Efficiency Axioms for simplicial complexes

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.Comment: 12 pages, 1 figur

Probabilistic values for simplicial complexes

In this manuscript, we define and study probabilistic values for cooperative games on simplicial complexes. Inspired by the work of Weber "Probabilistic values for games", we establish the new theory step by step, following the classical axiomatization, i.e. using the linearity axiom, the dummy axiom, etc. Furthermore, we define Shapley values on simplicial complexes generalizing the classical notion in literature. Remarkably, the traditional axiomatization of Shapley values can be extended to this general setting for a rather interesting class of complexes that generalize the notion of vertex-transitive graphs and vertex-homogeneous simplicial complexes. These combinatorial objects are very popular in the literature because of the study of Evasiveness Conjecture in Complexity Theory.Comment: 23 pages, 1 figur