{"type":"ScholarlyArticle","title":"Leadership in Congestion Games: Multiple User Classes and Non-Singleton Actions","authors":[{"name":"Alberto Marchesi","slug":"alberto-marchesi","given":"Alberto","family":"Marchesi","affiliation":"Politecnico di Milano , Piazza Leonardo da Vinci 32 , Milano , Italy"},{"name":"Matteo Castiglioni","slug":"matteo-castiglioni","given":"Matteo","family":"Castiglioni","affiliation":"Politecnico di Milano , Piazza Leonardo da Vinci 32 , Milano , Italy"},{"name":"Nicola Gatti","slug":"nicola-gatti","given":"Nicola","family":"Gatti","affiliation":"Politecnico di Milano , Piazza Leonardo da Vinci 32 , Milano , Italy"}],"abstract":"We study the problem of finding Stackelberg equilibria in games with a massive number of players. So far, the only known game instances in which the problem is solved in polynomial time are some particular congestion games. However, a complete characterization of hard and easy instances is still lacking. In this paper, we extend the state of the art along two main directions. First, we focus on games where players' actions are made of multiple resources, and we prove that the problem is NP-hard and not in Poly-APX unless P = NP, even in the basic case in which players are symmetric, their actions are made of only two resources, and the cost functions are monotonic. Second, we focus on games with singleton actions where the players are partitioned into classes, depending on which actions they have available. In this case, we provide a dynamic programming algorithm that finds an equilibrium in polynomial time, when the number of classes is fixed and the leader plays pure strategies. Moreover, we prove that, if we allow for leader's mixed strategies, then the problem becomes NP-hard even with only four classes and monotonic costs. Finally, for both settings, we provide mixed-integer linear programming formulations, and we experimentally evaluate their scalability on both random game instances and worst-case instances based on our hardness reductions.","doi":"10.24963/ijcai.2019/69","url":"https://ijcai.org/proceedings/2019/69","pdf":"https://ijcai.org/proceedings/2019/0069.pdf","markdown":"https://ijcai.org/proceedings/2019/69.md","bibtex":"https://ijcai.org/proceedings/2019/bibtex/69","volume":{"slug":"2019","title":"Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2019","url":"https://ijcai.org/proceedings/2019/","year":2019,"location":"Macao","dates":"10-16 August 2019","editors":"Sarit Kraus","isbn":"978-0-9992411-4-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2019"},"section":"Agent-based and Multi-agent Systems","track":"Main track","number":69,"pages":{"first":485,"last":491},"keywords":["Agent-based and Multi-agent Systems: Algorithmic Game Theory","Agent-based and Multi-agent Systems: Noncooperative Games","Agent-based and Multi-agent Systems: Economic Paradigms, Auctions and Market-Based Systems"],"videos":[],"references":[{"position":1,"raw":"[Ackermann et al., 2008] H. 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