Incentive Ratios for Fairly Allocating Indivisible Goods: Simple Mechanisms Prevail

We study the problem of fairly allocating indivisible goods among strategic agents. Amanatidis et al. show that truthfulness is incompatible with any meaningful fairness notions. Thus we adopt the notion of incentive ratio, which is defined as the ratio between the largest possible utility that an agent can gain by manipulation and his utility in honest behavior under a given mechanism. We select four of the most fundamental mechanisms in the literature on discrete fair division, which are Round-Robin, a cut-and-choose mechanism of Plaut and Roughgarden, Maximum-Nash-Welfare and Envy-Graph Procedure, and obtain extensive results regarding the incentive ratios of them and their variants. For Round-Robin, we establish the incentive ratio of $2$ for additive and subadditive cancelable valuations, the unbounded incentive ratio for cancelable valuations, and the incentive ratios of $n$ and $\lceil m / n \rceil$ for submodular and XOS valuations, respectively. Moreover, the incentive ratio is unbounded for a variant that provides the $1/n$-approximate maximum social welfare guarantee. For the algorithm of Plaut and Roughgarden, the incentive ratio is either unbounded or $3$ with lexicographic tie-breaking and is $2$ with welfare maximizing tie-breaking. This separation exhibits the essential role of tie-breaking rules in the design of mechanisms with low incentive ratios. For Maximum-Nash-Welfare, the incentive ratio is unbounded. Furthermore, the unboundedness can be bypassed by restricting agents to have a strictly positive value for each good. For Envy-Graph Procedure, both of the two possible ways of implementation lead to an unbounded incentive ratio. Finally, we complement our results with a proof that the incentive ratio of every mechanism satisfying envy-freeness up to one good is at least $1.074$, and thus is larger than $1$ by a constant.