The Least Square Values and the Shapley Value
Irinel Dragan, University of Texas, Math., Arlington, Texas 76019-0408
dragan@uta.edu

A b s t r a c t

The Least Square Values of cooperative TU games (briefly
LS-values), were found as solutions of an optimization
problem on the preimputation set (M.Keane, 1969).  L.Ruiz,
F.Valenciano and J.Zarzuelo have axiomatized the LS-values
and discussed many properties (GEB, 1998). In earlier papers,
we have shown that the Banzhaf value (Dragan, 1996) and all
Semivalues (Dragan, 1999) are Shapley values of games easily
derived from a given game. In the present paper, we prove
that an LS-value of a given game is also a Shapley value of
a game easily derived from the given one. To do this we
derive an Average per capita formula for LS-values, very
similar to that obtained earlier for the Shapley value
(Dragan, 1991). A computational algorithm for Semivalues
follows and,  a theoretical result, the relationship with
the Shapley value is obtained. Further, a potential basis
for the space of  TU games is shown and from this, like in
the case of the Shapley value, we solve the inverse problem:
given an n- vector, find out the set of games for which the
LS-value equals the given vector. The inverse problem was
earlier solved for the Shapley value and the Weighted
Shapley value (Dragan, 1991).

Note: the full paper, will be available as a Technical
Report of this university within a month, and can be sent.