Resumen de Bilateral Coalescence and Superadditivity of Partition Functions
Ben McQuillin
I consider transformations on the partition function of an underlying cooperative game that correspond with coalitional bargaining procedures entailing periodic (sequential) opportunities for bilateral coalescence. I define two notions of superadditivity (bilateral superadditivity and multilateral superadditivity) on games in partition function form. They equate for games in characteristic function form, but mulitilateral superadditivity is stronger for games in partition function form. Both notions have the same intuitive appeal with respect to partition functions as superadditivity (conventionally defined) has for games in characteristic function form. But I show that the transformation of the partition function associated with any coalitional bargaining procedure entailing periodic (sequential) opportunities for bilateral coalescence fails to preserve the bilateral superadditivity of all underlying partition functions. Furthermore, it looks unlikely that we can find a useful sub-class of games in partition function form such that a property I call �meta-superadditivity� (superadditivity of the underlying partition function transformed by any t periods of coalitional bargaining) holds for some such coalitional bargaining procedure as the number of players becomes large. This result suggests that there is a conflict between superadditivity as an intuitively appealing characteristic of cooperative games, and sequential bilateral coalescence as an descriptively appealing procedure for coalition formation. It also means that it is harder than would otherwise have been the case to corroborate cooperative game solutions that rest on a linearity axiom with non-cooperative game solutions. My paper has a close bearing on Gul�s (1989, 1999) non-cooperative foundation for the Shapley value, which rests on meta-superadditivity (�value additivity� in Gul�s papers). It transpires that procedures, such Gul�s, that satisfy the null-player axiom in their solution, are less likely than others to satisfy the meta-superadditivity condition. And the supermodularity (or convexity) condition imposed by Gul (1999) on the underlying game (so that value additivity equates to efficiency) reduces the likelihood of meta-superadditivity further.
