Computing the Strategy to Commit to in Polymatrix Games (Extended Version)

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games-- polymatrix like--which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies. Introduction Leadership games have recently received a lot of attention in the Artificial Intelligence literature, also thanks to their use in many real-world applications, e.g., security and protection (Basilico, De Nittis, and Gatti 2017; Kar et al. 2017a; Kar et al. 2017b). In principle, the paradigm is simple--one or more leaders commit to a potentially mixed strategy, the followers observe the commitments, and then they play their best-responses--, but it can be declined in many different ways. The crucial issue is the computational study of the problem of finding the best leaders' strategy. In this paper, we provide new computational complexity results and algorithms for games with one leader and two or more followers.