A small party can fail to win a seat without any formal exclusion rule.
By the end: Explain eligibility thresholds, effective barriers, and the denominator used after exclusions.
Eligibility and allocation are different stages
A legal threshold can exclude a party from a seat-allocation tier unless it satisfies a specified condition. The percentage denominator and any exemptions are part of the law. This lab uses a simple national percentage of all valid party votes and no exemption. A party exactly at the threshold qualifies.
With votes 46,30,20,4, a 5% threshold excludes D even though D’s votes were valid. Those votes remain part of the original vote total. They do not become votes for A, B, or C; the eligible parties compete for all seats under the specified allocation procedure.
Test eligibility before comparing quotients
A threshold answers whether a list may compete in an allocation. The divisor formula answers how eligible lists divide the available seats. These are separate operations, even if a results table displays only their combined effect. In this lesson, imagine one district electing a legislature or a delegation within it, not one officeholder. The district has four invented lists with 46, 30, 20, and 4 valid party votes. These hundred vote units describe ballots cast for lists, not population counts or candidate rankings.
Set the threshold at five percent of all valid party votes. A list qualifies when its votes multiplied by one hundred are at least the threshold percentage multiplied by the total valid votes. D fails because four hundred is less than five hundred. A, B, and C pass. D's four ballots remain valid historical observations, but D cannot receive a seat in this allocation. The eligibility test does not transfer those ballots to another list or ask whom their voters would prefer instead.
Equality matters at the boundary. At a four percent threshold, D qualifies because four hundred equals four hundred. At five percent it fails. The available slider changes thresholds in whole percentage points, so it directly exposes this contrast without estimating anything between the two settings. A report that rounds a party's measured share to four percent might conceal a real boundary issue. Our constructed totals have no such uncertainty, and eligibility uses the exact totals rather than a rounded label.
D crosses an eligibility boundary, not a rounding boundary
Threshold of all valid votes
Exact test for D
Eligibility
0%
4 × 100 ≥ 0 × 100
Eligible
4%
4 × 100 = 4 × 100
Eligible at equality
5%
4 × 100 < 5 × 100
Ineligible
Constructed votes A/B/C/D = 46/30/20/4. Tests use all 100 valid party votes. No exemptions apply; passing this test does not promise a seat.
Now compare the two useful denominators after exclusion. A still received 46 of all 100 valid party votes. Among votes cast for eligible parties, it received 46 of 96, about 47.92 percent. The latter fraction describes A's relative size within the eligible pool. It does not mean that additional voters chose A. Divisor allocation can use the original eligible vote counts directly, because multiplying all of them by the same positive normalization factor leaves every quotient comparison unchanged.
For example, compare A's next quotient with B's when their divisors are three and one. Using original counts gives 46 divided by three against 30. Using eligible-vote proportions gives 46 divided by 288 against 30 divided by 96. Both comparisons favor B, since normalization divides both original quotients by 96. This equivalence explains why excluding D requires an eligibility decision but not a fictional redistribution of D's votes. Keep the denominator visible when describing shares, even when the allocation calculation does not need normalization.
Before reading the final seats, write two predictions for D. With three seats and no threshold, D is eligible but might lose every quotient contest. With twenty seats and a five percent threshold, it is ineligible before any contest occurs. These statements identify different reasons for zero seats. Check the lab's eligibility status as well as its seat count. In a real investigation, request the threshold denominator, exemptions, and tier to which it applies before treating the percentage printed beside a rule's name as a complete specification.
Exclusion changes the comparison pool, not the ballots
Quantity after D is excluded
Numerator / denominator
Meaning
A share of all valid votes
46 / 100 = 46%
Original party-vote share
A share of eligible votes
46 / 96 ≈ 47.92%
Relative size among eligible lists
A next quotient, divisor 3
46 / 3; normalized 46 / 288
Both normalized claims divide by 96
B next quotient, divisor 1
30 / 1; normalized 30 / 96
B beats A under either scaling
The quotient comparison holds the hypothetical divisors fixed. Normalization rescales every eligible quotient equally; it does not move D ballots to A, B, or C.
Whole seats create an effective barrier
Even without a legal threshold, a four-percent party may not obtain one of three seats. With twenty seats it may win one. This comes from finite seat size and competition among quotients. District magnitude can therefore matter as much as the formal eligibility percentage.
The effective barrier is not universally one divided by district magnitude. Allocation formula, vote distribution, fragmentation, and geographic concentration matter. A rule applied separately in many small districts is not the same as applying it once to a national vote total.
Find the seat that a small list cannot reach
A whole seat occupies one third of a three-seat delegation but only one twentieth of a twenty-seat delegation. These are seat shares of about 33.33 percent and five percent. They are not legal vote thresholds. A list can win a seat with a vote share below one seat's share, because whole-seat rounding compares competing claims. Conversely, eligibility alone promises no seat. To understand D's result, examine the last winning quotient and its strongest unsuccessful competitor rather than equating seat size with a universal minimum vote share.
With three seats, no threshold, and unmodified Sainte-Laguë, the first quotients are 46, 30, 20, and 4. A wins first. A's next divisor becomes three, so its next quotient is 46 divided by three, about 15.33. B then wins with 30, and its next quotient becomes ten. C wins the third seat with 20. D's first quotient of four never approaches these winning claims. The final allocation is one seat each for A, B, and C, with none for eligible D.
Three awards with four eligible lists
Seat to award
Next A / B / C / D quotients
Award
1
46 / 1; 30 / 1; 20 / 1; 4 / 1
A
2
46 / 3; 30 / 1; 20 / 1; 4 / 1
B
3
46 / 3; 30 / 3; 20 / 1; 4 / 1
C
Final seats
1 / 1 / 1 / 0
D eligible, no winning claim
Unmodified Sainte-Laguë and no threshold. Fractions are exact quotient expressions, not shares of votes. Only the awarded list advances its odd divisor.
Extending the same contest to twenty seats gives later claims a chance. A's successive quotients fall as its divisors grow, and the same happens to B and C. D's first quotient stays four until D receives a seat. In this profile it wins the fourteenth award, after A has six seats, B four, and C three. Before the thirteenth award, their next quotients are 46 divided by thirteen, 30 divided by nine, and 20 divided by five. C and D both offer exactly four, so C receives that seat under the alphabetical tie rule.
Immediately after that tied contest, C's divisor rises to seven. D still offers four, above A's 46 divided by thirteen, B's 30 divided by nine, and C's 20 divided by seven. D therefore wins the fourteenth seat. This is an exact comparison, not a judgment based on rounded decimal labels. The complete twenty-seat result is 9, 6, 4, and 1. D now holds five percent of seats with four percent of votes, a small rounding gain rather than exact proportional representation.
Compare what the additional seventeen seats changed. They did not make D more eligible, because it was already eligible. They lowered the competing quotient claims far enough for D's first claim to win. This is why magnitude can create an effective barrier without any formal exclusion. It also explains why one example cannot provide a universal effective-threshold percentage. Change the other lists' votes or change the divisor sequence, and the winning boundary can move even while D's own share and the number of seats stay fixed.
Use the small-district preset, then move only magnitude upward with the threshold left at zero. Before reaching fourteen, predict that D remains eligible without a seat. At fourteen, verify its first award and then compare the twenty-seat endpoint. The bars use percentage of seats rather than a common raw-seat scale, so one D seat at fourteen and one at twenty have different lengths. Read the printed seat counts alongside the panel totals. The unchanged reference prevents a larger assembly from looking automatically more favorable simply because its raw counts are larger.
D waits through an exact tie before its first seat
Next award
Seats already A/B/C/D
Competing next quotients
Winner
13
6 / 4 / 2 / 0
46/13; 30/9; 20/5; 4/1
C: 20/5 = 4/1, alphabetical tie
14
6 / 4 / 3 / 0
46/13; 30/9; 20/7; 4/1
D: 4 exceeds every rival
After 20 awards
9 / 6 / 4 / 1
D holds 1 of 20 seats
D seat share 5%, vote share 4%
Same votes, no legal exclusion. At thirteen seats D has zero; at fourteen it has one. Exact rational comparison, not display rounding, decides the claims.
Work through the mechanism
Change magnitude and eligibility separately
Presets replace both settings. Each slider changes only its named setting. Votes remain 46 / 30 / 20 / 4. The reference is always twenty seats with no threshold; there is no playback.
A / B / C / D: 10 / 6 / 4 / 0. D is ineligible. Excluded vote share: 4%.
Both bar scales run from 0% to 100% of the panel’s seats. Panel totals and exact seat counts remain visible; displayed percentages are rounded. A party exactly at the threshold qualifies. These national-list examples use unmodified Sainte-Laguë, no exemptions, and no geographic tiers.
Setting
Seats A/B/C/D
D’s status
3 seats, no threshold
1 / 1 / 1 / 0
Eligible, no seat
20 seats, no threshold
9 / 6 / 4 / 1
Eligible, one seat
20 seats, 5% threshold
10 / 6 / 4 / 0
Ineligible
Vote shares remain 46%, 30%, 20%, and 4% in every setting.
Separate the effects of magnitude and exclusion
The three named presets form a controlled comparison, not three independent electoral histories. Moving from three seats without a threshold to twenty seats without one changes magnitude only. Moving from twenty seats without a threshold to twenty with five percent changes eligibility only. The vote vector and allocation formula stay fixed in both comparisons. If you compare the first and third settings directly, two rules change together. You can calculate the combined result, but that comparison alone cannot tell you which change explains D's absence.
At twenty seats without exclusion, the allocation is 9, 6, 4, and 1. Set the threshold to five percent and D cannot compete. The remaining lists divide all twenty seats, producing 10, 6, 4, and 0. A gains the seat D previously held, but this does not mean D's voters transferred to A. It means the ordered list of eligible quotient claims changed. The next unselected claim from an eligible list now receives an award that an ineligible claim cannot receive.
Change one rule at a time
Setting
Seats A/B/C/D
D status
Comparison
3 seats, 0% threshold
1 / 1 / 1 / 0
Eligible
Starting small district
20 seats, 0% threshold
9 / 6 / 4 / 1
Eligible
Magnitude changes only
20 seats, 4% threshold
9 / 6 / 4 / 1
Eligible at equality
Eligibility unchanged
20 seats, 5% threshold
10 / 6 / 4 / 0
Ineligible
Eligibility changes only
All rows hold votes 46/30/20/4 and unmodified Sainte-Laguë fixed. The four-percent setting is available through the existing threshold slider.
Inspect the boundary exactly. With no threshold, the twentieth award gives A its ninth seat at 46 divided by seventeen. Its next claim, 46 divided by nineteen, is lower. Once D is excluded, the highest twenty claims from A, B, and C include that additional A claim. A's tenth seat beats B's seventh claim of 30 divided by thirteen and C's fifth claim of 20 divided by nine. Cross multiplication gives 598 greater than 570 for the A versus B comparison, without relying on displayed decimals.
Measure the outcome against both vote bases. A's ten of twenty seats are fifty percent. Relative to all valid votes, its seat share exceeds its 46 percent vote share by four percentage points. Relative to eligible votes, fifty exceeds about 47.92 by about 2.08 percentage points. These are different comparisons, and neither should silently replace the other. The first records representation relative to all valid ballots; the second helps examine whole-seat rounding after exclusion. Neither measures whether the elected candidates resemble residents socially or advance their interests.
Try the twenty-seat no-threshold preset and predict the result at four percent before moving the threshold slider. D should remain eligible and the allocation should remain unchanged. Moving to five percent should then produce the single-seat exchange just derived. Continue to ten percent and explain why nothing further changes in this profile. D was already excluded, while C's twenty percent still qualifies. A threshold can move without changing seats when it crosses no eligibility boundary. This is a conditional result about these votes, not a general claim of harmlessness.
The experiment isolates the mechanical effect of a rule on reported ballots. A reform might also alter nominations, alliances, turnout, or party choices, but those responses require a different model and evidence. You have not measured them by moving a slider over fixed votes. Keep the counterfactual statement narrow enough to reproduce: with these four lists, these votes, twenty seats, and this divisor method, removing D's eligibility replaces its seat with A's tenth. That claim is fully explained by the award order, without a forecast about a future election.
Which claim replaces D in a twenty-seat allocation?
Claim
Exact quotient
Relevant comparison
A ninth seat
46 / 17
Last award with no threshold
A tenth seat
46 / 19
Wins additional eligible claim after D exclusion
B seventh seat
30 / 13
46 × 13 = 598 > 30 × 19 = 570
C fifth seat
20 / 9
46 × 9 = 414 > 20 × 19 = 380
After excluding D, A/B/C receive 10/6/4. The extra A seat follows quotient order. It is not a transfer of D voters to A.
Measure exclusion and proportionality separately
In the twenty-seat example, unmodified Sainte-Laguë gives 9,6,4,1 without a threshold and 10,6,4,0 with the five-percent threshold. A gains a seat after D becomes ineligible. The result is a conditional arithmetic example, not a prediction that every threshold benefits the largest party in that exact way.
Votes cast for excluded lists can affect perceived representation, but government formation also depends on coalition choices and institutional powers. Claims that thresholds necessarily create stability, or that more parties necessarily prevent it, need evidence beyond an allocation table.
Carry the allocation into institutional questions
An allocation table describes a legislature's composition, not who automatically holds a single executive office. In the twenty-seat no-threshold result, A has nine seats and B six. If an ordinary decision requires more than half of all twenty seats, eleven affirmative votes suffice. A cannot pass that decision alone. A and D together have ten seats, still short. A and C have thirteen and could pass it if every member supports the proposal. This arithmetic identifies possible majorities under a stipulated voting requirement, not actual coalition agreements.
After D is excluded, A has ten seats, B six, and C four. A still lacks eleven, despite gaining a seat. B and C together also have ten, so neither grouping alone clears the stipulated requirement. A with C has fourteen. These calculations show why losing one list and awarding its seat elsewhere need not produce a majority for the largest party. Abstentions, attendance rules, executive appointment procedures, and constitutional requirements would change the relevant decision problem. None is encoded in the party-vote allocation formula used here.
Possible voting blocs under an eleven-vote requirement
Proposed bloc
No threshold: 9/6/4/1
5% threshold: 10/6/4/0
A alone
9: short
10: short
A with D
10: short
10: short; D has no members
B with C
10: short
10: short
A with C
13: enough if united
14: enough if united
Both assemblies contain twenty seats. Assume a decision needs more than half of all seats and every member of the named bloc votes yes. These are possible vote totals, not predicted coalitions or government-appointment rules.
Geographic tiers require another distinction. Imagine twenty seats elected as several small delegations instead of one twenty-seat district. The formula must then run within each district using its own votes and seat count. Adding local results need not reproduce a national allocation, even if the national vote totals match. A geographically concentrated small party can compete differently from one with dispersed support. National totals alone cannot reconstruct those local quotient contests. The current lab has one allocation pool, so it cannot test that geographic comparison without additional district-level data.
Population equality also remains separate. Increasing district magnitude from three to twenty in this experiment changes available seats while holding the hundred vote units fixed. It does not establish a feasible national apportionment or equal residents per representative. Real population comparisons need the appropriate population measure divided by seats, whereas this experiment uses valid party votes for eligibility and vote totals divided by odd divisors for allocation. Calling each denominator by name prevents a turnout pattern from becoming an unsupported conclusion about population representation or legal compliance.
A threshold may give a small party reasons to seek an alliance or give voters reasons to choose a larger list. Whether they do so depends on expectations, organization, and available alternatives. The fixed ballots establish no rate of such behavior. Claims about stability or turnout need observed outcomes, a definition of the outcome, and a credible comparison that addresses other institutional differences. The selected historical handbook discussion motivates these questions, but historical country examples do not establish current statutory thresholds or causal effects of adopting them elsewhere.
For a final paper comparison, describe what evidence would distinguish two reports of zero seats for D. Request its eligibility status, the seat count, the divisor rule, and the award boundary. If D was eligible in a three-seat contest, the binding obstacle was competing claims for scarce whole seats. If D was barred from a twenty-seat contest, eligibility prevented any claim from winning. Whether either arrangement is preferable depends on explicit objectives about inclusion, local representation, and decision making. The calculation identifies the mechanism, not the objectives.
What would explain a report of zero seats?
Proposed explanation
Evidence needed
What zero alone cannot establish
Legal exclusion
Threshold base, exemptions, eligibility decision
That the list failed eligibility
Finite-seat competition
Magnitude, formula, votes, final quotient boundary
That a formal threshold existed
Geographic aggregation
District votes and local allocation rules
That national totals were one allocation pool
Unequal population representation
Relevant population divided by seats
Population equality from turnout counts
Effects on government
Voting requirement and observed coalition evidence
A stable majority from fewer lists alone
The first two mechanisms are isolated in the lab. Geographic, population, and political claims require information outside its one-pool fixed-vote allocation.
Exercises
A missing seat does not prove exclusion
D has 4% of votes and receives zero of three seats. Must there have been a legal threshold?
Hint
Compare eligibility with the size of a whole seat.
Worked answer
Mechanism
Possible reason for zero seats
Formal threshold
Party is ineligible
Small magnitude
Eligible party loses the few seat awards
The same outcome can arise through different mechanisms.
No. D can be eligible but fail to obtain a sufficiently high quotient in a three-seat allocation. Zero seats alone does not identify a legal exclusion. You need the stated threshold, magnitude, formula, and actual votes to explain the result.
Which denominator did you use?
After D’s 4% is excluded, A has 46 of the 96 eligible vote units. Is A’s original vote share now 47.92%?
Hint
Compute an eligible-vote share without rewriting the original electorate.
Worked answer
Base
A share
All valid party votes
46%
Votes for eligible lists
47.92%
Renormalization must name its denominator.
A’s share among eligible votes is 46/96 ≈ 47.92%. Its share of all valid party votes remains 46/100 = 46%. Both can be useful, but they answer different questions. Exclusion changes the allocation pool, not the historical ballots.
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.