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Cooperation Tendencies and Evaluation of Games
Francesc Carreras
1
and Maria Albina Puente
2
1
ETSEIAT, Technical University of Catalonia, Colom 11, Terrassa, Spain
2
EPSEM, Technical University of Catalonia, Manresa, Spain
Keywords:
Cooperative Game, Shapley Value, Probabilistic Value, Binomial Semivalue.
Abstract:
Multinomial probabilistic values were first introduced by one of us in reliability and later on by the other,
independently, as power indices. Here we study them on cooperative games from several viewpoints, and es-
pecially as a powerful generalization of binomial semivalues. We establish a dimensional comparison between
multinomial values and binomial semivalues and provide two characterizations within the class of probabilis-
tic values: one for each multinomial value and another for the whole family. An example illustrates their use
in practice as power indices.
1 INTRODUCTION
Weber’s general model for assessing cooperative
games (Weber, 1988) is based on probabilistic val-
ues, a family of values axiomatically characterized by
means of linearity, positivity, and the dummy player
property. Every probabilistic value allocates, to each
player in each game of its domain, a weighted (con-
vex) sum of the marginal contributions of the player
in the game. These allocations can be interpreted as a
measure of players’ bargaining relative strength. The
most conspicuous member of this family (in fact, the
inspiring one) is the Shapley value (Shapley, 1953).
In the present paper we study a subfamily of proba-
bilistic values that we call multinomial (probabilistic)
values.
1
Technically, their main characteristic is the
systematic generation of the weighting coefficients in
terms of a few parameters (one parameter per player).
Our research group has been studying semival-
ues (Dubey et al., 1981), a subfamily of probabilis-
tic values characterized by anonymity and including
the Shapley value as the only efficient member. In
the analysis of certain cooperative problems we have
successfully used binomial semivalues (Freixas and
Puente, 2002), a monoparametric subfamily that in-
cludes the Banzhaf value (Owen, 1975).
From this experience, we feel that multinomial
1
They were introduced in reliability (Freixas and
Puente, 2002) with the name of “multibinary probabilistic
values” and independently defined for simple games only
—i.e. as power indices— in a work on decisiveness (Car-
reras, 2004), where they were called “Banzhaf α–indices.”
values (n parameters, n being the number of players)
offer a deal of flexibility clearly greater than binomial
semivalues (one parameter), and hence many more
possibilities to introduce additional information when
evaluating a game. Fig. 1 describes the relationships
between the above values and families of values and
the main characteristics of each one of them.
Probabilistic values provide tools to study not
only games in abstracto (i.e. from a merely structural
viewpoint) but also the influence of players’ person-
ality on the issue. They are assessment techniques
that do not modify the game but only the criteria by
which payoffs are allocated. Parameters will be ad-
dressed here to cope with different attitudes the play-
ers may hold when playing a given game, even if they
are not individuals but countries, enterprises, parties,
trade unions, or collectivities of any other kind. We
will attach to parameter p
i
the meaning of generical
tendency of player i to form coalitions, assuming p
i
and p
j
independent of each other if i 6= j.
Multinomial values are a consistent alternative or
complement to classical values (Shapley, Banzhaf).
They represent a wide generalization of binomial
semivalues, whose monoparametric condition implies
a quite limited capability of analysis of cooperation
tendencies. Of course, these tendencies can neither
be analyzed, without modifying the game, by means
of the classical values, which can be concerned only
with the structure of the game.
The organization of the paper is as follows. Sec-
tion 2 includes a minimum of preliminaries. In Sec-
tion 3, we introduce multinomial values. Section 4
415
Carreras F. and Puente M..
Cooperation Tendencies and Evaluation of Games.
DOI: 10.5220/0004414104150422
In Proceedings of the 15th International Conference on Enterprise Information Systems (ICEIS-2013), pages 415-422
ISBN: 978-989-8565-59-4
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)