Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Spring 2016 MATH 1300-101
Last updated: February 2016
Comments and suggestions: Email wimr@mscs.mu.edu
Power from chapter 2
From the chapter we can learn:
- Voting power distribution systems where all parties P_i have a vote
weight w_i, and where it takes quota amount q to pass a motion.
- Examples of possible anarchy.
- Examples of guaranteed gridlock.
- When does a system force unanimity?
- What is dictatorial power?
- What is veto power?
- What is a dummy party?
Here are examples of attempts to quantify the power of the parties.
- The Banzhaf power index.
List all winning coalitions.
If there are 4 parties, then there are 2^4 = 2*2*2*2 = 16 possible coalitions.
Inside each winning coalition underline the critical parties.
For each party P_i count the number of times it occurs underlined, to get sum
B_i.
Add the total number of all underlined parties in all winning coalitions to get
sum T.
Note that T also equals the sum of the B_i.
Then party P_i has as power index quotient β_i = B_i / T (the Greek letter
beta).
- The Shapley-Shubik power index.
List all winning coalition formations.
If there are 4 parties, then there are 4! = 4*3*2*1 = 24 possible coalition
formations.
Compute the SS_i as we did for the B_i before, except that there is another
definition of being a critical party.
Now T equals the number of possible coalition formations as well as equals the
sum of the SS_i.
Party P_i has as power index quotient σ_i = SS_i / T (the Greek letter
sigma).
We did some examples in class.
Another one:
- The system [11; 8, 7, 3, 2].
The Banzhaf power indices and the Shapley-Shubik indices gave the same numbers
in this special case, namely:
- β_1 = 5/12, β_2 = 1/4, β_3 = 1/4,
and β_4 = 1/12.
- σ_1 = 5/12, σ_2 = 1/4, σ_3 = 1/4,
and σ_4 = 1/12.
- The homework below contains examples where the Banzhaf power indices and
the Shapley-Shubik indices are not equal.
Example Problem(s)
- Recommended problems from Chapter 2 of the book:
1, 4, 6, 7, 10
- Recommended problems from Chapter 2 of the book:
15, 19a
- Recommended problems from Chapter 2 of the book:
29, 31
- Give the Banzhaf power distribution of the weighted voting system
[5; 4, 1, 1, 1].
- Give the Shapley-Shubik power distribution of the weighted voting system
[5; 4, 1, 1, 1].