TITLE:
On the Coalitional Rationality and the Inverse Problem for Shapley Value and the Semivalues
AUTHORS:
Irinel Dragan
KEYWORDS:
Shapley Value, Banzhaf Value, Semivalues, Inverse Problem, Power Game, Power Core, Coalitional Rationality
JOURNAL NAME:
Applied Mathematics,
Vol.8 No.11,
November
24,
2017
ABSTRACT: In cooperative game theory, a central problem is to allocate fairly the win of
the grand coalition to the players who agreed to cooperate and form the grand
coalition. Such allocations are obtained by means of values, having some fairness
properties, expressed in most cases by groups of axioms. In an earlier
work, we solved what we called the Inverse Problem for Semivalues, in which
the main result was offering an explicit formula providing the set of all games
with an a priori given Semivalue, associated with a given weight vector. However,
in this set there is an infinite set of games for which the Semivalues are
not coalitional rational, perhaps not efficient, so that these are not fair practical
solutions of the above fundamental problem. Among the Semivalues,
coalitional rational solutions for the Shapley Value and the Banzhaf Value
have been given in two more recent works. In the present paper, based upon a
general potential basis, relative to Semivalues, for a given game and a given
Semivalue, we solve the connected problem: in the Inverse Set, find out a
game with the same Semivalue, which is also coalitional rational. Several examples
will illustrate the corresponding numerical technique.