Weighted Voting
Systems
Chapter 2
Objective: Calculate the Shapley-Shubik Power Index for a
weighted voting system. Learn additional notation and
terminology for sequential coalitions.

Write two or three sentences
comparing and contrasting the
Banzhaf vs. Shapley-Shubik Power
Distributions.
Closing Product
Exit Ticket
The set of all players in which order
matters.
 Denoted by 𝑃1 , 𝑃2 , 𝑃3
 The # calculated by

 N! = N x (N-1) x (N-2) x … 1
 N = number of players
Sequential Coalitions

Consider the following weighted voting
system [4:3,2,1].
◦ How many sequential coalitions are there?
◦ List all of the sequential coalitions.
Example # 1

Find 11!

Find 11!/10!

Find 101!/99!

Find (9!+11!)/10!
Example # 2
A player in a sequential coalition that
turns it from a losing to a winning
coalition.
 There is only one per coalition.
 Find it by adding up weights from Left to
Right.

Pivotal Player

Consider the following weighted voting
system [4:3,2,1].
◦ Find the pivotal player in each sequential
coalition.
Example # 3

A mathematical measurement of a
player’s power in a weighted voting
system based on how many times that
player is the pivotal player in a sequential
coalition.

σ1 = # of time Player 1 is the pivotal player
N!
Pronounced ‘sigma-one’
Shapely-Shubik Power Index

A complete list of the power indexes

σ1 , σ2 ,σ3, …… σn

The sum of all the σ’s is equal to 1. (or
100% if using percentages.
Shapley-Shubik Power
Distribution
i.
ii.
iii.
iv.
v.
Make a list of all sequential coalitions.
Determine the pivotal player of each
coalition.
Count the number of times each player
is pivotal. (SS1 for P1,SS2 for P2, etc.)
Find the total number of sequential
coaltions (N!)
Find the ratio for each player (σ1 =
SS1/N!)
Calculation
Do the next one on your own.
Consider the weighted voting system
[95: 65, 35, 30, 25]. Find the following:
a)
List all of the sequential coalitions.
b)
Circle the pivotal player in each coalition.
c)
Calculate the Shapley-Shubik Power Distribution.
Example #4
Discuss with a neighbor some of the
similarities and differences between
Banzhaf and S-S Power Index.
Then write a couple of sentences that
summarizes what you discussed.
Consider the following ….