Topics: Game theory, Nash equilibrium, Non-cooperative game Pages: 169 (38920 words) Published: March 10, 2013
Network Connection Games
Near-Optimal Networks designed with Selfish Agents
Martin Hoefer
Institut f¨r Informatik
Technische Universit¨t Clausthal
September 2004



Prof. Dr. Ingbert Kupka, TU Clausthal
Prof. Dr. Klaus Ecker, TU Clausthal
Dr. Piotr Krysta, Universit¨t Dortmund

Hiermit versichere ich, die vorliegende Arbeit selbst¨ndig verfaßt und keine anderen als die angea gebenen Quellen und Hilfsmittel benutzt zu haben.

Clausthal-Zellerfeld, 20. September 2004

Martin Hoefer

I am indebted to a number of people, who supported me in the preparation of this work, especially
• Dr. Piotr Krysta, for suggesting the topic, for scientifically supporting the thesis, and enabling me to stay in Dortmund. I really enjoyed the numerous hours we spent discussing. • Prof. Dr. Ingbert Kupka and Prof. Dr. Klaus Ecker for their encouragement, their willingness to review this thesis, and their great support throughout the years of my studies. • Dr. Jan Preusser, for proof-reading this work, his valuable advice, and for his friendship.

• my family, for supporting me in my work, my studies, my hobbies, and everything else I do in my life.
• my friends, who were there to tolerate and endure the talking about games, players and algorithms, and who took care of distracting me successfully when it was necessary. Thank you !


This thesis is concerned with the study of network creation games - a new research direction in the area of algorithmic game theory. A simple model called the connection game is considered, which was recently introduced by Anshelevich et al [3]. In this game selfish agents are to build or maintain a network. Every agent has a certain connectivity requirement, i.e. a set of terminals that he strives to connect with a network. Possible edges have costs, and the goal of the agent is to minimize his payments. In general it is NP -complete to determine, whether a given game has a Nash equilibrium. Instead a problem is considered, in which one needs to find cost allocations of socially optimal networks. The allocations shall give any player the least incentive to remove his payments from the network. The results of this thesis are:

• For games with 2 players 2-approximate Nash equilibria exist. New combinatorial insights suggest that this continues to hold for the general case for any number of players and 2 terminals per player.

• The algorithm presented in [3] for the general case was shown to compute 3-approximate Nash equilibria. In this thesis the approximation guarantee is shown to be tight. • If edge costs are given by the Euclidean norm, small games with 2 players and 2 terminals per player allow cheap (1 + )-approximate Nash equilibria. This does not hold for games with more players.

• Important tools established for the analysis of connection games fail to provide significant insight for more elaborate network creation games.


In dieser Arbeit werden Netzwerkverbindungsspiele betrachtet, ein neues Forschungsthema im Bereich der algorithmischen Spieltheorie. Es wird ein einfaches Modell (connection game) n¨her a
untersucht, das vor kurzem von Anshelevich et al [3] formuliert und analysiert wurde. Darin wird modelliert, wie unabh¨ngige, eigenn¨tzige Agenten ein Netzwerk erstellen oder betreiben. Jeder a
Agent besitzt eine Menge von Terminals und hat einen Verbindungsbedarf, d.h. er versucht, seine Terminals durch ein Netzwerk zu verbinden. M¨gliche Kanten m¨ssen bezahlt werden, und jeder o
Agent versucht, seine Kosten zu minimieren. Das Entscheidungsproblem, ob ein Verbindungsspiel ein Nash Equilibrium zul¨ßt, ist NP -vollst¨ndig. Stattdessen wird ein anderes Problem betrachtet. a
Die Kosten eines f¨r die Gesamtheit der Terminals optimalen Netwerkes sollen auf die Spieler u
verteilt werden. Dabei soll jedem Spieler m¨glichst wenig Anlaß gegeben werden, vom...
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