The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector

Abstract

Abstract:The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoffvector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.

The least square nucleolus is a normalized Banzhaf value

“…We are able to show that it corresponds to the Least Square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem (Ruiz et al, 1996). Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus.…”

“…(N, v) and it is an imputation. Then, by Ruiz et al (1996) it is the LSnucleolus. Now, assume that there is some l ∈ N with (N, v).…”

“…Formally, B a is the value on G defined for every (N, v) ∈ G and i ∈ N by Ruiz et al (1996) proved the equivalence between the additive normalization of the Banzhaf value and the Least Square prenucleolus, LS-prenucleolus. The LS-prenucleolus is defined for every (N, v) ∈ G as the optimal solution of the optimization problem…”

“…In order to solve this drawback, Ruiz et al (1996) defined the Least Square nucleolus, LS-nucleolus, for every (N, v) ∈ G as the optimal solution of the optimization problem…”

“…If the LS-prenucleolus is an imputation, it coincides with the LS-nucleolus, but in general, both concepts provide different allocations. Ruiz et al (1996) proposed the following algorithm to obtain the LS-nucleolus.…”