Welcome to Chapter 8: Voting and Social Choice!

Get ready to dive into the math behind decision-making! This chapter explores how mathematical principles impact elections, fair division, and apportionment. Understanding these concepts can empower you to be a more informed citizen and negotiator.

Sections Covered:

  • 8.1: Measuring Voting Power: Does My Vote Count? This section explores how to quantify the influence of individual voters and coalitions.
  • 8.2: Voting Systems: How Do We Choose a Winner? We'll examine different voting methods and their potential strengths and weaknesses, including plurality voting, instant runoff, and the Condorcet winner criterion.
  • 8.3: Fair Division: What is a Fair Share? Learn about methods for dividing assets fairly, like the divide-and-choose procedure, adjusted winner procedure, and the Knaster procedure.

Key Concepts and Examples:

8.1: Measuring Voting Power

  • Voting Coalition: A group of voters who vote the same way.
  • Winning Coalition: A set of voters with enough votes to determine the outcome.
  • Quota: The number of votes needed to win.
  • Critical Voter: A member of a winning coalition whose removal would cause the coalition to lose.

For example, consider a committee with Abe (4 votes), Ben (3 votes), and Condi (1 vote). A simple majority (5 votes) is required to win. If Abe, Ben, and Condi vote together, the coalition has 8 votes, making it a winning coalition. If Condi were to leave, that is 4 + 3 = 7 votes so Abe and Ben could still win without her. In that coalition, Abe and Ben are both critical voters.

Banzhaf Power Index: This index measures a voter's power by calculating the proportion of times they are a critical voter in winning coalitions. The formula is:
$$ \text{Banzhaf Power Index} = \frac{\text{Number of times voter is critical}}{\text{Total number of instances of critical voters}} $$

Shapley-Shubik Power Index: Calculated as the fraction (or percentage) of all permutations of the voters in which that voter is the swing.

8.2: Voting Systems

  • Plurality Voting: The candidate with the most votes wins, even if they don't have a majority.
  • Instant Runoff Voting: Voters rank candidates; if no candidate receives a majority, the candidate with the fewest votes is eliminated, and their votes are redistributed based on voters' next choices.
  • Spoiler Effect: A candidate with little chance of winning can impact the outcome by drawing votes away from a major candidate.
  • Condorcet Winner: A candidate who would win in a head-to-head election against every other candidate.

Let's say we have candidates A, B, C, and D. A Borda count assigns points based on ranking. If there are 100 total voters with 3 points for a first-place vote, 2 for a second-place, 1 for third, and 0 for last, the totals from each candidate can be computed, and then the person with the highest point total wins.

8.3: Fair Division

  • Divide-and-Choose: One person divides the items, and the other person chooses which part they want. Fair and simple!
  • Adjusted Winner Procedure: Two people assign points to each item; items are initially awarded to the highest bidder, then adjusted until both parties are equally satisfied.
  • Knaster Procedure: Each party secretly bids on each item; the highest bidder gets the item, but pays the others their fair share based on their bids.

Imagine siblings dividing an inheritance using the Adjusted Winner Procedure. They each assign 100 points to items like a guitar, car, and cash. By carefully transferring items (or portions thereof), they can achieve a fair outcome where everyone feels they received an equitable share.

Chapter 6 Quiz Reminder

Don't forget to complete the Chapter 6 quiz! Review your notes and practice problems to ensure you're prepared. Good luck!

Let's make math fun and engaging. Together, we can conquer any challenge!