A ballot format determines which information is expressed, but not how honestly or comparably it is reported.
By the end: Count approval ballots and distinguish ordinal preference from acceptance thresholds and cardinal scores.
Approval counts supported candidates
Under approval voting for one office, a voter may approve multiple candidates and the candidate with the most approvals wins. Each approved candidate gets one approval from that voter. These are separate candidate totals, so their sum can exceed the number of voters. The winning total is still measured against the number of voters.
Using the same underlying complete rankings, approving only the first choice gives A 40, B 35, C 25. If everybody approves their first two choices, A gets forty, B sixty, and C one hundred. The lab changes the approval policy, not the rankings. Neither policy is automatically implied by an ordinal ranking.
Locate the approval boundary
A ranking answers which candidate a voter prefers to which other candidate. An approval ballot answers which candidates the voter marks as approved. These reports contain different information. Someone who ranks A above C above B might approve A alone or approve A and C. Both sets put an approval boundary between adjacent ranks. Neither tells the counter how large the preference gaps are. For the single office considered here, the counter adds approvals separately for each candidate and selects the largest total, subject to a tie rule.
Work through the complete profile by ballot group. Forty voters rank A above C above B. Under the top-two policy they contribute forty approvals to A and forty to C. Thirty-five rank B above C above A, contributing thirty-five to B and thirty-five to C. Twenty-five rank C above B above A, contributing twenty-five to C and twenty-five to B. The totals are A forty, B sixty, C one hundred. C's total includes all three groups, although only twenty-five voters put C first.
Top-two approvals on actual ballot choices
Each drawn ballot represents its labeled group of identical reports, not one voter. The ranking is supplied separately; the approval marks do not encode the order of the two approved candidates. All 100 ballots include C.
Compare that count with the first-only policy. Each group now approves its highest-ranked candidate alone, giving forty, thirty-five, and twenty-five. A wins instead of C. No voter changed their stipulated ranking. The added assumption about where approvals stop changed the reports that the rule counted. It would therefore be inaccurate to describe this as two approval counts of identical ballots. It is one approval rule applied to two different ballot profiles generated from the same rankings by two explicit policies.
Each voter can contribute at most one approval to any particular candidate in this example. Across candidates, however, that voter can contribute several approvals. Under top-two reporting, one hundred people produce two hundred approval marks. C's one hundred approvals mean that every submitting voter approved C. Dividing C's total by two hundred instead answers what fraction of all approval marks went to C. That fraction is fifty percent, but it is not the fraction of voters who approved C. Keep both numerator and denominator attached to the statement.
Approval also does not preserve every distinction in a ranking. A ballot approving A and C treats both alike for this count even if its voter prefers A. Reversing their positions within the approved set would leave the approval ballot unchanged. A first-only ballot similarly treats its two unapproved candidates alike. This loss of ordering information is compatible with gaining a different distinction, the reported boundary of acceptability. The format changes which questions the count can answer rather than giving a complete description of a person's preferences.
Try a paper variation before using the lab. Keep the forty A-first voters and thirty-five B-first voters at first-only approval. Let the twenty-five C-first voters approve C and B. A then has forty approvals, B sixty, and C twenty-five, so B wins. This uses the same complete rankings as both earlier cases. The three possible winners demonstrate why the rankings alone cannot determine an approval result. This mixed policy is a worked counterfactual, not a third control in the two-policy lab below.
Three approval reports from the same rankings
Approval policy
A approvals
B approvals
C approvals
One office
Everyone first-only
40
35
25
A
Everyone top-two
40
60
100
C
Only C-first voters add second choice
40
60
25
B
Each row uses the complete 40 A>C>B, 35 B>C>A, 25 C>B>A rankings. The reports change, not the rankings. Each candidate total counts people approving that candidate.
Scores express another kind of report
Score voting lets voters assign numbers on an allowed scale and aggregates them under a specified rule. A scale needs a defined range and treatment of omissions. A score can express more than rank, but interpersonal comparability is an assumption, not a fact created by printing the same numbers on ballots.
For an illustrative 0–5 scale, assign scores 5,4,0 to each group’s first, second, and last choices. In our profile A totals 200, B 275, and C 425. This is one stipulated scoring policy. Changing intensity reports while preserving every ranking can change the score totals.
Account for score units
A score ballot supplies numbers as well as an order. To count one, specify the permitted scale and aggregation rule. Here use a separate hypothetical rule in which every voter scores every candidate between zero and five and the counter sums each candidate's points. All one hundred ballots are complete. With complete ballots, dividing every total by the same one hundred would produce candidate averages with the same ordering. That equivalence can fail if averages use different numbers of ratings.
Under the stated five, four, zero rank policy, the forty A-first ballots contribute A two hundred points, C one hundred sixty, and B zero. The thirty-five B-first ballots contribute B one hundred seventy-five, C one hundred forty, and A zero. The twenty-five C-first ballots contribute C one hundred twenty-five, B one hundred, and A zero. Add by candidate rather than by group to get A two hundred, B two hundred seventy-five, and C four hundred twenty-five. The resulting winner is C under this stipulated scoring policy.
Sum score points by candidate
Ballot group
A points
B points
C points
40 voters: A>C>B
40 × 5 = 200
40 × 0 = 0
40 × 4 = 160
35 voters: B>C>A
35 × 0 = 0
35 × 5 = 175
35 × 4 = 140
25 voters: C>B>A
25 × 0 = 0
25 × 4 = 100
25 × 5 = 125
Candidate totals
200
275
425
The complete ballots use 5 points for first, 4 for second, and 0 for last. Column sums select C. The grand total is 900 points from 100 ballots, not 900 people.
Every ballot in that policy contributes nine points across candidates, so the grand total is nine hundred points. That is an accounting check, not the number of voters. C's average is 425 divided by 100, or 4.25 points per submitting voter. Relative to the maximum possible five hundred points for one candidate, C received eighty-five percent. Neither figure says that eighty-five percent of voters approved C. Scores do not define approval unless a separate rule specifies which ratings count as approval for that different question.
Now preserve every ranking but change the second-rank score to one. First receives five and last receives zero. A still gets two hundred points because A is first for forty people and last for the other sixty. B receives one hundred seventy-five plus twenty-five, or two hundred. C receives forty plus thirty-five plus one hundred twenty-five, also two hundred. All three tie; alphabetical priority would select A. To see that even the winner can change, give second rank one-half point instead. Totals become two hundred, 187.5, and 162.5, and A wins outright.
The changed results do not require any change in who prefers whom. They follow from changing the distances between reported numbers. A common numerical scale makes arithmetic possible, but does not prove that a four-point difference represents the same welfare difference for every person. Someone may reserve five for an exceptional candidate while another uses five for whichever candidate they like best. Adding their reports still produces a defined score total. Calling that total a measured sum of well-being requires a further interpretation and evidence that the ballot design itself cannot supply.
Rankings do not fix score distances
First / second / last points
A total
B total
C total
Result
5 / 4 / 0
200
275
425
C
5 / 1 / 0
200
200
200
Three-way tie; A by priority
5 / 0.5 / 0
200
187.5
162.5
A
Every policy preserves all strict rankings in the same 100-voter profile. Alphabetical priority selects A from the exact three-way tie. These policies are written examples, not score controls in the approval lab.
Distinguish a mechanical re-expression from changing relative weights. If every score on every ballot is doubled, every candidate total doubles and the ordering stays unchanged. If only one voter's reports are doubled, that person's contribution gains weight relative to others, and the result can change. Allowed scales may forbid that report. This thought experiment explains why preserving an individual's order is insufficient to preserve a summed result. The lab below changes approval cutoffs only. It does not let you enter these score policies or test interpersonal welfare comparisons.
Work through the mechanism
Change the approval cutoff, not the rankings
Stipulated ballots, not inferred from first-choice totals
Voters
Ranking, best first
40
A > C > B
35
B > C > A
25
C > B > A
C wins with 100 approvals from 100 voters. Total approvals: 200.
Both panels use 0–100 approvals per candidate. Changing the preset replaces every voter’s stated approval policy immediately. The rankings alone do not specify an approval cutoff or interpersonal intensity.
Candidate
First-only approvals
Top-two approvals
Illustrative score total
A
40
40
200
B
35
60
275
C
25
100
425
Scores use 5 for first rank, 4 for second, and 0 for last. These scores are additional assumptions, not inferred intensities.
Explore reports without changing preferences
Use the approval lab to separate its reference from its selected panel. The left panel always displays first-only approvals, forty for A, thirty-five for B, and twenty-five for C. Select the first-only button and both panels should agree. Predict the right panel under top-two before choosing that button. A stays at forty because the other groups rank A last. B gains the twenty-five C-first voters. C gains both other groups. Check that the selected totals become forty, sixty, and one hundred without any ranking changing.
The bars share a zero-to-one-hundred scale for each candidate. That common scale is appropriate because there are one hundred submitting voters and no candidate can receive more than one approval per ballot. Do not expect the three bar lengths to sum to a fixed length. Their sum rises when voters approve more candidates. A pie chart of the two hundred top-two marks would answer a different question and hide the fact that C appears on every ballot. The displayed comparison is between candidate support counts, not shares of a legislative seat pool.
Different units answer different questions
Question under top-two approval
Numerator / denominator
Meaning
Which voters approve C?
100 / 100 voters = 100%
Every ballot includes C
Which approval marks name C?
100 / 200 marks = 50%
Half the marks name C
What is C’s average score under 5/4/0?
425 points / 100 voters = 4.25
Mean reported points
What fraction of C’s maximum points?
425 / 500 points = 85%
Not an approval percentage
All four computations are defined, but their denominators differ. The score rows add a scoring policy; they are not quantities the approval count alone can reveal.
On paper, start at first-only approval and let exactly ten of the forty A-first voters also approve C. Totals become A forty, B thirty-five, C thirty-five. C has gained ten approvals but A still wins. If sixteen of those voters add C instead, totals become forty, thirty-five, forty-one and C wins. Each changed ballot leaves its approval for A intact. The outcome changes because the extra approvals increase C's total past A's, not because the counter transferred votes away from A.
At exactly fifteen added approvals, A and C tie on forty. A complete implementation must then specify its tie procedure. Under the course's declared alphabetical priority for a hypothetical tied selection, A would take the office. The lab's two policies have no tie. State the tie convention when analyzing this paper variation rather than silently awarding the office on a displayed rounding difference. Approval counts here are integers, so the equality is exact. A real election's tie procedure must come from its own governing specification.
Contrast these additions with a ranked transfer. In a single-winner instant-runoff count, a ballot contributes to one continuing candidate at a time, and elimination can change that destination. Adding C to an approval ballot does not wait for A's elimination. Both approvals count immediately in separate candidate totals. There is no exhausted-ballot stage in this complete approval example. It would therefore be wrong to animate an approval addition as a full vote leaving A for C. The ballot formats implement different operations even when they originate from identical underlying rankings.
An extra approval does not remove the first
A-first voters also approving C
A approvals
B approvals
C approvals
Selection
0 of 40
40
35
25
A
10 of 40
40
35
35
A
15 of 40
40
35
40
A by declared tie priority
16 of 40
40
35
41
C
Start from first-only reports. Only the indicated A-first voters add C. A’s total never decreases. Alphabetical priority resolves the constructed exact tie; this is not a transfer count or an additional lab preset.
Finish the comparison by recording the object selected. The lab chooses one officeholder, not several representatives with vote-proportional seat shares. Electing the top five approval totals to five seats would be a different multiwinner rule. A cohesive majority might approve the same five candidates and win every seat, leaving a large minority without a winner. That possibility is enough to show why a single-office approval count cannot certify legislative proportionality. Multiwinner designs need their own specification of ballots, seat allocation, and the criterion used to evaluate representation.
More expressive does not mean incentive-free
A voter’s approval cutoff may depend on acceptability, expectations about close contenders, or instructions for the decision. Score voters may also have incentives to use the scale’s extremes. These are questions about behavior under a rule, not defects that can be detected from a ballot total alone.
Comparisons should hold constant what they claim to hold constant. Ranking information does not uniquely determine approval or score ballots. Approval totals should not be normalized into a fictitious one-vote-per-person party share unless that normalization is explicitly part of another institution’s rule.
Separate incentives from conclusions
An approval cutoff can reflect a fixed acceptability standard, but it can also depend on expectations about the contest. A voter might approve a less-preferred contender to help that contender defeat someone they like even less. Another might withhold approval from a tolerable rival to protect their favorite. The same approval set could come from a fixed standard or strategic calculation. Totals alone therefore do not identify motives or establish how people would respond to a reform.
Consider one voter who prefers A above C above B. Suppose the other ballots give A twenty approvals, B forty, and C forty. Declare alphabetical priority for an exact tie. If this voter approves only A, the final totals are twenty-one, forty, forty and B wins the tied lead. If the voter instead approves A and C, the totals become twenty-one, forty, forty-one and C wins. This voter prefers C to B, so the added approval improves the outcome under the stipulated beliefs and tie convention.
The extra approval can also harm this voter. Change the other totals to A forty, B twenty, C forty-one. Approving only A yields forty-one, twenty, forty-one. Alphabetical priority elects A from the tied lead. Approving A and C instead yields forty-one, twenty, forty-two, electing C, which this voter likes less than A. The same cutoff change now worsens the outcome. Strategic advice needs information about other ballots, preferences over winners, and the complete rule, not merely an ordinal ranking.
One additional approval can help or harm
Other ballots: A / B / C
Voter approves A only
Voter approves A and C
Effect for A>C>B voter
20 / 40 / 40
21 / 40 / 40: B wins tie
21 / 40 / 41: C wins
Improves B to C
40 / 20 / 41
41 / 20 / 41: A wins tie
41 / 20 / 42: C wins
Worsens A to C
Both cases use alphabetical priority for exact tied leads. The other candidate totals can overlap because voters may approve several candidates. These are conditional calculations with stipulated other ballots, not forecasts of strategic behavior.
These local calculations should not be confused with an impossibility theorem. Arrow's result concerns a rule producing a complete, transitive social ranking from unrestricted individual rankings of at least three alternatives. It rules out jointly satisfying unanimity, independence of irrelevant alternatives, and no dictator in that framework. The Gibbard-Satterthwaite theorem instead concerns deterministic single-winner rules on unrestricted strict preferences, at least three possible outcomes, and every outcome attainable. Under those assumptions, strategy-proofness requires dictatorship. Its conclusion is about the existence of profitable misreporting somewhere for a non-dictatorial rule, not every ballot in every election.
An approval ballot is a set rather than a complete strict ranking. Applying a theorem to a proposed approval mechanism requires stating how underlying preferences, available reports, outcomes, and ties fit the theorem's assumptions. Checkboxes do not establish strategy-proofness. Nor does a theorem about social rankings prove that two particular approval policies must select the same candidate. The worked examples establish their own narrow arithmetic claims. They do not estimate how often voters find useful strategies, how difficult coordination is, or how costly mistakes would be.
When evaluating a claim that approval or score voting is better, first ask what better means. More voters approving the selected candidate is a criterion about the reported approval ballots. Measured welfare, easier participation, and more representative legislatures are different claims requiring different evidence. A proposal also needs ballot-validity, omission, and tie rules before its count is reproducible. Selected discussions in the Stanford social-choice entry support these information distinctions, not an endorsement of one design. The invented profiles here demonstrate mechanisms and possibilities rather than predictions about any electorate or instructions under current election law.
A ballot result and a theorem have different objects
Analysis
Input and output
What its conclusion establishes
Approval calculation
Approval sets → one officeholder
Winner under the given reports and tie rule
Arrow framework
Unrestricted individual rankings → complete transitive social ranking
Unanimity, independence of irrelevant alternatives, and no dictator cannot all hold for at least 3 alternatives
Gibbard-Satterthwaite framework
Unrestricted strict preferences, deterministic single winner, at least 3 attainable outcomes
A non-dictatorial rule cannot prevent profitable misreporting at every profile for every voter
The strategy theorem gives an existence result about an opportunity somewhere under its assumptions. It does not say every voter can gain in every election. Neither theorem assigns a practical frequency or recommends a ballot format.
Exercises
Can approvals exceed voters?
One hundred voters produce 200 approvals under the top-two policy. Has the count necessarily duplicated ballots incorrectly?
Hint
Distinguish a candidate’s approval total from the sum across candidates.
Worked answer
Quantity
Meaning
100 voters
People submitting ballots
200 approvals
Two approved candidates per voter
C gets 100 approvals
100% of voters approve C
The denominator for C’s voter support remains one hundred.
No. Each voter was allowed to approve two candidates, so the sum is two hundred. No individual candidate can receive more than one hundred approvals in this example. C’s one hundred means every voter approved C, not that C received half of a pool of two hundred people.
Rank does not specify a cutoff
Two voters both rank A>C>B. One approves only A; the other approves A and C. Are their ballots necessarily inconsistent with those rankings?
Hint
A ranking orders candidates but does not locate an acceptance boundary.
Worked answer
Approval set
Possible cutoff
A only
Between first and second
A and C
Between second and third
The lab makes its added cutoff policy explicit.
No. Both approval sets can agree with the ranking. One cutoff falls between A and C, the other between C and B. To convert a ranking into approval automatically, you must add a cutoff rule or information about acceptability. The same issue arises when inventing cardinal scores from ranks.
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.