Manipulation (COMSOC)

Definition

Manipulation ( COMSOC)

Manipulation is a false preference report that gives the reporting voter a strictly preferred outcome, holding all other reports fixed. For a social choice function $f$ and a truthful profile $R=({\succ }_i,R_{-i})$, voter $i$ can manipulate if

$$\exists {\succ }_i^{\prime }\in \mathcal{L}(C):\underset{\text{winner after false report}}{\underbrace{f({\succ }_i^{\prime },R_{-i})}}{\succ }_i\underset{\text{winner after truthful report}}{\underbrace{f({\succ }_i,R_{-i})}}.$$

Here $R_{-i}$ contains the other voters’ reports, and $\mathcal{L}(C)$ is the set of strict rankings. The improvement over truthful voting is evaluated using the voter’s true preference ${\succ }_i$, not the reported ranking ${\succ }_i^{\prime }$. It is a successful instance of strategic voting.

Coalitional manipulation

A group of voters coordinates false reports, keeping outsiders’ reports fixed, so that every member strictly prefers the resulting winner to the truthful winner. The plurality example uses this form.

Set-valued outcomes

For a social choice correspondence, the definition additionally needs a way to compare winner sets, or a specified tie-breaking rule. Rankings of individual alternatives alone do not determine whether one winner set is better than another.