The classic notion of \emph{truthfulness} requires that no agent has a profitable manipulation -- an untruthful report that, for \emph{some} combination of reports of the other agents, increases her utility. This strong notion implicitly assumes that the manipulating agent either knows what all other agents are going to report, or is willing to take the risk and act as-if she knows their reports. Without knowledge of the others' reports, most manipulations are \emph{risky} -- they might decrease the manipulator's utility for some other combinations of reports by the other agents. Accordingly, a recent paper (Bu, Song and Tao, ``On the existence of truthful fair cake cutting mechanisms'', Artificial Intelligence 319 (2023), 103904) suggests a relaxed notion, which we refer to as \emph{risk-avoiding truthfulness (RAT)}, which requires only that no agent can gain from a \emph{safe} manipulation -- one that is sometimes beneficial and never harmful. Truthfulness and RAT are two extremes: the former considers manipulators with complete knowledge of others, whereas the latter considers manipulators with no knowledge at all. In reality, agents often know about some -- but not all -- of the other agents. This paper introduces the \emph{RAT-degree} of a mechanism, defined as the smallest number of agents whose reports, if known, may allow another agent to safely manipulate, or $n$ if there is no such number. This notion interpolates between classic truthfulness (degree $n$) and RAT (degree at least $1$): a mechanism with a higher RAT-degree is harder to manipulate safely. To illustrate the generality and applicability of this concept, we analyze the RAT-degree of prominent mechanisms across various social choice settings, including auctions, indivisible goods allocations, cake-cutting, voting, and two-sided matching.

翻译：经典的\emph{真实性}概念要求任何主体均无法通过不真实的报告（即针对\emph{某些}其他主体的报告组合能提升其自身效用的虚假报告）获得收益。这一强概念隐含地假设操纵主体要么知晓所有其他主体将报告的内容，要么愿意承担风险并表现得仿佛知晓其报告。在缺乏其他主体报告信息的情况下，大多数操纵行为具有\emph{风险性}——它们可能在某些其他主体的报告组合下降低操纵者的效用。相应地，近期一篇论文（Bu, Song and Tao, ``On the existence of truthful fair cake cutting mechanisms'', Artificial Intelligence 319 (2023), 103904）提出了一种宽松的概念，我们称之为\emph{风险规避真实性（RAT）}，该概念仅要求任何主体均无法通过\emph{安全}操纵（即有时有益且绝无害处的操纵）获利。真实性与RAT是两个极端：前者考虑完全知晓他人信息的操纵者，而后者考虑完全不知晓他人信息的操纵者。现实中，主体通常了解部分（而非全部）其他主体的信息。本文引入了机制的\emph{RAT度}，定义为：若已知其报告可能使另一主体进行安全操纵的最小主体数量；若不存在这样的数量，则定义为$n$。这一概念在经典真实性（度为$n$）与RAT（度至少为$1$）之间建立了连续谱：具有更高RAT度的机制更难以被安全操纵。为阐明该概念的普适性与适用性，我们分析了多种社会选择场景中重要机制的RAT度，包括拍卖、不可分割物品分配、蛋糕切割、投票及双边匹配。