Minimum coloring problem: the core and beyond

The minimum coloring game allows to model economic situations where costly and potentially conflicting tasks have to be executed. The primitives of the game are a graph (describing conflicts between the respective tasks) and a cost function (giving the cost of completing any number of pairwise conflicting tasks). This setting gives rise to a cooperative game where agents have to split the minimum cost of completing all tasks. Our study of the core allows to find a necessary and sufficient condition guaranteeing its non-vacuity; and we describe a subset of allocations that are always in the core (when it is nonempty). These allocations put weights on the largest incompatible groups, called maximum cliques. We then propose two cost sharing rules and their axiomatizations. The first rule assigns shares in proportion to the number of maximum cliques an agent belongs to; and the key property in its axiomatization prevents splitting and merging manipulations. The second rule is an extension of the well-known airport rule; and it requires every agent to pay a minimal fraction of the total cost.