It's Not All Black and White: Degree of Truthfulness for Risk-Avoiding Agents

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.