Voting methods and apportionment are tools for turning individual preferences or population counts into fair group decisions. This cheat sheet helps students compare election systems, spot when methods give different winners, and understand how seats are divided among states or groups. It is useful for applied math, civics, statistics, and real-world decision making.
The goal is to make each method easy to follow with clear steps and formulas.
The main voting methods include plurality, instant runoff, Borda count, Condorcet comparisons, and approval voting. Apportionment methods start with a standard divisor, calculate quotas, and assign whole-number seats. Hamilton's method uses lower quotas and largest remainders, while divisor methods adjust the divisor until the total number of seats is correct.
Fairness criteria help evaluate whether a method behaves reasonably in different situations.
Key Facts
- In plurality voting, the candidate with the most first-place votes wins, even if that candidate does not have a majority.
- A majority means more than half of the votes, so a candidate needs votes greater than total votes / 2.
- In Borda count with n candidates, a first-place ranking earns n - 1 points, second place earns n - 2 points, and last place earns 0 points.
- In a Condorcet comparison, compare two candidates at a time, and the candidate preferred by more voters wins that head-to-head matchup.
- In Hamilton's method, standard divisor = total population / total seats, and standard quota = group population / standard divisor.
- In Hamilton's method, each group first receives its lower quota, then leftover seats go to the groups with the largest fractional remainders.
- In Jefferson's divisor method, use modified quotas and round each quota down until the rounded total equals the number of seats.
- In Webster's divisor method, use modified quotas and round each quota to the nearest whole number until the rounded total equals the number of seats.
Vocabulary
- Preference schedule
- A table that shows how many voters ranked the candidates in each possible order.
- Plurality winner
- The candidate who receives the greatest number of first-place votes.
- Majority
- More than half of the total votes, calculated as a number greater than total votes / 2.
- Condorcet winner
- A candidate who beats every other candidate in one-on-one head-to-head comparisons.
- Standard quota
- The exact number of seats a group should receive before rounding, found by group population / standard divisor.
- Modified divisor
- An adjusted divisor used in divisor methods so rounded quotas add to the required number of seats.
Common Mistakes to Avoid
- Confusing plurality with majority is wrong because the plurality winner only needs the most first-place votes, not more than half of all votes.
- Giving the wrong Borda points is wrong because the point scale depends on the number of candidates, with n - 1 points for first place and 0 points for last place.
- Skipping head-to-head comparisons in Condorcet voting is wrong because the method requires each candidate to be compared directly against every other candidate.
- Rounding standard quotas immediately in Hamilton's method is wrong because Hamilton first uses lower quotas, then assigns remaining seats by largest remainders.
- Using the same divisor for every divisor method without checking the seat total is wrong because Jefferson, Webster, and other divisor methods may require a modified divisor.
Practice Questions
- 1 In an election with 100 voters, Candidate A gets 42 first-place votes, Candidate B gets 35, and Candidate C gets 23. Who is the plurality winner, and does that candidate have a majority?
- 2 Four candidates are ranked in a Borda count election. How many points should a candidate receive for first place, second place, third place, and fourth place?
- 3 A country has 50 seats and a total population of 2,000,000. Region X has population 360,000. Find the standard divisor and Region X's standard quota.
- 4 Explain why two fair-looking voting methods can produce different winners from the same preference schedule.
Understanding Voting Methods & Apportionment
The ballot format controls what information an election can use. A single-choice ballot records only each voter’s top pick. A ranked ballot records a full order of preference, so later choices can matter if an early choice is eliminated.
In instant runoff, the last-place candidate is removed after a count, then those ballots transfer to the next remaining choice. A ballot can become exhausted when it lists no candidate still in the race.
This matters because the final winner may have a majority of active ballots but not of every ballot originally cast. Voters may also vote strategically when they fear that their sincere favorite has little chance.
Ranking methods reward different kinds of support. Borda count gives credit for being placed near the top by many voters, which can favor a broadly acceptable compromise candidate. It may not favor the candidate with the most first-place support.
Condorcet reasoning tests whether one candidate can defeat every rival in separate head-to-head contests. Sometimes no such candidate exists. Preferences can form a cycle where one candidate beats a second, the second beats a third, and the third beats the first.
Approval voting asks voters to decide which candidates are acceptable rather than rank them. Its result depends heavily on where voters draw that approval line.
Apportionment has a different starting point because people cannot be divided into fractions of a seat. A quota describes the share of seats a population would receive in a perfectly divisible system. The hard part is converting those shares into whole seats while keeping the total fixed.
Hamilton’s method pays close attention to fractional leftovers. This can produce surprising changes when the total number of seats changes or when a population changes slightly. These outcomes are called paradoxes.
Divisor methods avoid some of these problems, but their rounding rules create their own patterns. Jefferson tends to favor larger populations because every modified quota is rounded down. Webster aims for more balanced rounding, though it can still give a result that differs from Hamilton.
When working through examples, make an organized table before making any decision. For elections, list each ballot group, its number of voters, and the ranking or approvals it gives. For apportionment, list every population, quota, initial seat assignment, and remainder or modified quota.
Check totals after every round. A useful final step is to compare representation ratios. Divide each group’s population by its assigned seats and see how similar those values are.
No method makes every fairness goal true at once. The important skill is identifying which goal a method protects and which tradeoff it accepts.