The lattice of embedded subsets

In cooperative game theory, games in partition function form are real-valued function on the set of so-called embedded coalitions, that is, pairs $(S,\pi)$ where $S$ is a subset (coalition) of the set $N$ of players, and $\pi$ is a partition of $N$ containing $S$. Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.

💡 Research Summary

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The paper addresses a fundamental gap in the theory of cooperative games in partition function form (PFF). While such games are defined on “embedded coalitions” – pairs ((S,\pi)) where (S\subseteq N) is a coalition and (\pi) is a partition of the player set (N) that contains (S) as one of its blocks – the literature has never provided a rigorous order structure on these objects. Consequently, different authors have used incompatible notions of inclusion, marginal contribution, and solution concepts, which hampers a unified treatment of PFF games.

The authors begin by formalising the set of embedded subsets as
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