Welcome to Chapter 8: Voting and Social Choice!
Chapter 8 can be one of the most challenging, but also one of the most rewarding, chapters in the book. We'll be exploring the math behind voting systems and fair division, providing you with tools to analyze how decisions are made in various contexts. Don't hesitate to come to class prepared with questions; your active participation is key to mastering these concepts!
Section 8-1: Measuring Voting Power
In Section 8-1, we'll focus on understanding how to measure voting power. This involves understanding the relationship between the number of votes a person has and their actual influence on the outcome. Key concepts include:
- Voting Coalition: A group of voters who vote the same way.
- Winning Coalition: A set of voters with enough votes to determine the outcome. A coalition that does not have enough votes is a Losing Coalition.
- Quota: The minimum number of votes required to win an election.
- Critical Voter: A member of a winning coalition is considered a critical voter if their removal would cause the coalition to become a losing one.
For $n$ voters, the total number of possible coalitions is given by $2^n - 1$.
We will also delve into methods for measuring true voting power:
- Banzhaf Power Index: The Banzhaf power index is calculated by determining the number of times a voter is critical in a winning coalition and dividing that by the total number of instances in which any voter is critical. Expressed as a fraction or percentage. For instance, if Abe is critical 3 out of 5 times, his Banzhaf power index would be $\frac{3}{5}$ or 60%.
- 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.
For example, consider the 1988 Democratic National Convention. Dukakis had 1401 votes, Jackson had 1218, Gore had 325, and Babbitt had 197. To determine the Banzhaf index, we first need to calculate the quota (the number of votes for simple majority). In this case it is 1571 votes.
Section 8-2: Voting Systems
Section 8-2 shifts our focus to different voting systems and how they determine a winner. One crucial concept is recognizing that with three or more candidates, no perfect voting system exists. We'll explore various systems, including:
- Plurality Voting: The candidate with the most votes wins, even if they don't have a majority.
- Preferential Voting System: Voters rank candidates in order of preference.
- Top-Two Runoff System: If no candidate receives a majority, a second election is held with only the top two vote-getters.
- Elimination Runoff System: The candidate with the fewest votes is eliminated, and the process repeats until a majority is reached.
- Borda Count: Points are assigned based on ranking (e.g., last place gets 0 points, the next gets 1, and so on), and the candidate with the most points wins.
- Condorcet Winner: A candidate who would win in a one-on-one election against each of the other candidates.
Related concepts:
- Spoiler: A candidate who has no realistic chance of winning but whose presence affects the outcome.
- Condorcet winner criterion: If there is a Condorcet winner, that candidate should win the election.
- Independence of irrelevant alternatives: If candidate A wins against B, and another election is held, A should still win.
- Arrow's Impossibility Theorem: If there are three or more candidates, there is no voting system other than dictatorship for which the Condorcet winner criterion and Independence of irrelevant alternatives hold.
Don't forget to review the class notes and quizzes provided for each section. Good luck, and remember, asking questions is a sign of a sharp mind!