Preference Ballot: This isn’t your usual ‘pick one’ ballot. Here, you rank your choices, giving a fuller picture of what you really want.

Preference Schedule: Think of it as a cheat sheet that organizes everyone’s rankings, making it easier to see group preferences.

In the realm of voting, the plurality method and the Condorcet winner are two key concepts that often come into play. The plurality method is the most straightforward: the candidate with the most first-choice votes wins. However, this method doesn’t always reflect the majority’s preference, as a candidate can win with a plurality but not a majority of votes. On the other hand, the Condorcet winner is a candidate who would win in every one-to-one comparison against all other candidates. This concept introduces a fairness criterion, aiming to identify a candidate that truly represents the majority’s preference.

Plurality Method: Focus only on the first-choice votes. The candidate with the most first-choice votes wins, even if they don’t have an absolute majority.

Condorcet Winner: This is the candidate who would beat every other candidate in a one-on-one vote. To find the Condorcet Winner, compare each candidate against all others in one-to-one matchups and see who comes out on top in each.

Instant Runoff Voting (IRV), also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting.

IRV involves preference ballots and a process of eliminating candidates with the fewest first-place votes. Votes for the eliminated candidate are redistributed to voters’ next choices. This process continues until a choice has a majority (over [latex]50\%[/latex]).

Monotonicity criterion refers to the principle that if voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning.

The criterion is violated in some elections, but this doesn’t mean IRV always violates it; it depends on the election context.

The Borda count method adds a new layer to voting by assigning points to candidates based on their ranking. While this system can sometimes yield a more “consensus-based” winner, it has its drawbacks. For instance, a candidate with a majority of first-place votes might not win, violating the majority criterion. This method is often used in sports awards and other scenarios where a more nuanced view of preferences is beneficial.

Borda Count: In this method, each rank you give to a candidate earns them points. The candidate with the most points wins, but this may not always align with the majority’s first-choice preference.

Majority Criterion: If a candidate has a majority of first-place votes, they should ideally be the winner. Borda Count can sometimes violate this criterion.

Consensus-Based Voting: Borda Count aims for a more broadly acceptable option rather than focusing solely on first-choice votes. It considers every voter’s entire ranking to determine the outcome.

Copeland’s Method: A voting method that aims to satisfy the Condorcet Criterion by conducting pairwise comparisons between candidates. The more preferred candidate in each pair gets [latex]1[/latex] point, and in case of a tie, each gets [latex]\frac{1}{2}[/latex] point. The candidate with the most points wins.

IIA Criterion: Removing a non-winning choice from the ballot should not change the election’s winner.

Things to consider when applying Copeland’s method:

• Pairwise Comparisons: Always compare candidates in pairs to determine the more preferred one.
• Point Allocation: Award [latex]1[/latex] point to the more preferred candidate and [latex]\frac{1}{2}[/latex] point to each in case of a tie.
• Totaling Points: Sum up the points for each candidate; the one with the most points wins.
• Check for IIA: Ensure that removing a non-winning candidate doesn’t change the election outcome.
• Tie-Breaking: Copeland’s method can often result in ties, requiring more advanced methods for resolution.

Approval Voting: A voting method where you mark all choices you find acceptable. The option with the most approval wins. Approval voting can easily violate the majority criterion, meaning the candidate with the most first-choice votes may not win.

Strategic Insincere Voting: This method is susceptible to voters not voting their true preference to increase the chances of their choice winning.

Things to consider when applying the approval voting method:

• Marking Choices: In approval voting, you’re not ranking candidates; you’re simply marking all that you find acceptable.
• Tallying Votes: Count the number of approvals each option receives. The one with the most wins.
• Strategic Voting: Some voters may mark options they find less preferable to increase the chances of their true preference winning.

Which Method is Fair? The search for a perfect voting method is elusive, as no method satisfies all fairness criteria.

Arrow’s Impossibility Theorem: Proven by Kenneth Arrow in 1949, this theorem states that no voting method can satisfy all fairness criteria.

Condorcet’s Voting Paradox: This paradox shows that voting preferences are not transitive, meaning that if A is preferred over B, and B over C, it doesn’t necessarily mean A is preferred over C.

Method Selection: The choice of voting method often depends on what seems most fair for the specific situation.

Things to consider when evaluating voting methods and interpreting results:

• Fairness Criteria: Always consider the fairness criteria when evaluating a voting method.
• Transitivity: Be cautious of the Condorcet’s Voting Paradox when interpreting voting results.
• Method Choice: The “best” method may vary depending on the context and what fairness criteria are most important.