A rule is manipulable at a profile when some voters can get a result they prefer by reporting a ranking other than their own. Under the Borda count, voters who rank a rival second can bury it at the bottom, and under plurality voters whose favourite cannot win can vote for their second choice. The Gibbard–Satterthwaite theorem says that with three or more candidates every non-dictatorial rule that can elect any candidate is manipulable somewhere. The wrong model is that strategic voting is lying about facts.
Core ideas
Every rule so far has been run on the voters' real rankings. This lesson asks what happens when the voters know the rule and know one another's rankings.
Voters are asked for a ranking of the candidates, which only they hold. A misreport is a submitted ranking that differs from that private one. No fact about the world is stated falsely, and nobody else can check it.
A group of voters gains from a misreport when the winner it produces is ranked above the sincere winner by those voters' true rankings. The question is always asked of the true ranking, never of the submitted one.
The lab takes each group of voters who share a ranking, lets them change together to every other ranking, and reports the first change that elects someone they rank above the sincere winner. It searches blocs, and a lone voter is a smaller case.
The material
A profile where the Borda count can be gamed, a profile where plurality can, and the one setting in which no misreport helps.
A voting rule is a procedure, and a procedure invites the question of whether it can be used against its own purpose. If the rule is run on the ballots the voters hand in, a voter who knows the rule and the likely ballots of the others can choose a ballot to get the best result, and the best ballot is not always the true one.
Take seven voters: two rank the candidates A B C, two rank them B C A, and three rank them C B A. Under the Borda count, with 2, 1 and 0 points, A gets 4, B gets 9 and C gets 8, so B wins. The three voters who rank C first prefer C to B, and C is 1 point short.
Suppose they report C A B, not C B A. They still give C its 2 points, but they now give B nothing and give A a point. If two of the three do this, B falls to 7, A rises to 6 and C keeps 8, so C wins; if all three do it, which is the switch the lab tries, the totals are 7, 6 and 8 and the winner is the same. Nothing about C's support increased; B's total was lowered by ranking it where it did not belong. This is burying, and the lab reports it as voters ranking C B A who gain by C A B.
A rule is manipulable at a profile when some group of voters can get an outcome they prefer, by their true rankings, by reporting rankings other than their true ones.
Burying works because the Borda count takes every position into account, and a voter controls where the rival is placed on their own ballot. Plurality offers a different route. Take seven voters: two rank the candidates A C B, three B C A and two C B A. Plurality elects B with 3 first places. The two voters who rank A first prefer C to B. Their favourite cannot win, but by reporting C A B they give C four first places and elect it, which they prefer to B.
That is the spoiler voter's dilemma: voting for your favourite and getting your worst, or voting for your second choice. The lab shows it when you set the rule to plurality on that profile. Under the Borda count the same profile elects C without any misreport, and the lab finds a different bloc that gains.
With two candidates there is nothing to bury and no one to switch to. A voter who ranks A above B can report that, or the reverse, and the reverse can only help B. With three voters at A B and two at B A, the lab finds no group that gains by misreporting, under any rule.
Suppose there are at least three candidates, and a rule picks one winner from every profile of rankings. If every candidate can win under some profile, and no one voter is a dictator, then there is some profile at which some voter gains by reporting a ranking other than their own.
The theorem is the companion of Arrow's, and its proof is not reproduced here. It says that manipulability is not a defect of plurality or Borda in particular. What it leaves open is how often a profile is manipulable, how much information the manipulators need, and how costly the manipulation is to find. The lab assumes the manipulators know every ballot, which is the assumption most favourable to the manipulators.
A note about reading the tile. “None found” means that no bloc of identical voters, switching together to any other ranking, elects a candidate they rank above the sincere winner on this one profile. It does not say the rule is safe.
Interactive
six voting rules, the majority relation, judgment aggregation and the jury theorem
Method
Five moves for a small profile. The method is a search, and the lab does the same search over every ranking.
Count the true ballots. This is the outcome that any manipulating group has to beat.
Choose those who rank the sincere winner below another candidate. A group that already has its favourite elected has nothing to gain.
Try the obvious ones: under Borda, push the sincere winner to the bottom; under plurality, put a second choice first.
Leave every other group's ballot as it was. If the new winner is one the group ranks above the sincere winner by their true ranking, they have gained.
Compare the new winner with the sincere one in the group's true order, not the reported one. A misreport that changes the winner to someone the group likes less is not a manipulation.
Two voters rank A B C, two B C A, three C B A. Borda gives 2, 1 and 0.
No one stated a false fact. The voters submitted a ranking they did not hold, and the rule treated it as sincere. One of the three alone cannot do it, because B would still tie C at 8; two of the three suffice.
Given a profile and a rule, name a group whose true ranking puts another candidate above the sincere winner, give the ranking they report to elect that candidate, and state what the theorem guarantees.
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