Credibility of group manipulation in random assignments

For the random assignment problem, we introduce the notion of collusion-proofness that happens if the rule is immune to any collusion on misreporting preferences as it is either not profitable to form or profitable to betray. We characterize collusion-proofnessand prove that an assignment rule is collusion-proof if and only if it is strategy-proof and non-bossy. Therefore, the random serial dictatorship rule is immune to any collusive agreement between agents on preference misrepresentation, though it is not (weakly) group strategy-proof. This result gives us a possibility between collusion-proofness, ex-post Pareto efficiency, and equal treatment of equals. Furthermore, we proved that a large family of strategy-proof mechanisms called rank exchange mechanisms, à la Shende and Purohit (2023), are collusion-proof but group manipulable.