# Augmenting and Decreasing Systems

• J. M. Bilbao
• M. Ordóñez
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

## Abstract

This paper deals with cooperative games in which there exist a feasible coalition structure. Augmenting and decreasing systems are set systems specially introduced for analyzing certain situations of partial cooperation. They are dual structures in the sense that a coalition is feasible in one of them if and only if the complement is feasible in the another one, so if an augmenting system represents the feasible options of the players to cooperate in a game then its dual decreasing system analyze the options of the players out each coalition. Augmenting systems are generalizations of antimatroids and systems of connected subgraphs in a graph. This fact means to relate two interesting structures in games. We study the core and the Weber set for games on augmenting systems. Later, two very known classical solutions for games are defined on augmenting systems: the Shapley value and the Banzhaf one. Finally we leverage the duality relationship to analyze these values for decreasing systems.

## Keywords

Augmenting system Decreasing system Core Weber set Shapley value Banzahf value

90C27 90D12

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• J. M. Bilbao
• 1