  Despite many distributed resource allocation (DRA) algorithms have been
reported in literature, it is still unknown how to allocate the resource
optimally over multiple interacting coalitions. One major challenge in solving
such a problem is that, the relevance of the decision on resource allocation in
a coalition to the benefit of others may lead to conflicts of interest among
these coalitions. Under this context, a new type of multi-coalition game is
formulated in this paper, termed as resource allocation game, where each
coalition contains multiple agents that cooperate to maximize the
coalition-level benefit while subject to the resource constraint described by a
coupled equality. Inspired by techniques such as variable replacement, gradient
tracking and leader-following consensus, two new kinds of DRA algorithms are
developed respectively for the scenarios where the individual benefit of each
agent explicitly depends on the states of itself and some agents in other
coalitions, and on the states of all the game participants. It is shown that
the proposed algorithms can converge linearly to the Nash equilibrium (NE) of
the multi-coalition game while satisfying the resource constraint during the
whole NE-seeking process. Finally, the validity of the present allocation
algorithms is verified by numerical simulations.
