5, 5, 2]
a. # of players                                     a. # of players
c. weight of P3                                     c. weight of P3

d. minimum # of votes needed to pass a motion       d. minimum # of votes needed to pass a motion

   Properties for Size of the Quota

o The quota must be __________ ____________ ___________ the total number of votes.
w1  w2  ...  w N
q
2

o The quota ______________________________________ the total number of votes.
q      w1  w2  ...  w N
Example 3: [31: 12, 8, 6, 5, 5, 5, 2]                    Example 4: [q: 12, 8, 6, 5, 5, 5, 2]

If a 2/3 majority is needed to pass a motion, what is    What is the minimum quota we can have to pass a
the minimum # of votes needed?                           motion?

   Power Definitions (which votes and therefore players are most important )

o Dictator: Player’s weight is greater than or equal to the quota, all the power
there can only be one dictator in any weighted voting system

o Dummy: Any player’s weight won’t affect the outcome, powerless
When a system has a dictator, all other players are dummies

o Veto Power: Motion cannot be passed unless the player votes in favor
All other players can’t meet quota with their votes

o Example: [11: 12, 5, 4]                                         Veto power:

   Dictator:

   Dummies:                                            o Example: [6: 5, 5, 1]

   Veto power:                                                Dictator:

   Dummies:

o Example: [15: 14, 8, 7]                                         Veto power:

   Dictator:

   Dummies:
o Example: [6: 4, 1, 1]
   Veto power:
   Dictator:

o Example: [6: 4, 2, 1]                                           Dummies:

   Dictator:                                                  Veto power:

   Dummies:

HOMEWORK: p. 72 #1-5, 7, 9
Section 2.2 and 2.3 Banzhaf Power Index Applications
 Coalitions:

o Grand Coalition:

o Set Notation for Coalition of players:

o Winning Coalition:

Is the dictator a winning coalition?

o Losing Coalition:

   Critical Player:

o All players, some, or no players can be critical within a coalition

   EXAMPLE: [101: 99, 98, 3]
   Banzhaf’s Power                    COALITIONS # of Votes               Win or Lose   Critical Player(s)
Interpretation:
P1
Player’s Power Should be
measured by how often player       P2
is ________________
P3
   Calculating the Banzhaf          P1, P2
Power Index:
1) Find all winning           P1, P3
coalitions.                P2, P3
2) Identify which players
are critical in each       P1, P2, P3
winning coalition.
3) Count the total number of times that each player, Pi, is critical, Bi.
4) Find the total number of times all players are critical, T.
N
T=   B
i 1
i

5) Banzhaf Power Index, β (beta): ratio of how often a player is critical to all players being
critical
Bi
i       is the Banzhaff Power Index for player Pi
T
o   EXAMPLE#1: [101: 99, 98, 3]
P1                        P2                          P3
# times player is
critical, Bi
Banzhaf Power
Index, βi

o EXAMPLE #2: [4: 3, 2, 1]
COALITIONS       # of Votes                               Win or Lose                  Critical Player(s)
P1
P2
P3
P1, P2
P1, P3
P2, P3
P1, P2, P3

P1                       P2                       P3
# times player is
critical, Bi

Banzhaf Power
Index, βi

o Banzhaf Power Distribution: complete list of all Banzhaf Power Indexes
β1 , β2, β3,…, βN

   The sum of all Banzhaf Power Indexes = 1 = β1 + β2 + β3 + … + βN

o Total Number of Possible Coalitions from N Players:

   It may be easier to write out all coalitions and then eliminate losing coalitions instead of
specifically create winning coalitions only

HOMEWORK:               Day One p72 #11, 12 p.75# 41, 42, 45
Day Two p. 73 #15, 17, 19-21 p.76 #48
Section 2.4 and 2.5 Shapley-Shubik Power Index and Applications

   Banzhof Coalitions = ________________________________________

Shapley-Shubik Coalitions = __________________________________

   Sequential Coalitions:

o Notation: <P1, P2, P3> is ____________________ from <P1, P3, P2>

   What are all sequential coalitions that can be made with exactly 3 players P1, P2, P3?
(Hint you should get 6)

   How many sequential coalitions with N players are there? (ordered sequence containing N items)

Multiplication Rule: If there _______ ways to do X and _______ ways to do Y, then there is ______
ways to do X and Y.

o Factorial, N!: The __________________________ of the first N positive integers
3! =

5! =

5!/ 3! =

The number of sequential coalitions N players is _________________________

   Pivotal Player:

o Pivotal Player’s Weight + Previous Coalition Players’ Votes ____________________ quota

o Previous Coalition Players’ Votes __________________ quota

   Players to the Left in order are a ___________________ coalition

o __________________________________________ pivotal player per coalition
   Shapley – Shubik Interpretation of Power: Player’s Power Should be measured by how often player
is _____________________________

o Calculating the Shapley – Shubik Power
1. Find all possible sequential coalitions for the N players
 total # of coalitions = T
2. Find pivotal players in each sequential coalition
3. Count number of times each player, Pi, is pivotal = SSi
4. Shapley-Shubik Power Index, σ, (sigma): Ratio of how often a player is pivotal to the
number of sequential coalitions
SSi
i 
T
o Shapley- Shubik Power Distribution: Complete list of σ for each player

   Example 1: [4: 3, 2, 1]
Underline or circle each pivotal player and give the SS Power Distribution as a simplified fraction and
percent.

Sequential Coalitions               Pivotal Player

   Example 2: [16: 9, 8, 7]
Underline or circle each pivotal player and give the SS Power Distribution as a simplified fraction and
percent.

Sequential Coalitions               Pivotal Player

HOMEWORK: DAY ONE p.75 #37, 43, 49ab p. 74# 23-25
DAY TWO p.74#29, 34, 35 p.76 #51

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