We consider the situation that the grand set of all players, in a cooperative
game in characteristic function form with a nonempty core, consists of
recognizable subsets of players. The underlying assumption is that the players
of each subset have something in common between themselves which is not shared
by the players of other subsets. For example, the members of these subsets are
distinguishable by their nationalities or by their geographic locations or by
some other market-related economic or social property. In addition to the
agreement of all players to play the game to reach a point in the core, the
members of each subset want to steer the iterative orbit of the bargaining
process to a point in the core that will reflect the strength of their
solidarity within their subset. This is why we call such subsets solidarity
sets. The solidarity means that players of a solidarity set are willing to
participate in coalitions that include only members of their solidarity set and
are not willing to participate in coalitions of other solidarity sets. The
question is how can such a game be played? Is there an iterative bargaining
scheme that will generate a sequence of payoffs that converges to a point in
the core which reflects the solidarity of the members of the solidarity sets?
We propose bargaining schemes that allow the members of each solidarity set to
perform, independently from other solidarity sets and, therefore,
simultaneously in time, a full but finite sweep of intercoalitional
collaboration. Then the end-points of all these sweeps are combined into a
single payoff vector via a convex combination wherein the coefficients of the
convex combination may reflect some a priori knowledge about the relative
strengths of the different solidarity sets. These bargaining schemes are
applied to a new type of cooperative game that we call solidarity cooperative
game (SCG).