The Degree of Strategy-Proofness for Risk-Averse Committee Selection
The classic notion of strategyproofness implicitly assumes that a manipulating agent either possesses complete knowledge of what all other agents are going to report, or is willing to take the risk and act as if they know these reports. To capture the profound uncertainty of real-world voters, recent work introduced \emph{risk-avoiding truthfulness (RAT)} and the \emph{RAT-degree}, which quantifies the exact number of known reports required for a manipulation to be strictly safe. While the RAT-degree has been analyzed in settings such as single-winner elections, its implications for multi-winner voting remain unexplored. In this paper, we bridge this gap by extending the RAT-degree framework to approval-based committee (ABC) selection, focusing initially on the prominent Proportional Approval Voting (PAV) rule. We establish tight bounds on its susceptibility to safe subset manipulations, proving that PAV is immune to superset risk-avoiding manipulations given knowledge of at most $f = \lfloor \frac{n}{k+1} \rfloor - 1$ voters, but vulnerable when $f = \lceil \frac{n}{k} \rceil$. Recognizing that this degree of immunity may be insufficient in practice, we explore how to enhance strategic robustness by relaxing the proportionality requirement. We introduce a novel parameterized generalization of PAV, the family of $d$-RPAV rules, which encapsulates this inherent trade-off: a higher parameter $d$ yields stronger truthfulness and strategic robustness at the expense of weaker, relaxed proportionality guarantees. Specifically, we establish a generalized tight lower bound, proving that $d$-RPAV is completely immune to safe manipulation given knowledge of at most $f = \lfloor \frac{dn}{k+2d-1} \rfloor - 1$ voters.