The lattice of embedded subsets

TL;DR

This paper introduces a structured lattice framework for embedded subsets in cooperative game theory, unifying the understanding of games in partition function form and analyzing their properties.

Contribution

It defines a lattice structure on embedded coalitions, providing a unified framework for analyzing games in partition function form.

Findings

Established a lattice structure for embedded coalitions

Unified the analysis of partition function form games

Provided insights into properties of such games

Abstract

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.