New Mathematical Properties of the Least Square Value
The Least Square Values for cooperative TU games (briefly, LS-values) represent a family of solutions of the following family of optimization problems associated with a cooperative TU game: minimize the sum of weighted squares of deviations of the excesses from their average, on the preimputation set; the weights were positive numbers associated to the coalitions, depending only on their size. For each system of weights a LS-value is obtained by the Lagrange multipliers method; the Shapley value belongs to the family for specified weights. L. Ruiz, F. Valenciano and J. Zarzuelo who introduced this class of values (see), gave two axiomatic characterizations of the LS-values: one is based upon a set of axioms including linearity, efficiency, symmetry, inessential games additivity and weighted average marginal contribution monotonicity; the other is based upon a reduced game property relative to a Davis/Maschler type of reduced game (see ), and standardness for two person games. In the present paper, we start by showing that for whatever system of weights we can express the corresponding LS-value by means of average per capita formulas similar to those proved earlier by the author for the Shapley value (see ). These formulas provide an algorithm for computing in parallel the LS-values with one operation more than in the computation of the Shapley value and very little memory capacity. Theoretically, the same formulas show that for each system of weights the corresponding LS-value is the Shapley value of a game easily obtained from the given game. This fact suggests that the LS-values have more properties analogue to the Shapley value than those considered in . Therefore, we introduce a potential function for the LS-values and determine a potential basis for GN, the space of TU games with the set of players N, relative to the LS-values. The potential basis is allowing us to solve for LS-values what we call the inverse problem: given a vector [see pdf for notation] and a system of weights ?, find out an explicit formula giving all games v € GN for which the LS-value corresponding to ? is f. In particular, for f=0, we find the null space of the LS-value; the nullity equals 2n - n — 1. The inverse problem for the Shapley value, the weighted Shapley value and the Banzhaf value have been solved earlier by the author (see , , and ). Further, we introduce a new concept of consistency. Usually, a group of players in a game v € GN, who agreed upon a division rule, either leave the game with their payoffs or have a contract for getting later those payoffs. The set T C N of remaining players would like to play a game (the reduced game) in which the same division rule would offer the same payoffs as in the initial game. Then, the division rule is said consistent relative to the reduced game. Now, we allow to the set of remaining players to use a new division rule if this rule gives them in the reduced game the same payoffs as in the initial game. The pair of rules will be called consistent relative to the reduced game. To develop such a consistency scheme, we introduce a new value, associated with a system of weights, called the Extended Least Square Value, and define a new reduced game of Hart/Mas-Colell type (see  and ), and prove that the pair (LS, ELS) is consistent relative to this reduced game. Then, we give an axiomatic characterization of ELS-values based upon the reduced game axiom and a weighted standardness axiom.