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Essential Concepts

• Preference ballots allow voters to rank candidates or options in order of preference, rather than selecting just one option. This method provides a more detailed view of voter preferences.
• A preference schedule is a table used to organize how people rank different options in an election. It helps in understanding the collective preferences of the voters.
• Various methods can be used to determine the winner of an election based on preference ballots. These methods include the Instant Runoff method, Borda count, Copeland’s method, and Approval Voting method.
• The plurality method is commonly used in the United States, where the candidate with the most first-preference votes wins, regardless of whether they have an absolute majority. This method can result in a winner who does not have a majority (over [latex]50\%[/latex]) of the votes.
• The plurality method can be problematic in elections with three or more choices, as it may not accurately reflect the overall preferences of the voters. A candidate can win with a plurality of votes but still be less preferred by the majority when compared to other candidates in one-to-one matchups.
• The Condorcet Winner is a candidate who would win in a one-to-one comparison against every other candidate. This criterion suggests that if such a candidate exists, they should be the winner of the election. However, the plurality method does not always result in the Condorcet Winner being elected.
• Insincere voting occurs when voters cast ballots counter to their actual preferences for strategic reasons, especially in situations where similar candidates may split the vote. This strategy is used to prevent a less preferred candidate from winning due to vote splitting among similar candidates.
• 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 is a voting method where points are assigned to candidates based on their ranking on each ballot. The last choice gets 1 point, the second-to-last gets 2 points, and so on, with the highest-ranked choice receiving the most points. The points from all ballots are totaled, and the candidate with the highest point total wins the election.
• The Borda count method can sometimes result in a candidate winning without having a majority of first-place votes. This situation can violate the majority criterion, which states that if a choice has a majority of first-place votes, it should be the winner.
• The Borda count can also violate the Condorcet Criterion, as it may not select a candidate who would be preferred in head-to-head comparisons against all other candidates.
• The Borda count is described as a consensus-based voting system because it considers every voter’s entire ranking to determine the outcome. This approach can sometimes choose a more broadly acceptable option over the one with majority support.
• Copeland’s Method focuses on pairwise comparisons between candidates, awarding points based on which candidate is more preferred in each comparison. The candidate with the most points, indicating the most pairwise wins, is declared the winner.
• Copeland’s Method satisfies the Condorcet Criterion, as it directly compares each candidate against every other candidate. However, Copeland’s Method can result in ties and may not always select a Condorcet Winner if one does not exist.
• The IIA Criterion states that the removal of a non-winning choice should not change the winner of the election. Copeland’s Method can violate the IIA Criterion, as shown in examples where the removal of a candidate changes the outcome of the election.
• Approval Voting allows voters to mark all choices they find acceptable, with the option receiving the most approval declared the winner. This method can be more appropriate in some decision-making scenarios where ranking candidates is not feasible.
• Approval Voting can violate the majority criterion, as it may result in the election of a candidate who is not the majority winner. The method is susceptible to strategic insincere voting, where voters do not vote for their true preference to increase the chances of their preferred choice winning.
• Arrow’s Impossibility Theorem states that no voting method can satisfy all fairness criteria simultaneously. This theorem highlights the inherent limitations and complexities in designing a completely fair voting system.
• Condorcet’s Voting Paradox demonstrates a scenario where voting preferences are not transitive. For example, if A is preferred over B, and B over C, it does not necessarily mean A is preferred over C. This paradox shows that no matter whom we choose as the winner, a significant portion of voters would prefer someone else.
• Due to the impossibility of a totally fair method, various voting systems like Plurality, Instant Runoff Voting (IRV), Borda Count, and Copeland’s Method are used in different contexts. The decision on which voting method to use is often based on the context and the specific requirements of the election, considering factors like the number of candidates, the nature of the electorate, and the desired outcomes of the voting process.
• Apportionment is the process of dividing a fixed number of things, like legislative seats, among groups of different sizes, typically based on population. In politics, apportionment is crucial for fair representation, determining how many elected officials represent a given area.
• The apportionment process must adhere to certain rules: only whole numbers can be used, all items must be divided, each group must receive at least one item, and the distribution should be approximately proportional to the population.
• Different methods of apportionment include Hamilton’s method, Jefferson’s method, Webster’s method, Huntington-Hill method, and Lowndes’ method. These methods have been used at various times to apportion the U.S. Congress, with the current method being the Huntington-Hill method.
• Hamilton’s method involves determining a divisor (total population divided by the number of representatives) and then calculating each state’s quota (state population divided by the divisor). The lower quotas (whole numbers from the quotas) are summed, and any remaining representatives are assigned to states with the largest decimal parts of the quota.
• Hamilton’s method obeys the quota rule, ensuring that the final number of representatives a state gets is within one of its quota.
• Hamilton’s method is subject to several paradoxes, including the Alabama Paradox, New States Paradox, and Population Paradox. These paradoxes demonstrate situations where increasing the total number of representatives or changing state populations can lead to counterintuitive results, like a state losing a representative when the total number of representatives increases.
• Jefferson’s method for apportionment starts similarly to Hamilton’s method but adjusts the divisor if the initial total is less than the required total. The method involves reducing the divisor and recalculating quotas until the total number of representatives matches the required total.
• Jefferson’s method tends to favor larger states and does not always follow the quota rule, which states that the final number of representatives should be within one of the state’s quota.
• Webster’s method, like Jefferson’s, involves adjusting the divisor but starts by rounding quotas to the nearest whole number instead of truncating them. If the initial total of representatives does not match the required total, the divisor is adjusted (either increased or decreased) and quotas are recalculated.
• Webster’s method is less biased towards larger states compared to Jefferson’s method and follows the quota rule more often, though not always.
• The Balinski-Young Impossibility Theorem states that any apportionment method that always follows the quota rule will be subject to paradoxes like the Alabama, New States, or Population paradoxes. This theorem implies a trade-off in apportionment methods: avoiding paradoxes may require giving up the guarantee of following the quota rule.
• The Huntington-Hill Method of apportionment is the current method used in the U.S. Congress, focusing on minimizing the percent differences in representation. It starts similarly to Webster’s method but uses the geometric mean to determine whether to round up or down the quota. The geometric mean of the lower quota and one value higher is calculated, and the quota is rounded up if it’s larger than this mean, or down if smaller. The method continues adjusting the divisor until the total number of representatives matches the required total.
• Lowndes’ Method, proposed by William Lowndes, was designed to favor smaller states, though it has never been used to apportion Congress. Like Hamilton’s method, it follows the quota rule and drops the decimal parts of quotas. The remaining representatives are assigned based on the ratio of the decimal part of each state’s quota to its whole number part, favoring states where an additional representative would make a more significant impact. This method emphasizes the value of additional representatives more in smaller states than in larger ones.
• In most states, the apportionment process involves drawing legislative districts so that each legislator represents approximately the same number of constituents. This process aims to ensure fair representation, where a small city may have several representatives, while a large rural region may be represented by one legislator.
• When populations change, it becomes necessary to redistrict the regions each legislator represents. This is also true for federal legislators.
• The redistricting process is typically done by the legislature itself, which can lead to the practice of gerrymandering.
• Gerrymandering occurs when districts are drawn based on the political affiliation of the constituents to the advantage of those drawing the boundary. It involves manipulating district boundaries to create an electoral advantage for a particular party or group.
• Weighted voting is a system where each vote has a weight attached to it, often proportional to the amount of shares owned in a corporate setting or other factors in different contexts. 
• In weighted voting, the power each voter has in influencing the outcome is a key consideration.
• The weighted voting system is represented in a shorthand form, where ‘[latex]q[/latex]‘ is the quota (minimum weight needed for a proposal to be approved) and ‘[latex]w[/latex]‘ represents the weight for each player.
• The quota must be more than half the total number of votes to ensure that only one decision can reach the quota at a time, preventing conflicting outcomes.
• The quota cannot be larger than the total number of votes, as this would make it impossible for any group of voters to reach the quota, rendering decision-making ineffective.
• Weighted voting systems can be applied to various scenarios, such as corporate decisions, committee voting, and other group decision-making processes. The method involves determining the quota and assigning weights to each voter based on their stake or representation.
• The Banzhaf Power Index is a numerical method to assess the power of individual voters in a weighted voting situation, originally created by Lionel Penrose and reintroduced by John Banzhaf.
• To calculate the Banzhaf Power Index, list all winning coalitions, identify critical players in each coalition, count how many times each player is critical, and convert these counts to fractions or decimals.
• This index measures a player’s ability to influence the outcome of the vote, with a higher index indicating greater power or influence in the voting system.
• The Banzhaf Power Index can reveal interesting dynamics in parliamentary governments and other decision-making bodies where forming coalitions is essential.
• The Banzhaf Power Index helps to quantify the influence of each district or player in a weighted voting system, highlighting disparities in power among them.
• The Electoral College system in the United States is an example of a weighted voting system where states are the players, and the Banzhaf Power Index can be used to analyze the power distribution among states. Due to the complexity of the Electoral College, the power index for this system often requires computation by a computer using approximation techniques.
• The Shapley-Shubik power index is a method for calculating the power of individual voters in a weighted voting system. It was introduced by economists Lloyd Shapley and Martin Shubik. This index focuses on the likelihood of a player being pivotal in a sequential coalition, where the order of joining the coalition is considered important.
• Sequential coalitions list players in the order they join the coalition. The order is significant as it reflects the sequence of decision-making or alliance formation.
• A pivotal player in a sequential coalition is the one who changes the coalition from a losing to a winning one. There can only be one pivotal player in any sequential coalition.
• To calculate the Shapley-Shubik power index, list all possible sequential coalitions, identify the pivotal player in each, count how many times each player is pivotal, and convert these counts to fractions or decimals.
• The total number of sequential coalitions is determined by the factorial of the number of players, represented as [latex]N![/latex], where [latex]N[/latex] is the number of players. The factorial, denoted as [latex]N![/latex] , is the product of all positive whole numbers up to [latex]N[/latex] and is used to calculate the total number of sequential coalitions in a voting system.
• Computing the Shapley-Shubik power index by hand can be complex for larger voting systems due to the large number of sequential coalitions.

Glossary

apportionment

the process of allocating seats in a legislative body, such as a parliament or congress, based on a specific formula that usually takes into account population size or other demographic factors

approval voting

where ballot asks you to mark all choices that you find acceptable and the option with the most approval is the winner

Borda count

a voting method where points are assigned to a candidate based on their ranking

coalition

any group of players voting the same way

Condorcet Winner

a candidate who would win in a one-to-one comparison against every other candidate in a multi-candidate election

dictator

if one’s weight is equal to or greater than the quota

dummy

if one’s vote is never essential for a group to reach quota

fairness criteria

statements that seem like they should be true in a fair election

gerrymandering

when districts are drawn based on the political affiliation of the constituents to the advantage of those drawing the boundary

instant runoff voting (IRV) method

voters rank their preferred candidates or options in order of preference. In each round of counting, the candidate with the fewest first-place votes is eliminated, and the votes for that candidate are redistributed to the voters’ next choice. This process continues until one candidate obtains an absolute majority of votes, usually exceeding [latex]50\%[/latex], and is declared the winner of the election.

majority criterion

if a choice has a majority of first-place votes, that choice should be the winner

monotonicity criterion

If voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning

plurality method

voters choose their preferred candidate or option, and the candidate with the most votes, even if it’s not an absolute majority, wins the election

preference ballot

a ballot in which the voter ranks the choices in order of preference

preference schedule

a table used to organize how people rank different options or choices in an election

quota rule

the final number of representatives a state gets should be within one of that state’s quota

veto power

if one’s support is necessary for the quota to be reached