Efficiency Axioms for simplicial complexes

Abstract: We study the notion of efficiency for cooperative games on simplicial complexes. In such games, the grand coalition $[n]$ may be forbidden, and, thus, it is a non-trivial problem to study the total number of payoff $v_{\Delta}$ of a cooperative game $(\Delta, v)$. We address this question in the more general setting, by characterizing the individual values that satisfy the general efficient requirement $v_{\Delta}^{gen}$ for a generic efficiency assignment. The traditional and the probabilistic efficiency are treated as a special case of this general efficiency. Finally, we introduce a new notion of efficiency arising from the combinatorial and topological property of the simplicial complex $\Delta$. The efficiency in this scenario is called simplicial and we characterize the individual values fulfilling this constraint.