2 research outputs found
Efficiency Axioms for simplicial complexes
We study the notion of efficiency for cooperative games on simplicial
complexes. In such games, the grand coalition may be forbidden, and,
thus, it is a non-trivial problem to study the total number of payoff
of a cooperative game .
We address this question in the more general setting, by characterizing the
individual values that satisfy the general efficient requirement
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 . 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