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.

翻译：我们研究在策略性代理人之间公平分配不可分割物品的问题。Amanatidis等人表明，诚实性与任何有意义的公平概念都不相容。因此，我们采用激励比率的概念，定义为在给定机制下，代理人通过操纵能获得的最大可能效用与其诚实行为下的效用之比。我们选取离散公平分配文献中四种最基础的机制——轮询机制、Plaut和Roughgarden的切分选择机制、最大纳什福利机制和嫉妒图程序——并对其及其变体的激励比率进行了广泛研究。对于轮询机制，我们建立了在加性和次加性可取消估价下的激励比率为2，在可取消估价下无界，在次模和XOS估价下分别为n和⌈m/n⌉。此外，对于提供1/n近似最大社会福利保证的变体，激励比率无界。对于Plaut和Roughgarden的算法，激励比率在使用词典序打破平局时要么无界要么为3，在使用福利最大化打破平局时为2。这种分离展示了打破平局规则在设计低激励比率机制中的关键作用。对于最大纳什福利机制，激励比率无界。而且，通过限制代理人对每个物品有严格正价值可以绕过这种无界性。对于嫉妒图程序，两种可能的实现方式都导致无界激励比率。最后，我们证明了满足至多一个物品无嫉妒性的每个机制的激励比率至少为1.074，因此恒大于1。