# On Decomposition of Solutions for Coalitional Games

## TL;DR

This paper explores how solutions to coalitional games can be decomposed into elementary components, providing a clearer understanding of solution structures like the core and Shapley value.

## Contribution

It introduces a novel decomposition framework that factors solutions through elementary game components, simplifying analysis of solution concepts.

## Key findings

- Decomposition maps have simple structures.
- Solutions can be expressed via elementary game components.
- Focus on polyhedral cones of zero-normalized games.

## Abstract

A solution concept on a class of transferable utility coalitional games is a multifunction satisfying given criteria of economic rationality. Every solution associates a set of payoff allocations with a coalitional game. This general definition specializes to a number of well-known concepts such as the core, Shapley value, nucleolus etc. In this note it is shown that in many cases a solution factors through a set of games whose members can be viewed as elementary building blocks for the solution. Two factoring maps have a very simply structure. The first decomposes a game into its elementary components and the second one combines the output of the first map into the respective solution outcome. The decomposition is then studied mainly for certain polyhedral cones of zero-normalized games.