Incentive Analysis of Collusion in Fair Division

We study fair division problems with strategic agents capable of gaining advantages by manipulating their reported preferences. Although several impossibility results have revealed the incompatibility of truthfulness with standard fairness criteria, subsequent works have circumvented this limitation through the incentive ratio framework. Previous studies demonstrate that fundamental mechanisms like Maximum Nash Welfare (MNW) and Probabilistic Serial (PS) for divisible goods, and Round-Robin (RR) for indivisible goods achieve an incentive ratio of $2$, implying that no individual agent can gain more than double his truthful utility through manipulation. However, collusive manipulation by agent groups remains unexplored. In this work, we define strong group incentive ratio (SGIR) and group incentive ratio (GIR) to measure the gain of collusive manipulation, where SGIR and GIR are respectively the maximum and minimum of the incentive ratios of corrupted agents. Then, we tightly characterize the SGIRs and GIRs of MNW, PS, and RR. In particular, the GIR of MNW is $2$ regardless of the coalition size. Moreover, for coalition size $c \geq 1$, the SGIRs of MNW and PS, and the GIRs of PS and RR are $c + 1$. Finally, the SGIR of RR is unbounded for coalition size $c \geq 2$. Our results reveal fundamental differences of these three mechanisms in their vulnerability to collusion.