Specify the rule and criteria, reproduce the arithmetic, then ask what evidence supports broader claims.
By the end: Audit claims about fairness, majority support, proportionality, and institutional consequences.
Write down the complete specification
Before reproducing a result, identify the office, seat count, districts, ballot format, validity rules, allocation formula, threshold denominator, exhaustion rules, tie procedure, and any compensating tiers. Similar names can hide consequential variants. The original ballots or appropriate aggregates must contain the information the calculation uses.
Then verify conservation and totals. Ranked transfers must not create votes. A fixed-size allocation must fill exactly the available seats. Expanded assemblies need a changed denominator. Reproduce intermediate steps rather than trust a final chart with no stated method.
Reconstruct the count before judging the result
Treat an electoral claim as a sequence of reproducible decisions. Start with the object being selected. A single office needs one winner; a legislature needs a seat allocation that may leave coalition formation unresolved. Next identify what each submitted ballot reports. One candidate mark cannot supply later rankings, and a party vote does not identify how that voter ranked individual candidates. Ask whether the available data contain every input the claimed procedure uses. Inventing missing preferences would fail that audit.
For the complete compromise profile, the available rankings are forty A above C above B, thirty-five B above C above A, and twenty-five C above B above A. The first-choice totals are 40, 35, 25. Plurality elects A. Instant-runoff eliminates C, transfers its twenty-five ballots to B, and elects B with sixty. Pairwise counting instead finds C preferred to A by sixty and to B by sixty-five. Three different outcomes follow from three specified questions about the same complete reports. No voter changes their mind here.
Now erase the later rankings and keep only the first-choice totals. Plurality remains reproducible, but the other two results are no longer identified. C voters might next prefer A rather than B, and other groups' pairwise comparisons are also missing. The audit should state that limit instead of silently restoring the earlier profile. If reports are incomplete because the rule permits optional ranking, explicitly define treatment of omissions. Under the separate C-only example, twenty-five ballots exhaust and A wins with forty of seventy-five continuing votes, not sixty of one hundred.
Can these inputs reproduce the claimed winner?
Available data
Plurality
Instant-runoff or pairwise
Only first totals 40 / 35 / 25
A wins
Later comparisons not identified
Complete 40 A>C>B, 35 B>C>A, 25 C>B>A
A wins
IRV: B; Condorcet: C
Same first totals, but 25 C-only ballots
A wins
Optional-ranking IRV: A with 40/75 continuing
The incomplete example is a separate hypothetical optional-ranking contract. It is not an instruction for Australian House ballot validity. Pairwise counting here requires complete rankings.
Transfer counts need an account of where value goes. In the bounded two-seat STV example, A's fifty initially unit-value ballots retain 34 units and transfer 16. With thirty next preferring B, the added values are 9.6 to B and 6.4 to C. The resulting accounts are A retained 34, B 35.6, C 30.4, totaling 100. Those are value units, not counts of new physical papers. B fills the second seat and the example stops, so a further redistribution of B's surplus would violate its stated contract.
An audit should preserve intermediate decisions rather than compare only final winners. Check the elimination candidate, the continuing total, and the destination of transferred value at each stage. For divisor allocation, check the winner's next divisor rather than divide every party by the same round number. Exact quotient ties in the bounded model favor the earlier input party. An instant-runoff elimination tie instead removes the alphabetically last tied candidate. Similar-looking tie statements govern different events, and neither is a substitute for checking the tie rules in an actual legal specification.
Audit retained value as well as transfers
Bounded STV account
Before A surplus
After A surplus
A initial / retained value
50
34
B continuing value
26
26 + 30 × 0.32 = 35.6
C continuing value
24
24 + 20 × 0.32 = 30.4
Total value
100
100
Decision
A takes seat 1
B takes seat 2; stop
100 complete initially unit-value ballots, two seats, quota 34. Fifty A papers retain 0.68 each and transfer 0.32 each. No final-winner surplus is redistributed.
Use the complete offline traces in the earlier runoff and transfer views to perform these checks without relying on animation timing. First inspect the original totals, then compare each later account against the prior one. In instant-runoff, active contributions plus exhausted ballots remain equal to original ballots. In the STV illustration, retained plus continuing value remains one hundred. These are different conservation statements because an elected candidate's retained value is neither an exhausted paper nor continuing support for another candidate. Labeling each account prevents a superficially plausible total from hiding duplication.
Name the criterion before declaring a winner
Majority support, Condorcet consistency, proportional party shares, local accountability, candidate choice, and participation are different criteria. Arrow’s theorem addresses aggregating unrestricted individual rankings over at least three alternatives into a complete transitive social ranking. Complete means every pair can be compared. Unanimity requires society to rank A above B when everyone does. Independence of irrelevant alternatives requires the social A/B comparison to depend only on individuals’ A/B comparisons, not their other rankings. No dictator means no individual’s ranking always becomes the social order regardless of everyone else. The theorem says these requirements cannot all be met.
That is not the same mathematical problem as allocating proportional legislative seats, nor exactly the same as the single-winner strategy theorem. Changing the ballot format or the preference domain changes which assumptions apply. Saying that no rule can meet a particular set of requirements is not proof that all rules perform equally well on every objective.
Compare criteria without merging their objects
A criterion evaluates a specified property of a rule or outcome. Condorcet consistency concerns selecting a candidate who defeats every rival when such a candidate exists. Party proportionality concerns how a legislature's party seat shares compare with a specified party-vote denominator. Their objects differ. A single office cannot be divided into 40%, 31%, and 29% officeholders under an ordinary deterministic election. Conversely, a legislature with those approximate seat shares does not establish that each elected individual would defeat every rival in a head-to-head contest.
Arrow's theorem concerns a different output again, a complete transitive social ranking over at least three alternatives. Complete means every pair can be compared. Transitive means that ranking A above B and B above C requires A above C. In the standard framework, the aggregation rule accepts all admissible individual preference orderings, respects unanimous strict comparisons, and satisfies independence of irrelevant alternatives. That independence condition says the social comparison of A and B depends only on individuals' comparisons of A and B, not their placements of other alternatives.
Non-dictatorship rules out any fixed individual whose strict comparisons always determine society's comparisons regardless of everyone else. Arrow's result says that this requirement cannot coexist with the other stated requirements on that unrestricted domain. A majority cycle supplies a concrete reason to care about the ordering condition. With 35 A above B above C, 33 B above C above A, and 32 C above A above B, the majorities support A over B, B over C, and C over A. The supporting majority changes between comparisons despite consistent individual rankings.
A social order is more than a winner label
Requirement
What to check across admitted profiles
Complete and transitive output
Every pair comparable; A>B and B>C require A>C
Unanimity
If everyone strictly prefers A to B, so does society
Pairwise independence
Unchanged individual A/B comparisons keep the social A/B comparison unchanged
Non-dictatorship
No fixed person always imposes their strict comparisons
Arrow concerns an unrestricted domain of preference orderings over at least three alternatives. These requirements cannot all hold there. This table does not assert that a seat allocation is a social order.
Follow those coalition changes numerically. A over B has support from the 35 A-first and 32 C-first voters, totaling 67. B over C has support from the 35 A-first and 33 B-first voters, totaling 68. C over A has support from the 33 B-first and 32 C-first voters, totaling 65. No single group supplies all three majorities. Adding these counts would count some people repeatedly, because the entries describe separate hypothetical comparisons. The result is a cyclic majority relation, not evidence that one hundred people submitted inconsistent ballots.
The strategy theorem does not require a social ranking as its output. Gibbard-Satterthwaite instead concerns a deterministic single-winner choice on unrestricted strict preferences, with at least three alternatives and every alternative attainable in the version used here. Requiring strategy-proofness for every voter at every profile forces dictatorship. Arrow's independence condition is about how social pairwise comparisons respond across profiles. Strategy-proofness is about whether one person's changed report can produce an outcome they genuinely prefer. Their formal objects and requirements differ.
Different majorities support each comparison
Pair
Supporting ranking groups
Winning count out of 100
Opposite count
A over B
35 A>B>C + 32 C>A>B
67
33
B over C
35 A>B>C + 33 B>C>A
68
32
C over A
33 B>C>A + 32 C>A>B
65
35
Each ranking group is individually transitive. The majority relation cycles. Counts belong to separate hypothetical comparisons and must not be added as distinct people.
Neither theorem ranks all systems. Approval cutoffs and cardinal scores add information or reporting assumptions beyond an ordinal ranking, so they need their own analysis rather than a claim that a different ballot escapes every tradeoff. A restricted preference domain also changes the question. When assessing a proposal, specify the property you want to preserve and identify the assumption you relaxed. Failure of one universal guarantee does not establish failure of every useful criterion, and a desirable numerical outcome does not establish that a rule has a universal guarantee.
Work through the mechanism
Independent versus compensating list seats
Party votes remain 40 / 31 / 29. Whole-assembly target at ten seats: 4 / 3 / 3. Changing the scenario replaces district wins in both panels.
Both bar scales run from 0% to 100% of each panel’s stated assembly size. Counts are exact; displayed percentages are rounded. The second model keeps district wins, tops other parties up to their original targets, and retains overhang without leveling. It is not a specification of a current country’s complete mixed system.
Claim
Required check
"The winner has a majority"
Majority of which population or ballot total?
"The system is proportional"
Which vote and seat totals, at which tier?
"The map caused the result"
Fixed voters, valid alternatives, other constraints?
"Reform will improve government"
Behavioral evidence, causal assumptions, and objectives?
A reproducible count is necessary for some claims but insufficient for all of them.
Audit the assembly and its denominator
In the mixed-system view, party votes stay 40, 31, 29 while district wins can change. The whole-assembly ten-seat target under unmodified Sainte-Laguë is 4, 3, 3. With district seats 4, 1, 0, compensation adds 0, 2, 3. The final seats match the target because the top-up calculation subtracts what each party already holds. The independent list tier instead awards 2, 2, 1 without that subtraction. Adding the same district seats yields 6, 3, 1. A proportional component does not establish combined proportionality.
Compute the differences using common denominators. In the parallel ten-seat result, seat shares are 60%, 30%, 10%, against vote shares 40%, 31%, 29%. The seat-share minus vote-share gaps are plus 20, minus 1, and minus 19 percentage points. They sum to zero because both sets of shares sum to one hundred. The compensatory target has seat shares 40%, 30%, 30%, so the gaps are zero, minus 1, and plus 1. Whole seats still prevent exact representation of every share even when compensation reaches the specified target.
Account for each tier before adding seats
Normal districts 4 / 1 / 0
Parallel
Compensatory
Party votes
40 / 31 / 29
40 / 31 / 29
List additions
2 / 2 / 1
0 / 2 / 3
Final seats
6 / 3 / 1
4 / 3 / 3
Seat shares
60% / 30% / 10%
40% / 30% / 30%
Seat minus vote share
+20 / −1 / −19 points
0 / −1 / +1 points
Both assemblies have ten seats. Compensation targets the whole assembly; the independent list tier does not subtract district victories. Sainte-Laguë is unmodified in this example.
Select the sweep preset and keep party votes fixed. A now holds five district seats despite a ten-seat target of four. Under the declared policy, A keeps those five and B and C each receive three. The assembly therefore has eleven seats, not ten. A's share is 5/11, about 45.45%, and each other party has 3/11, about 27.27%. Comparing five seats with the original denominator ten would overstate A's actual seat share. Retaining an overhang also does not supply the additional leveling that another hypothetical design might specify.
The changed denominator affects coalition arithmetic as well as party percentages. In a ten-seat assembly, a strict majority needs six seats. In an eleven-seat assembly, it also needs six, because more than 5.5 requires an integer of at least six. That equal threshold does not make the assemblies interchangeable. With 4, 3, 3, B and C together have exactly six. With 5, 3, 3, they still have six but face a different distribution of the remaining seats. A seat count identifies possible majorities without predicting which parties will cooperate.
One retained seat changes the denominator
One rectangle per seat in both rows. Party votes stay 40/31/29. Keeping A’s fifth district seat without leveling produces an eleven-seat assembly, not a ten-seat percentage base.
Use the view's simultaneous panels to check one claim at a time. Start with normal district wins and record both final vectors before changing anything. Then select the sweep and record each panel's stated assembly size. The parallel panel remains ten seats because its five list seats stay independent, giving 7, 2, 1 after A's sweep. The compensatory panel expands to eleven. The bars compare seat shares on a common percentage scale, so use their count labels too. Equal bar lengths across differently sized chambers would not necessarily mean equal seat counts.
A real audit would inspect thresholds, regional allocations, candidate qualification, and overhang and leveling law. The selected historical handbook passages cannot establish a current jurisdiction's complete specification. Here the justified conclusion is narrower. Under these invented votes and declared policies, independent allocation preserves a district bonus, ordinary compensation reaches its whole-assembly target, and retained overhang without leveling leaves a different share distribution. Whether one tradeoff is acceptable depends on representation objectives that should be stated separately from the successful reproduction of the arithmetic.
Separate mechanics, behavior, and values
Suppose a proposed assembly has five district seats and five independent list seats, party votes 40/31/29, and district wins 4/1/0. The model gives final seats 6/3/1, not proportional compensation. The factual correction is mechanical. Whether that tradeoff is acceptable is a further judgment about objectives.
Claims about stable governments, corruption, turnout, policy quality, or social inclusion need empirical evidence and a causal argument. Countries that use different rules differ in many other ways, and reforms may change several institutions together. A useful audit states what is known, what follows only under a model, what remains uncertain, and whose values determine the recommendation.
Build a counterfactual that supports the claim
A counterfactual comparison asks what changes and what stays fixed. The district experiment fixes twenty-five voters, their A or B choices, and the plurality rule. It changes the partition between five row districts and five column districts. Each contiguous group contains five voters. Row A counts are 3, 3, 2, 2, 2, giving two A seats. Column A counts are 3, 3, 3, 3, 0, giving four. A remains twelve of twenty-five voters, or 48%, under both partitions. This identifies sensitivity to grouping.
The same calculation does not establish malicious intent or legal invalidity. Neither partition is real, and the cells supply no separate resident-population data. Equal observed votes per district do not establish equal residents per representative. For an illustrative population contrast, two one-seat districts may each cast one hundred votes while containing two hundred and four hundred residents. Their turnout counts are equal but their residents per seat differ twofold. Whether residents are the legally relevant base would require the applicable law, not an inference from this arithmetic.
For the grid, inspect the middle-district preset in the earlier boundary view. It highlights row three and column three without moving a voter. Row three has A 2 and B 3, while column three has A 3 and B 2. Then inspect the last district. Row five has A 2 and B 3, while column five has A 0 and B 5. The local comparisons explain how concentrating five B voters in one column can coexist with A winning the other four columns. They do not require a change in turnout or party preference.
Audit the actual groups of fixed voters
Every outlined district contains five voters and elects one representative by plurality. Counts run top to bottom for rows and left to right for columns. The grid is schematic, not a geographic or resident-population map.
Broader causal claims need a comparison that addresses other differences. Suppose one set of countries has proportional systems and frequent coalition governments. That association does not isolate the effect of electoral rules, because party histories, executive selection, social divisions, and reform choices may differ too. Even a before-and-after reform comparison can mix the rule change with concurrent events. A credible study must explain why its comparison estimates the outcome without the reform and which assumptions permit that inference. Reproducing a seat formula does not provide that missing counterfactual evidence.
The outcome also needs a definition. Stable government might mean cabinet duration, legislative success, or continuity of policy, which can move differently. Inclusive representation might mean party shares, elected members' backgrounds, or whose interests affect decisions. Ask which measure was observed, when, and for whom. A normative judgment then assigns weight to those outcomes and to procedures or rights. No algebraic identity determines that weighting. An empirically supported benefit can still conflict with another objective that a voter or constitution treats as important.
What comparison could support which conclusion?
Claim type
Comparison or evidence needed
Not established by it
Mechanical boundary effect
Same 25 voters and rule, rows versus columns
Intent or legal validity
Behavioral response
Reports or participation under changed incentives
A causal effect without a credible counterfactual
Causal reform effect
Outcome with reform versus credible estimate without it
Universal desirability
Normative or legal verdict
Explicit values or applicable legal requirements
A verdict from seat shares alone
Selected cached source passages support specific mechanisms and theorem statements. They are not a full literature review, a current statutory audit, or an empirical estimate for these invented examples.
Finish by writing a bounded conclusion rather than a universal verdict. For the grid, the demonstrated result is a two-seat change for A when this fixed electorate moves between these two partitions. For the mixed example, it is the difference between independent list awards and compensating top-ups under the specified overhang policy. Predicting turnout or coalition behavior requires more evidence. Reference descriptions explain selected mechanisms and theorem statements. Applying them to a real election still requires its current legal specification. State what evidence could change the conclusion.
Exercises
Audit the two-tier proposal
A proposal adds district wins 4,1,0 to independent list seats 2,2,1 and calls the ten-seat result proportional to party votes 40%,31%,29%. What should an audit say?
Hint
Compute the combined seats before evaluating the label.
Worked answer
Party
Vote share
Parallel seat share
Compensatory seats
A
40%
60%
4
B
31%
30%
3
C
29%
10%
3
The arithmetic distinguishes the mechanisms without deciding every normative question.
The combined allocation is 6,3,1, giving seat shares 60%,30%,10%. That is a parallel system in this example, not compensation to whole-assembly party entitlements. A compensatory target of 4,3,3 would need top-ups 0,2,3. Whether to prefer it depends on the stated representational objectives.
Separate an identity from a prediction
A report correctly calculates a seat allocation, then concludes the reform will necessarily produce stable governments. Which part is established?
Hint
A seat formula does not contain a model of coalition behavior or institutional conflict.
Worked answer
Statement
Evidence type
These votes yield these seats
Specified counting procedure
The reform changes future voting
Behavioral and causal evidence
The reform is desirable
Evidence plus explicit objectives
Do not let a correct calculation carry an unsupported political forecast.
The calculation establishes the allocation under the stipulated ballots and rules. Stability requires a definition and evidence about coalition formation, party organization, executive-legislative relations, and other conditions. A causal claim also needs a credible comparison, not just countries grouped by their current electoral systems.
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 discussions of majority cycles, preference aggregation, Arrow’s theorem, approval/cardinal information, and the Gibbard-Satterthwaite theorem. The lessons state assumptions rather than treating impossibility results as a universal ranking of systems. Its choice-set discussion allows weak pairwise winners; this course uses strict victories for a Condorcet winner.
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.