Dominance of capacities by k-additive belief functions

Abstract

In this paper we deal with the set of $k$-additive belieffunctions dominating a given capacity. We follow the lineintroduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general $k$-additive measures.First, we show that the conditions for the general $k$-additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid. On the other hand, we show that these conditions cannot be improved.We solve this situation by imposing additional constraints on the dominating functions. Then, we consider the more restrictive case of $k$-additive belief functions. In this case, a similar result with stronger conditions is proved. Although better, this result is not completely satisfactory and, as before, the conditionscannot be strengthened. However, when the initial capacity is a belief function, we find a subfamily of the set of dominating $k$-additive belief functions from which it is possible to derive any other dominant $k$-additive belief function, and such that theconditions are even more restrictive, obtaining the natural extension of the result for probabilities. Finally, we apply these results in the fields of Social Welfare Theory and Decision Under Risk.Linear programming, decision analysis, capacity,dominance, k-additivity, belief functions