Manipulable Voting Rule (COMSOC)

Definition

Manipulable Voting Rule ( COMSOC)

A voting rule is manipulable if some voter can obtain a strictly preferred outcome by reporting a false ranking while all other reports remain fixed. One successful manipulation suffices; not every voter or profile must permit one.

Single-winner output

For a rule $f$ returning one alternative, there exist a profile $R=({\succ }_i,R_{-i})$ of true rankings, a voter $i$ and a false report ${\hat{\succ }}_i\ne {\succ }_i$ such that

$$\underset{\text{winner after lying}}{\underbrace{f({\hat{\succ }}_i,R_{-i})}}{\succ }_i\underset{\text{winner when truthful}}{\underbrace{f({\succ }_i,R_{-i})}}.$$

The unchanged true ranking ${\succ }_i$ evaluates both outcomes. Singleton-valued outputs are compared through their sole alternatives.

Set-valued output

Fix an extension ${\succ }_i^{\mathrm{set}}$ of voter $i$‘s preferences to non-empty winner sets. Manipulability requires some profile, voter and false report satisfying

$$\lambda ({\hat{\succ }}_i,R_{-i}){\succ }_i^{\mathrm{set}}\lambda ({\succ }_i,R_{-i}).$$

The preference extension must be specified: rankings of individual alternatives alone do not determine whether one set is better than another. Alternatively, fix a tie-breaking rule and apply the single-winner definition to the resulting rule.