Robust Popular Matchings

We study popularity for matchings under preferences. This solution concept captures matchings that do not lose against any other matching in a majority vote by the agents. A popular matching is said to be robust if it is popular among multiple instances. We present a polynomial-time algorithm for deciding whether there exists a robust popular matching if instances only differ with respect to the preferences of a single agent while obtaining NP-completeness if two instances differ only by a downward shift of one alternative by four agents. Moreover, we find a complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable.Comment: Appears in: Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2024

The Flow Game: Leximin and Leximax Core Imputations

Recently [Vaz24] gave mechanisms for finding leximin and leximax core imputations for the assignment game and remarked, "Within the area of algorithm design, the "right" technique for solving several types of algorithmic questions was first discovered in the context of matching and later these insights were applied to other problems. We expect a similar phenomenon here." One of the games explicitly mentioned in this context was the flow game of Kalai and Zemel [KZ82]. In this paper, we give strongly polynomial time mechanisms for computing the leximin and leximax core imputations for the flow game, among the set of core imputations that are captured as optimal solutions to the dual LP. We address two versions: 1. The imputations are leximin and leximax with respect to the distance labels of edges. 2. The imputations are leximin and leximax with respect to the product of capacities of edges and their distance labels.Comment: 10 page