Section 8.1: Measuring Voting Power

Welcome to the notes for Section 8-1, where we begin our exploration of Chapter 8: Voting and Social Choice. In this section, we'll focus on measuring voting power. Let's dive in and understand the crucial concepts that help us determine the real impact of each vote!

Key Concepts

  • Voting Coalition: A group of voters who vote the same way.
  • Winning Coalition: A set of voters with enough votes to determine the outcome of an election; otherwise, it's a Losing Coalition.
  • Quota: The number of votes necessary to win the election in a voting system.

Example: County Convention

Consider a scenario: Abe has 4 votes, Ben has 3 votes, and Condi has 1 vote at a county convention. A simple majority wins. Let's find the quota and determine the winning coalitions.

  1. What is the quota?

    There are $4 + 3 + 1 = 8$ votes in total. Therefore, a simple majority requires $\frac{8}{2} + 1 = 5$ votes. The quota is 5.

  2. List all coalitions and identify winning coalitions.
    Abe (4) Ben (3) Condi (1) Total Votes Winning Coalition?
    Yes Yes Yes 8 Yes
    Yes Yes No 7 Yes
    Yes No Yes 5 Yes
    Yes No No 4 No
    No Yes Yes 4 No
    No Yes No 3 No
    No No Yes 1 No

Critical Voter

A Critical Voter is a member of a winning coalition whose removal would cause the coalition to become a losing one.

Using the winning coalitions from our previous example:

  • In the {Abe, Ben, Condi} coalition, Abe is critical.
  • In the {Abe, Ben} coalition, both Abe and Ben are critical.
  • In the {Abe, Condi} coalition, both Abe and Condi are critical.

Counting Coalitions

For $n$ voters, the total number of possible coalitions is given by the formula $2^n - 1$ (excluding the empty coalition).

Banzhaf Power Index

The Banzhaf Power Index measures a voter's power by calculating the number of times a voter is critical in a winning coalition, divided by the total number of times any voter is critical. It's expressed as a fraction or a percentage.

Using our previous example:

  • Abe is critical 3 times.
  • Ben is critical 1 time.
  • Condi is critical 1 time.

The total number of times any voter is critical is $3 + 1 + 1 = 5$. Therefore:

  • Abe's Banzhaf Power Index is $\frac{3}{5} = 60\%$.
  • Ben's Banzhaf Power Index is $\frac{1}{5} = 20\%$.
  • Condi's Banzhaf Power Index is $\frac{1}{5} = 20\%$.

Shapley-Shubik Power Index

The Shapley-Shubik Power Index is calculated as the fraction (or percentage) of all permutations of the voters in which that voter is the swing voter. The swing voter is the one whose vote makes the total meet the quota, thus deciding the outcome.

That's it for Section 8-1! Keep practicing, and you'll master these concepts in no time. Understanding voting power is essential for informed participation in democratic processes. Good luck with your assignment!