Entropy of capacities on lattices and set systems

We propose a definition for the entropy of capacities defined on lattices. Classical capacities are monotone set functions and can be seen as a generalization of probability measures. Capacities on lattices address the general case where the family of subsets is not necessarily the Boolean lattice of all subsets. Our definition encompasses the classical definition of Shannon for probability measures, as well as the entropy of Marichal defined for classical capacities. Some properties and examples are given.

💡 Research Summary

The paper addresses a fundamental gap in the theory of capacities (monotone set functions) by extending the notion of entropy from the classical Boolean setting to arbitrary lattices and set systems. Classical capacities are defined on the power set 2^N, which forms a Boolean lattice; the Shannon entropy for probability measures and the Marichal entropy for capacities are both special cases of an entropy defined on this lattice. However, many applications—such as decision making on hierarchical categories, cooperative games on restricted coalitions, or information aggregation on graph cliques—require a domain that is not a full Boolean lattice. The authors therefore propose a unified entropy definition that works for any finite lattice L equipped with a partial order ≤ and a distinguished bottom ⊥ and top ⊤ element.

Core Construction
The key idea is to replace the usual probability mass p_i, which corresponds to the increment of a measure when moving from a set to a larger set, with the “increment of a capacity” along each cover relation of the lattice. For any two elements x≺y (i.e., y covers x), the increment is Δv(x,y)=v(y)−v(x), where v:L→