Wim Ruitenburg's Fall 2024 MATH 1300-101

Marquette University

Department of Mathematical and Statistical Sciences

Wim Ruitenburg's Fall 2024 MATH 1300-101

Last updated: 19 September 2024
Comments and suggestions: Email wim.ruitenburg@marquette.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. The parties P_1, P_2, ..., P_n are ordered so that w_1 >= w_2 >= ... >= w_n.
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 with more then 3 parties:

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 Problems

Without using Banzhaf or Shapley-Shubik, determine in each of the following examples what the power indices of the parties should be.

[8: 9, 4, 2].
[12: 7, 4, 2].
[10: 7, 5, 2].

Compute the Banzhaf power index in the following cases.

[12: 6, 4, 3, 1].
[11: 6, 4, 3, 1].

Recommended problems from Chapter 2 of the book:
23a, 3, 5, 7
Recommended problems from Chapter 2 of the book:
21
Recommended problems from Chapter 2 of the book:
33
Give the Banzhaf power distribution of the weighted voting system [8; 7, 4, 2].
Give the Shapley-Shubik power distribution of the weighted voting system [8; 7, 4, 2].
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].
Give the Shapley-Shubik power distribution of the weighted voting system [4; 2, 1, 1, 1, 1, 1]. This is like where the president can break a tie vote. Please be smart and don't list all 6! = 720 sequential coalitions.