Monotone cooperative games and their threshold versions

Cooperative games provide an appropriate framework for fair and stable resource allocation in multiagent systems. This paper focusses on monotone cooperative games, a class which comprises a variety of games that have enjoyed special attention within AI, in particular, skill games, connectivity games, flow games, voting games, and matching games. Given a threshold, each monotone cooperative game naturally corresponds to a simple game. The core of a threshold version may be empty, even if that is not the case in the monotonic game itself. For each of the subclasses of monotonic games mentioned above, we conduct a computational analysis of problems concerning some relaxations of the core such as the least-core and the cost of stability. It is shown that threshold versions of monotonic games are generally at least as hard to handle computationally. We also introduce the length of a simple game as the size of the smallest winning coalition and study its computational complexity in various classes of simple games and its relationship with computing core-based solutions. A number of computational hardness results are contrasted with polynomial time algorithms to compute the length of threshold matching games and the cost of stability of matching games, spanning connectivity games, and simple coalitional skill games with a constant number of skills.