How rational voters game the rules — and how agenda-setters engineer outcomes even when all votes are sincere.
Three Modes of Voting
Sincere Voting
Vote for the most-preferred option at each stage, regardless of what others do or what the likely outcome is. True preferences are revealed directly.
Strategic Voting
Vote against a nominal preference when doing so produces a better reachable outcome — given what other voters are expected to do. Common in single-round elections.
Sophisticated Voting
A form of strategic voting specific to sequential agendas. Apply backward induction over the full sequence: vote against an immediate preference at an early stage to secure a better final outcome.
Gibbard–Satterthwaite Theorem
The Theorem. For any group choosing among at least three alternatives: if the voting rule is non-dictatorial and admits any preference ordering, then there will always exist some situation in which at least one voter has an incentive to misrepresent her true preferences — that is, to vote strategically.
Corollary: No voting rule is immune to manipulation. Changing the rule does not eliminate strategic incentives — it only changes who benefits and how.
Scenario. A student union elects a president by plurality voting (first-past-the-post): each voter casts one vote; the candidate with the most votes wins. There are three candidates — A (left), B (centre), C (right) — and three voter groups of roughly equal size.
True Preferences
Group
1st choice
2nd choice
3rd choice
Share
Progressives
A
B
C
34%
Moderates
B
A
C
33%
Conservatives
C
B
A
33%
If all vote sincerely: A gets 34%, B 33%, C 33% → A wins.
Yet B is the Condorcet winner: B beats A (66–34) and beats C (67–33) in direct pairwise votes. B is the majority-preferred candidate — but does not win under sincere plurality voting.
Who has an incentive to vote strategically?
Conservatives (true order: C ≻ B ≻ A): their first choice C cannot win — tied at 33% while A leads at 34%. If they vote sincerely, C and B split the vote and A wins — their worst outcome.
Strategic move: Conservatives abandon C and vote B instead → B gets 66%, A 34%, C 0% → B wins.
Voting for their second choice is rational: it prevents their worst outcome. This is the logic of the wasted vote — under plurality, a vote for an unviable candidate has no effect on the outcome.
Simulate the Election
Choose how each group votes, then run the election.
Select voting behaviour above and click Run Election.
Agenda Control: The ability to determine which alternatives are considered and in what order. Whoever sets the agenda can often engineer their preferred outcome — even when all voters vote sincerely. Agenda power is most potent when a majority cycle exists: because every alternative can be beaten by some other, the agenda setter can work backwards from their desired winner to construct a sequence that delivers it.
Voting Agenda: The sequence in which alternatives are paired and voted on. For example, agenda ABC means A vs B first, then the winner vs C. Different agendas applied to the same preferences can — and under a cycle, will — produce different winners.
Example — You are the agenda setter
Pick which two alternatives face off first — the winner then faces the third. Same preferences, different order, different outcome.
Using the classic Condorcet cycle: A beats B (2–1), B beats C (2–1), C beats A (2–1). Under a cycle, whoever controls the agenda controls the outcome — each of the three possible agendas produces a different outcome.
Step 1 — Pick your Round 1 matchup
Results
Scenario (four alternatives). A committee of three equal factions must choose among four budget options: W (Welfare), X (Research), Y (Teaching), Z (Status quo). Voting proceeds by binary majority rule: two alternatives are compared at a time; the winner advances to face the next.
Faction Preferences
Faction
1st
2nd
3rd
4th
F1 — Welfare
W
X
Y
Z
F2 — Research
X
Y
Z
W
F3 — Teaching
Y
Z
W
X
These preferences produce a majority rule cycle: W beats X (F1+F3), X beats Y (F1+F2), Y beats Z (F1+F3), Z beats W (F2+F3) — so W ≻ X ≻ Y ≻ Z ≻ W. There is no Condorcet winner. The outcome depends entirely on the order of votes.
Build a Voting Agenda
Click alternatives in order to set the agenda, then run the vote.
Each round: the first alternative faces the second; the winner faces the third; and so on. Select as many alternatives as you like — the vote runs only over those selected, in the order you chose.
Agenda: —
Set an agenda and run the vote to see the result.
What this shows. Any of the four alternatives can be made the winner by choosing the right sequence — even though all votes are sincere. Try the following agendas to verify:
· X → W → Y → Z → Y wins (F3's preferred agenda)
· Y → Z → X → W → W wins (F1's preferred agenda)
· W → Y → Z → X → X wins (F2's preferred agenda)
· X → Y → W → Z → Z wins (status quo preserved)
The outcome is not determined by preferences alone — it is determined by who sets the agenda.