No-show Paradox (COMSOC)

Definition

No-show Paradox ( COMSOC)

A voting rule exhibits the no-show paradox if a voter can obtain a strictly preferred outcome by abstaining rather than reporting their true preferences, with all other votes unchanged. For a single-winner rule $f$ defined for varying numbers of voters, this means that some profile $R=({\succ }_i,R_{-i})$ satisfies

$$\underset{\text{winner without voter }i}{\underbrace{f(R_{-i})}}{\succ }_i\underset{\text{winner with voter }i}{\underbrace{f({\succ }_i,R_{-i})}}.$$

Here $R_{-i}$ removes voter $i$‘s ballot, and ${\succ }_i$ is their true preference ranking. Participating hurts this voter according to their own preferences; they benefit by not showing up.

Unlike manipulation by a false report, the voter submits no ballot. For a set-valued rule, comparing outcomes additionally requires a preference over winner sets or a fixed tie-breaking convention.