Choosing one officeholder and composing a legislature are different collective decisions.
By the end: Separate institutional objectives, ballot rules, counting rules, and evidence about political outcomes.
This course assumes arithmetic and percentages, not political science, probability, or prior knowledge of any country’s electoral law. It introduces the institutions and counting rules needed to follow expert discussions. Every numerical electorate is hypothetical. No example is a forecast, voting instruction for a real election, or endorsement of a party or electoral reform.
Start with the institution
A single office has one winner. A legislature divides many seats among representatives who may need to form a governing coalition. A rule that selects a locally popular candidate does not automatically produce a legislature whose party shares match national vote shares. Before comparing rules, identify what is being elected and which decisions that body can make.
Representation can mean a connection to a geographic constituency, proportional party influence, inclusion of social groups, responsiveness to policy preferences, or the ability to remove an incumbent. These objectives can conflict. A mathematical calculation can establish how a rule allocates seats, but cannot decide how much weight you should give each objective.
One office and a share of a chamber
Suppose one hundred people choose a mayor from A, B, and C. The institution needs one person, so allocating forty percent of the office to A is not an available result. Now suppose the same people elect ten legislators. Four A representatives, three B representatives, and three C representatives is a possible composition. The first question asks which candidate should hold an indivisible office. The second asks how influence should be divided among several officeholders. Neither question determines how the executive is selected afterward.
Consider a ten-seat chamber containing A four, B three, and C three. Under a stipulated rule requiring more than half of all seats for a decision, six votes suffice. A cannot pass the decision alone despite being the largest party. A and B have seven seats, A and C have seven, and B and C have six. All three combinations meet the numerical requirement. Arithmetic identifies feasible majority combinations, but it does not tell us whether their members agree on a policy or will cooperate.
A largest party is not a majority coalition
Quantitative seat units for a ten-member chamber. Party cohesion and cooperation are not assumed. Decision thresholds are stipulated separately from the election count.
Change the decision requirement to eight of ten seats while leaving the election result untouched. None of those two-party combinations now suffices. All three parties together would have ten. This contrast locates an institutional assumption outside the election count. A statement about who can govern needs rules for appointment, confidence, or executive powers, as applicable. A statement about who can enact a particular decision needs its voting requirement. Winning the largest legislative delegation is not, by itself, winning an independently elected executive office.
Party shares also leave individual representation unresolved. Imagine that A's four seats come from an ordered list. A party-controlled list and a voter-influenced list can return the same number of A legislators but different people. To judge candidate choice, inspect the nomination and selection procedures rather than the party totals alone. To judge descriptive representation, inspect who actually holds those seats and which population is being compared. A chamber can match party vote shares while differing from its population in age, occupation, or other characteristics.
Use the seat units below to separate these questions on paper. First circle a combination that reaches six seats. Then identify a combination that fails the eight-seat requirement even though it exceeds half. Finally, replace one A legislator with another member of A. The party arithmetic stays fixed, while the identity of a representative changes. This exercise isolates the level at which each claim operates. It supplies no assumption that legislators always vote together or that party labels fully describe their policy preferences.
A comparison therefore needs an objective stated in operational terms. Matching party vote shares requires vote and seat totals. Giving a locality an identifiable representative requires a constituency assignment. Letting voters reject a particular incumbent requires information about candidate selection and replacement. These objectives can overlap, but evidence for one does not establish the others. The selected representation discussion in International IDEA's handbook distinguishes several such aims. Its design framework helps organize questions here; it does not turn any chosen aim into a universally binding definition of fairness.
Different representation claims need different observations
Claim to examine
Required observation
What party seat totals omit
Party proportionality
Party votes and final party seats
Which individuals hold the seats
Local representation
Constituency assignments and elected members
Whether a locality has its own representative
Candidate choice
Nomination, ballot, and replacement procedures
Who selected each elected person
Descriptive representation
Member characteristics and a named comparison population
How membership compares with that population
These are distinct objects of comparison. Matching a party share does not establish the other forms of representation.
Separate the components
An electoral system specifies who is eligible, which choices appear on the ballot, how voters may express them, how votes are counted, and how seats are distributed across districts or tiers. District magnitude means the number of seats elected in one district, not its area or population. A five-member district has magnitude five even if its population is smaller than another district.
Other institutions matter too. Candidate nomination, campaign rules, election administration, legislative powers, executive selection, and the franchise can change the meaning of a result. Voting rules are neither the whole political system nor a sufficient test of whether an election is free, competitive, or credibly administered. Party proportionality also does not guarantee that representatives reflect the electorate’s social backgrounds. Candidate nomination, list order, and representation quotas can change who is elected without changing party seat shares. Whether representatives advance a group’s interests is a further empirical question.
Specify what the ballot tells the count
Begin a count specification with its input. A single mark beside A reports a choice of A among the candidates offered. It does not say whether that voter prefers B to C, regards either as acceptable, or thinks A only slightly better. A complete ranking A above C above B supplies an order for every pair. It still supplies no numerical distances between them. Two people can submit that identical ranking even if one feels almost indifferent between A and C and the other strongly favors A.
The complete profile used later stipulates forty A above C above B ballots, thirty-five B above C above A ballots, and twenty-five C above B above A ballots. Its first choices are forty, thirty-five, and twenty-five. Another complete profile could keep those first choices but change the later ranks. A one-choice count could not distinguish them. A pairwise comparison or transfer count might. This is why replacing a ballot format cannot be modeled merely by relabeling the same three aggregate first-choice numbers.
Approval requires another piece of information. Someone who ranks A above C above B might approve A alone or approve both A and C. Both reports can respect the same order. A cutoff specifying which candidates count as acceptable distinguishes them. In the stipulated complete profile, approving only first choices produces forty, thirty-five, and twenty-five approvals. Approving the top two produces forty, sixty, and one hundred. That difference follows from an added reporting policy, not from a discovery that the original rankings secretly contained an acceptance boundary.
The same order permits different reports
Report or policy
Example or A / B / C totals
Information supplied or added
One candidate mark
A
Does not order B against C
Complete ranking
A > C > B
Orders every pair; no intensity or acceptance cutoff
First-only approval policy
40 / 35 / 25 approvals
Adds a cutoff after first place
Top-two approval policy
40 / 60 / 100 approvals
Adds a cutoff after second place
Scores 5 / 4 / 0 by rank
200 / 275 / 425 score units
Adds scores, not inferred utility distances
Aggregate rows use 40 A>C>B, 35 B>C>A, and 25 C>B>A ballots. There are 100 people. The top-two policy produces 200 approvals, not 200 voters.
Scores require still more assumptions about how a voter uses a numerical scale. Assigning five to first place, four to second, and zero to last gives totals A two hundred, B two hundred seventy-five, and C four hundred twenty-five in this profile. Those totals follow from the specified conversion. The ordinal ballots alone do not identify those numbers or establish comparable welfare units across people. Another score assignment could preserve every order while changing the totals. A calculation must identify the reported quantity before comparing its aggregate.
Next specify how many seats the district fills. A five-seat district can occupy a small area or a large one. Its magnitude is five in either case. Population per seat answers a different question. With five hundred thousand residents and five seats, that ratio is one hundred thousand residents per seat. One million residents and one seat gives one million per seat. Changing magnitude does not, without further population information, tell us whether representation per resident became more equal across districts or whether the district boundaries changed.
Before following any later model, annotate its input and output yourself. Write whether the input is candidate marks, complete ranks, approvals, or party votes. Write whether the output is one winner or a vector of whole seat counts. Check the missing-information column in the comparison below before trying to reuse an input for another rule. In particular, optional incomplete rankings need an explicit treatment of omissions. They must not enter a calculation requiring complete comparisons through an unstated assumption about how unlisted candidates compare.
Seats per district and residents per seat
District
Residents
Magnitude, in seats
Residents per seat
X
1,000,000
1
1,000,000 ÷ 1 = 1,000,000
Y
500,000
5
500,000 ÷ 5 = 100,000
Magnitude counts seats filled within a district. These stipulated resident totals permit a separate population-per-seat comparison; turnout totals would not replace them.
Work through the mechanism
Seat share as a percentage
s is the party’s seat count; S is the whole assembly’s seat count. Vote share instead uses the relevant vote counts. If the assembly expands, update S.
Question
Relevant object
Who holds one office?
One candidate
How is legislative influence divided?
Seat allocation across representatives
Can voters remove a particular person?
Candidate choice and accountability rules
Was the election administered credibly?
Registration, access, counting, audit, and dispute procedures
These are related questions, not interchangeable definitions of representation.
Keep the denominator attached to the claim
A share divides a specified numerator by a specified total. Suppose A receives forty of one hundred valid votes and six of ten seats. Its vote share is forty percent; its seat share is sixty percent. Subtracting gives a twenty-percentage-point seat bonus relative to those votes. Dividing sixty by forty instead gives a ratio of one and a half. The gap and the ratio describe the same allocation differently. Neither calculation tells us which people voted, whether the seats were legally allocated, or whether the bonus is desirable.
Now distinguish the people who could participate from the ballots included in the count. Imagine two hundred eligible people, one hundred twenty returned ballots, twenty invalid ballots, and one hundred valid ballots. A's forty valid votes are forty percent of valid ballots, one third of returned ballots, and twenty percent of eligible people. These percentages are compatible because their denominators differ. Calling the first number support from forty percent of the eligible population would silently replace a count of valid choices with a count of eligible persons.
Transfers can create another denominator without creating or destroying physical papers. In the separate optional-ranking example, twenty-five C-only ballots exhaust when C leaves the count. Seventy-five ballots continue. A's forty then exceed half of continuing votes because half of seventy-five is thirty-seven and a half. Forty remains below half of the original hundred. Exhaustion means that the stipulated instructions no longer provide an active choice. It does not mean those people disappeared, became ineligible, or submitted invalid papers at the start of this hypothetical count.
Forty votes, four different support fractions
Named base
A numerator
Denominator
A share
Valid ballots
40 votes
100 valid ballots
40%
Returned ballots
40 votes
120 returned ballots
33⅓%
Eligible people
40 voters choosing A
200 eligible people
20%
Continuing ballots in a separate optional-ranking example
40 active votes
75 continuing ballots
53⅓%
The first three rows share one hypothetical participation record. The last row is a separate count with 100 original valid ballots and 25 exhausted C-only ballots. Exhausted is not the same as initially invalid.
Multi-seat transfers require a further distinction between papers and current value. In the bounded two-seat example, fifty initially unit-value A ballots retain thirty-four units for A and transfer sixteen units. Each paper transfers 0.32 units and retains 0.68. There are still fifty papers in that group. Counting fifty retained full votes plus sixteen transferred units would create value that the model never supplied. Its accounting identity tracks retained and continuing value together, while its declared stopping rule ends redistribution once both available seats have been filled.
Seat denominators can change too. A ten-seat target of four, three, three would give A forty percent of seats. If the stated mixed-system policy retains five A district wins and gives B and C three each, the final assembly contains eleven seats. A then has five elevenths, about 45.45 percent, not fifty percent. This is retained overhang without leveling. Keeping the old ten-seat denominator would falsely report a distribution totaling more than one hundred percent and conceal the remaining gap from the original party vote shares.
For a practical check, cover the denominator labels in the table and try to reconstruct each one from the sentence being tested. A claim about an active transfer round needs continuing votes. A claim about party proportionality needs the relevant party votes and the final assembly seats. A claim about population equality needs population counts, not merely turnout. Restore the labels and reject any comparison that changed its base without saying so. A percentage is informative only after its counted units and reference population are clear.
Two accounts whose units must not be mixed
Account
Parts
Correct total or share
Initial A ballot papers in bounded STV
50 papers, each initially worth 1
50 initial value units
Value retained and transferred from those papers
34 retained + 16 transferred
50 value units, not 66
Per A paper after this initial surplus
0.68 retained + 0.32 transferred
1 initial value unit
Ten-seat target
A 4, B 3, C 3
A share 4/10 = 40%
Retained overhang without leveling
A 5, B 3, C 3
11 seats; A share 5/11 ≈ 45.45%
The STV account uses the stated one-surplus, two-seat model and stops once seats are filled. The mixed-system account retains district wins without leveling. These are separate teaching models, not statutory procedures.
Keep the comparison honest
The numerical examples use invented candidates and parties A, B, C, and D. Unless stated otherwise, votes are valid, counts are exact, and ties use a declared alphabetical priority. Holding ballots fixed isolates the counting rule. It does not predict how people would vote, which candidates would run, or which coalitions would form under another rule.
A real comparison needs the current legal specification and evidence about behavior. Historical examples in the optional electoral-design handbook are not a guide to current law. The lessons explain mechanisms without endorsing a party, a jurisdiction’s arrangements, or a uniquely fair system.
Separate a result from an explanation or judgment
A mechanical claim states what a specified procedure does to specified inputs. If the ballots, district assignments, and count rules are complete, you can reproduce the output and examine intermediate steps. A behavioral claim instead concerns how people respond to the rule. Voters might coordinate, candidates might withdraw, or parties might change nominations. Those responses alter the inputs. Holding ballots fixed is useful precisely because it excludes them. It estimates no response size and supplies no forecast of a future election under a changed institution.
Take the statement that a new rule would produce more proportional seats and more durable governments. The first part may be checked mechanically for the stipulated votes. The second needs a definition of durability, such as duration until a government changes, and evidence about actual institutions and political behavior. Even an observed association between electoral rules and government duration is not automatically causal. Countries may differ in constitutional powers, party organization, economic conditions, or reform histories that affect both the adopted rule and the measured outcome.
An empirical comparison should therefore describe its counterfactual. What would have happened to the same polity at the same time without the reform? We cannot directly observe both histories. Comparing earlier and later elections leaves other changes unresolved. Comparing different places introduces differences between places. Evidence can still be useful, but its design and assumptions determine which explanation it supports. A fixed-ballot simulation answers a narrower counterfactual, asking how those particular reports would count under another fully specified procedure, without assuming that people would submit them there.
Holding reports fixed answers a narrower question
A schematic comparison of research questions, not a quantitative estimate. Changing the count on fixed reports does not predict how a reform changes those reports.
A normative claim assigns importance to outcomes or procedures. Someone may value proportional party shares, another may prioritize an identifiable local representative, and both may care about candidate choice. Calculating a seat gap does not determine the weights among those objectives. A legal claim asks a different question again, whether the actual specification and administration comply with the applicable law. The handbook's historical institutional examples cannot establish current statutory requirements. That task needs the relevant current legal text and evidence about the election being assessed.
Try auditing the sentence, 'A received forty percent of votes and sixty percent of seats, so the system must be replaced.' The first clause states two quantities that need their denominators and source. Explaining the gap requires the allocation rule and district results. The recommendation adds a criterion and a comparison with feasible alternatives. Nothing in subtraction alone supplies those additions. Conversely, refusing to infer the recommendation does not erase the twenty-point gap. You can establish the arithmetic clearly while leaving the policy judgment open.
Use the counterfactual comparison below to write two separate conclusions. One should say exactly what changed when only the counting procedure changed. The other should identify what evidence would be needed before expecting that change in practice. This prevents a correct calculation from carrying an unsupported prediction. It also makes disagreement more precise. People can agree on the vote-to-seat calculation and disagree about objectives, or agree on objectives and disagree about likely behavior. Those are different disputes that call for different evidence or explicit value judgments.
What follows from forty percent of votes and sixty percent of seats?
Statement
Kind of claim
Additional requirement
The seat-share gap is +20 percentage points
Arithmetic
Named vote and seat denominators
This count follows the specified formula
Mechanical
Complete rule and reproducible intermediate steps
The result complied with applicable law
Legal
Current applicable law and procedural evidence
A reform will change future participation
Behavioral and causal
Evidence and an explicit counterfactual design
The system should be replaced
Normative recommendation
Criteria, weights, and feasible alternatives
A calculation can support the first row without establishing the other rows. A recommendation needs explicit objectives as well as evidence.
Exercises
Two different objectives
A party gets 40% of national votes and wins 60% of district seats. Is the arithmetic alone enough to call the election illegitimate?
Hint
Distinguish disproportionality from the legal rule and the broader democratic process.
Worked answer
Observation
What follows
40% votes, 60% seats
A 20-percentage-point seat bonus
Counting rule
Needed to explain the conversion
Legitimacy judgment
Needs additional criteria and evidence
A measurable gap and a normative conclusion are different claims.
No. It establishes a gap between vote share and seat share. Whether the rule is desirable requires criteria; whether the election followed law requires legal and procedural evidence; legitimacy also involves participation and political rights. You can criticize disproportionality without pretending that one statistic settles every question.
Magnitude is not size on a map
District X elects one representative from one million residents. District Y elects five from half a million. Which has larger district magnitude?
Hint
Count seats elected within the district.
Worked answer
District
Magnitude
Residents per seat
X
1
1,000,000
Y
5
100,000
Magnitude and population per representative measure different properties.
Y has magnitude five; X has magnitude one. Population per seat is a separate question: one million for X and one hundred thousand for Y. A comparison of population equality must use the appropriate legal population base, not confuse magnitude with population.
Sources and further reading
The explanations, counts, and worked answers are complete without these optional links. Reference annotations identify the selected material reviewed. Historical country examples are not statements of current law, and the bounded models are not certified election-counting software. Mechanical results, behavioral evidence, and judgments about desirable representation remain separate.
Selected explanations of system families, district magnitude, thresholds, proportional formulas, mixed systems, and design objectives. Country examples and legal thresholds are historical, not statements of current law.
Overview of how magnitude and other design choices affect the translation of votes into seats. Not evidence that one institutional objective is universally preferable.