Capacities and Games on Lattices: A Survey of Results

Abstract

We provide a survey of recent developments about capacities (or fuzzy measures) and ccoperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.

Computing Shapley Effects for Sensitivity Analysis

“…One can interpret the Shapley value either as payoff from joining a coalition or from leaving the anti-coalition. An alternative route to computation is offered by the formula of [10,23], for which one needs to calculate the Möbius inverses mob(α) of the value functions val(α). These are defined implicitly by val(α) = β⊆α mob(β) and therefore mob(α) = β⊆α (−1) |α|+|β| val(β) [21,Chap.…”

Comments on: Transversality of the Shapley value

“…There is a book by Bilbao (2000) entirely devoted to this topic, gathering many of his personal works on the topic. I also wrote a recent survey for games defined on lattices (Grabisch 2006), with emphasis on the Shapley value (see also Grabisch and Lange 2007). Apart mathematical fun, the interest of these works is twofold: first, they can be used to define games where the actions a player can take are described in a more sophisticated way than the simple participation/non participation alternative.…”