Separate a counting counterfactual from a behavioral prediction
Voters, candidates, and parties can respond to rules, so fixed-ballot comparisons answer a limited question.
By the end: Identify strategic incentives, turnout effects, and assumptions behind counterfactual election claims.
Rules can change the ballots they receive
A sincere ballot is not defined identically across formats. A strict ranking can report an order, while an approval ballot also needs an acceptance threshold and a score ballot needs scale choices. A strategic report is chosen to obtain a preferred outcome given beliefs about others’ behavior, rather than simply report the information a sincerity convention asks for.
Coordination can also happen before voting through nominations, alliances, withdrawals, or joint lists. Electoral thresholds may create incentives to combine lists; district contests may encourage avoiding competing candidates. These are possible mechanisms, not universal behavioral laws.
A strategic choice needs a comparison
A strategic voting claim needs a voter, a genuine preference, an alternative report, and a comparison with what would happen otherwise. Consider one office filled by plurality. Other voters have already supplied A with 4 votes, B with 3, and C with 2. One remaining voter ranks C above B above A. A sincere first-choice report adds one to C, producing 4, 3, 3. A wins. The voter's favorite loses, but that fact alone does not establish a useful strategic alternative.
Now let that same voter report B instead. The totals become 4, 4, 2. For this paper example, declare that an exact winning tie favors the alphabetically earlier candidate. A still wins, so this particular change gives the voter no improvement. Change the other voters' totals to A 4, B 4, C 1. A C report gives 4, 4, 2 and elects A by that tie rule. A B report instead gives 4, 5, 1 and elects B, which this voter prefers to A.
One changed report, two different opportunities
Other votes A/B/C
Report C
Report B
Improves C>B>A?
4 / 3 / 2
4 / 3 / 3: A wins
4 / 4 / 2: A wins
No
4 / 4 / 1
4 / 4 / 2: A wins
4 / 5 / 1: B wins
Yes
One additional voter chooses a report. Exact winning ties favor the alphabetically earlier candidate in this paper example. True preferences remain C>B>A.
The second comparison establishes a profitable unilateral change under known other votes. It does not require the voter to regard B as their favorite. The relevant comparison is B against the outcome A obtained by the sincere report. Keep the voter's true ranking separate from their submitted mark throughout the calculation. Replacing the true ranking with the submitted mark would make it impossible to say whether the change helped. The example concerns one person choosing one report, rather than a coordinated agreement among several voters.
Knowledge changes the decision. If the remaining voter cannot tell whether the other totals are 4, 3, 2 or 4, 4, 1, the benefit of reporting B is uncertain. A strategic decision then depends on beliefs about those situations and perhaps other possible situations. A demonstration made after seeing every ballot establishes an opportunity in that stipulated profile. It does not establish that an ordinary voter had the information needed beforehand. A forecast of strategic behavior would need an account of information as well as preferences.
Ballot formats also change what counts as the sincere report in this comparison. On a complete ranking ballot, C above B above A specifies every pair. On a one-mark plurality ballot, the direct first-choice report is C. On an approval ballot, the ranking permits either C alone or C with B under different acceptance cutoffs. A score report needs still more information. You cannot diagnose misreporting in approval or score voting merely because a ballot differs from the rank-based conversion used in an earlier illustration.
What information makes a report sincere?
Ballot for one office
Information supplied
What remains unspecified by C>B>A
Plurality
One mark for a candidate
Direct first-choice report is C
Complete ranking
Every strict comparison
Report C>B>A
Approval
A set of approved candidates
Cutoff: C alone or C and B
Score
Numbers on an allowed scale
Intensity reports and use of scale
A ranking does not locate an acceptance cutoff or assign scores. These are report formats, not four interchangeable encodings of measured utility.
Work through the two plurality situations on paper before reading their outcomes in the table. Hold the voter and tie procedure fixed, then ask which change in the other totals makes the B report effective. Next reverse the declared winning-tie priority so that B defeats A in an exact tie. In the second situation, the C report now already elects B, so the B report no longer improves the result. This comparison explains why a complete strategic example must specify tie handling rather than hide it behind rounded percentages.
Turnout and participation change denominators
A second round can have different turnout from the first. Surveys of first-round voters do not automatically describe everyone who votes later. A party’s percentage can rise while its raw vote total falls if the overall turnout declines more. Always keep counts and denominators together.
For example, forty of one hundred votes is 40%. Thirty-six of eighty votes is 45%. Support among votes cast increased in percentage terms, but the raw count fell by four. Neither fact alone explains persuasion, abstention, entry of new voters, or changed eligibility.
Separate changing people from changing shares
Suppose A receives 40 of 100 votes in an earlier election and 36 of 80 in a later election. The observed vote share rises from 40% to 45%, while the observed count falls by four. Those observations allow several individual histories. In one history, four former A voters and sixteen former non-A voters abstain, while nobody switches sides. The later electorate then contains 36 A voters and 44 others. The share increase arises entirely through unequal withdrawal from the two groups, without any conversion to A.
A second history yields the same aggregates through different behavior. Let ten former A voters abstain, six former non-A voters switch to A, and ten other former non-A voters abstain. A finishes with 40 minus 10 plus 6, or 36. Other parties finish with 60 minus 6 minus 10, or 44. Both histories have eighty votes and the same party shares. In this simplified comparison, all voters belong to the original hundred and every participating person casts one valid party vote. There are no new entrants.
Identical totals do not identify voter switching
Flow from earlier 40 A, 60 others
No switching
Six switch toward A
A voters abstain
4
10
Other voters abstain
16
10
Other voters switch to A
0
6
Later A votes
40 − 4 = 36
40 − 10 + 6 = 36
Later other votes
60 − 16 = 44
60 − 10 − 6 = 44
Later turnout and A share
80; 36/80 = 45%
80; 36/80 = 45%
Constructed histories, with no new entrants and one valid party vote per participant. Both start with A at 40/100 = 40%.
Aggregate totals therefore do not identify individual persuasion. The first history has no switching and the second has six switches toward A, despite identical published results. Allowing new voters, migration, invalid ballots, or changes in eligibility creates further possibilities. To distinguish these mechanisms, an investigator needs evidence that follows people or otherwise constrains the flows. Even a panel survey needs care about nonresponse and inaccurate reports. The numerical ambiguity exists before any survey problem appears, because many different flow patterns produce the same final totals.
A rule change introduces another comparison. Holding the submitted ballots fixed and recounting them under another procedure isolates a mechanical conversion. Letting people respond to the new procedure asks a behavioral question. A runoff can change who participates and what they know. A threshold can encourage joint lists or discourage a small-party vote. These are possible responses with directions that depend on expectations and institutions. Neither the fixed-ballot result nor the turnout example supplies a measured estimate of how large those responses would be after an actual reform.
Use the earlier approval view to practice separating an input change from a rule change. Select first-only approval, record A 40, B 35, C 25, then select top-two approval and record A 40, B 60, C 100. The aggregation rule still counts approvals. The preset changes the stipulated reporting policy while the complete rankings stay fixed. The changed winner therefore does not demonstrate that the same approval ballots elect different candidates under identical rules. It demonstrates that rankings alone leave an outcome-relevant cutoff unspecified.
Approval totals use two different units
Fixed rankings, changed policy
A / B / C approvals
Approvals summed
C support among voters
First only
40 / 35 / 25
100
25/100 = 25%
Top two
40 / 60 / 100
200
100/100 = 100%
Both rows have 100 voters. The policy changes reports, not the approval-counting rule. Summed approvals are not a count of distinct people.
Finally compare the approval total with turnout. Under the top-two policy, 100 voters supply 200 approvals because each approves two candidates. C's 100 approvals mean approval from every participating voter, not support from half a population of 200 people. Under the turnout comparison, 36 of 80 is a share of ballots cast, not a share of all residents. Naming the unit prevents two different errors. Multiple approvals do not create people, and a smaller participating electorate does not establish that the eligible population shrank.
Work through the mechanism
Election
Votes for a party
Total votes
Share
First
40
100
40%
Later
36
80
45%
The totals are invented and compare counts with shares, not estimated effects of an electoral reform.
Read the strategy theorem's quantifiers
The Gibbard-Satterthwaite theorem concerns a rule that takes reported preferences and returns one winning alternative. In the version used here, preferences are unrestricted strict rankings, the rule is deterministic, and every alternative in a set of at least three can win for some reports. Strategy-proofness would mean that truthful reporting protects every voter against a profitable unilateral misreport for every preference profile and every configuration of others' reports. This guarantee is stronger than observing honest behavior in a few elections.
Under those assumptions, a strategy-proof rule must be dictatorial. A dictator here is a fixed individual whose top alternative the rule always selects, regardless of everyone else's reports. The term names a mathematical property of the function, not a diagnosis of a country's political regime. The conclusion rules out combining non-dictatorship with the other stated requirements. It does not recommend dictatorship. A designer who rejects that property must examine which other assumptions or guarantees they are willing to change and what the resulting procedure does.
Negating the universal guarantee gives an existence statement. For a non-dictatorial rule meeting the remaining assumptions, there is some profile and some voter with a profitable misreport. It does not follow that every profile contains such a voter, that each voter benefits from lying, or that the gain is large. Our two plurality situations illustrate why instances differ. Reporting B helps the C-preferring voter in one situation but not the other under the declared tie priority. One successful example establishes vulnerability, not a frequency distribution over elections.
A universal protection and its failure
Statement
Scope
What would establish it?
Strategy-proofness
Every voter, every admitted profile, every unilateral alternative report
No report produces a strictly preferred winner
Failure of strategy-proofness
Some voter at some admitted profile
One profitable unilateral misreport
Frequent strategic voting
People behaving across observed elections
Behavioral data and a defined frequency measure
Existence of an opportunity neither estimates its frequency nor guarantees that anyone can recognize it before voting.
The theorem also does not supply a forecasting model. It contains no probability distribution over preference profiles, no account of polling accuracy, and no cost of calculating a useful report. It therefore cannot estimate how often voters will recognize or exploit opportunities. Nor does unilateral strategy-proofness automatically answer every question about organized coalitions of voters or candidate entry. Those questions change who acts and what action is available. A claim about coordinated withdrawals needs its own comparison even if the voting rule is already known to permit individual manipulation.
Restricted domains alter the guarantee being requested. If alternatives lie on a common line and each voter's preference falls away on either side of one favorite, the admitted profiles exclude some otherwise possible rankings. This is the single-peaked restriction, an assumption about preferences rather than an interface control in these lessons. Random selection also changes a deterministic theorem's problem, because lotteries need rules for comparing uncertain outcomes. Neither change by itself proves that the resulting institution meets every practical objective. It explains why the original assumptions must accompany the conclusion.
Different outputs, different requirements
Theorem
Output
Requirements relevant here
Arrow
Complete transitive social ranking
At least 3 alternatives, unrestricted domain, unanimity, pairwise independence, non-dictatorship
Gibbard-Satterthwaite
One deterministic winner
Unrestricted strict rankings, at least 3 alternatives, every alternative attainable, non-dictatorship, strategy-proofness
Each listed package is incompatible. Arrow is about aggregating an order; Gibbard-Satterthwaite is about protecting truthful reports in a single-winner choice rule.
Test your interpretation with two statements. The first says that no admitted profile permits a profitable unilateral misreport. That is strategy-proofness. The second says that most people in one survey reported their actual favorite. That is an empirical observation about a sample, even if measured perfectly. The observation cannot establish the universal property, and failure of the property cannot refute the observation. Keep the distinction when reading reform arguments. A theorem can exclude a perfect guarantee while leaving substantial differences in practical incentives open to investigation with behavioral evidence.
Use theorem statements with their assumptions
The Gibbard-Satterthwaite result concerns deterministic single-winner rules over unrestricted strict preferences with at least three possible outcomes, requiring every outcome to be attainable. Under those assumptions, a strategy-proof rule must be dictatorial. It does not say every voter can profitably manipulate every election.
Restricted preference domains, randomization, and different information or institutional goals change the mathematical problem. Impossibility results identify incompatible requirements under stated assumptions. They do not identify the best practical rule, estimate manipulation frequency, or justify ignoring every useful safeguard.
Trace coordination without forecasting it
Strategic responses can occur before any ballot reaches a counter. Suppose two small invented lists each receive 3 of 100 valid party votes and the eligibility threshold is 5% of all such votes. Each list fails separately. If the same six voters support a joint list and all other votes stay fixed, the joint list qualifies with 6%. This is a conditional arithmetic statement. A real agreement would also need decisions about candidates and list order, and some supporters might reject it. The original six votes cannot simply be assumed after coordination.
Eligibility is only the first stage. Passing 5% permits entry into allocation but does not guarantee a seat for every district magnitude. Compare votes 46, 30, 18, 6 under unmodified Sainte-Laguë. With three seats, the first awards go to A, B, and C, because their first quotients exceed D's 6 and A's next quotient is only 46/3. D qualifies yet receives no seat. With ten seats, the same eligible D can win one. A coordination incentive therefore depends on the allocation opportunity beyond the threshold itself.
To verify the ten-seat comparison on paper, begin with A, B, C, A, B, A, giving counts 3, 2, 1, 0. The next competing quotients are 46/7, 30/5, 18/3, and 6/1. A takes seat seven. For seat eight, B, C, and D all have exact quotient 6, while A has 46/9. The model's earlier-input priority gives B seat eight, then C seat nine, then D seat ten. Final counts are 4, 3, 2, 1. Rounded decimals must not replace these exact comparisons.
Passing a threshold still leaves seat competition
Stipulated party votes
5% eligibility
3-seat result
10-seat result
Separate 3% lists
Each fails
Not allocated here
Not allocated here
Joint votes 46 / 30 / 18 / 6
All qualify
1 / 1 / 1 / 0
4 / 3 / 2 / 1
Seats 8, 9, 10 in joint case
B: 30/5 = 6; C: 18/3 = 6; D: 6/1 = 6
Not applicable
B, then C, then D
Unmodified Sainte-Laguë, exact ties favor earlier input parties A, B, C, D. The joint case assumes all six small-list votes remain with the joint list. This is not a prediction of coordination.
The earlier threshold view offers a separate controlled exploration with its own fixed votes 46, 30, 20, 4. Select the large-district preset without a threshold, then increase only the threshold to 5%. D changes from eligible with one seat to ineligible with none. Return to no threshold and reduce only magnitude to three. D is now eligible but still has no seat. Record those two paths separately. They show different mechanisms behind an identical zero-seat result, rather than evidence that actual voters coordinated in response to either setting.
Coordination for a single office differs from coalition formation in a legislature. With seats A 4, B 3, C 3 in a ten-seat chamber, a majority requires six seats. A with B has seven, A with C has seven, and B with C has six. None is selected by the seat formula. Even knowing each party's vote total leaves bargaining preferences and institutional procedures unspecified. Calling one coalition inevitable would add behavioral assumptions that the allocation model does not contain. Majority arithmetic identifies feasible seat combinations, not agreements to govern.
Seat majorities do not select a governing agreement
Each rectangle is one seat. The bars compare possible two-party majorities on the same seat scale, not probabilities of cooperation.
An evidence-based prediction would connect these mechanisms to observations. It might ask whether voters knew the threshold, whether parties could legally coordinate nominations, and whether agreements changed participation. A causal claim would also need a credible comparison for what those same actors would have done without the institutional change. The handbook discusses design considerations, not measured effects for these invented parties. Separate that source role from the exact count. We can verify seat awards and majority combinations without predicting any real political response.
Exercises
A rising percentage with fewer voters
A party falls from forty votes out of one hundred to thirty-six out of eighty. What increased and what decreased?
Hint
Compute both the numerator change and the new fraction.
Worked answer
Measure
Change
Raw votes
−4
Relative change in raw votes
−10%
Vote share
+5 percentage points
Percentage change and percentage-point change are distinct.
Its raw count decreased by four votes, or 10% of its former count. Its share rose from 40% to 45%, an increase of five percentage points. Those observations do not identify which individuals changed behavior or why.
Do not overread a theorem
Someone says that a strategy-proof dictatorship theorem proves any ordinary voter can improve the result by lying in every election. What is wrong?
Hint
Distinguish a universal guarantee about a rule from every individual election instance.
Worked answer
Theorem-level claim
Not implied
Some profitable misreport exists under the assumptions
Every voter can always gain
No universal strategy-proof guarantee
Manipulation is frequent or easy
Quantifiers and assumptions are part of the result.
The theorem rules out a non-dictatorial deterministic rule satisfying all its assumptions and protecting everyone against profitable misreporting in every profile. It does not say every voter has such an opportunity in every profile, can recognize one, or knows others’ votes. It also does not assign a practical frequency or magnitude to the problem.
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.