Maximal and stable proper sub-coalitions in TU coalitional games

In this paper, we characterize the coalition worth function in a generalized TU coalitional game to explain the formation, size, and composition of two particular classes of sub-coalitions, defined as the maximal proper subcoalition (MPS) and stable proper sub-coalition (SPS). In the case of MPS, the coalition worth function provides sufficient incentive to each of some (but not all) players to form the coalition, not to integrate with player(s) outside the coalition, and prevents disintegration into a smaller coalition. We label this as the definition of MPS from a micro-perspective. On the other hand, SPS is defined based on the distribution of coalition worth among the members. We also define an MPS from the macro perspective, which is based on the sub-coalition (group) worth function. We show that in order to be an MPS, the coalition worth function has to be cohesive over the set of players forming the coalition and sub-additive over the set of players outside the coalition. However, this is only necessary but not sufficient for forming an MPS. Interestingly, we find that there can be TU coalitional games where there are MPS, but there is no maximal proper sub-coalition structure (MPSS). Next, we show that an MPS is always an SPS, but the converse is not true. Thus, the results of MPS automatically extend to SPS. Finally, we show that an MPSS and/or stable proper sub-coalition structure (SPSS) has higher welfare than the grand coalition.