Capacities satisfying the buoyancy property on average

Buoyancy and its dual property antibuoyancy relate to the Pigou-Dalton principal of progressive transfers. When capacities satisfying the antibuoyancy property are used in Choquet integration, the effective weights applied are decreasing with respect to the inputs when arranged in increasing order. As well as this property having useful interpretations for measures of welfare and species diversity, it also allows for efficient solutions when the Choquet integral is used as an objective function in optimisation. In this contribution, we explore two weaker properties, which can be interpreted in terms of overall behavior with the capacity being buoyant or antibuoyant on average. One of these is expressed simply in terms of the Shapley interaction indices and amounts to buoyancy holding on average between pairs of weights, while the other looks at buoyancy of the cardinality indices. We explore the relationship between these and other capacity properties, formulate linear constraints that can be used when learning capacity weights from data, propose some random generation methods and finally present some numerical experiments to validate their usefulness.