Geometry of Set Functions in Game Theory: Combinatorial and Computational Aspects

TL;DR

This thesis explores the geometric and combinatorial structures of cooperative game theory, introducing new polyhedral tools, algorithms, and theoretical insights to analyze core stability and coalition properties.

Contribution

It develops a new family of polyhedra called basic polyhedra, generalizes the Bondareva-Shapley Theorem, and links combinatorics with game theory to analyze core stability and coalition structures.

Findings

Introduction of basic polyhedra for coalition analysis

Generalization of Bondareva-Shapley Theorem for computational checks

Algorithms for core stability and coalition generation

Abstract

The main ambition of this thesis is to contribute to the development of cooperative game theory towards combinatorics, algorithmics and discrete geometry. Therefore, the first chapter of this manuscript is devoted to highlighting the geometric nature of the coalition functions of transferable utility games and spotlights the existing connections with the theory of submodular set functions and polyhedral geometry. To deepen the links with polyhedral geometry, we define a new family of polyhedra, called the basic polyhedra, on which we can apply a generalized version of the Bondareva-Shapley Theorem to check their nonemptiness. To allow a practical use of these computational tools, we present an algorithmic procedure generating the minimal balanced collections, based on Peleg's method. Subsequently, we apply the generalization of the Bondareva-Shapley Theorem to design a collection of algorithmic procedures able to check properties or generate specific sets of coalitions. In the next chapter, the connections with combinatorics are investigated. First, we prove that the balanced collections form a combinatorial species, and we construct the one of k-uniform hypergraphs of size p, as an intermediary step to construct the species of balanced collections. Afterwards, a few results concerning resonance arrangements distorted by games are introduced, which gives new information about the space of preimputations and the facial configuration of the core. Finally, we address the question of core stability using the results from the previous chapters. Firstly, we present an algorithm based on Grabisch and Sudh\"olter's nested balancedness characterization of games with a stable core, which extensively uses the generalization of the Bondareva-Shapley Theorem introduced in the second chapter. Secondly, a new necessary condition for core stability is described, based on the application ...