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Network Connection Games Near-Optimal Networks designed with Selfish Agents Diplomarbeit Martin Hoefer Institut f¨ ur Informatik Technische Universit¨ at Clausthal September 2004

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Network Connection GamesNear-Optimal Networks designed with Selfish Agents

Diplomarbeit

Martin Hoefer

Institut fur InformatikTechnische Universitat Clausthal

September 2004

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Erstgutachter: Prof. Dr. Ingbert Kupka, TU ClausthalZweitgutachter: Prof. Dr. Klaus Ecker, TU ClausthalBetreuer: Dr. Piotr Krysta, Universitat Dortmund

Hiermit versichere ich, die vorliegende Arbeit selbstandig verfaßt und keine anderen als die ange-gebenen Quellen und Hilfsmittel benutzt zu haben.

Clausthal-Zellerfeld, 20. September 2004 Martin Hoefer

Acknowledgement

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 enablingme 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 willingnessto 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 doin my life.

• my friends, who were there to tolerate and endure the talking about games, players andalgorithms, and who took care of distracting me successfully when it was necessary. Thankyou !

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Abstract

This thesis is concerned with the study of network creation games - a new research direction inthe 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 buildor maintain a network. Every agent has a certain connectivity requirement, i.e. a set of terminalsthat he strives to connect with a network. Possible edges have costs, and the goal of the agent isto minimize his payments. In general it is NP -complete to determine, whether a given game has aNash equilibrium. Instead a problem is considered, in which one needs to find cost allocations ofsocially optimal networks. The allocations shall give any player the least incentive to remove hispayments from the network. The results of this thesis are:

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

• The algorithm presented in [3] for the general case was shown to compute 3-approximateNash 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 terminalsper player allow cheap (1 + ε)-approximate Nash equilibria. This does not hold for gameswith more players.

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

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Zusammenfassung

In dieser Arbeit werden Netzwerkverbindungsspiele betrachtet, ein neues Forschungsthema imBereich der algorithmischen Spieltheorie. Es wird ein einfaches Modell (connection game) naheruntersucht, das vor kurzem von Anshelevich et al [3] formuliert und analysiert wurde. Darin wirdmodelliert, wie unabhangige, eigennutzige Agenten ein Netzwerk erstellen oder betreiben. JederAgent besitzt eine Menge von Terminals und hat einen Verbindungsbedarf, d.h. er versucht, seineTerminals durch ein Netzwerk zu verbinden. Mogliche Kanten mussen bezahlt werden, und jederAgent versucht, seine Kosten zu minimieren. Das Entscheidungsproblem, ob ein Verbindungsspielein Nash Equilibrium zulaßt, ist NP -vollstandig. Stattdessen wird ein anderes Problem betrachtet.Die Kosten eines fur die Gesamtheit der Terminals optimalen Netwerkes sollen auf die Spielerverteilt werden. Dabei soll jedem Spieler moglichst wenig Anlaß gegeben werden, vom optimalenNetzwerk abzuweichen. Die Resultate dieser Arbeit ergeben sich wie folgt:

• In jedem Verbindungsspiel fur 2 Spieler existieren 2-approximative Nash Equilibria. Neuekombinatorische Betrachtungen lassen die Vermutung zu, daß dies auch fur Spiele mit beliebigvielen Spielern und 2 Terminals pro Spieler gilt.

• Fur den Algorithmus in [3] wurde gezeigt, daß er fur jedes Spiel maximal ein 3-approximativesNash Equilibrium berechnet. Es gibt Spiele, auf denen dieser Faktor tatsachlich erreicht wird.

• Wenn das Verbindungsspiel in die euklidische Ebene verlegt wird, besitzen kleine Spiele fur2 Spieler mit je 2 Terminals gunstige (1 + ε)-approximative Nash Equilibria. Fur Spiele mitmehr Spielern ist dies nicht der Fall.

• Wichtige Werkzeuge zur Analyse von Verbindungsspielen lassen sich nicht auf erweiterte undkomplexere Varianten ubertragen.

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Contents

1 Introduction 91.1 The Connection Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Previous and Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.2 Outline and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Nash Equilibria and Prices of Anarchy . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Computational Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Steiner Tree Problems 212.1 The Steiner Tree Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Definition and Computational Hardness . . . . . . . . . . . . . . . . . . . . 212.1.2 Minimum Spanning Tree Heuristic . . . . . . . . . . . . . . . . . . . . . . . 232.1.3 The Greedy Loss Contracting Algorithm . . . . . . . . . . . . . . . . . . . . 24

2.2 The Geometric Steiner Tree Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.1 Hardness, Properties and Minimum Spanning Trees . . . . . . . . . . . . . 272.2.2 A polynomial time approximation scheme . . . . . . . . . . . . . . . . . . . 27

2.3 The Generalized Steiner Tree Problem . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Connection Games 333.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Connection Games for 2 Players and 2 Terminals . . . . . . . . . . . . . . . . . . . 34

3.2.1 Exact Nash Equilibria and Prices of Anarchy . . . . . . . . . . . . . . . . . 343.2.2 Approximate Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.3 Why Connection Sets? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Connection Games for 2 Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 General Connection Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.1 3-approximate Nash Equilibria with Connection Sets . . . . . . . . . . . . . 423.4.2 2-approximate Nash Equilibria with Deviation Sets . . . . . . . . . . . . . . 463.4.3 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Geometric Connection Games 514.1 Geometric Connection Games for 2 Players and 2 Terminals . . . . . . . . . . . . . 51

4.1.1 Exact Nash Equilibria and Prices of Anarchy . . . . . . . . . . . . . . . . . 514.1.2 Cheap Approximate Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Games with more Players and more Terminals . . . . . . . . . . . . . . . . . . . . 684.2.1 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.2 A Negative Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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6 CONTENTS

4.2.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Discussion 735.1 The 2-Connection Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 The Bounded Distance Connection Game . . . . . . . . . . . . . . . . . . . . . . . 755.3 Final remarks and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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List of Figures

2.1 Tight example for the MST-approximation. . . . . . . . . . . . . . . . . . . . . . . 232.2 New edges added in a loss contraction. . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Levels of squares and lines inside the bounding box. . . . . . . . . . . . . . . . . . 282.4 Location of level 1 squares after a dissection shift by (a, b). . . . . . . . . . . . . . 29

3.1 Connection game with 2 players and 2 terminals. . . . . . . . . . . . . . . . . . . . 343.2 Extending Tp and assigning costs of the path. . . . . . . . . . . . . . . . . . . . . . 413.3 Lower bound example for 2 players and k terminals per player. . . . . . . . . . . . 413.4 a) A decomposition of T ∗ into paths R(t); b) c) The paths Q(t) for a single player i. 443.5 Tightness examples for a 3-approximation using connection sets. . . . . . . . . . . 453.6 Two deviations cost as much as three connection sets. . . . . . . . . . . . . . . . . 473.7 Lower bound example for approximate Nash purchasing the optimal network. . . . 50

4.1 Geometric games with maximum price of anarchy. . . . . . . . . . . . . . . . . . . 524.2 d1 stays the cheapest connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Network types with one edge per connection set. . . . . . . . . . . . . . . . . . . . 554.4 Network types with connections sets consisting of two edges. . . . . . . . . . . . . 564.5 T ∗ and TE after removing e1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Behavior of the ratio r(0.5, α) for angles between 120 ◦ and 180 ◦. . . . . . . . . . . 574.7 Transformations increasing the upper bound for r. . . . . . . . . . . . . . . . . . . 584.8 Network structure to bound the ratio in games, for which d2 ∈ TE does not hit e3. 584.9 d′ is more profitable than d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.10 d and d′ are better than da and db. . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.11 Possible deviation types for player 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 654.12 Geometric lower bound example for approximate Nash purchasing the optimal net-

work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.13 Network after a) five improvement steps, b) player 1 has removed most parts of d1. 70

5.1 A game with no 2-connection sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 A game with no connection sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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8 LIST OF FIGURES

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Chapter 1

Introduction

The analysis of networks delivers a broad range of interesting questions, which are important formany areas in research and application domains. One of the unique artefacts and dynamic drivingforces in modern society is the Internet - a platform that offers unique opportunities in humanhistory in all parts of everyday life. Therefore an interesting research direction is to understandhow to analyze and influence the development of the Internet. For computer scientists, it isstraightforward to model and analyze the Internet as a graph, which allows to study networkingproblems. This has been done intensively for many networks and network problems during the lastdecades, which came from fields like Operations Research, Computer Science, and Mathematics.The results were applied to applications in domains like public traffic, supply chains, scheduling,programming, parallel computing, etc. However, unlike most of these networks the Internet as anetwork is not centrally built or maintained. Instead there are a number of decentralized publicand private parties that jointly create, develop and improve it. These parties will be called agents,autonomous systems or players. Each of them is primarily interested in achieving a limited goalfor itself and is assumed to act selfishly. This leads to a game-theoretic approach to study thebehavior of agents and the results of their actions on the network.In particular, an interesting question is to characterize the topology of the Internet. At present thetopology is still subject to change and development. A lot of independent players are concernedwith establishing and maintaining connections. Some of them like global players in business andindustry hold a lot of servers and connections throughout different parts of the world. Someinteresting questions with respect to the topology development of the Internet are the following:

• Why are certain links in the Internet topology established, and others are left unconnected?

• Who has an interest in building these links, and who will profit from it?

• Who has an interest in maintaining certain links?

• How do these interests change if the topology changes?

• Are players motivated to cooperate and to join their forces in the creation of a link?

• What would be the most profitable topology, and who is motivated to establish and maintainthe required connections ?

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10 CHAPTER 1. INTRODUCTION

Answers to some of these questions could be very important for improving the structure and thetopology of the Internet. Insights could be used to guide public authorities, telecommunicationsfirms and other parties concerned with building computer networks. They would eventually get anidea how to establish cheap networks that other selfish parties and companies will be motivatedto pay for, maintain and use. This could eventually lead to an improved topology. But perhapscheap networks are not possible to be stable - i.e. some parties might always be motivated to moveaway from a given network and establish other links. One can certainly think of a lot of differentresults that might yield important insights and implications. At present there is still only verylittle understanding about this development and the underlying dynamics.

In the process of network creation there are a lot of different issues to consider like building andmaintenance costs, bandwidth constraints, speed, fault-tolerance, security, etc. All these mightinfluence and complicate the decision of a company whether to build and maintain a certain con-nection or not. However, if one is striving to get some first glimpses on the dynamics of selfishnetwork creation, the models should be kept simple. Hence, in this work a very simple model ofnetwork creation will be considered. This network creation game is called connection game andhas been introduced by Anshelevich et al [3]. In a graph each agent has a specific connectivityrequirement, i.e. holds a number of terminals and wants to build a network that connects them.Every possible edge has a certain cost, and the agents goal is to pay as little as possible for hisconnection. Apart from the network creation aspect the game also provides insight for a central-ized authority on how selfish agents might be motivated to build and maintain certain parts ofan existing network. Agents have the possibility to share the cost of the edges, which provides aconcept of agent cooperation in the game. The analysis of this game involves game-theoretic aswell as graph-theoretic and computational concepts.

Since computer scientists started to work on research related to selfish agents and computationalproblems in the mid-1990s, there has been quite significant progress. Apart from the Internet thereexist a variety of application domains of game-theoretic frameworks for computational problems.Broadly speaking today there are two major research directions.On the one hand researchers consider computational problems in the presence of selfishly motivatedagents. Such settings are modelled with a non-cooperative game. In this context stable outcomes(e.g. networks, allocations, connections) of interactions of the non-cooperative selfish agents cor-respond to Nash equilibria [77, 82]. Typically, introducing the dynamics of selfish behavior givesrise to a variety of new issues. In particular, outcomes representing Nash equilibria can be muchmore costly than socially optimum solutions. Papadimitriou [84] refers to this cost increase dueto selfish behavior as the price of anarchy. The price of anarchy has been studied in a number ofgames dealing especially with networking issues, such as load balancing [25, 26, 69, 97], routing[20, 21, 95, 98, 101], facility location [31, 107], and flow control [2, 33, 102]. As Nash equilibriain games might behave quite differently, two types of the price of anarchy have been proposed inthe literature [3]: the optimistic price of anarchy measuring the best Nash equilibrium and thepessimistic price of anarchy measuring the worst Nash equilibrium compared to the optimal cen-tralized solution. Here the term price of anarchy will be used equivalently to the pessimistic priceof anarchy in accordance with a great majority of the present literature. More relevant backgroundon important results and developments in this field will be provided in section 1.2.The results on non-cooperative games and Nash equilibria build the basis for what is called com-putational mechanism design, the second major research direction. Mechanism design is a conceptfrom classic game theory with important applications in economics and business. An institution,

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1.1. THE CONNECTION GAME 11

called the mechanism, collects private information from the agents, determines an outcome, andspecifies a payment for each agent. The agents have private valuation functions, which determinethe value of the outcome to them. They are hoping to get an outcome, which maximizes theirpersonal utility, which depends on the valuation and the payments. The mechanism, however,tries to maximize some social goal, which can be contrary to the agents private interests. Thus,the agents can be motivated to report wrong or biased information to influence the decision ofthe mechanism. The use of this concept was motivated by the problem to design efficient proto-cols for Internet applications with selfish participants. Moreover, the concept can be applied tocomputational problems from a diverse set of domains like combinatorial auctions and e-commerce[29, 86], scheduling [81], broadcasting network design [39, 61, 12], or routing [38, 81]. In section1.3 some important developments in this field will be presented with a focus on network creationand multicast transmissions as these topics are closely related to network creation games.

This work will be pointed towards the first direction, i.e. it will be concerned with the connectiongame and present results about Nash equilibria and the price of anarchy. For most of this thesisit is assumed that the reader is familiar with some basic notions from the field of approximationalgorithms like the O-notation, NP -completeness or approximation factors. For an introduction tothese concepts see Hochbaum [58] or Vazirani [106]. Furthermore in some occasions basic knowledgeof important properties and solution methods of linear programming is needed. For a cumulativeintroduction to various aspects of linear programming see e.g. Chvatal [18] or Nemhauser andWolsey [78]. For an introduction to applications of linear programming and duality in the field ofapproximation see the book by Vazirani [106].

1.1 The Connection Game

The connection game for N players works as follows. For each game instance, an undirected graphG = (V,E) is given, for which a cost function c : E → R+ specifies non-negative edge costs. Ter-minals of the players are located at the nodes, which a player tries to connect by purchasing somesubgraph of G. Each player then offers payments indicating how much he is willing to contributetowards each edge in G. If these payments sum up to at least the cost c(e), the edge is bought,i.e. added to the network. Here the cost can be shared in any way between a deliberate numberof players. Once an edge is bought, it can be used by any player to connect the terminals - evenby players, who did not contribute to the cost of that edge. Every player tries to minimize hispayments, but insists on connecting his terminals. For simplicity the connection game has beenformulated with the connectivity requirement just to keep the terminals connected no matter e.g.how long the path between two terminals might be.The connection game has a lot of similarities with some classic network design problems. It canbe seen as a game-theoretic generalization of the generalized Steiner tree problem in networks[1, 49, 91]. More precisely, finding an optimal centralized solution, i.e. the social optimum networkminimizing the sum of all player’s payments, is the generalized Steiner tree problem. Finding aminimum cost network for one player is the Steiner tree problem. These problems have been stud-ied quite intensively, and there are well-understood properties and solution techniques available.Chapter 2 will provide a review of some important issues about these problems.

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12 CHAPTER 1. INTRODUCTION

1.1.1 Previous and Related Work

The connection game has been introduced and studied recently by Anshelevich et al in [3]. Nat-urally, there will be numerous references to their paper, because it is the only work about theconnection game so far. The authors describe several properties of deterministic Nash equilibria.From a game-theoretic perspective it might also seem straightforward to consider mixed strategyequilibria. However, the game should be seen as a model how players construct and maintainlarge-scale networks, hence randomizing over strategies is not a realistic option. Furthermore anotion of approximate Nash equilibria is considered. It is explored, how far from an exact Nashequilibrium one has to go, if a specific network is bought, i.e. how unhappy a player is and byhow much he can decrease his payments. A k-approximate Nash equilibrium is a situation, inwhich each player can only decrease his payments by a factor of k at most. This gives rise to atwo-parameter optimization problem - on the one hand the cost of the solution network shouldbe minimized, on the other hand the incentives for the players to deviate should be as small aspossible.The particular results of [3] are:

• In a special case where all players share one terminal and each player holds exactly 2 termi-nals, there is a Nash equilibrium, which is as cheap as the optimum centralized solution. Inthis case the optimistic price of anarchy is 1. In practice calculating the optimum networkis NP -hard. However, there is a polynomial time algorithm starting from a k-approximationfor the optimum network that computes a (1 + ε)-approximate Nash equilibrium, which is atmost as costly as the k-approximate starting network.This result generalizes to hold for directed graphs. This is rather unusual, because prob-lems in directed graphs are often more difficult than the corresponding undirected problems[15, 36]. Another generalization, for which the results hold, is an incorporation of maximumpayment limits for agents, i.e. each agent has a maximum payment he is willing to offer forall edges. Once the assigned costs exceed this limit, he is not willing to connect his terminals.

• For the general case without any restrictions on players and/or terminals, both prices ofanarchy are close to N . In an exact Nash equilibrium eventually the agents buy a network,which isN times more costly than the optimal network. Furthermore it is not guaranteed thata game allows any Nash equilibrium at all. If instead the agents purchase the social optimumnetwork, a 3-approximate Nash equilibrium is possible. Again, in general determining theoptimum solution can be NP -hard. However, the results can be combined with currentapproximation techniques for Steiner problems. There is a polynomial time algorithm, whichprovides a (4.65 + ε)-approximate Nash with a cost within a factor of 2 from the optimumnetwork.

• In general, determining whether a given game has a Nash equilibrium is NP -complete. Thesame problem, however, is solvable in polynomial time for games with 2 players and 2 termi-nals each. In terms of lower bounds, there are games, in which each equilibrium purchasingthe optimum solution is at least a

(32 − ε

)-approximate Nash.

As the main focus of this work is to extend the understanding of connection games, some of theseresults will be presented in more detail in the context of metric connection games in chapter 3.The study of connection games should provide understanding of how different service providersbuild and maintain the Internet topology. As this game is a generalization of Steiner networkproblems, it is related to a lot of literature in this area, especially [1, 49, 91].

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1.1. THE CONNECTION GAME 13

Fabrikant et al [34] study a different network creation game, which has a more restricted model ofagents and their possible behavior. Similar network games have also been proposed for the creationand maintenance of social networks [8, 56]. In these network games each agent corresponds to asingle node, and each agent is only allowed to buy edges adjacent to the node. Being well suitedin the context of social networks this model has some deficiencies when related to communicationnetworks like the Internet. Here agents are not directly associated with nodes, and their interestmight be to care for more complex networks, which are not necessarily locally clustered. Oftentimes it might be in the interest of agents to share the costs of parts of the network. This is aninteresting feature of the connection game, which is not present in the network games of [8, 34, 56].Jain and Vazirani [61] present a cost-sharing game related to Steiner trees. However, their workis focused on formulating a mechanism for multicast transmission. In their game each player i hasa utility ui for belonging to the Steiner tree. The goal is to provide a mechanism, which builds aSteiner tree and decides on the cost-shares for each player. The utility ui places an upper boundon the cost a player is willing to pay for a connection to the tree. In their work Jain and Vaziranidesign a truthful mechanism - a mechanism, in which selfish agents will find it optimal to tellthe mechanism their true valuation. The tree bought is the minimum spanning tree, which is a2-approximation for the Steiner tree problem. This game has some similarities to the connectiongame - especially the special case considered in [3], in which all players share a terminal. In theconnection game, however, Anshelevich et al studied the cost and structure of Nash equilibria.Furthermore there is no central authority like a mechanism to collect all payments. It is ratherspecified in detail, on which edge each player pays how much to reach an equilibrium.

1.1.2 Outline and Results

In this thesis the connection game is studied in terms of deterministic exact and approximate Nashequilibria. After revisiting and extending the previous results, a restricted version of connectiongames is analyzed, in which edge costs are given by distances in the Euclidean plane. In particular,this thesis will be organized as follows:

• As mentioned before, sections 1.2 and 1.3 conclude this chapter with a review of importantrelated developments in the field of algorithmic game theory and computational mechanismdesign.

• Chapter 2 is devoted to simple and generalized Steiner tree problems in graphs. Importantresults from the literature about NP -hardness and hardness of approximation are brieflysummarized, and important algorithmic techniques are highlighted. In terms of edge costscases with general costs as well as costs due to Euclidean distances in the plane are considered.

• In chapter 3 the focus lies on general connection games, especially on finding good approx-imate Nash equilibria purchasing the optimal solution. Throughout the chapter the mainnotion of a connection set from [3] will be used and analyzed regarding analytical advantagesand limits.In sections 3.2 and 3.3 games with 2 players are considered as an introduction. We presenta new lower bound of

(65 − ε

)and an algorithm for finding 2-approximate Nash purchasing

the optimum solution for 2 players and any number of terminals per player. Section 3.4 isdevoted to the case with any number of players and terminals per player. The algorithmfor 3-approximate Nash equilibria is analyzed, and tightness of the approximation factor isshown. Afterwards a new method for solving the problem is presented, which is based on a

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14 CHAPTER 1. INTRODUCTION

parametric linear programming formulation. This method is conjectured to lower the factorto 2 for games, in which each player holds 2 terminals.

• Chapter 4 is devoted to games, in which the edge costs are given by Euclidean distances in theplane, as this is a reasonable assumption for many large-scale networks [28]. For the specialcase with 2 players and 2 terminals, the results for general edge costs can be improved. Section4.1 outlines a method, which finds exact Nash equilibria with a cost within a factor of

√2 to

the optimal network. Based on a discretization technique, even better (1 + ε)-approximateNash equilibria can be achieved, which are as cheap as the optimum network. For more playersand more terminals, a lower bound of

(43 − ε

)for approximate Nash equilibria purchasing

the optimal solution is derived in section 4.2.

• Finally, in chapter 5 we provide an outlook on issues related to general and geometric connec-tion games. In particular, the applicability of connections sets to other variants of connectiongames is discussed.

1.2 Nash Equilibria and Prices of Anarchy

The selfish behavior of independent, non-cooperative agents in network environments like the In-ternet has attracted the interest of theoretical computer scientists for some years. The classicdescription of rational behavior from game theory [82] is the concept of a Nash equilibrium. In anenvironment, in which each agent is aware of the situation facing all other agents, a Nash equilib-rium is a combination of deterministic or randomized choices (one for each agent), from which noagent has an incentive to unilaterally move away. Studying equilibria seems natural as they pro-vide stable conditions in an uncoordinated environment. Clearly, such an equilibrium can be muchmore costly than a socially optimal solution. Conditions, under which Nash equilibria can achieveor approximate the social optimum, have been studied extensively [67, 85]. In these cases thebest equilibrium was considered, which gave insights on methods to influence independent selfishagents to behave socially near-optimal. In opposite [69] Koutsoupias and Papadimitriou proposedto analyze the worst equilibrium to bound the loss in the performance due to selfish behavior of theagents. The worst-case cost increase due to lack of coordination was introduced as a research direc-tion. It was a natural extension of the field of theoretical computer science and algorithm design,as for instance considering the worst-case cost increase due to the lack of unlimited computationalresources led to the study of approximation algorithms [58, 106] or the lack of perfect informationto the study of online algorithms [13, 42]. The ratio of the cost of a worst equilibrium versus thecost of the optimal coordinated solution was introduced as a measure similar to performance orcompetitive ratios. This measure was called coordination ratio in accordance with the terminologyfor approximation and online algorithms. Papadimitriou [84] later named this measure the priceof anarchy, which will also be used here as it has been widely accepted in the literature.The price of anarchy was analyzed in [69] for a traffic optimization game in a network with 2 nodesand a number of parallel links. Each player wants to route traffic from his source to his sink alongthe links. Each link has a certain latency function depending on the traffic already present on it.Koutsoupias and Papadimitrou derived lower and upper bounds for the price of anarchy. Theysolved the case of 2 links and presented partial results for any number of links. This work was thenextended, most notably by Mavronicolas and Spirakis [72], who considered mixed Nash equilibriaand derived tight bounds for a special case of the game and Czumaj and Vocking [26], who derivedtight bounds for the price of anarchy for the general case of this traffic game.

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1.2. NASH EQUILIBRIA AND PRICES OF ANARCHY 15

Subsequently a number of researchers started to work on games and the price of anarchy. Espe-cially congestion games were subject of increased interest, because they offer natural applicationsto resource allocation problems in the Internet. In a congestion game players jointly use one ormore resources, which upon increased use deteriorate the profit for the players using them. It caneasily be verified that the initial example in [69] as well as a lot of Internet routing games fallinto this category. In their widely recognized paper, Roughgarden and Tardos [98] derived theprice of anarchy of a simple selfish routing game. They consider an undirected graph, in whicheach player has a source node and a sink node and tries to route his traffic selfishly across edgesof the network. Each player is assumed to have an infinitesimal non-splittable amount of trafficto route like a package in a communication network or a vehicle in a traffic network. Each edgehas a latency function, which specifies the delay depending on the traffic routed on the edge. Ifthe latency functions are linear, the price of anarchy is 4

3 . If the latency functions are generallynon-decreasing and positive, the price of anarchy is unbounded, however the performance of a Nashequilibrium is better than the optimal solution to the problem instance, in which each player hastwice the traffic to route.These rather immediate fundamental results fostered the chances of achieving a deeper understand-ing of the behavior of congestion games. Thus, thereafter researchers increased efforts to developsignificant results on selfish routing. Although the connection game is no congestion game, wewill overview the results in this important and currently most lively area. Roughgarden extendedhis work by considering different issues and the effects on the price of anarchy [94] like networktopology [95], mixed situations, in which a fraction of the traffic is carefully routed by a centralnetwork manager [97], or the problem of designing networks, which admit efficient Nash flows [93].In a more game-theoretic direction [99] identified types of congestion games with equilibria of acost close to the social optimum. Czumaj, Krysta and Vocking [26] analyzed the routing gamein terms of the latency functions used and derived results for the price of anarchy for familiesof general, monotone cost functions. Their results deliver insight on performance guarantees forqueueing situations of service requests at servers and server farms.Recently there have been successful attempts to analyze the selfish routing game in terms of themaximum latency experienced by any player. Here the maximum latency in an equilibrium versusthe maximum latency in a social optimum is measured. Results about computational complexity,fairness criteria and the price of anarchy were derived [22, 96]. Furthermore different adjustmentslike taxing of flows and pricing of edges and their effects on Nash equilibria were treated in [20, 21].Actually, similar problems of unregulated traffic networks have already been studied since the early1950s [9, 109], even algorithmic aspects like efficient computation of Nash equilibria were treated[44, 45]. A fundamental result in this field is known as Braess paradox [14]. With a small networkexample it is possible to show that adding an additional edge, which leaves the optimal solutionunchanged, can change the unique Nash equilibrium in a way that all players face a worse equi-librium latency. This result has inspired a lot of subsequent work, which classified the structureof networks allowing this behavior and identified similar counterintuitive observations [43, 46, 27].Recently the work on prices of anarchy, congestion games and routing yielded new results on com-putational hardness of detecting Braess paradox [93] and the severeness of this phenomenon [71].For more advanced routing models the work of Schulz and Stier-Moses [101] has provided someinsight on the performance of equilibria. They extended the congested selfish routing game to acapacitated network, which has latency functions that are non-convex, non-differentiable and non-continuous. Although the capacity extension heavily deteriorates the cost of the worst equilibrium,the cost of the best equilibrium stays the same as in the model without capacities. Thus, this workconnects to earlier results about equilibria and reintroduces considerations about the efficiency of

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16 CHAPTER 1. INTRODUCTION

the best equilibrium and the optimistic price of anarchy.More on the theoretical side, research has tried to bridge the gap between theoretical computerscience and game theory. Here algorithmic issues of finding and computing pure Nash equilibriaare considered. The classic result of Nash states that any game has a randomized equilibrium, i.e.an equilibrium under the assumption that any player plays each strategy with a certain probabil-ity. Unfortunately Nash’s result is only an (inefficient) proof of existence and does not provideinsights how to compute Nash equilibria efficiently in polynomial time. Furthermore the concept ofrandomized equilibria offers very limited applicability for many practical situations. Very recently,Fabrikant et al [35] pointed out some connections between pure Nash equilibria and local search.For congestion games it was already known from Rosenthal [92] that the Nash dynamics converge.A graph was considered, which has a node for every state - every possible combination of playerstrategies. If in such a state a player can improve his payoff by switching his strategy, a directededge is drawn between the corresponding nodes. This is only done for single player switches, i.e.edges are only present between nodes of states that differ in the strategy of exactly one player.Rosenthal showed that this graph contains no cycle, i.e. if the game is played iteratively by allow-ing each time one player to deviate, they ultimately arrive at a pure Nash equilibrium. Thus, therealways exists a pure strategy Nash equilibrium for congestion games. The existence proof relies onan argument using a potential function, with which the payoff for a player, who deviates from agiven state, can be calculated. Note that this construction implies a local search problem to finda pure strategy Nash equilibrium, which is in the class PLS [63]. Fabrikant et al show that if thecongestion game is described in a setting of networks (like a selfish routing game) and all playershave the same source and sink in the state-network described above, the pure strategy equilibriumcan be calculated in polynomial time. This holds even for nonatomic congestion games, whichconsist of an infinite number of players (e.g. the aforementioned selfish routing games, in whicheach player controls only an infinitesimal amount of traffic). In general, however, computing purestrategy Nash equilibria is PLS-complete [63].In terms of existence Monderer and Shapley [75] have shown that every congestion game has sucha potential function, which gives the payoffs for the deviations. Fabrikant et al [35] weaken theirassumptions slightly. For the application of a local search procedure in an acyclic state graph,the potential function is not required to provide the exact payoff but only the sign (positive ornegative payoff). Any local search problem in the class PLS can be coached as a game, for whichan analogous existence proof with the adjusted sign-correct potential function can be given. Thus,all of these games are guaranteed to have pure strategy Nash equilibria. This exemplifies some ofthe inherent connections between the notions of Nash equilibrium and local optimum.Another such connection is drawn by Devanur et al [31]. They consider a network service providergame, which they could base on a facility location problem. All Nash equilibria of this gamecorrespond to local optima in the instance of the facility location problem. Thus, they can applya previous result [7] about the approximation guarantee of local optima to bound the price ofanarchy to 5.Another interesting notion to consider is the question how to transform any given non-equilibriumstate into a Nash equilibrium. Feldmann et al [41] consider the selfish routing game with 2 nodesand parallel links. They provide a polynomial time algorithm to transform any routing in anequilibrium routing without increasing the social cost. As a result, they give a polynomial timeapproximation scheme (PTAS) for the computation of the best Nash equilibrium and provide ex-ponential upper and lower bounds for the worst-case number of greedy selfish (local improvement)steps for identical link capacities.Vetta [107] considered a class of games characterized by a utility system. The main ingredient of

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1.3. COMPUTATIONAL MECHANISM DESIGN 17

this concept is the assumption of submodularity of the cost function, which measures the socialvalue of a state. In economy, submodularity implements a concept of decreasing marginal util-ity, i.e. the benefit of a single additional piece of a good is smaller, if already a large number ofpieces are present. For this class of games, Vetta could prove that the price of anarchy with mixedstrategies is 2. For some restricted class the existence of pure strategy Nash equilibria was proven.Furthermore several meaningful applications to facility location, routing and combinatorial auc-tions were given.Finally, some aspects of the network creation game considered by Fabrikant et al [34] will be men-tioned. In their game agents are single nodes that choose to build connections to neighboringnodes. The final cost for an agent is determined by a linear combination of the number of edgesbought by him and the distance of his node to all other nodes in the network. Thus, the agent hastwo contrary interests: buying the least number of edges and being connected closely to all otherplayers. With a parameter the significance of one interest over the other can be controlled. Notethat this game is not a congestion game, thus tools like potential functions are not immediatelyavailable. Depending on the choice of the interest parameter, Fabrikant et al determined the priceof anarchy. If the interest in being connected is strong, the worst-case equilibrium network is a starand the price of anarchy is 4

3 . If the interest in buying less edges is becoming stronger, the analysisof Nash equilibria gets more complicated. From experimental results a conjecture about the treestructure of Nash equilibria is derived and interesting extensions of the game are proposed.In recent years the analysis of Nash equilibria and game-theoretic extensions of combinatorialproblems has become a very vivid research direction. From the great amount of results available,only a small collection of results were presented here, which are more or less directly related to theconnection game. For a more thorough introduction to the price of anarchy and selfish routing thereader is referred to [24].

1.3 Computational Mechanism Design

The theory of mechanisms (also called implementation theory) is a standard field in classic gametheory. We will give a quick introduction to explain some properties of this concept. This servesto understand some of the computational problems behind this theory and the type of applicationsthis concept has found for algorithmic problems concerning the Internet. For a more cumulativeand formal introduction to this topic the reader is referred to [60, 81, 86, 105].The mechanism is a central institution obligated to collect private data from selfish agents. Thisdata is called the type of the agent. Based on an objective function the mechanism then calculatesan outcome and assigns payments to the agents depending on the reported types of possibly allagents. Furthermore the agents have valuation functions specifying the value of an outcome to theagent given his type. The utility of an agent is then given by the valuation minus the payments1.A strategy for an agent is a choice rule, what type to report to the mechanism. Since the agentmight be able to influence the outcome with his reported type, he might be willing to lie to themechanism to achieve some outcome and payments, which are more favorable in the light of hisutility function. An interesting question is how to design a truthful mechanism, i.e. how to designthe rules, by which the mechanism calculates outcome and payments in a way that no selfish agentmight find it in his interest to misreport his type.For clarification consider the sealed Vickrey auction [108]. Here the mechanism tries to sell an itemvia an auction. It collects the bids from the agents (their types) once in a sealed form, such that

1This is termed quasi-linear utility. The presentation will be limited to this type of utilities.

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18 CHAPTER 1. INTRODUCTION

each agent knows only his own bid. Then it gives the item to an agent (the outcome), for which itcharges a payment. It only collects payments from the bidder that receives the item. The valuationfunction of an agent i is equal to some value ui if the agent receives the item and 0 otherwise.As the mechanism charges only the agent, which receives the item, the final utility functions ofthe agents are 0 if an agent does not get the item. If he gets the item, the function returns thevaluation minus the assigned payments. The objective function of the mechanism is such that itassigns the item to the highest bidder. Now assume the mechanism charges the bid as payment.Then for an agent it is not optimal to bid his true value. An agent will naturally never bid more forthe item than his value, because this makes the utility function become negative, and not gettingthe item yields a utility of 0. In case he is the highest bidder, he might also get the item with abid, which is slightly higher than the second highest bid. This will increase his utility function,because he gets the item at a lower payment. He will therefore try to guess the second highest bidand place a bid, which is slightly more. It is easy to see that once the mechanism incorporates thisnotion, i.e. assigns the item to the highest bidder at the cost of the second highest bid, it will betruthful.A mechanism is said to be truthful or strategyproof if truth telling is a dominant strategy foragents, i.e. telling the truth is always best - independent of the behavior of other agents. A muchcelebrated result in game theory are the generalized Vickrey-Clarke-Groves (VCG) mechanisms[108, 19, 52], for which truthfulness is guaranteed. A mechanism belongs to the weighted VCGfamily if the outcome is generated to maximize the objective function, and the payment of anagent is the weighted sum of the valuations of the other agents. In this way an agent can neverdirectly influence his payments. VCG mechanisms provide solutions if the optimization problemof the mechanism is utilitarian, i.e. if the objective function for an outcome is the weighted sumof the agents utilities using the reported types.The study of mechanism design in computational environments has been motivated by scientistsin the mid-1990s, who were concerned with observations and ideas about efficient protocols for theInternet [103]. In their seminal papers, Nisan and Ronen [79, 81] observed the wide applicabilityof mechanism design to computational and algorithmic problems concerning the Internet. Theyintroduced the topic to a greater research community in theoretical computer science and pointedout a range of example scenarios, in which game-theoretic problems need to be considered, e.g.in resource allocation, routing, or electronic trade. This attracted a variety of researchers, whosubsequently began to study algorithmic problems connected to mechanisms. These computationalproblems (e.g. computing the outcome, which maximizes the objective of a mechanism) need ef-ficient algorithms for practical use, however, in many cases they are NP -hard. This implies theneed to consider approximations and their effects on the properties of the mechanisms. If, forinstance, computing the optimal outcome of a VCG mechanism is an NP -hard problem, and themechanism uses an approximation algorithm, then in general truthfulness cannot be guaranteedto hold [80]. However, in this case the agents are also facing an NP -hard problem to optimallybias their reported types. In [80] some approaches were proposed to deal with these problemssatisfactorily.Apart from these rather fundamental issues many application areas have received attention. Withan interesting scenario Nisan and Ronen described a mechanism for a routing problem in a di-rected graph [81]. Here each edge is an agent. A package should be routed on the shortest pathbetween two fixed points u and v, and each agent can report a cost for carrying the package. Theyadjusted the payment function accordingly to get a truthful mechanism. It included a calculationfor each edge e of the shortest path between u and v not including e. An efficient algorithm forthis algorithmic problem was presented in [57] with the same asymptotical time complexity as the

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1.3. COMPUTATIONAL MECHANISM DESIGN 19

best algorithm for the single-source shortest path problem. Recently it has been observed that thepayments of the mechanism for ensuring truthfulness in this game can be very high. Archer etal [4] therefore consider the frugal path problem, in which the mechanism shall first select a pathand then induce truthful cost revelation without high payments. However, they provide argumentsthat on large classes of graphs no mechanism can avoid high payments for truthful cost revelation.Another area of strong research activity is multicast transmissions or multicast routing [30], forwhich cost-sharing methods were considered that allocate the cost of a transmission. Cost-sharinghas been extensively studied in economics literature [110]. In the area of computational mech-anism design the algorithmic issues of cost-sharing mechanisms for multicast transmissions havebecome important due to their applicability to digital information distribution services using theInternet or other digital networks. In a simple model agents are nodes in a tree graph and arewilling to receive a message from a fixed source node. They report their valuation of the messageto the mechanism. The mechanism then decides, which agent should receive the message. Thecost of the subtree consisting of the source and all agents picked for reception is then distributedand charged as payments to the included agents. Moulin and Shenker [76] designed strategyproofcost-sharing mechanisms satisfying a variety of economical properties like budget balance and ef-ficiency. Budget balance is the requirement that the sum of all payments pays exactly for thecost of the transmissions. Efficiency requires the created welfare to be maximized, i.e. the sum ofthe utilities of agents receiving the message minus the cost of the broadcasting tree is maximized.A classic result from game theory states that budget balance and efficiency cannot both be metwith a strategyproof mechanism [50, 90]. Hence, Moulin and Shenker designed two mechanismscalled marginal cost and Shapley value, which satisfy budget balance and efficiency, respectively.Feigenbaum et al [39] then proposed efficient algorithmic techniques for a distributed realization ofthese mechanisms. They also found that the maximum welfare is NP -hard to approximate withinany constant factor.In the following the notion of approximation was extended from the context of the solution ofcombinatorial problems to the context of economical properties of mechanisms. In [37] the authorsshowed that there is no mechanism, which is approximately budget balanced and approximatelyefficient. Jain and Vazirani [61] used approximation in both contexts. They adjusted the modelfor multicast transmissions to build cheaper networks. Instead of considering a tree graph, theyconsidered an arbitrary connected graph as the basis for the transmission. This is much more ap-propriate for today’s Internet, because the present topology is certainly not a tree. The mechanismthen has to pick the agents to receive the message and the lowest cost tree for broadcasting. This,of course, involves the solution of the NP -hard Steiner problem. Therefore they switched to theminimum spanning tree as a 2-approximation for the Steiner problem and designed a strategyproof,budget balanced mechanism for assigning the costs. Furthermore they considered the economicalnotion of fairness in a way that each subset of picked agents should not be assigned to pay morethan the cost of connecting them alone. They adjusted this notion, which subsequently allowedthem to derive a large class of mechanisms satisfying a wide range of fairness criteria [62]. Themechanisms they considered were actually group strategyproof, which means that even if agentsbuild coalitions, truthtelling remains a dominant strategy for all of them.More recently, Bilo et al [12] considered an application of the multicast transmission model to wire-less ad hoc networking. They constructed strategyproof mechanisms for cost-sharing, which areeither budget-balanced or efficient under some assumptions for the distance-power gradient. Theyprovide efficient algorithms to compute approximately budget balanced mechanisms for agents inany d-dimensional space.Building on this relatively simple network design and cost-sharing model more elaborate problem

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20 CHAPTER 1. INTRODUCTION

settings like the single source rent-or-buy game were considered. This setting depends on the singlesource rent-or-buy problem, which is an extension of the Steiner tree problem with economies ofscale. There is a single source providing a good to be transported to the agents. Each agent againis a node, but here he has a demand of 1. Apart from constructing the broadcasting tree onemust also decide, whether an edge is rented or bought. After an edge is bought all demand can berouted over the edge for free. However, for less heavily used edges there is also the possibility torent them and pay a cheaper cost depending linearly on the demand routed through it. For thisproblem Gupta et al [54] proposed a simple approximation method. It was recently used in [55]to construct a group strategyproof mechanism, which is 4.6-approximately budget balanced. Thiswas a significant improvement over the previously constructed mechanism in [83], which was only15-approximately budget balanced.Another interesting development in this area is that insights and techniques from game theory andmechanism design could be used to design better approximation algorithms. The multicommodityrent-or-buy problem is the extension of the single source rent-or-buy problem to multiple sourceswith multiple goods. For this problem cost-sharing techniques allowed Gupta et al [53] to derive asimple to analyze 12-approximation algorithm. This was a significant improvement over the pre-vious best constant factor approximation algorithm of Kumar et al [70], which used a primal-dualscheme and a rather complicated analysis.Algorithmic issues of mechanism design problems are still a very young research direction, howeverthere is already a great amount of literature on a variety of aspects and problems available. Wehave only presented a small collection of results that are rather closely related to the topic of thisthesis. For further details and a more complete treatment of research in computational aspects ofmechanism design, the reader is referred to [40].

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Chapter 2

Steiner Tree Problems

In this chapter some relevant previous work on Steiner tree network design problems is presented.Actually, simple variants of the classic Steiner tree problem in graphs have been under considerationfor several hundred years. There is a vast amount of literature available, however, here only materialwith implications for the connection game is considered, i.e. regarding computational hardness andpossibilities for polynomial-time approximation algorithms. First in section 2.1 the NP -hardnessresult is given and the relationships between Steiner trees and minimum spanning trees are pointedout. The recently discovered 1.55-approximation algorithm provides some insight into fundamentalcombinatorial properties of the problem. Section 2.2 deals with the special case of geometric edgecosts. Results about computational hardness, properties of optimal solutions and a polynomialtime approximation scheme (PTAS) are sketched. Finally, in section 2.3 the insights on Steinertrees and minimum spanning trees are embedded into a more general framework. The well-known2-approximation algorithm for this problem is presented. Note that this chapter is intended togive an overview over results and techniques employed in solving Steiner tree problems. Thereforerather main ideas are presented, and full proofs are only included in cases, which are able to fosterthe understanding of connection games. Most of the material of this chapter is based on Vazirani[106] and Gropl et al. [51].

2.1 The Steiner Tree Problem

2.1.1 Definition and Computational Hardness

The Steiner tree problem has its roots in a problem posed by Fermat (1601-1665). If three pointsare located in the plane, where is the point that minimizes the overall distances to all of them? Inthe present general definition it first appeared in a letter by Gauss, which he wrote to Schumacheron March 21, 1836. Courant and Robbins [23] referred to the problem under the name of Steiner,a well known geometer from the 19th century. (see e.g. [100] for more history). Today it is acentral problem in the field of efficient algorithm design and approximation with a wide range ofapplications in VLSI design, computational biology or networking. Formally it can be stated asfollows:Suppose we are given an undirected graph G = (V,E) and a cost function c : E → R+

0 specifyingnonnegative edge costs. The nodes of the graph are partitioned into two sets, which are namedterminals and Steiner. The problem is to find a minimum cost tree in G that contains all the

21

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22 CHAPTER 2. STEINER TREE PROBLEMS

terminals and may use any subset of Steiner points as well. We will denote the optimum solutiontree by T ∗, which will be called Steiner minimal tree (SMT). Furthermore, let c(T ) =

∑e∈T c(e)

be the cost of the tree T ⊆ E.In terms of computational hardness, Karp has shown Theorem 2.1 in his seminal work [64].

Theorem 2.1 The Steiner tree problem in graphs is NP-hard, even for unweighted graphs.

The proof of this theorem can be given for example by a reduction from the problem ”Exact Coverby 3 Sets” (X3C). A polynomial transformation is constructed that transforms an instance of X3Cto an instance of the Steiner tree. The elements and sets are translated into nodes of a tree, andedges are drawn to indicate, whether a certain element is part of a set. This instance then has acheap Steiner tree if and only if the corresponding instance of X3C has a solution.Hence, in general there is very little hope that an efficient algorithm solving the Steiner treeproblem will ever be found. For some special cases of the Steiner tree problem efficient algorithmsare available. For instance the case without any Steiner nodes can be solved by computing theminimum spanning tree, the case with only two terminals by computing the shortest path. For thegeneral case efficient approximation algorithms can be found, but they cannot deliver arbitrarilygood factors. Specifically, no PTAS is possible for the Steiner tree problem, because the PCP-theorem in the seminal contribution by Arora et al [6] together with a reduction due to Bern andPlassmann [11] delivered the following result.

Theorem 2.2 There exists some constant k > 1 such that no polynomial time approximationalgorithm for the Steiner tree problem can have a performance ratio smaller than k, unless P =NP.

When the first version of this theorem appeared, the proof of the PCP-theorem and the reductionby Bern and Plassmann involved big constants that yielded a lower bound on k, which was onlyvery slightly above 1. Since then there has been some improvement on the ratio, the latest byChlebık and Chlebıkova [16]. They use the non-approximability result of Hastad [104] on theproblem of the maximum satisfiability of linear equations modulo 2 with three unknowns perequation (Max-E3-Lin-2). Their main statement is given in Theorem 2.3.

Theorem 2.3 It is NP-hard to approximate the Steiner tree problem within a ratio of 1.01063 >9695 .

Before presenting approximation algorithms we will reason that the core of the Steiner problemlies in the restriction to metric edge costs satisfying the triangle inequality. For every instance ofthe Steiner problem an instance of the metric Steiner tree problem is constructed, in which G is acomplete graph and for any nodes u, v and w the triangle inequality c((u, v)) ≤ c((u,w))+c((w, v))holds.

Theorem 2.4 There is an approximation factor preserving reduction from the Steiner tree problemto the metric Steiner tree problem.

Proof. For any instance I of the Steiner tree problem we construct the metric closure, which willbe an instance I ′ of the metric Steiner tree problem. Let G = (V,E) and c be given for I. Thenthe graph for I ′ is the complete undirected graph G′ = (V,E′), E′ = V × V , and for any edgee = (u, v) ∈ E′ the costs are given by the shortest path between u and v in I. The partition intoterminals and Steiner nodes stays the same.

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2.1. THE STEINER TREE PROBLEM 23

For any edge the cost in I ′ is not greater than the cost in the original instance I, so the optimalsolution of I ′ is not more costly. For any Steiner tree in I ′, we can construct a Steiner tree in I byreplacing the edges with the corresponding shortest paths. The tree in the original instance willnever be more costly but eventually cheaper, because we can remove edges from cycles and edgesincluded twice. Hence, if we have a Steiner tree, which is k times the optimum in I ′, then thiscarries over to I, because the optimum solution will not be cheaper and our transformed tree willnot be more expensive. 2

Hence, any algorithm approximating the metric Steiner tree problem will always deliver the samefactor for the entire Steiner tree problem.

2.1.2 Minimum Spanning Tree Heuristic

In the following approximation algorithms for the metric version of the Steiner tree problem arepresented. At first, the minimum spanning tree (MST) for the terminals (denoted by Tms) will beused to approximate the cost of an optimum Steiner tree. The following theorem was independentlyobserved by many authors [17, 59, 68, 87], it was already observed in [48].

Theorem 2.5 The cost of Tms is a 2-approximation for the cost of T ∗. The factor of 2 is asymp-totically tight.

Proof. Consider an optimal Steiner tree T ∗. Suppose we double the edges of the tree. Then it ispossible to build an Eulerian cycle through all nodes of T ∗ by traversing the edges in depth-firstsearch order. The cost of this cycle is exactly 2c(T ∗). Suppose now we start at a terminal andrun once around the cycle. Each time a new terminal v is discovered, the path along the cycleis ’shortcut’. The direct connection between v and terminal v ′, which was newly discovered justbefore v, is added. This allows us to remove the path from v to v ′ from the cycle. The exchangewill decrease the cost, because the triangle inequality imposes that the direct connection is thecheapest. It will furthermore remove all Steiner vertices from the cycle and form a spanning treeTs of the terminals. Hence,

c(Tms) ≤ c(Ts) ≤ 2c(T ∗).

2

2

2

2

2

1

1

1 11

1

Figure 2.1: Tight example for the MST-approximation.

To see that the approximation can actually be as bad as factor of 2 away from optimum considerthe metric closure of the network in Figure 2.1. Each edge to the middle node has cost 1, eachedge between the outer nodes has cost 2. Only the middle node is a Steiner node, all outer nodes

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24 CHAPTER 2. STEINER TREE PROBLEMS

are terminals. If there are n required nodes, T ∗ will be the star of cost n. A minimum spanningtree, however, has cost 2(n− 1), which gives a factor of 2− 2

n . 2

Although there are algorithms with better approximation factors available already, this algorithmis appealing, because it can be implemented to run rather quickly. The best implementation is dueto Mehlhorn [73] and runs in time O(|E|+ |V | log |V |).

2.1.3 The Greedy Loss Contracting Algorithm

Over several decades there had not been any improvement on the approximation factor untilthe idea of k-Steiner trees was explored in the early 1990s. The development led to the greedyloss contracting algorithm by Robins and Zelikovsky [91], whose analysis has resulted in the bestapproximation factor presently known. We will present the algorithm and give some glimpses ofthe analysis.A Steiner tree, in which all required vertices are leaves, is called a full Steiner tree. If a Steinertree has some terminals as interior nodes, it can be decomposed in full components by splittingthe interior terminals. Finally, a k-Steiner tree is a connected tree spanning all terminals, whichcan be covered by full components with at most k terminals. Note that this implies that terminalsas well as edges or Steiner nodes can be present in more than one full component. An optimumk-Steiner tree will be denoted by T (k)∗. Obviously, every Steiner tree is a k-Steiner tree, if k isthe number of terminals.A k-Steiner tree is an approximation for a Steiner tree. The worst-case approximation factor akslowly approaches 1 as k goes to infinity. Unfortunately, finding T (k)∗ is NP -hard for k ≥ 4,for k = 3 and ε > 0 there exists a randomized polynomial time (1 + ε)-approximation algorithm[89]. The greedy loss contracting algorithm will approximate the optimum k-Steiner tree T (k)∗

instead of the optimum Steiner tree T ∗. The approximation ratio of the algorithm then needs tobe multiplied with ak to yield the final guarantee for the Steiner problem.The high-level algorithm can be stated as follows:

Algorithm 1 High-Level Greedy Algorithm

1. Let W be the set of all full components of up to k terminals.

2. i← 0

3. while an improving full component exists do

4. Choose wi+1 ∈W that minimizes a selection function φi.

5. i← i+ 1

6. imax ← i

7. output a Steiner tree using w1, . . . , wimax .

Note that this algorithm has polynomial running time if k is a fixed constant. |W | ≤ |V k|, andan optimal Steiner tree for only k terminals can be computed in polynomial time if k is a fixedconstant. To be able to specify φi, the Contraction Lemma 2.1.1 is needed. Let Tms(R) be aminimum spanning tree for the set R ⊆ V of terminals. Suppose we add a new (possibly parallel)edge e to the graph G, then let

Tms(R/e)� a minimum spanning tree for R in G+ e

The difference c(Tms(R)) − c(Tms(R/e)) is what one can gain in the minimum spanning tree byadding an edge e. If c(e) = 0, then this corresponds to the case, in which the edge e is contracted.

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2.1. THE STEINER TREE PROBLEM 25

For some sets of edges E0, E1, E2, how does the gain of e.g. E2 change, if E0 and/or E1 are addedto the graph ? Will it be possible to decrease the minimum spanning tree by putting in the setE2 more after having added some other edges? The answer is: No! In the following lemma, whichwas derived in [111, 10], notation of the form E0E1 is short for E0 ∪ E1.

Lemma 2.1.1 Let E0, E1, E2 be sets of (new) edges between terminals. Then

c(Tms(R/E0))− c(Tms(R/E0E2)) ≥ c(Tms(R/E0E1))− c(Tms(R/E0E1E2))

Considering the Steiner problem again it is easy to notice that it reduces to find the right subsetof Steiner nodes. Once the Steiner nodes of T ∗ are known, the tree can be found by calculatingthe minimum spanning tree for all required and included Steiner nodes. The idea is to excludeunprofitable Steiner nodes from consideration by measuring their minimum distance to terminalnodes. This is done with the notion of the loss. The loss of a set of Steiner nodes S ⊂ V isa minimum length forest denoted by L(S) ⊆ E, in which every Steiner node is connected to aterminal. The loss of a collection of full components or a tree is defined to be the loss of theincluded Steiner nodes with respect to tree edges. A notion of contraction is introduced to applythe Contraction Lemma 2.1.1 in this case. Consider the components of the loss of a Steiner tree. Ineach component of this forest there is exactly one terminal, however in the corresponding tree (onwhich the loss is based) all components are connected. Contracting the loss of a Steiner tree meansthat for each edge between two components a new edge is inserted between the correspondingterminals with the same cost. Note that this has the same effect on the minimum spanning treeas if all edges of the loss were actually contracted. See Figure 2.2 for a depicted loss contraction.Thick edges are part of the loss. For every thin edge of the tree outside the loss a dashed edgewith the same cost is added between the terminals of the components of the loss.

Figure 2.2: New edges added in a loss contraction.

To apply the lemma 2.1.1, the lengths of the newly inserted edges must be independent of the pre-vious loss contractions with the same Steiner nodes. Therefore the instance must be preprocessedby duplicating Steiner nodes such that no two full components of the graph share a Steiner node.This will increase the instance only polynomially, while c(T (k)∗) will not increase. The set W offull components will then refer to the preprocessed instance.The following lemma was established in [65]:

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26 CHAPTER 2. STEINER TREE PROBLEMS

Lemma 2.1.2 The cost of the loss of a Steiner tree is at most half its total cost.

Finally, we are ready to describe the selection function of the loss contracting algorithm. Ititeratively contracts losses of Steiner vertices. Let

cl(·) = c(Tms(R/L(·)))

be the cost of a minimum spanning tree after the loss of certain full components has been contracted.Due to the preprocessing, the effect of a loss contraction can be modelled by inserting new edgesbetween terminals. The use of the contraction lemma can be justified. After components w1, . . . , wihave been chosen, the cost of the Steiner tree is

c(w1 . . . wi) = cl(w1 . . . wi) + c(L(w1 . . . wi))

The selection function

φi �c(L(w))

cl(w1 . . . wi)− cl(w1 . . . wiw)

measures the cost of the loss of a new full component w versus its reduction of the minimumspanning tree.

Theorem 2.6 The greedy loss contracting algorithm computes a 1 + ln 32 ≤ 1.55 approximation for

T (k)∗.

The proof of this theorem relies on the two presented Lemmas 2.1.1 and 2.1.2 and was given byRobins and Zelikovsky [91]. Although this algorithm seems to come significantly below the boundof 2, some drawbacks must be pointed out. Note that the 1.55 ratio is given for approximating theminimum k-Steiner tree, thus it must still be multiplied with the ratio of the k-Steiner tree to yieldthe final approximation guarantee. k must be very large to come close to 1.55. This makes thealgorithm quite impractical as it works with the set of all full components with up to k vertices,which size is exponential in k. However, the analysis of the algorithm is not tight, and the givenbound is still far away from the lower bound of 1.01063 in Theorem 2.3. It is widely believed thatthere is significant improvement possible, so most theoretical research has focused on improvingthe bounds instead of finding practically efficient methods. For issues on practically solving theSteiner tree problem see for instance Polzin [88].

2.2 The Geometric Steiner Tree Problem

Steiner tree problems with geometric edge costs were the first ones considered. Actually, the ge-ometric version has its roots in the above mentioned problem of Fermat with three points in theplane. It can be stated as follows:Given a number of required terminals at points in the plane, find a set of Steiner points, such thatthe minimum spanning tree for all terminals and Steiner points is minimal. Note that any line inthe plane can possibly be part of a Steiner tree, thus there is no restriction to certain nodes andedges as it is in the general metric problem. Note that possibly any point in the plane can be usedas a Steiner point.As most results of this section require a variety of technical details or the analysis of different cases,the presentation will be restricted to the outline of some important ideas of the proof and solutiontechniques.

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2.2. THE GEOMETRIC STEINER TREE PROBLEM 27

2.2.1 Hardness, Properties and Minimum Spanning Trees

Some well-known basic properties of optimal geometric Steiner trees will be of interest for connec-tion games, therefore the following three statements from [48, 74] are given.

Lemma 2.2.1 Any 2 adjacent edges in an optimal geometric Steiner tree connect with an innerangle of at least 120 ◦.

Lemma 2.2.2 Every Steiner point of an optimal geometric Steiner tree has degree 3, and each ofthe 3 edges meeting at it makes angles of 120 ◦ with the other two.

Corollary 2.2.1 Let R be the set of terminals, then an optimal geometric Steiner tree has at most|R| − 2 Steiner points.

Unfortunately, solving the problem to optimality is just as hard as in the general case. Theorem2.7 states the result. Regarding approximation hardness, the problem admits a polynomial timeapproximation scheme, hence there is no such statement as Theorem 2.2 possible.

Theorem 2.7 The geometric Steiner tree problem using Euclidean distances is NP-hard.

A proof of this theorem was given by Garey, Graham and Johnson [47]. They prove the NP -completeness result for both the decision problems with edge costs given by Euclidean and Man-hattan norms. The reduction is designed from the X3C problem, which was also used for theproof of the general case in Theorem 2.1. One of the fundamental problems with the Euclideannorm is that irrational edge costs are often unavoidable. This complicates computational issueswith standard decision problems using statements like ’Is there a tree with cost less than K?’.This problem can be avoided by using edge costs rounded to the next integer, which is also donewhen solving the problem in practical situations. The exact value of the Euclidean norm can beapproximated by scaling the instance appropriately. However, it was shown that this is not thekey ingredient for the hardness of the problem.For the reduction from X3C, the elements of the sets in an X3C instance are supposed to beintegers. For every set geometric gadgets are assembled according to the integers present in theset. The assembled gadgets are then combined to yield the geometric Steiner tree instance. Thegadgets are only used to specify the locations of the terminals. Using basic properties of geometricSteiner trees, the authors provided arguments that in such an instance an optimum Steiner treealso yields a solution to the X3C problem and vice versa.For the geometric version of the Steiner tree problem the minimum spanning tree heuristic wasexamined as well. With similar arguments one can derive a factor of 2 here, however, provingtightness of the approximation factor turned out to be much more complicated than in the generalmetric case. In 1968 Gilbert and Pollack [48] conjectured that the minimum spanning tree heuristichas an approximation ratio of 2√

3, which is achieved on a geometric instance of the type depicted

in Figure 2.1 with 3 terminals. The inter-terminal cost in this instance is√

3 instead of 2. At theSteiner point the edges connect with 120 ◦ angles. The conjecture remained open for over 20 yearsuntil Du and Hwang [32] presented a proof that established 2√

3as the worst-case factor for the

MST-approximation.

2.2.2 A polynomial time approximation scheme

For geometric Steiner trees a randomized polynomial time approximation scheme was presented byArora [5] that calculates a (1+ε)-approximation. The running time is bounded byO(n(logn)O(1/ε)).

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28 CHAPTER 2. STEINER TREE PROBLEMS

The algorithm uses a rounding technique to limit the number of points, which can be used forterminals or Steiner nodes. It constructs a grid in the plane. All terminals are moved to gridpoints. Edges are only allowed if they use certain points (”portals”) on the grid lines. Steiner pointsare located only at portals, too. A dynamic programming procedure is applied to enumerate allmeaningful tree structures and constructs an optimal Steiner tree under the mentioned restrictions.Finally, using some geometric arguments it is shown that the cost increase due to the use of roundingand portals is at most a factor of (1 + ε). Here we will only sketch some important details of thealgorithm.Let the n terminals be given in the plane (assume wlog that n = 2k for k ∈ N). Define a boundingbox around the points to be the smallest axis-parallel square of length L, which includes all nterminal points. With a technical adjustment, one can ensure that L = 4n2 and that there isa unit grid defined on the square such that each terminal point lies on a gridpoint. The basicdissection of the bounding box is a partition, which works recursively and splits a L × L squareinto an L/2 × L/2 square and so on. Let us view this as a 4-ary tree T , in which the root is thebounding box and each node has four children - one for each square after splitting. A square isassigned a level, which is the level in the tree (the bounding box has level 0). This is done untilunit squares are achieved. Then T has depth k = O(logn). We will also refer to levels of lines, i.e.the borders of each square of level i are lines of level i. (see Figure 2.3.

Bounding Box − Level 0

Square − Level 1

Square − Level 1Square − Level 1

Level 1

Level 3

Level 2

Figure 2.3: Levels of squares and lines inside the bounding box.

A line contains a number of m ∈ [k/ε, 2k/ε] equidistant points called portals. Lines are only allowedto be crossed by edges at portals and Steiner points can only be located here. Each square has4m portals, where m is chosen to be a power of 2. Thus any portal from lower level squares is aportal for all higher level squares it lies in. Clearly m = O(logn/ε). Now an (m, r)-light Steinertree is a Steiner tree that crosses each edge at most r times and each time at a portal. We chooser = O(1/ε) constant.The algorithm first calculates the basic dissection and then applies a dissection shift. It shifts thedissection by some values a in horizontal and b in vertical direction. Squares that are (partly)moved over the right or upper border of the bounding box are ’wrapped around’ and attached atthe left and lower border. However, they will still be thought of as the same square. See Figure2.4 for an illustration of the four squares on level 1 after a dissection shift.

After this shift the tree T will be computed. If all children of a square do neither contain aterminal nor a Steiner point, they can be deleted from the tree and do not need to be considered.

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2.2. THE GEOMETRIC STEINER TREE PROBLEM 29

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b

a

Square 2

Square 4

Square 1

Square 3

Figure 2.4: Location of level 1 squares after a dissection shift by (a, b).

The squares containing Steiner nodes have to be ”guessed” by the algorithm. However, the Steinertree can only enter and leave a square for 4r times, where r is constant. So it can be treated asan instance of the Steiner tree problem of size at most 4r, which can be solved in constant time[74]. With this adjustment the number of leaves of T will be O(n), and the size of T is boundedby O(n logn).Finally, a dynamic programming procedure starts. For each leaf square in T every possible setP of portals with at most r portals from each line is considered. For each of these sets P everypossible partition of portals must be considered. The partition specifies, which portals must beincluded in one component. Then for any given square, any set P of portals, and any partition ofP an optimal Steiner forest including the terminals and Steiner nodes in the square is computed.This can be done in constant time, because m and r are fixed. After having computed all forestsfor the leaf squares, the dynamic programming algorithm uses a bottom-up approach. For squaresat interior nodes of T it assembles the best tree from the corresponding best trees in the childrensquares. Again an optimal forest for any possible set of P portals with at most r portals from eachline and any possible partition is assembled. Of course here connectivity issues have to be takencare of. Furthermore in the parent square only half of the portals of the child squares are present.Once the algorithm has tested all trees and squares, it stops with the optimal (m, r)-light Steinertree as output.Finally, the approximation ratio is given by the following theorem.

Theorem 2.8 Suppose we are given a constant ε > 0 and an instance of the geometric Steinertree problem that yields certain dimensional properties. If L is the size of the bounding box andthe shifts 0 < a, b < L are picked randomly, then with probability at least 1

2 the dissection shift hasan associated (m, r)-light Steiner tree, which is a (1 + ε)-approximation for the optimum, wherem = O(logn/ε) and r = O(1/ε).

Any instance of the geometric Steiner tree problem can be adjusted to yield the dimensionalproperties of Theorem 2.8. The increase due to this adjustment can be compensated by loweringε accordingly. For the proof on this theorem, correctness of the algorithm, running time issues,derandomization and further details we refer the reader to [5].

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30 CHAPTER 2. STEINER TREE PROBLEMS

2.3 The Generalized Steiner Tree Problem

In the generalized Steiner tree problem not all terminals must be connected with a tree. Instead itis specified, which terminals must be connected in a component. The solution can be a forest, hencethe problem is sometimes also referred to as the Steiner forest problem. The generalized Steinerproblem can be stated as follows: Suppose we are given an undirected graph G = (V,E) and acost function c : E → Q+

0 specifying nonnegative edge costs and a collection of disjoint subsetsR1, . . . , Rp ⊆ V . The problem is to find a minimum cost forest, which for each i connects all nodesof Ri. We will denote the optimum solution forest by T ∗. Furthermore, let c(T ) =

∑e∈T c(e) be

the cost of the forest T ⊆ E and n = |R1 ∪ . . . ∪Rp| the number of terminal nodes.The results for the Steiner problem about NP -hardness and hardness of approximation carry overto this problem as it contains the Steiner problem as a special case. Furthermore metric edge costsagain form the main case of the problem.Interestingly, the performance guarantee of the minimum spanning tree heuristic also carries overfrom the Steiner tree problem. This was first observed by Agrawal et al [1], who presented a combi-natorial algorithm that constructs (generalized) minimum spanning trees connecting the terminals.Goemans and Williamson [49] translated this algorithm into a more general setting. They formu-lated the greedy procedure as a primal-dual method based on a linear programming formulation.Actually, the formulation and the algorithm is applicable to a whole class of constrained forestproblems.For an instance of the generalized Steiner tree problem define a function f : 2V → {0, 1} in thefollowing way. If there is a terminal t ∈ S ⊆ V , which needs to be connected to a terminal in V −Slet f(S) = 1 and 0 otherwise. With this function the problem can be formulated as an integerproblem. We will present the linear relaxation in Table 2.1 (left), which is given by changing theintegrality constraints xe ∈ {0, 1} to xe ∈ [0, 1], and the dual program (right). Here δ(S) are theedges between the cut of (S, V − S) in G.

Min∑

e∈Ecexe, Max

S⊆Vf(S)yS ,

subject to∑

e:e∈δ(S)

xe ≥ f(S), S ⊆ V , subject to∑

S:e∈δ(S)

yS ≤ ce, e ∈ E,

xe ∈ [0, 1] e ∈ E yS ≥ 0 S ⊆ V

Table 2.1: Linear relaxation and the dual for the generalized Steiner problem.

Observe that in general every feasible dual solution puts a lower bound on every feasible primalsolution - especially every feasible integer solution of the primal. To bound the approximationratio, the algorithm produces a feasible integral primal solution and a feasible dual solution. Thenthe approximation factor achieved by the algorithm can be derived by upper bounding the fractionof the cost of the integral primal solution over the cost of the dual solution. This is the classicapproach of primal-dual algorithms.The algorithm now works in a greedy fashion. In terms of the dual variables the algorithm startswith all yS = 0 and increases them in a synchronized manner. Based on these values an edge e ispicked to be included into the solution, if it is tight, i.e. if

∑S:e∈δ(S) yS = ce. Then the connectivity

requirement of some sets yS has been fulfilled by the edge picked, and the algorithm stops increasingtheir dual variables yS . The algorithm terminates if no variable yS can be increased anymore.

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2.3. THE GENERALIZED STEINER TREE PROBLEM 31

Turning to the primal view we see the analogy to the minimum spanning tree algorithm. Thealgorithm starts with an empty set of edges T . Initially each node is a component. A componentC is called active, if f(C) = 1, however there is no edge from the cut (C, V − C) in T yet.Otherwise it is called inactive. The algorithm picks an edge e in every iteration, which connectstwo distinct active components. The edge is chosen to have minimum ”reduced” cost, i.e. that putsthe strongest constraint on the increases of all the dual variables of active components. Thus, if theedge is picked, its dual constraint is tight. This is reasonable, because for a small approximationfactor the value of the dual should be as high as possible (but the solution should not be infeasible).Therefore, the edge with the strongest upper bound on the increase is picked. The componentsare then joined - hence the old components disappear and their corresponding dual variables stayuntouched for the rest of the algorithm. As the components either become connected and disappearor become inactive because f(C) = 0, no dual constraint goes overtight and becomes infeasible.The feasibility of the dual is always guaranteed with this choice of the edge. This is a classicfeature of primal-dual algorithms - on the one hand the primal solution is made feasible, on theother hand, the dual solution is feasibly maximized in order to become near-optimal.At the end of the algorithm the greedy augmentation has eventually over-augmented the solution,i.e. there might be redundant edges in the forest T , which can be deleted without destroying thesolution. T is acyclic because the algorithm always adds edges between distinct components. Theonly redundant edges are edges between components that actually do not need to be connected.Thus, any redundant edges can simply be deleted without violating the feasibility of the solution.After using this pruning step, the algorithm outputs its final solution. With the following theoremthe approximation factor was established in [49].

Theorem 2.9 The greedy algorithm delivers a solution, which is a(2− 2

n

)-approximation for the

optimum Steiner forest. The approximation factor is tight.

For the proof of this theorem it is shown that the increase of the primal versus the increase ofthe dual is bounded in each iteration by the average degree of a node in a component. This is aresult of the relaxed dual complementary slackness conditions, which state that either yC = 0 or∑e:e∈δ(S) xe = f(S). In the final solution these conditions are relaxed to

∑e:e∈δ(S) xe = kf(S)

for any k, however, k will be considered for all components, thus a bound on the average value isneeded. The average degree of a node in a forest with at most n nodes is bounded by

(2− 2

n

),

which then establishes the approximation ratio. This analysis is tight, which can be seen with thelower bound example of the minimum spanning tree heuristic for Steiner trees (see Figure 2.1).In terms of running time, this algorithm is again quite fast. It can be implemented to run inO(min(|V |2 log |V |, |E||V |α(|E|, |V |))) time, where α is the inverse of the Ackermann function.

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32 CHAPTER 2. STEINER TREE PROBLEMS

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Chapter 3

Connection Games

In this chapter we will present several results for connection games with unrestricted cost functions.In section 3.2 an introduction is given and some foundational observations are presented withconnection games for 2 players and 2 terminals. Some of the important results from [3] regardingexistence of Nash equilibria and prices of anarchy will be stated and extended by deriving aconstant-factor lower bound for approximate Nash equilibria buying the optimal solution. Thisresult will become more significant when compared to the geometric version of the connection gamein chapter 4.In the subsequent sections the focus is put on the problem of finding approximate Nash equilibriabuying the optimal solution T ∗ for more players and more terminals. In section 3.3 an optimalalgorithm for 2 players and any number of terminals is presented that achieves a 2-approximateNash equilibrium if players are not permitted to share the costs of edges. This factor cannot easilybe improved with cost-sharing. Section 3.4 deals with the problem for any number of players andany number of terminals per player. The general algorithm from [3] yielding a 3-approximate Nashis outlined, and games are analyzed, for which the analysis of the algorithm of [3] is tight. Inaddition these games clarify that the algorithm cannot be improved with cost-sharing.Afterwards a new approach for finding approximate Nash equilibria buying T ∗ for games with 2terminals per player is presented. It solves the problem with a parameterized linear programmingformulation in polynomial time. This approach is conjectured to yield a 2-approximate Nashequilibrium buying the optimal network for N players and 2 terminals per player. A local Steinertree observation is used to support this hypothesis.

3.1 The Model

In the following we will formally define the connection game for N players as it was presented in[3]. Suppose we are given an undirected graph G = (V,E) and a cost function c : E → R+

0 , whichassigns each edge e a non-negative cost c(e). In addition, there are N players, and each playeri has a set of terminal nodes he wants to connect. Terminals of different players do not have tobe distinct. A strategy of a player is a payment function pi, where pi(e) is the amount of moneyplayer i is offering to contribute to the cost c(e). Any edge e, for which

∑i pi(e) ≥ c(e) holds,

is considered bought, and Gp denotes the graph of bought edges. p here is the array of paymentfunctions p = (p1, . . . , pN ), which specify how much each player pays for the edges in Gp. Sinceeach player must connect all of his terminals, they must be connected in Gp. However, each player

33

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34 CHAPTER 3. CONNECTION GAMES

is trying to minimize his total payments∑e∈E pi(e).

A Nash equilibrium of the connection game is a payment function p such that if players offerpayments p, no player i has an interest to deviate from his payments pi. This is equivalent torequiring that if pj for all j 6= i are fixed, then pi minimizes the payments of player i. A k-approximate Nash equilibrium is a function p such that no player i could decrease his paymentsby more than a factor of k by deviating, i.e. by using some other payment function p′i.Some useful properties of Nash equilibria in the connection game are as follows. Suppose we havea Nash equilibrium p, and let T i be the smallest tree in Gp connecting all terminals of player i.From the definitions follows that

1. Gp is a forest,

2. each player i only contributes to costs of edges on T i,

3. each edge is either paid for fully or not at all.

3.2 Connection Games for 2 Players and 2 Terminals

In this section important notions for analyzing payment functions purchasing the optimal solutionT ∗ are introduced. The focus will be on the game depicted in Figure 3.1, in which all edges havecost 1. Any three edges can form an optimal solution, however, in the following suppose wlog thatthe optimum solution in this game is given by e1, e2 and e3.

1

1

e e21

e3

e4

2

2

Figure 3.1: Connection game with 2 players and 2 terminals.

3.2.1 Exact Nash Equilibria and Prices of Anarchy

Most of the results in this subsection (i.e. Lemma 3.2.1, Theorems 3.1, 3.2, 3.3 and 3.4) are dueto [3]. When we start dealing with connection games, one of the first and significant observationsis that Nash equilibria might not exist.

Lemma 3.2.1 The game in Figure 3.1 has no Nash equilibrium.

This is easily verified, because every Nash equilibrium is a path connecting the terminals of eachplayer. It would have cost at least 3. Let wlog e1, e3, e2 be this path. Hence in a Nash equilibrium,player 1 is not willing to pay for e2, player 2 is not willing to pay for e1. So e1 is paid fully byplayer 1, e2 fully by player 2. Suppose now player 1 pays some share of e3. Then his paymentsamount to more than 1, but he can deviate by removing all his payments and buying only e4.

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3.2. CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 35

Then a connection between his terminals is established with e2, which is gratis for him. The costof this deviation would be 1, thus in an equilibrium player 1 is not willing to pay anything for e3.For player 2 the symmetric argument holds. However, with e1 and e2 alone no player is satisfied,therefore the network will never yield a stable Nash equilibrium.

As a game might not have a Nash equilibrium, the next question is to decide, whether a givengame allows an equilibrium or not. Theorems 3.1 and 3.2 provide some insight on this question.

Theorem 3.1 For connection games with 2 players and 2 terminals per player it is decidable inpolynomial time, whether a given game allows a Nash equilibrium.

Theorem 3.2 For general connection games with N players and at least 2 terminals per player itis NP-complete to decide, whether a given game allows a Nash equilibrium.

The algorithm for the 2-player-case simply enumerates on all possible Nash structures. As thenumber of players is fixed, there is only a constant number of possible Nash structures, which aretested to fit into the underlying graph. This way the algorithm is able to find an equilibrium inpolynomial time. For the general case a polynomial reduction from 3-SAT can be given to createan instance of the connection game, which has a Nash equilibrium if and only if the correspondinginstance of 3-SAT allows a feasible solution.Although a game might not have a Nash equilibrium, it might be cheap if it exists. How costlycan a Nash equilibrium be? Theorems 3.3 and 3.4 answer this question and characterize the pricesof anarchy.

Theorem 3.3 The price of anarchy in the connection game with N players is N .

Proof. This follows from the network that has 2 nodes and 2 parallel edges between them. Eachplayer has one terminal at each node. One of the edges has cost 1, the other one cost N . Ifeverybody pays a share of 1 on the costly edge, nobody has an incentive to deviate to fully buythe cheap edge. The final solution will be N times the optimal solution (which has cost 1). Thisgives a lower bound on the price of anarchy. However, N is always an upper bound, because oncethe equilibrium costs more than N times the optimal solution, there is one player that can deviateby buying the whole optimal solution. Thus, the situation would not be a Nash equilibrium. 2

Actually, this proof applies even to connection games with 2 players and 2 terminals per player.

Theorem 3.4 The optimistic price of anarchy in the connection game with N players is at leastN − 2.

The proof of this Theorem is slightly more complicated. It contains an instance of Figure 3.1, inwhich all edges have a very cheap cost of ε > 0. This instance is extended by two nodes that areconnected to this 2-player network with edges of cost of N

2 − 1 − ε. These two additional nodescontain all terminals for the remaining N − 2 players. In addition, there is an edge connectingthem directly with a of cost 1. If the N − 2 players do not pay some share of the small 2 playerinstance, there will be no Nash equilibrium. To do his, however, they must use the two very costlyedges to connect to the instance. Any equilibrium will then be N − 2 times more costly than theoptimum solution, which includes the direct connection and 3 edges from the 2 player instance.Note that in this case the theorem relies on the extension for more players and does not directlytranslate to the case with only 2 players.

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36 CHAPTER 3. CONNECTION GAMES

3.2.2 Approximate Nash Equilibria

Let us now turn to the problem of finding good approximate Nash equilibria buying T ∗. A valuabletool for this task is the notion of a connection set, which was introduced in [3] for connection gameswith any number of players and terminals.

Definition 3.2.1 A connection set S of player i is a subset of edges of T i, such that for eachconnected component C in T ∗ \ S either

1. there is a terminal of i in C, or

2. any player that has a terminal in C has all of its terminals in C.

Intuitively, after removing a connection set from T ∗ and somehow reconnecting the terminals ofplayer i the terminals of all players will be connected in the resulting solution. As T ∗ is the optimalsolution, the maximum cost of any connection set S is a lower bound for the cost of any deviationof player i. This observation leads to the following lemma for any number of players and terminals:

Lemma 3.2.2 Let p be a payment function purchasing T ∗ with the following properties

1. if p(e) > 0 then e is bought fully by a player,

2. each player i only buys edges from T i.

If a player buys a set of edges, which is the union of k connection sets, then p is an k-approximateNash equilibrium.

Connection sets in a game with 2 terminals per player are easy to determine. Each T i forms apath inside T ∗, and two edges belong to the same connection set for player i if they both lie onthe paths T j for the same set of players. Reconsider the game in Figure 3.1 with T ∗ given above.There are two connection sets {e3}, {e2} for player 1 and two connection sets {e1}, {e2} for player2. Every connection set consists of exactly one edge. Suppose e4 has a cost of (1 + ε), then T ∗

is the unique optimal solution. Under the assumption that every edge is bought fully by a playerthere is a lower bound of (2− ε) for approximate Nash equilibria buying T ∗ in this game.

Theorem 3.5 If players are not allowed to share the costs of edges, there is a connection gamefor 2 players and 2 terminals per player, in which any equilibrium purchasing the optimal networkis at least a (2− ε)-approximate Nash equilibrium.

This lower bound applies to all connection games, especially to the payment functions determinedby the algorithm of [3] for games with any number of players and terminals. These issues arefurther discussed in the context of general connection games in section 3.3.

Returning to the game of Figure 3.1 observe that there are only 3 edges that must be paid for,the second connection sets for player 1 and 2 coincide on the same edge. Only one player can beassigned to pay 2 connection sets and have a factor of (2−ε). We will use the notion of edge-disjointconnection sets to capture this observation.

Definition 3.2.2 In a game with N players and 2 terminals per player, T ∗ is said to have kedge-disjoint connection sets, if k is the minimum number of connection sets that the edges of T ∗

can be partitioned into.

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3.2. CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 37

If players hold only 2 terminals, it is unimportant for which player the edges form a connectionset, because a connection set for a player always consists of edges that are present in exactly thesame collection of subtrees T j . However, then this is a connection set for all players j that havethese edges in their subtree. The edge-disjoint connection sets can be identified by grouping edgesthat are located in the same collection of trees T j . In this case the edge-disjoint connection setsare unique and well-defined.The payments of the players amount only to the cost of the edge-disjoint connection sets. Bearingthis in mind it is possible to strengthen Lemma 3.2.2 for the case with 2 terminals per player.

Lemma 3.2.3 In a connection game with N players and 2 terminals per player, let p be a paymentfunction purchasing T ∗ such that each player i purchases only shares of edges from T i. If a playerbuys shares rj ∈ [0, 1], j = 1, . . . , k of k edge-disjoint connection sets, then the player can only

improve his payments by a factor of(∑

j rj

).

Lemma 3.2.3 formalizes the fact that the lower bounding argument of connection sets also workswith partial costs of connection sets and partial costs of deviations. For connection games with 2players and 2 terminals per player there are only 3 edge-disjoint connection sets possible, becausethe potential set of the player set {1, 2} has only 3 non-empty elements. Hence it follows that

Theorem 3.6 For any optimal centralized solution T ∗ of a connection game with 2 players and 2terminals per player, there exists a 1.5-approximate Nash equilibrium purchasing exactly T ∗.

Proof. Each player i fully pays for the connection set of edges that are only in his tree T i andno other tree T j . If trees T 1 and T 2 overlap and the third edge-disjoint connection set is present,the cost of each edge in this connection set will be shared equally between players 1 and 2. WithLemma 3.2.3 the theorem follows. 2

This result is an upper bound on the smallest achievable incentives to deviate from the cheapestnetwork. But what about a lower bound - how much can the approximation improve? As thegame in Figure 3.1 has no Nash equilibrium, in general we cannot achieve an exact Nash buyingT ∗. The following theorem answers this question.

Theorem 3.7 For any ε > 0 there is a game with 2 players and 2 terminals per player suchthat any equilibrium, which purchases the optimal network, is at least a

(65 − ε

)-approximate Nash

equilibrium.

Proof. Consider the quadratic game in Figure 3.1 with the optimal solution T ∗. Every playeris only interested in establishing a path between his terminals. First consider player 1. Supposehe pays an amount of x1 on e1, y1 on e3 and z1 on e2. There are only two deviations player 1considers: Either he satisfies his connection requirement by paying only for the path (e1, e3) andremoves his payments from e2, or he will choose to deviate to (e4, e2) and removes his paymentsfrom e1 and e3. His approximation factor will be

max

{x1 + y1 + z1

x1 + y1,x1 + y1 + z1

z1 + 1

}.

With x2, y2 and z2 defined accordingly for player 2, his factor will be

max

{x2 + y2 + z2

z2 + y2,x2 + y2 + z2

x2 + 1

}.

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38 CHAPTER 3. CONNECTION GAMES

These expressions are minimized if

x1 + y1 = z1 + 1 and x2 + 1 = z2 + y2. (3.1)

As the final factor will be the maximum factor of any player, we see that it is minimized if theadditional constraint

x2 + y2 + z2

x2 + 1=x1 + y1 + z1

z1 + 1

holds, which implies using (3.1) that x2 = z1. As the optimum solution is bought,

x1 + x2 = y1 + y2 = z1 + z2 = 1

follows. Using these equations one can see that each player buys exactly half of the network andthe payments are located symmetrically. Furthermore x2 = z1 = 1

4 , and therefore

x1 + y1 + z1

z1 + 1=

1.5

1.25=

6

5.

If we assign a cost of (1 + ε) to e4, T ∗ becomes the unique optimum and the theorem follows. 2

This lower bound payment function explicitly assigns each player to pay for parts of each edge.Hence, it cannot directly be combined with the notion of connection sets that was derived before,because a connection set for player i is always completely located inside T i. Let us instead considerthe special case of payment functions ps assigning player i only to pay for edges inside his subtree T i.This is a slight generalization of payment functions that are constructed by considering connectionsets only. For this class of functions the following theorem holds.

Theorem 3.8 There is a game with 2 players and 2 terminals such that for any optimal centralizedsolution T ∗ every payment function ps is at least a 1.5-approximate Nash equilibrium purchasingexactly T ∗.

Proof. The proof follows directly with the game in Figure 3.1. 2

3.2.3 Why Connection Sets?

The analysis using connection sets has a couple of striking advantages in the treatment of hugenetworks like Internet graphs. Specifically

• it allows to reduce the time and space complexity of algorithms, because one can disregardhuge parts of the graph. One needs to pay attention only to the optimal tree T ∗.

• it allows easy and simple analysis of algorithms for Nash equilibria buying the optimal solu-tion.

• the algorithms can be combined with approximation algorithms to yield polynomial timealgorithms. The performance can be analyzed using the performance guarantees of theapproximation algorithms.

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3.3. CONNECTION GAMES FOR 2 PLAYERS 39

To clarify the last point observe that it is possible to start with an approximate solution forestand to identify connection sets. If a player is found that has a cheaper deviation for one ofhis connection sets, the connection set can be replaced by this deviation. This results in a newcheaper feasible solution forest. Thus, the selfish improvement on connection sets helps to improvethe solution network. However, as the problem of finding the best deviation is the Steiner treeproblem, it is at present only possible to find the best deviation within a guaranteed factor of 1.55(c.f. section 2.1). This means any rational player (with unlimited computing power) might beable to improve on each connection set by a factor of 1.55. Hence, any algorithm that assigns kconnection sets in the optimal solution results a factor of 1.55k in the presence of approximation.Furthermore it is not guaranteed that the cost of the solution forest is significantly improved, so iteventually might still be twice the optimum (c.f. the approximation algorithm for Steiner forestsin section 2.3). Another important fact to notice is that the algorithm might be continue to makeconnection sets improvement steps for an exponential number of iterations. A way to prohibit thisis to replace a connection set by a better deviation only if this deviation offers some minimum costimprovement. If this cost improvement is based on a parameter ε, polynomial running time can beachieved. An algorithm, which makes each player buy k connection sets in T ∗, can with presentapproximation algorithms be turned into a polynomial time algorithm. The adjusted version willprovide a (1.55k−ε)-approximate Nash equilibrium buying a solution forest of cost at most 2c(T ∗).In this case the algorithm naturally must use the information about the whole graph and not onlyabout T ∗.These elegant features of connection sets do not come without a price. The minimum achievablefactor using connection sets is substantially higher than the lower bound. Notice that in the lowerbound payment we explicitly consider the deviation using e4 and assign payments to players outsidetheir subtree T i. The tradeoff for disregarding the rest of the graph in the analysis will becomeeven more significant in more general cases (see Theorems 3.10 and 3.13).

3.3 Connection Games for 2 Players

If we turn to the generalization for 2 players and any number of terminals, the results from the casewith 2 terminals per player provide some orientation. Theorem 3.3 describing the price of anarchyand Theorems 3.5 and 3.7 about lower bounds for approximate Nash purchasing T ∗ hold for thiscase as well. Therefore again the problem of determining cheap approximate Nash equilibria isconsidered. Using the notion of connection sets a method is derived that calculates a paymentscheme purchasing the optimum solution such that no cost sharing of edges is needed, and eachplayer purchases only 2 connection sets. It will provide a 2-approximate Nash for any game with2 players. This is optimal assuming no cost sharing is allowed (c.f. the lower bound in Theorem3.5).

Theorem 3.9 For any optimal centralized solution T ∗ in a connection game with 2 players, thereexists a 2-approximate Nash equilibrium purchasing exactly T ∗.

Proof. Consider the optimal solution T ∗. Suppose T ∗ is a connected tree. If this is not the case,apply the algorithm to any component.At first, each player pays the edges that belong only to his tree T i and no other tree T j . Thisforms one connection set for each player. Contract the edges paid for and consider the resultingtree T ′. The edges of T ′ must be assigned such that for each player only one additional connectionset is created. Note that with more than 2 terminals per player the subtrees T i are not necessarily

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40 CHAPTER 3. CONNECTION GAMES

paths. The recognition of connection sets by sets of players described in the context of Definition3.2.2 is not possible.Consider the 4 types of nodes in T ′:

1. no terminal is located at v,

2. only player 1 has terminals at v,

3. only player 2 has terminals at v,

4. player 1 and 2 have terminals at v.

Taking a closer look at property 1 of Definition 3.2.1 of connection sets notice that nodes of type1 and 4 do not pose any requirements on the distribution of the payments. They are not ableto contradict the definition of a connection set. This is different with nodes of types 2 and 3.If the edges paid for by player 1 are to form a connection set, we must ensure that he does not’disconnect’ one terminal of player 2 from the others. The edges would violate Definition 3.2.1 ifplayer 1 buys all edges incident to a node v of type 3, but player 2 still has other terminals outsidev.

Condition 3.3.1 If only player i has terminals at a node u ∈ T ′, then player i completely pur-chases a path in T ′ from u to a node v ∈ T ′, at which the other player 3− i has a terminal.

Lemma 3.3.1 If the payment scheme p satisfies Condition 3.3.1, every player buys only oneconnection set in T ′.

Proof. Suppose we remove the payments of player 1. Every component that is composed only ofnodes of type 1 satisfies property 2 of Definition 3.2.1. Every other component must include atleast one node from types 2 - 4. Every component that contains a node of 2 or 4 satisfies property 1of Definition 3.2.1. Every component that contains a node of 3 also contains a path paid by player2 to a node with a terminal of player 1. Thus, every such component also contains a terminal ofplayer 1 and satisfies property 1. With all possible components covered the payment of player 1forms one connection set. The lemma follows with the similar observation for player 2. 2

Now we will show how to find a payment scheme that satisfies Condition 3.3.1 and purchases alledges of T ′. The method works incrementally and starts with a leaf u of T ′. We will denote by Tpthe subgraph of T ′ consisting of purchased edges. At first Tp consists only of the leaf u. Note thateach leaf is a node of type 4.

Algorithm 2 An algorithm determining a 2-approximate Nash equilibrium purchasing T ∗

1. Pick a path P between a node u ∈ Tp and a leaf v 6∈ Tp that has only unpaid edges.

2. Process P node by node from u to v. Depending on the type of the node under considerationthe subsequent edges of the path are assigned as follows:

• If a node of type 2 is encountered, the edges are paid by player 1 until a node with aterminal of player 2 is encountered.

• If a node of type 3 is encountered, the edges are paid by player 2 until a node with aterminal of player 1 is encountered.

• All other edges are assigned to be purchased by a deliberate player.

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3.3. CONNECTION GAMES FOR 2 PLAYERS 41

Tp

1,22 1,2 1

2 2 10 0

Figure 3.2: Extending Tp and assigning costs of the path.

3. Add P to Tp and continue at 1 until T ′ is purchased.

Figure 3.2 shows an example of an extension step. The assignment starts at the leftmost node,which is part of Tp. The edges labelled with 0 can be purchased by a deliberate player.As T ′ is a tree, the way of picking the paths will always ensure that the whole tree is processed.At the final node of every path terminals of both players are located, which is the reason whythe assignments described in step 2 can always be feasibly completed on P . It is easy to see thatthe final payment scheme satisfies condition 3.3.1. Thus, T ∗ is bought with 2 connection sets perplayer and Theorem 3.9 follows. 2

For connection games with 2 players and 2 terminals per player we saw that by sharing the costsof connection sets it was possible to significantly improve the factor. Here, however, it will becomeobvious that without considering all deviations explicitly the bound of 2 is asymptotically tight.

Theorem 3.10 For any ε > 0 there is a game with 2 players such that any deterministic algorithmusing only the optimum solution T ∗ as input constructs a payment function, which is at least a(2− ε)-approximate Nash equilibrium.

Proof. Consider a game with the optimum network T ∗ in Figure 3.3 with 2 players and the samenumber of k terminals for each player. The algorithm only knows this tree, the optimality of it,and nothing about the rest of the graph.

1

2

1

2

1

2

1

2 2

1

Figure 3.3: Lower bound example for 2 players and k terminals per player.

Each edge has a cost of 1, so the total cost of the network amounts to 4k − 5. There is at leastone player that has to pay 2k − 5

2 . Observe the following fact:

Lemma 3.3.2 In any game with T ∗ depicted in Figure 3.3 each connection set consists of at mostk − 1 edges.

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42 CHAPTER 3. CONNECTION GAMES

Proof. Consider the connection sets for player 1. Suppose for contradiction one connection sethas strictly more than k − 1 edges. Then after removing the edges there must be at least k + 1components, so at least one component does not contain a terminal of player 1. This componentthen must either consist only of Steiner nodes or include all terminals of player 2. First, observethat to every Steiner node a terminal of 2 is connected with an edge that is only in T 2. As aconnection set for player 1 uses only edges from T 1, any component with a Steiner node will alsocontain a terminal of player 2. Second, by leaving the terminals of 2 connected, we can just removeedges, which are outside the subtree T 2. This are only k−1 edges. Hence, there is a contradiction,and with the symmetric argument for player 2 the lemma follows. 2

With the lemma follows that a player might be able to connect his terminals using an alternativetree of cost k − 1. Hence, in the worst case a player will eventually have a chance to reduce hispayments by a factor of

4k − 5

2(k − 1)= 2− 1

2(k − 1),

which approaches 2 as the number of terminals k goes to infinity. If the deviation tree has a costof (k− 1)(1 + ε), the optimum solution T ∗ becomes the unique optimum and the theorem follows.2

This bound also provides a lower bound on the observed factor of 1.5 for 2 terminals per player.This is natural, as the network for 2 terminals per player shrinks to the one from Figure 3.1, whichwas previously used to derive the factor. Together with Theorem 3.6 this factor is tight for thedescribed class of algorithms and connection games with 2 players and 2 terminals per player.Here the limit of the analysis with connection sets becomes visible. Using cost-sharing of edgesonly it is not possible to improve the worst-case bound on approximate Nash equilibria. Edgesand deviations outside T ∗ must be considered explicitly, eventually including the whole underlyinggraph G. As it is easy to see, not both players can have the described deviation of cost k−1, whichestablishes the factor - otherwise the described network T ∗ would not be optimal. This is a strongindicator that further improvement is possible! However, we must sacrifice the advantages in theanalysis with connection sets, namely the efficient reduction of huge graphs by only consideringT ∗ and the simplicity of the analysis of algorithm performance. This issue will be revisited in thetreatment of the case with more players below.

3.4 General Connection Games

In the previous sections it has become apparent that games for 2 players offer cheap and goodapproximate Nash equilibria. The situation becomes more complicated if any number of playersand any number of terminals per player are allowed. Some valuable results for this situation havealready been presented, i.e. the NP -completeness in Theorem 3.2 and the prices of anarchy inTheorems 3.3 and 3.4. Hence, again the task will be to find cheap approximate Nash equilibria,for which the lower bounds of Theorems 3.5 and 3.7 apply. Most of these results are derived forthe case of 2 terminals per player, however, they trivially continue to hold for the generalizationof any number of terminals per player.

3.4.1 3-approximate Nash Equilibria with Connection Sets

Next, we will present the algorithm from [3], which uses the notion of connection sets to bound thepossible deviation costs. It assigns each player at most 3 connection sets in any connection game.

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3.4. GENERAL CONNECTION GAMES 43

The algorithm provides the proof for the following very general Theorem 3.11.

Theorem 3.11 For any optimal centralized solution T ∗ of a connection game there exists a 3-approximate Nash equilibrium purchasing exactly T ∗.

Proof (sketch). Here the algorithm will be reproduced. For the proof that it assigns at most 3connection sets to any player the reader is referred to the treatment in [3].

Algorithm 3 Let us assume that the forest T ∗ consists only of a tree, otherwise the procedurecan be applied to every component of T ∗. Obviously, consecutive edges that are incident at aSteiner node of degree 2 will always belong to the same connection set, thus we can simplify T ∗

by contracting the Steiner node. In the following let us assume that T ∗ has no Steiner nodes ofdegree 2.The algorithm starts by creating the tree T ′. For all players it assigns the edges to a player i, whichbelong only to the subtree T i of player i. This trivially assigns each player at most one connectionset, thus in T ′ every player must not get assigned more than two additional connection sets. Thealgorithm creates a decomposition of T ′ into directed paths and processes these paths recursivelyin stages (see Figure 3.4a). In each stage a set R of directed paths is processed, which connectterminals via unpaid edges to the purchased part Tp of T ′. The paths are denoted by R(t), wheret is the terminal that is connected to Tp. By r(t) the other end of R(t) (the one in Tp) is denoted.The cost of the edges will be assigned to be purchased by the terminals connecting to it, whichthen provide the paths for the next stage. A player will purchase the union of the edges purchasedby his terminals. In the beginning we select an arbitrary path R(t) from a terminal t to anotherterminal of the same player. The path is directed away from t. Let R = {R(t)}.Step 1: Consider R generated in the previous phase. Each path R(s) from R is processed asfollows. Let v1, v2, . . . be the nodes of R(s) numbered away from terminal s. Each vk is eithera terminal or has a subtree outside R(s) connecting to it. Consider the component of vk, i.e.the component created by removing edges (vk−1, vk) and (vk, vk+1). Next, consider the terminalsin the components belonging to players that also have terminals outside this component. Theseplayers have an incentive to connect this component to the rest of the tree. Sk will be the set ofterminals of such players that are located in the component of vk. Do not include s in S1, and letthe set of r(s) be empty. Now let t ∈ Sk be a terminal and i be the player, who owns t. Form apath Q(t) for t as follows:

• If i does not own s, pick the smallest l > k such that Sl contains a terminal of player i.

– If such a node vl exists, set Q(t) to be the path from vk to vl.

– If no such node vl exists, and T i contains r(s), set Q(t) to be the path from vk to r(s).

– If no such node vl exists, and T i does not contain r(s), set Q(t) to be the path from vk′to vk, where vk′ is the first node of R(s) such that Sk′ contains a terminal of player i.

• If i owns s, pick the largest l < k such that Sl contains a terminal of player i.

– If such a node vl exists, set Q(t) to be the path from vl to vk.

– If no such node vl exists, set Q(t) to be the path from s to vk.

Figure 3.4b shows the paths Q(t) for terminals t1, . . . , t4 of player i, if i does not own s. Figure3.4c) presents the same case, however here player i does own terminal s. Edges belonging to thesame connection set are identified using the notion of links.

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44 CHAPTER 3. CONNECTION GAMES

s r(s)

t1 t2 t3 t4

Q(t Q(t Q(t Q(t1 2 3 4) ) ))

sQ(t Q(t Q(t Q(t1 2 3 4) ) ))

2tR(t2 )

r(t2 )

t1

t6t7

t3

t4

t8

9t

5t

r(s)

t1 t2 t3 t4

b)

c)a)

Figure 3.4: a) A decomposition of T ∗ into paths R(t); b) c) The paths Q(t) for a single player i.

Definition 3.4.1 A link L is a maximal set of edges of R(s) such that for every edge e ∈ L, theset of paths Q(t) that contain e is exactly the same, for t ∈ dvk∈R(s)Sk.

A link is a connection set of any terminal t with L ⊆ Q(t). If L is removed and the endpoints ofQ(t) are reconnected, the endpoints of all other paths Q(t′) become connected again and a feasiblesolution is achieved. To pay for R(s) we choose one terminal from each Sk. These terminals payfor the complete path R(s), where each terminal t pays for at most one link L with L ⊆ Q(t). Thisis done in Step 2.Step 2: In this step a bipartite graph (VR, VL) is constructed, where VR are the vertices of R(s)and VL contains a vertex for each link on R(s). There is an edge between a node vk ∈ VR anda node L ∈ VL if some terminal t ∈ Sk exists such that L is a connection set for t (L ⊆ Q(t)).Now the Hungerian Algorithm can be used to compute a perfect matching in this graph to get oneterminal for each link that is assigned to purchase it. See [3] for the proof that such a matching isalways possible.Step 3: Finally, all the payments for the complete path R(s) are assigned. Consider the terminalsthat were picked to pay for links in Step 2. The paths from the terminals tk to vk ∈ R(s) are stillunpaid, thus these paths compose the set R for the next phase. Eventually there is a terminaltk ∈ Sk, whose path still needs to be purchased, but there has not been any terminal from Skpurchasing any link. In this case we choose tk ∈ Sk and include its path R(tk) into R. If all pathsR(s) of the current phase are processed, R becomes the set of all paths R(tk) that resulted fromany of the paths R(s) considered in the current phase.

2

It is easy to see that this algorithm does not involve any cost-sharing between players. Each edgeis paid fully by a player. Due to Theorem 3.5 it could be possible that the algorithm actuallyalways achieves a 2-approximate Nash equilibrium. Theorem 3.12 declines these hopes and showsthat factor 3 is tight.

Theorem 3.12 In the worst case Algorithm 3 calculates a payment function p, which is a 3-approximate Nash equilibrium.

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3.4. GENERAL CONNECTION GAMES 45

3

N

1

N−1

2

N−1

1,2N

3

4

a) b)

13

2

3

d

1

2

Figure 3.5: Tightness examples for a 3-approximation using connection sets.

Proof. Consider a game with the unique optimal network T ∗ depicted in Figure 3.5a, in which alledges have cost 1. At first the network T ′ is created. For the recursive part let wlog the initialpath be the path between the terminals of player 1. Then we have 2 links on this path, which mustbe paid for by player 2 and 3 respectively. In the second phase assume the terminal of player 2 ispicked to pay the last remaining edge, which forms the third connection set for him. Thus, player2 pays 3 connection sets. Now there could be an edge d 6∈ T ∗ between the terminals of player 2 ofcost (1 + ε) with an arbitrary small ε > 0. It creates a deviation that is at least a factor of (3− ε)cheaper. 2

In this tightness example there are only 6 edge-disjoint connection sets. By assigning one connectionset to player 1 instead of player 2, the factor can be lowered to 2 for every player. If more playersare involved, the number of connection sets can increase. With partial payments to connectionsets, however, it might seem possible to equilibrate the costs of the connection sets feasibly to staybelow the bound of 3 for all players. Unfortunately, Theorem 3.13 provides an argument that anydeterministic algorithm using only T ∗ as input cannot significantly profit from cost sharing (c.f.Theorem 3.10). In this way the algorithm represents an optimal algorithm.

Theorem 3.13 For any ε > 0 there is a game such that any deterministic algorithm using only theoptimum solution T ∗ as input constructs a payment function, which is at least a (3−ε)-approximateNash equilibrium.

Proof. Consider an optimal solution T ∗ shown in Figure 3.5b, in which all edges have cost 1.The proof works similar to the proof of Theorem 3.10. If each edge has cost 1, the tree has cost3N − 3. So there is at least one player that pays a cost of 3− 3

N . Taking a closer look at this gameit becomes obvious that each edge is a distinct connection set, because it is shared by a distinctset of player subtrees T i. Hence, each deviation will just be lower bounded by a cost of 1. Thismeans that a player can have an alternative of cost (1 + ε), which the algorithm does not know.It is possible that the algorithm assigns this player a cost of 3− 3

N . As N approaches infinity, thedeviation factor becomes (3− ε), and the theorem is proven. 2

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46 CHAPTER 3. CONNECTION GAMES

Again the limits of the analysis with connection set are visible. In the next section a way ofachieving better approximate Nash equilibria will be considered.

3.4.2 2-approximate Nash Equilibria with Deviation Sets

To achieve better approximate Nash equilibria up to all the edges for possible deviations outside T ∗

must be taken into account. The problem will be formalized in the context of linear programming.In the following we will not go deeply into the theory of linear programming, but rather formulatethe problem and refer to standard techniques to obtain a solution. The task to find a minimumapproximate Nash equilibrium buying the optimal solution can be formulated as a parameterizedlinear program:

Min k,

subject to

N∑

i=1

xie = 1 for all e ∈ T ∗,

e∈T∗xiec(e) ≤ k

(cT +

e∈T∩T∗xiec(e)

)for all T ∈ Di, 0 ≤ i ≤ N ,

xie ∈ [0, 1], k ∈ [1, 3].

The first set of constraints ensures that exactly the optimum solution T ∗ is bought. xie is thepercentage player i pays for the cost of edge e ∈ T ∗. In the second set of constraints the set ofpossible trees Di is considered, which player i could choose to connect his terminals with. For eachsuch tree the cost is calculated for player i. If T has edges e in common with T ∗, player i will onlyneed to pay his share xie. For all edges in T outside T ∗ player i will have to buy the full cost c(e).Note that the costs for edges outside T ∗ form a constant, as they are not altered in the process ofadjusting xie. To point out this fact the expression

cT =∑

e∈T \ T∗c(e)

is used in the formulation. The contribution of player i to T ∗ must be at most k times the costof any deviation tree. If this holds for all players and all trees, the payment function specified bypi(e) = xiec(e) will be a k-approximate Nash equilibrium purchasing the optimal solution.Exactly speaking, this ’linear’ program is not linear. In the second set of constraints k is alwaysmultiplied by xie, although both are variables. However, using the algorithm in the last sectionthere is always a feasible solution to this program for k = 3. The optimization problem can berestated as a decision problem, whether there is a payment function, which is a k-approximateNash. Assuming k is fixed the parameterized linear program reduces to a standard linear program,for which a feasible solution needs to be found. If it is possible to solve this decision problem inpolynomial time, the minimum k can be found with binary search on the interval [1, 3].Another, more severe problem with the formulation in such a general form is the fact that one mightnot be able to test all trees from Di. There might be exponentially many resulting in an exponentialnumber of constraints. In the linear programming literature there have been approaches to dealwith an exponential number of constraints. Namely, the ellipsoid algorithm by Khachian [66, 78]

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3.4. GENERAL CONNECTION GAMES 47

is a famous example of a polynomial time algorithm for such linear programs. The crucial feature,however, is the possibility to determine a violated constraint in polynomial time. Given a set ofvalues for k and xie, one could in each iteration reduce the number of constraints using a procedureto determine the least cost deviation tree T ∈ Di for player i. If this tree satisfies its constraintwith factor k, then any tree for player i will satisfy it. Then a violated constraint can alwaysbe identified, or it can be assumed that all constraints are satisfied by considering only a linearnumber of constraints. Unfortunately, the problem of finding the best deviation tree is the NP -hardSteiner tree problem. Therefore the problem will be adjusted and restricted to the special casewith 2 terminals per player. In this case finding the violated constraint reduces to a calculation ofa shortest path, which can be done in at most time O(n2 logn), where n is the number of nodesin G. Using the ellipsoid algorithm it is decidable in polynomial time, whether for a given k ak-approximate Nash equilibrium purchasing T ∗ exists.With this procedure an algorithm exists that determines k-approximate Nash equilibria with aminimal factor k. The interesting question is how bad these approximations can be. Is there anupper bound on k that is better than 3 ? To give an understanding of how one might answer thisquestion, the following Conjecture 3.1 is stated and supported by some combinatorial observations.

Conjecture 3.1 For any optimal centralized solution T ∗ of a connection game with 2 terminalsper player there exists a 2-approximate Nash equilibrium purchasing exactly T ∗.

To attack the proof of this theorem it seems profitable to employ the understanding of the notionof connection sets as much as possible. Let us assume that the payment function belongs to thepreviously described class ps, i.e. each player is only assigned to pay for edges inside his subtreeT i. Then consider the following constellation in T ∗.

d

e12e

e3 d2313

C

C

C1C

d12

4

2

3

Figure 3.6: Two deviations cost as much as three connection sets.

Let the edges e1, e2 and e3 be distinct connection sets. If the distribution of the terminals requiresonly C1, C2 and C3 to be connected, then for the sum of any two deviations

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48 CHAPTER 3. CONNECTION GAMES

c(di) + c(dj) ≤ c(e1) + c(e2) + c(e3), i 6= j, i, j ∈ {12, 23, 13} (3.2)

holds. Suppose for instance d13 + d23 are cheaper, then the tree T ′ = T ∗−{e1, e2, e3})∪{d12, d23}using the deviations is a cheaper tree satisfying the connectivity requirements. This contradictsthe optimality of T ∗.It is easy to observe that the same argument can be applied, if k connection sets connect to the innercomponent, however, only the outer k components need to be connected in terms of connectivityrequirements. Then the cost of the k connection sets amounts for the k−1 deviation paths neededto reconnect the components.It is easy to see that three or more connection sets connecting to a Steiner node build a constellationlike this. The Steiner node can feasibly stay unconnected, only the rest of the nodes must beconnected. No player actually has a connectivity requirement that makes him incorporate theSteiner node in the network. The connections are just present because other connections aremore expensive. Hence, a Steiner point is just present because of the lack of cheap alternativeconnections. However, if no cheap alternative connections exist, no large deviation factors canbe achieved. In particular, in Figure 3.6 all possible combinations of two or more deviations areat least equally costly as the three connection sets. In the graph there must be players withconnectivity requirements such that the outer three components need to be connected. As theedges e1, e2 and e3 are part of the optimum solution, however, there is no combined deviation thatis cheaper than the sum of the costs of these edges. Hence, the cost of the three connection sets canalways be distributed to the involved players such that no player will find it profitable to deviate -just let each player pay the cost of the cheapest deviation path between the components, in whichhis terminals are located. Then at least the cost of 2 different deviations is charged, which pay forthe cost of the three connection sets. Obviously, the three edges form some notion of generalizedconnection set for a set of more players. This will be referred to as k-deviation set.

Definition 3.4.2 Let Ec be the union of at least m ≥ k edge-disjoint connection sets, which arepart of the same tree component T . Ec is called a k-deviation set if k of the components in T −Ecmust be reconnected together into a tree to construct a feasible solution network.

This captures the observation of the cost reduction. The k connection sets must be at least ascostly as k−1 alternative connections. After the removal of a k-deviation set, there are at least k−1players unconnected. The connection requirements of these players impose the need to reconnectthe created components. However, any reconnection that leads to a feasible solution must be morecostly than the cost of the k connection sets. This is due to the assumption that the connectionsets come from an optimum forest T ∗. Hence, the cost of the (at least) k connections set canalways be assigned to k−1 players such that no player has an incentive to deviate. Note that sucha specific assignment of payments relies on knowing the exact alternative deviations and eventuallyrequires to analyze the whole graph G. In the present case of 2 terminals per player the deviationscan be found by shortest path calculations.Now consider the lower bound example of Figure 3.5b. Actually, the edges incident at Steinerpoints of degree 2 always belong to the same connection set. This was already an observationmade in Algorithm 3, where Steiner points were simply dropped from consideration. We will referto a graph with dropped Steiner points of degree 2 as a connection set graph. Hence, in a connectionset graph every Steiner point has at least degree 3. It is easy to see using well-known arguments(c.f. Lemma 2.2.2 and Corollary 2.2.1) that any connection set graph has at most 2N − 2 Steinernodes and 4N − 3 edges in a game with N players and 2 terminals each. Each edge can be a

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3.4. GENERAL CONNECTION GAMES 49

separate connection set, as long as it is shared by a different set of players. In this case, however,two edges incident at the two terminals of each player form one connection set. Hence, there are atmost 3N−3 edge-disjoint connection sets in a game. The deviation factor for a player is maximizedif all connection sets have the same cost. Hence, one can see that in general no stronger lowerbound than the one derived in Theorem 3.13 can be achieved.In this bad example the notion of k-deviation sets allows to see that actually 2N − 1 connectionsets can be paid for by the players without any incentive to deviate! Consider the 2N − 1 edgeson the left half of the cycle involving all the Steiner points. These are all edges left of the interiorterminal of player N , which form 2N − 1 edge-disjoint connection sets. If these sets are removed,then only N+1 components must be reconnected - all N−2 Steiner points can remain unconnected.Therefore, the 2N − 1 edges can be paid for such that no player has an incentive to deviate. Notethat this is also true for any collection of players {N,N − 1, . . . , i} and the corresponding set of2N − 2i + 1 connection sets on the left half of the cycle. Hence, the payment function can besuch that on these edges any player pays only for edges located in his subtree. The remainingN − 2 connection sets can be distributed such that no player must pay for more than 1 connectionset. Hence, using a deviation set the analysis can be refined to yield only a 2-approximate Nashequilibrium.The conjecture now is that k-deviation sets allow such a reduction in any game for N players and 2terminals per player, which forces a player to pay for more than 2 connection sets. Unfortunately,a formal proof of this statement has not yet been derived.

3.4.3 Lower Bounds

In [3] a general lower bound for approximate Nash buying T ∗ has been established, which will bepresented next. In Theorem 3.7 we already exhibited a lower bound of

(65 − ε

)for a game with 2

players. This lower bound can be increased to(

32 − ε

)in the case of more players.

Theorem 3.14 For any ε > 0 there is a connection game such that any equilibrium, which pur-chases the optimal network, is at least a

(32 − ε

)-approximate Nash equilibrium.

Proof (sketch). The proof is analogous for the proof of Theorem 3.7, thus we will only sketch theimportant points. For further details consult [3]. Consider a graph HN , in which each edge hascost 1 and each player i has 2 terminals si and ti. Figure 3.7a shows the structure of the graph,which is presented for N = 5. Every optimal solution T ∗ is a connected minimum spanning treeof cost 2N − 1. Assume T ∗ is formed by the outer ring, in which the edge (s1, tN ) is missing.In a 3

2 -approximate Nash players 1 and N do not have incentives to pay more than a cost of 3 inthe solution. Suppose they do, then the interior players must only cover a cost of 2N − 7. Nowconsider an interior player i, for which in Figure 3.7b a simplified partial view is depicted.The payments can always be redistributed in the way that player i pays only x+ y+ z. Thus, thedeviation factor for player i becomes

max

{x+ y + z

x,x+ y + z

y + 1,x+ y + z

z + 1

}

This is minimized for x = y + 1 = z + 1. There is at least one player, for which the paymentsamount to

x+ y + z = 3x− 2 ≥ 2N − 7

N − 2

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50 CHAPTER 3. CONNECTION GAMES

5

5

1

1

42

4

3

2

3

b)a)

x

y z

s

t si i

i+1t i−1

Figure 3.7: Lower bound example for approximate Nash purchasing the optimal network.

and the deviation factor becomes

x+ y + z

x≥ 6N − 21

4N − 11

Obviously the payments of players 1 and N lead only to a negligible cost reduction for the interiorplayers. If we increase the cost of every edge outside T ∗ to 1 + ε, T ∗ becomes the only optimalsolution. If N approaches infinity, every payment scheme purchasing T ∗ becomes at least a

(32 − ε

)-

approximate Nash equilibrium, and the theorem follows. 2

The previous proof provides some further insight into the problem. A direct corollary deals withthe assignment inside the subtree T i. Let ps again be a payment function that assigns a player topay for edges located in his subtree T i only.

Corollary 3.4.1 For every ε > 0 there is a game, for which any ps is at least a (2−ε)-approximateNash equilibrium.

Consider the game from Figure 3.7 again. Now the payments of each player are located completelyon x. Players 1 and N can now pay a cost of 4. Hence there is at least one player that must pay

x ≥ 2N − 9

N − 2

This is also the value of his deviation factor, because he now has deviations of cost 1. If Napproaches infinity, and if all edges outside T ∗ have cost (1 + ε), the corollary follows.Hence, if an algorithm assigns payments for a player i only inside the subtree T i, it cannot achievea better factor than 2. Clearly, the analysis of the method presented in section 3.4.2 did rely onconsidering payment functions with this property, and hence would be tight in this sense. Observethat in the game of Figure 3.7a no k-deviation set is present because of the special ordering of theplayers on the optimum solution path.

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Chapter 4

Geometric Connection Games

In this chapter the analysis of connection games is extended to the special case, in which edge costsare given by Euclidean distances in the plane. It is natural that this restricted setting might offerimproved properties regarding the cost, existence and computational complexity of Nash equilibria.The geometric connection game is defined similar to the general connection game, however, herethe geometric Steiner tree problem is used as a foundation. Hence, it is assumed that any pointin the plane can be a Steiner point and any segment can be used as an edge or purchased as adeviation.In the following sections some interesting theorems for geometric connection games will be proven.In most cases the optimal Steiner tree T ∗ will be considered, and the properties given in Lemmas2.2.1, 2.2.2 and Corollary 2.2.1 will be used.

4.1 Geometric Connection Games for 2 Players and 2 Ter-minals

First the case for 2 players and 2 terminals per player will be considered. In the geometric environ-ment the high number of possible deviations provides a significant improvement over the generalcase. Although the price of anarchy is still N in the geometric case, the optimistic price of an-archy is small. For any geometric connection game with 2 players and 2 terminals it is possibleto find exact Nash equilibria that are only a factor of

√2 away from the centralized optimum.

This is an improvement over the general case, in which instances without any Nash equilibriumexist. Furthermore a discretization scheme is specified to derive arbitrary small deviation factorsfor the players for approximate Nash equilibria buying the optimum solution. This is a significantdifference to the general case, for which we established a lower bound of

(65 − ε

).

4.1.1 Exact Nash Equilibria and Prices of Anarchy

In the geometric environment, the pessimistic price of anarchy is still as bad as in the general case.

Theorem 4.1 The price of anarchy in the geometric connection game is N .

51

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52 CHAPTER 4. GEOMETRIC CONNECTION GAMES

1,2 1,2

1e3

e e21 22

e

a) b)

1,...,N1,...,N

e

e

e

e

3

1

2

N

eN−1

Figure 4.1: Geometric games with maximum price of anarchy.

Proof. At first, let us consider somewhat generally how a player in the geometric environment ismotivated to deviate from a given payment scheme. For simplicity it is assumed that each playerhas only 2 terminals.

Lemma 4.1.1 Suppose we are given a geometric connection game with 2 terminals per player anda feasible forest T , which satisfies the connection requirement for each player. Furthermore, let pbe a payment function, which specifies a payment for each player on each edge. If the deviation fora player i from p includes an edge e 6∈ T , this edge is a straight segment, with start and end eitherat a terminal or some other part of T . It is located completely inside the Euclidean convex hull ofT .

Proof. This statement is verified by showing that once an edge violates the given properties, thereis a cheaper deviation for i with an edge fulfilling them. The cheapest connection between twopoints in the plane is a straight line, therefore any curved edge bought fully by a player is morecostly than the straight connection between its endpoints.Now consider straight edges bought fully by a player that meet at a common point u. As everyplayer has only two terminals, he is only motivated to establish a path between the terminals.Thus, at any point u a player never has an incentive to (partially) buy more than two edgesmeeting at the point. Thus, if an edge does not start or end at some point of T , it is of no use forplayer i.It is easy to see that once a straight edge leaves the Euclidean convex hull of T , it will never hitany point of T again and therefore be of no use to player i. 2

With these observations it is easy to specify some properties of Nash equilibria for a geometricconnection game.

Lemma 4.1.2 In a Nash equilibrium of the geometric connection game for N players edges e1, e2

bought fully by one player are straight segments and meet with other, differently purchased edgeswith an inner angle of at least 90 ◦. In the case of 2 terminals per player e1 and e2 can only meetat a point if they have an inner angle of 180 ◦.

Proof. All properties follow directly from the triangle inequality. 2

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 53

Consider the game for 2 players and 2 terminals shown in Figure 4.1a. Let e2 = e21 ∪ e22. Thepayment scheme purchases T in the following way. Player 1 pays for e3 and e21. Player 2 paysfor e1 and e22. Let the costs be e1 = e3 = e21 = e22 = 1

2 . e1 and e2 as well as e2 and e3 areorthogonal. The optimal solution in this network is the direct connection between the terminals.To start the observation that this network forms a Nash equilibrium, note that the necessaryconditions of Lemma 4.1.2 are fulfilled. In addition, no player can deviate by simply removing anypayment from the network. Lemma 4.1.1 restricts the attention to straight segments inside therectangle, which is the Euclidean hull of T . The argument is given for player 1 - it can be appliedsymmetrically to player 2. Note that a deviation with an endpoint inside the same segment (orwith endpoints only in e21 and e22) is not profitable. Furthermore consider a segment connectingtwo points on segments paid for by player 2. It is always longer than the parts player 1 couldremove. However, this is also true for a deviation edge that starts and ends in segments boughtby player 1. Suppose he buys such an edge d = (u, v) with u ∈ e3 and v ∈ e21. Let ed3 be the partof e3 inside the cycle introduced by d in T (ed21 accordingly). Then with

|d|2 = |ed3|2 + (|ed21|+1

2)2

and |ed3|, |ed21| ≤ 12 the inequalities

|d| ≥ |ed21|+1

2≥ |ed3|+ |ed21|

holds. d is not profitable for player 1. Hence, all edges player 1 would consider for a deviationare unprofitable. With the symmetric argument for player 2 we see that the payment schemerepresents a Nash equilibrium. Since the optimum solution is half of the cost of T , the theoremfollows for games with 2 players and 2 terminals per player.In the network with more players assume that each player has 1 terminal at each node. Constructa path between the nodes, which approximates a cycle with N straight edges of cost 1 (see Figure4.1b). Each player i is assigned to pay for one edge ei of cost 1. Observe that the necessaryconditions of Lemma 4.1.2 are fulfilled. Now consider the deviations for a player i. He will neitherconsider segments that cost more than 1 nor segments that do not allow him to save on ei. Forthe remaining deviations it is easy to see that no segment will yield any profit, because the cyclicstructure makes the interior angles between the edges amount to at least 90 ◦. Any deviationd = (u, v) from a point u ∈ ei to any other point v will be longer than the corresponding part edithat it allows to save. This argument is valid for any player i. As the optimum solution is thedirect connection of cost 1, the theorem follows. 2

This result is contrasted with a method to find good Nash equilibria inside the Euclidean convexhull of the solution.

Theorem 4.2 In any geometric connection game with 2 players and 2 terminals exists a Nashequilibrium, which is a

√2-approximation for the optimum centralized solution.

Proof. The theorem will be proven using an algorithm that repeatedly applies a local improvementstep (c.f. section 1.2 and the treatment of local search).

Local Improvement Step:Allow one player to deviate to his cheapest deviation.

The algorithm starts with the empty network and applies the step 3 times. At first the player

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54 CHAPTER 4. GEOMETRIC CONNECTION GAMES

1 1d1

A AA1 2 3

Figure 4.2: d1 stays the cheapest connection.

is picked, who has the cheapest direct connection between his terminals. During the next stepsthe players are alternated. To show that this leads to a Nash equilibrium the following lemma isneeded.

Lemma 4.1.3 Suppose player 1 buys the direct connection d1 between his terminals and player 2buys straight segments si, i ∈ N that meet with d1 with inner angles of more than 90 ◦. Player 1will not be able to decrease his payments using any portion of the segments si.

Proof. This is rather obvious from the triangle inequality. Divide the plane into three regionsbased on d1 (see Figure 4.2).Assume the lemma holds for segments that hit the middle region A2. If a straight segment sof player 2 meets d1 at a terminal, it must never cross to the A2, because otherwise the innerangle would be less than 90 ◦. As we assume the lemma holds for A2, it suffices to consider onlydeviations that start at a point u ∈ d1. However, from u player 1 needs less cost to reach histerminal than any other point on s. Hence, he will never be motivated to use some portion of s.Turning to the middle region A2 consider a segment s that meets at an inner point of d1. It isobvious that s must be orthogonal to d1 to provide an inner angle of 90 ◦. Suppose at first thatthere is only one segment s, and that player 1 uses some portion of it in a deviation. Then he hasto pay the connection from two points on d1 to two points on s. This is obviously more costly thanthe part on d1 he can save. Assume now there are more segments si, i ∈ N connecting to d1. Againthe cheapest connections between any si and sj as well as the cheapest connections between theterminals and any segment si are parallel to d1. Therefore, d1 provides the cheapest connectionbetween the terminals of player 1 and the lemma is proven. 2

Now recall the local improvement steps. In the first step player 1 chooses to pay for his directconnection d1. For the second step let us at first consider the case that player 2 connects to d1 anduses some portion of it. He will connect in the cheapest way, obeying the properties described inLemma 4.1.1. His connections will be straight segments connecting to d1 with at least 90 ◦ innerangles when connecting to a terminal and exactly 90 ◦ angles when connecting to an interior pointof d1. Thus, Lemma 4.1.3 can be used to argue that a Nash equilibrium is achieved.Otherwise, it could happen that player 2 picks his direct connection d2 as well. Then player 1 couldpossibly improve his payments using the payments of player 2. If he can, his new connections willallow to argue that Lemma 4.1.3 holds for player 2 and an equilibrium is achieved. Otherwise we

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 55

obviously have a Nash equilibrium, because no player can decrease his payments anymore.

The observation about the social quality of the Nash equilibrium is stated in the following lemma.

Lemma 4.1.4 The Nash equilibrium network TE designed with the local improvement steps is atmost a

√2 approximation of the optimal centralized network T ∗.

Proof. Let the equilibrium network be denoted by TE . The lemma states that for the ratio

r =|TE ||T ∗| ≤

√2 (4.1)

holds. For the rest of the proof assume that player 2 starts the local improvements, i.e. thatd2 ≤ d1. Observe that |TE | ≤ |d1| + |d2| holds, i.e. the cost of the direct deviations is always anupper bound for the cost of the Nash equilibrium found. We will consider all possible geometricconnection games for 2 players and 2 terminals per player. The classification depends on thestructure of the optimal centralized solution. If T ∗ consists of two components or two edges, whichare exactly d1 and d2, the method finds an exact Nash equilibrium purchasing it. So suppose T ∗ isa connected tree, which has an edge e3 present in both subtrees T 1 and T 2. The different networktypes to consider are shown in Figures 4.3 and 4.4.

a) b)

e 1,2

e

3

2 e

2

1

e1

3

e2

e

1

1

2

2

1

Figure 4.3: Network types with one edge per connection set.

In the following T ∗ and TE are transformed into different networks for the different types of games.In each step this will increase the bound on r. In the figures thin lines indicate the parts thatamount for TE and thick lines the parts that amount for T ∗.

Type 4.2.1 Consider a game, in which the optimal network is of the type depicted in Figure 4.3a.It has no Steiner point, thus the angles between the segments are possibly greater than 120 ◦. Asd2 ≤ d1, TE contains d2. In the network in Figure 4.3a player 1 uses e1 in TE to connect histerminal. e1 is also part of T ∗. Thus, the ratio can be increased by dropping e1 from both T ∗ andTE . The result of this is depicted in Figure 4.5.Remember Figure 4.5 does not depict the initial game but a network construction to bound theratio in a game with an optimal network of Figure 4.3a. Notice that with the Cosine Theorem

|d2| =√|e2|2 + |e3|2 − 2|e2||e3| cosα,

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56 CHAPTER 4. GEOMETRIC CONNECTION GAMES

a)

e

b)

2

2

2

2e

e

1

1 e

ee

21

2

3

1

3

1

1

1

e12

e11

2

e21

2

e22

1

Figure 4.4: Network types with connections sets consisting of two edges.

11,2 e3

2

2e2d

d’

α

Figure 4.5: T ∗ and TE after removing e1.

where α ∈[

23Π,Π

]is the inner angle between e2 and e3. The length of the small segment d′ can

be given as

|d′| = |e2||e3| sinα√|e2|2 + |e3|2 − 2|e2||e3| cosα

.

Thus, the ratio is bounded by

r =|TE ||T ∗| ≤

|d2|+ |d′||e2|+ |e3|

.

Now we will analyze how large this bound can be. Assume the cost of T ∗ to be fixed to |T ∗| =|e2| + |e3| = 1 and look for a game, which distributes this cost to e2 and e3 as badly as possible.Let x = |e2| and 1− x = |e3|, then

r(x, α) ≤ 2x(sinα− 2 cosα− 1)(1− x) + 1√2x(1 + cosα)(x− 1) + 1

.

To maximize this expression set the derivative rx(x, α) = 0. This results in

2(sinα− 2 cosα− 1)(2x(1 + cosα)(x− 1) + 1)

(2x(sinα− 2 cosα− 1)(1− x) + 1)(1 + cosα)(1− 2x) = (1− 2x).

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 57

It follows that the square root results in a real value only for x = 12 . This represents a maximum

point. For any given value of α the ratio is maximized by setting |e2| = |e3|. Then

r(1

2, α) ≤ sinα

2√

2 + 2 cosα+

√2 + 2 cosα

2.

From a plot of this function in Figure 4.6 one can see that the function becomes maximized forα ≈ 122 ◦, which corresponds to a radial value of approximately 2.12. In any case the value of r isless than 1.125 <

√2.

1

1.025

1.05

1.075

1.1

1.125

2.0944 2.35619 2.61799 2.87979 3.14159

r(0.5, a)

Figure 4.6: Behavior of the ratio r(0.5, α) for angles between 120 ◦ and 180 ◦.

These steps to upper bound the ratio can be applied to a lot of games. The crucial point in theapplication is that both direct deviations d1, d2 must share a point with edge e3. Then d2, e3 anda third edge, which is only in the subtree of player 2, will always form at least one such triangledepicted in Figure 4.5. Then some parts of the the payments of player 1 can be upper bounded bythe cost of the edges, which are only present in his subtree. This will always enable to transformthe networks to arrive at a constellation depicted in Figure 4.5 and to upper bound r by 1.125.

To illustrate this fact and the possible transformations a little further two additional types of gamesare analyzed. Notice that both types have an optimal solution T ∗ such that both d1 and d2 sharea point with e3.

Type 4.2.2 Let us turn to games, which have optimal networks T ∗ including exactly one Steinerpoint and three edges (see Figure 4.3b). TE consists of d2 and the segment connecting the terminalof player 1. This segment can be upper bounded by e1 and the shortest connection between d2

and the Steiner point. Then using this upper bound for TE the ratio can be further increased bydropping e1. This leads directly to a network construction of Figure 4.5, thus in this case ratio ragain is bounded by 1.125.

Type 4.2.3 Consider a game, which has an optimal solution of the type depicted in Figure 4.4a.It has two Steiner points and the terminals of the players are located on different sides of the linethrough e3. Again we assume d2 ≤ d1. Then player 1 connects directly to d2 from his terminals.Suppose he buys e11 and e12 instead and connects to d2 from the Steiner points. This will bemore costly than the connections he actually chooses in TE , hence one can upper bound TE withthis assumption. However, now in the cost of T ∗ and in the upper bound of |TE | the costs of theedges e11 and e12 are present. Hence, we can increase the upper bound by removing these costs

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58 CHAPTER 4. GEOMETRIC CONNECTION GAMES

from both expressions. This is equivalent to adjusting the network as depicted in Figure 4.7. Thisnew network construction, however, can easily be identified to consist of two networks that werealready considered in the previous types. Thus, the ratio r again is bounded by 1.125.

1

2

2

1

Figure 4.7: Transformations increasing the upper bound for r.

Type 4.2.4 Finally, consider the type of games, in which at least one of the deviations does notmeet or cross e3. Assume that this is deviation d2. Thus the connection set of T ∗ that belongsonly to T 2 consists of two edges e21 and e22. Otherwise at least one terminal of player 2 wouldbe located on e3 and d2 would hit e3. Suppose the deviation d2 is part of the final network TE ,and let TE be such that player 1 chooses to connect to d2. This of course implies that at least oneterminal of player 1 is not located on e3. Otherwise again the transformations described above canbe used to bound the ratio by 1.125. So consider the case in which at least one terminal of player1 is not located on e3 (see Figure 4.4b for an example). Then these terminals connect to a Steinerpoint in T ∗. Here it is possible to transform similarly to the previous cases and bound the lengthof a direct connection from the terminal of 1 to d2. These direct connections are in length upperbounded by the lengths of edges e11 or e12 between the terminal and the Steiner point plus thelengths of the direct connections d′ and d′′ between the Steiner point and d2. Hence, it is againpossible to remove e11 or e12 from T ∗ and the upper bound of TE while increasing the upper boundfor r. Apparently the ratio r in any game with a tree TE of the described form (a direct deviationthat does not cross e3) can be upper bounded by a network construction depicted in Figure 4.8.

e3

2

2e

α

2

β

e22

1

d2

d’d’’

Figure 4.8: Network structure to bound the ratio in games, for which d2 ∈ TE does not hit e3.

With an easy observation we can see that the middle part of d2 between d′ and d′′ is always shorterthan e3. Thus, removing these parts from consideration further increases the upper bound on r.Finally, two right-angled triangles are left. Hence, our ratio r is upper bounded by the sum of thelengths of the two legs over the length of the hypotenuse in a right-angled triangle. With basiccalculus it can be verified that r ≤

√2. This concludes the proof of the lemma.

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 59

With the proof of Lemma 4.1.4 the proof of Theorem 4.2 is completed. 2

Note that the analysis is far from being tight. Actually,√

2 is only derived for the last type of games.The analysis suggests that r is very likely to be much smaller. The transformations and networkconstructions to upper bound the factor tend to increase it dramatically. This allows a feasible,complete and relatively simple analysis, which unfortunately tends to be way too pessimistic. Mostlikely it is possible to bound TE using the minimum spanning tree in some way. With the Gilbert-Pollack ratio a tighter analysis might lower the factor to the range of 2/

√3. Deriving a tight

analysis remains as an open problem.

4.1.2 Cheap Approximate Nash Equilibria

The optimal centralized solution for the geometric connection game is a geometric Steiner forest.Therefore, it obeys the Lemmas 2.2.1, 2.2.2 and the Corollary 2.2.1 (c.f. section 2.2). Purchasingthe optimal solution is very economical, which can be seen by the following Theorem 4.3.

Theorem 4.3 For any geometric connection game for 2 players and 2 terminals and any ε > 0there exists a (1 + ε)-approximate Nash equilibrium as cheap as the optimum solution.

Proof. Recall Figures 4.3 and 4.4, which depict the different structures of the optimal solutionto consider. Note that if the optimal solution has only two edge-disjoint connection sets, a Nashequilibrium is straightforward. We will therefore only consider games, in which T ∗ contains anedge e3 that is present in T 1 and T 2.

Type 4.3.1 This type is composed of games, for which the optimal network is a path (see Figure4.3a). The following lemma describes the structure of meaningful deviations. Edges e1, e2 and e3

are defined according to Figure 4.3.

Lemma 4.1.5 Given a game of the Type 4.3.1 and a payment function p that assigns player ionly to pay for edges in T i, the only deviations player 1 will pick are straight segments from apoint u ∈ e1 to a point v ∈ e3, player 2 only from u ∈ e2 to v ∈ e3.

Proof. We analyze the payments for player 1. The observation follows symmetrically for player2. With Lemma 4.1.1 the structure of meaningful deviations can be restricted to straight edgesbetween the segments. Consider a straight edge starting and ending at the same segment. Asthe segments of the optimal solution are also straight, it is of no use for any player. Noticing thesymmetry of edges there remain only three types of meaningful deviations:

1. u ∈ e1, v ∈ e3

2. u ∈ e1, v ∈ e2

3. u ∈ e2, v ∈ e3

Consider first case 3: If player 1 chooses to deviate to such an edge d, c(d) ≥ p1(ed3), where ed3 is thepart of e3 that is located inside the cycle imposed by edge d. As p1(e2) = 0, he could only reducehis payments by p1(ed3) and therefore cannot decrease his payments. Actually this is even easierto see, as the deviations of case 3 just allow player 1 to deviate from (parts of) one connection set.Payments on one connection set, however, always provide lower bounds for the deviations.If e1 and e2 are located on different sides of the line through e3, an edge of case 2 always crosses e3

and therefore decomposes into two edges of the other cases. If e1 and e2 are located on the same

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60 CHAPTER 4. GEOMETRIC CONNECTION GAMES

side (this is depicted in Figure 4.3a), then by easy observation for each edge of case 2 there is acheaper edge, which starts at the same point u ∈ e1 and goes directly to the terminal of player1 on e3. However, as player 1 does not pay for e2, the savings on e1 and e3 for player 1 are thesame for both edges. Hence an edge of case 1 provides a superior way to deviate. This proves thelemma. 2

An adjusted version of this lemma will be true for nearly all game types in the remaining proof.The Cosine Theorem as a generalization of the Pythagoras Theorem can be used to observe thatthe deviation lengths are minimized if the angle between segments is minimized, i.e. amounts to120 ◦ (c.f. Lemmas 2.2.1 and 2.2.2). Therefore a worst-case angle assumption is given and assumedto hold for the remaining part of the proof:

Angle Assumption:To minimize the deviation lengths all edges of the optimal solution are assumed to connect withinner angles of 120 ◦.

Let a payment scheme be as follows:Player 1 buys e1 and the consecutive half of segment e3 that connects to his terminal. Player 2buys e2 and the other half of segment e3. This payment scheme will be shown to form a Nashequilibrium for player 1 and player 2. Note first that the necessary conditions from lemma 4.1.2are fulfilled.Consider a deviation d = (u, v) for player 1 with u ∈ e1 and v ∈ e3. As the angle between e1 ande3 is exactly 120 ◦, the cost of this segment is

|d| =√|ed1|2 + |ed3|2 + |ed1||ed3|,

where ed1 and ed3 are the segments of e1 and e3 in the cycle, respectively. The payment of player

1, which can be removed when buying d, is p1(ed1) + p1(ed3) = |ed1| + max(|ed3| − |e3|2 , 0). Once v

lies in the segment paid for by player 2, |ed3| < |e3|2 and the deviation cannot be cheaper than |ed1|.

Otherwise, if |ed3| ≥ |e3|2 , one can see that

|ed1||e3|+ |ed3||e3| ≥|e3|2

4.

Then

|ed1|2 + |ed3|2 + |ed1||ed3| ≥ |ed1|2 + |ed3|2 +|e3|2

4− |ed1||e3| − |ed1||e3|+ 2|ed1||ed3|

follows, and finally

c(d) = |d| ≥ |ed1|+ |ed3| − ||e3|2

= p1(ed1) + p1(ed3)

is derived. Hence, player 1 has no way of improving his payments. By symmetry the same is truefor player 2, and the proof for this type is completed.

Type 4.3.2 Games of this type are characterized by optimal networks forming a star. All termi-nals are distributed onto the leaves (see Figure 4.3b). The existence of a strict Nash requires toassume a cost sharing property.

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 61

Cost-Sharing Property:The cost-sharing property is fulfilled, if between any two points an edge can connect to, the costof the segment can be shared between players.

Thus, either somehow the points in the plane considered for the game must be limited or costsharing of infinitesimal small segments must be allowed. A conflicting situation would be to allowcontinuous deviations but to require that it is specified where the payment of a player on e3 isexactly located. This would require to discretize the payment to small pieces of length l withoutlimiting the possible deviations. Then a factor of (1 + ε(l)) is lost in the final approximation.First, the argument is described that with the cost-sharing property it is possible to achieve anexact Nash equilibrium. Lemma 4.1.5 also holds for this type.

Lemma 4.1.6 Lemma 4.1.5 also holds for Type 4.3.2.

Proof. This is straightforward because every other deviation allows the corresponding player tosave on (parts of) only one connection set, but these parts provide lower bounds for the deviations.2

Now consider the following payment scheme:Player 1 buys e1 and half of any arbitrarily small piece of e3. Player 2 buys e2 and the other halfof e3.With a deviation d = (u, v) for player 1 with u ∈ e1 and v ∈ e3 the amount of payment player 1

can save is (p1(ed1) + p1(ed3) = |ed1|+ |ed3 |2 , and

(|ed1|+|ed3|2

)2 = |ed1|2 + |ed1||ed3|+|ed3|2

4= |d|2 − 3|ed3|2

4.

Hence, the deviation is always more costly than the possible cost saving for player 1. The existenceof a strict Nash follows from the symmetric argument for player 2.This result is applicable to any situation, in which cost sharing on e3 is allowed. In particular,between any two points of e3 that can be part of a deviation the players can share the cost ofthe corresponding segment of e3. Once this is not the case, suppose there is a discretization of

segment e3 into |e3|t segments e3,j , j = 1, . . . , |e3|t of length l, which must be bought by a singleplayer. The enumeration starts at the Steiner vertex. The connection point between e3,j ande3,j+1 will be referred to as the right end of e3,j and the left end of e3,j+1. One of the two playersmust buy the piece e3,1 of length l, with the left end being the Steiner point. This player thenbuys two consecutive straight segments with an inner angle of less than 180◦. Hence, he violatesthe conditions in Lemma 4.1.2. He can reduce his payments by connecting directly between theendpoints. In such a situation there is a payment function yielding a (1 + ε(l))-Nash equilibrium.Let us define the following payment function. The segments e3,j are assigned such that every twoneighboring segments are paid for by different players. Assume that player 1 buys the first andthe other ’odd’ segments. d again is the deviation from u ∈ e1 to v ∈ e3. First, note that once vis in a segment e3,j bought by player 2, there is a deviation to the left end of e3,j that is cheaperand allows player 1 the same savings. This implies that he will only deviate to points located onsegments e3,j , which he buys. Actually, a player can achieve the best cost reduction if he deviatesto e3,1 (see Figure 4.9).

Lemma 4.1.7 For every deviation d from u ∈ e1 to a point in segment e3,j, j 6= 1 there is adeviation d′ from u to a point in e3,1, which yields a higher approximation factor for player 1.

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62 CHAPTER 4. GEOMETRIC CONNECTION GAMES

Proof. Assume d = (u, v) with u ∈ e1, v ∈ e3,j 6= e3,1 and ed1, ed3 being the corresponding segments

in the cycle. e3,j is paid for by player 1. Let r be the number of full segments player 1 buys on ed3.Let e′ be the segment between the left end of e3,j and v. Thus p1(ed3) = rl+e′ and |ed3| = 2rl+ |e′|.v′ is to be the point on e3,1 that has distance |e′| to the left end of e3,1 and d′ = (u, v′) the deviationto this point.

1

ud

d

v’

’e

1,2v

e e

1

3,1 e3,2 3,j

Figure 4.9: d′ is more profitable than d.

The statement of the lemma can be formalized as

p1(ed1) + p1(e3)

|d′|+ p1(e3)− |e′| ≥p1(ed1) + p1(e3)

|d|+ p1(e3)− rl − |e′| .

Note that a fixed point of e1 is considered, however, two different points on e3. For calculatingthe factor the change in the total payments on e3 have to be taken into account. Note that thedeviation to v is more costly than d′, but allows player 1 to decrease the contribution to the costof e3 by an additional amount of rl. Using some calculus

3(rl)2 + 2|e′|+ |ed1| ≥ |d′|

is derived, which holds because all variable values are nonnegative. The lemma follows. 2

Having established this result, we continue to upper bound the approximation factor player 1 canachieve. With the triangle inequality it follows that player 1 will always deviate from his terminalto the right end of e3,1. Furthermore his factor will be maximized if he has no additional paymentsthan |e1|+ |e3,1|. Then with

ε1(x, l) =x+ l√

x2 + l2 + xl

the factor becomes

1 + ε(l) ≤ ε1(|e1|, l).

In the worst case |e1| = l this amounts to 2√3. To complete the proof it is immediately obvious

that player 2 pays at most half of e3 for any deviation from e2 to e3. He therefore has a strictequilibrium. It is only the player buying e3,1 and the odd segments that can improve his payments.The player can be picked in the beginning such that

ε(l) = min{ε1(|e1|, l), ε1(|e2|, l)} − 1.

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 63

Type 4.3.3 Here games have optimal networks with two Steiner points. The terminals of a playerare located on different sides of the line through e3 (see Figure 4.4a). The connection sets e1 ande2 now consist of two lines e11, e12 and e21, e22, respectively.

Lemma 4.1.8 Under the same assumptions as of Lemma 4.1.5 the only deviations player 1 willconsider in this network are straight segments from a point u ∈ e11 or a point u ∈ e12 to a pointv ∈ e3, player 2 only from u ∈ e21 or e22 to v ∈ e3.

Proof. The lemma can be proven by applying the corresponding arguments from the previousnetwork types to exclude nearly all other deviations. Note that a segment between u ∈ e11 andv ∈ e12 crosses e3 and therefore decomposes to two segments of the specified form. The statementfor player 2 follows by symmetry. 2

If the cost-sharing property holds, an equilibrium payment function can be designed by lettingplayer 1 pay for e11, e12, half of e3, and player 2 pay for the rest. Then for all possible deviationsthe cost is higher than the contribution to T ∗ a player could save. This follows from the proof ofType 4.3.2.With the discretization of e3 described for Type 4.3.2, a factor of (1 + ε(t)) is lost for this type,too. Now both players might deviate from both sides of e3, however, the most efficient deviationsagain are the ones to the edges e3,1 and e3,|e3|/t, which connect to the Steiner points. To seethis consider two deviation segments da = (ua, va) and db = (ub, vb) with ua ∈ e11, ub ∈ e12,va, vb ∈ e3. If the cycles introduced by da and db intersect on e3, the segments derived by changingthe endpoints (i.e. d = (ua, vb) and d′ = (ub, va)) lead to a better, non-intersecting deviation.However, once the cycles do not intersect on e3, the situation can be decomposed. Each side withe11 and e12 can be treated separately (see Figure 4.10). Now the rest of the proof can be givenby applying the analysis of Type 4.3.2 with e11 and e12. The cases for player 2 with e21 and e22

follow symmetrically.

1

γ

u

dd

u

v

va

ad

b

b

a

b

e 1

1

2e

1

1

d’

Figure 4.10: d and d′ are better than da and db.

Note that the bound for ε(l) can be adjusted to hold

ε(l) ≤ max{min{ε1(|e11|, l), ε1(|e21|, l)},min{ε1(|e12|, l), ε1(|e22|, l)}}if the segments e3,j are assigned alternating. The segments e3,1, e3,|e3|/t must be assigned to theplayer, which gives the smaller factor on the respective side of the tree. For this one might needan odd number of segments. Hence, the step length must be adjusted accordingly.

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64 CHAPTER 4. GEOMETRIC CONNECTION GAMES

Furthermore the proof of this type actually covers all previously considered game types, especiallyall games for which any of |e11|, |e12|, |e21|, |e22| = 0. Note that in these networks there might notbe Steiner points, and the angle between the edges could be greater than 120 ◦. However, this istaken care of by the angle assumption.

Type 4.3.4 The last game type considered is the one, in which T ∗ includes two Steiner points,and the terminals of each player are located on the same side of the line through e3 (see Figure4.3b). Here some additional deviations need to be incorporated that complicate the analysis.

Lemma 4.1.9 Suppose we are given a game of the specified type and a payment function thatassigns payments to player i only in his subtree T i. Then the only deviations player 1 considersare straight edges between u ∈ e11 or e12 and v ∈ e3 as well as the direct connection between histerminals. For player 2 the symmetric argument holds.

Proof. Again the previous arguments are used to exclude all other deviations. In opposite to type4.3.3 a direct connection between e11 and e12 does not cross e3 and the case does not decompose.If, however, player 1 picks such a deviation, by the triangle inequality he will be most profitableto pick the direct connection between his terminals. 2

First, we will notice that this situation allows a Nash equilibrium with the cost-sharing property.To realize the payment function scale the network such that e3 has length 1. Now treat e3 as theinterval [0, 1]. Let f(x, y) ∈ [0, 1], 0 ≤ x ≤ y ≤ 1 be a function that specifies the fraction player 1pays in the subinterval [x, y] of e3. Let wlog the Steiner point of e11 be point 0 of e3 and the otherSteiner point be point 1. Now we have to ensure that for every deviation from e11 or e12 to a pointx, y ∈ e3, the savings on the segments do not exceed the cost of the deviation. This requirementgives the following bounds:

|e11|+ yf(0, y) ≤√|e11|2 + y2 + |e11|y,

|e12|+ yf(1− y, 1) ≤√|e12|2 + y2 + |e12|y.

For player 2 symmetric requirements lead to

|e21|+ b(1− f(0, y)) ≤√|e21|2 + y2 + |e21|y,

|e22|+ b(1− f(1− y, 1)) ≤√|e22|2 + y2 + |e22|y.

Furthermore bounds can be derived from the direct connections between the terminals. They willbe denoted by d1 and d2 for players 1 and 2, respectively. From the optimality of T ∗ and |e3| = 1follows

|d1|+ |d2| ≥ |e11|+ |e12|+ |e21|+ |e22|+ 1. (4.2)

A Nash equilibrium payment postulates that d1 and d2 are not cheaper than the contribution ofthe players, thus

|d1| ≥ |e11|+ |e12|+ f(0, 1),

|d2| ≥ |e21|+ |e22|+ 1− f(0, 1).

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 65

1

e

1

12

d

de 1e

11

3

Figure 4.11: Possible deviation types for player 1.

The nature of these segments implies that their bounds only apply to the payment on the wholeedge e3, i.e. they do not restrict the partition of the payment inside the edge (see Figure 4.11).Using a function h(x) =

√x2 + x+ 1− x and solving for f results in

|e21|+ |e22|+ 1 ≤ f(0, 1) ≤ |d1| − |e11| − |e12|, (4.3)

1− h(|e21|y

) ≤ f(0, y) ≤ h(|e11|y

), (4.4)

1− h(|e22|y

) ≤ f(1− y, 1) ≤ h(|e12|y

). (4.5)

Now consider the behavior of h(x) in (4.4) and (4.5) when altering the constants |e11| and |e12|.Observe that for the derivative of h(x)

h′(x) =2x+ 1− 2

√x2 + x+ 1

2√x2 + x+ 1

< 0 (4.6)

holds. The function is monotonic decreasing in x, so increasing e11, e12, e21, e22 tightens lower andupper bounds. Hence, only deviations from terminals to e3 will be considered, as this results inthe strongest bounds for the equilibrium payment function.In addition to the bounds it is required that payments can be feasibly distributed to subintervals -i.e. one must ensure that the payment of player 1 for an interval [x, y] is the sum of the paymentsfor the two subintervals [x, z] and [z, y] for any v ∈ [x, y]. With this property f(x, y) can bespecified by using the functions f(0, y) and f(1− y, 1):

f(x, y) =yf(0, y)− xf(0, x)

y − x =yf(1− y, 1)− xf(1− x, 1)

y − x , 0 ≤ x < y ≤ 1

It follows that f(0, y) = f(1 − y, 1) and f(y, y) = f ′(0, y) = −f ′(1 − y, 1) for all y ∈ [0, 1]. Thus,for the rest of the proof we will strive to provide a feasible function f(0, y), which obviously mustobey all bounds (4.3)-(4.5).Next the feasibility of the bounds has to be examined:

Lemma 4.1.10 The bounds (4.3)-(4.5) do not imply a contradiction. In particular the interiorbounds (4.4), (4.5) can be fulfilled by f(0, y) = 1

2 .

Proof. It has already been shown that the upper bound function h(x) is monotonic decreasing inx. So for any x, x′ > 0

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66 CHAPTER 4. GEOMETRIC CONNECTION GAMES

limx→∞

1− h(x) = 12 = lim

x→∞h(x),

1− h(x) ≤ 12 ≤ h(x′)

holds. This proves the second part of the lemma.

With (4.2) it is obvious that (4.3) is no contradiction. Assume that the bounds in (4.3) and(4.4) form a contradiction. Then at least one of

1− h(|e21|) > |d1| − |e11| − |e12|,|e21|+ |e22|+ 1 > h(|e11|)

must hold. However, with (4.2) it follows

√|e21|2 + 1 + |e21| < −|e22|,√|e11|2 + 1 + |e11| < −|e12|,

and thus the bounds are feasible. The first part of the lemma follows with a similar argument forthe bounds of (4.3) and (4.5). 2

Interestingly this result supports the proofs for the previous Type 4.3.3. The function used thereis the constant function f(a, b) = 1

2 and satisfies the bounds (4.4) and (4.5), which are the onlyones that apply to that situation.For the present type a solution is possible, too. In the easiest case if

|e21|+ |e22|+ 1 ≤ 1

2≤ |d1| − |e12| − |e11| (4.7)

holds, f(0, y) = f(x, y) = 12 again is a solution. The remaining details and results for this case

then follow analogously to the previous Type 4.3.3. Therefore suppose for the remaining proofthat (4.7) is not valid. A solution for this more complicated situation will be presented in the nextlemma.

Lemma 4.1.11 One of the two functions

f1(0, y) = h(k

y) or f2(0, y) = 1− h(

k

y)

can be used to construct a payment function that is Nash equilibrium.

Proof. In the first case suppose

b = |e21|+ |e22|+ 1 >1

2.

Then construct a function for f(0, y) that behaves like the upper bounds and achieves a value ofb for y = 1:

f1(0, y) = h(k

y) =

√k2 + ky + y2 − k

y, k =

1− b22b− 1

.

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4.1. GEOMETRIC CONNECTION GAMES FOR 2 PLAYERS AND 2 TERMINALS 67

In the second case suppose

b = |d1| − |e11| − |e12| <1

2.

Then construct a function for f(0, b) that behaves like the lower bounds and achieves a value of bfor y = 1:

f2(0, y) = 1− h(k

y) = 1−

√k2 + ky + y2 − k

y, k =

1− b22b− 1

− 1.

To achieve a consistent definition of f(x, y) we define f(1, 1) = f(0, 0) := limy→0 f(0, y) = 12 . With

lemma 4.1.10 and the monotony of h(x) it is natural that the functions f1, f2 obey the bounds(4.3)-(4.5) for any y ∈ [0, 1]. They allow the construction of a Nash equilibrium payment function.f(x, y) can be given as

f1(x, y) =

√k2 + ky + y2 −

√k2 + kx+ x2

y − x , k =1− b22b− 1

,

f2(x, y) = 1−√k2 + ky + y2 −

√k2 + kx+ x2

y − x , k =1− b22b− 1

− 1.

The payment for player 2 is given by a function of the other type respectively using the sameconstant k. 2

This concludes the proof for the existence of a Nash equilibrium if the cost-sharing property holds.Turning to the dicretization case suppose e3 is dicretized into segments e3,j of length l, but thesesegments can be shared in any way between the players. We just require that the shares purchasedby a player form a continuous segment inside e3,j . Of course, one could then further refine thediscretization of the segments of e3,j with a smaller l to obey the stronger assumption that eachsegment has to be purchased fully by one player. However, this would obviously lead to negligiblyperturbed results but involve much more technical details.

Let the segments e3,j be defined as for the previous type, and let the enumeration start at anarbitrary Steiner point. Suppose that player 1 buys his share on the left part of each segment(connecting to the left end) and player 2 on the right. With a similar calculation the samestatement as in Lemma 4.1.7 is verified, however, now the corresponding point on segment e3,1

might be in the part paid for by player 2. Thus, the bound might be slightly lower, because player1 is only willing to deviate to the end of the part of the first segment bought by him. Intersectingdeviations as shown in Figure 4.10 can again be excluded from consideration. By assuming thateach player pays only one segment on e3, the following upper bound for ε(l) is established:

ε(t) ≤ max{ε1(a, tb) | a ∈ {|e11|, |e12|, |e21|, |e22|}, b ∈ {1, f(0, t), 1− f(0, t)}} − 1.

2

An algorithm to calculate the Nash equilibrium payment is straightforward. Once the discretizationcomes into play, it can be verified for all the types that the calculation of p eventually needs O(1)per segment, which then gives a running time of O(1/ε).

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68 CHAPTER 4. GEOMETRIC CONNECTION GAMES

4.2 Games with more Players and more Terminals

4.2.1 Lower Bounds

In this section a general setting will be considered. Unfortunately, most of the appealing results forthe small case with 2 players and 2 terminals per player do not translate to more players. Theorem4.3 cannot be generalized, i.e. there are games with N ≥ 3 players, in which the optimum solutionis at least a constant factor approximate Nash equilibrium. Furthermore the stepwise adjustmentusing the local improvement step results in cycling dynamics for some games.

Theorem 4.4 There exists a game with N players, for which every optimum solution is at leasta(

43 − ε

)-approximate Nash equilibrium.

Proof. The proof is analogous to the proofs of Theorems 3.14. It uses a geometric version of thesame game, i.e. again there is a circle of terminals with unit distance, and the optimal solutionwill be a minimum spanning tree of cost 2N −1. In the geometric environment, however, the edgescrossing the interior of the circle are not of interest, because their cost is always larger than 1.Actually, their cost exceeds 2 once the number of players is more than 4, which then is more thanthe asymptotical payment of each player in the best payment scheme. So no player will considerthem as a reasonable alternative.

5

5

1

1

42

4

3

2

3

b)a)

x

t si i

y z

Figure 4.12: Geometric lower bound example for approximate Nash purchasing the optimal net-work.

Consider the game in Figure 4.12, in which every edge of T ∗ has cost 1. T ∗ is depicted with anadditional edge of cost 1, which will be the only deviation edge considered. The situation for aplayer can be depicted with the simplified view of Figure 4.12b. Note that for players 1 and N ,z = 0 and y = 0, respectively. Therefore we do not have to treat them separately like in the generalcase. For every player there are at least two ways to deviate, either he just contributes to one halfof the cycle by paying x or he completes the other half of the cycle by paying y + z + 1. Thus hisdeviation factor will be at least

max

{x+ y + z

x,x+ y + z

y + z + 1

}.

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4.2. GAMES WITH MORE PLAYERS AND MORE TERMINALS 69

Minimizing this expression with x = y + z + 1 there is at least one player, who is assigned to payfor

x+ y + z = 2x− 1 ≥ 2N − 1

N.

Solving for x and combining with x = y + z + 1 results in

x+ y + z

x=

2x− 1

x≥ 4N − 2

3N − 1.

Now move the terminals s1 and tN a little further to the outside keeping the lengths of the edges(s1, s2) and (tN−1, tN ) to 1, but increasing the length of (s1, tN ) to length (1 + ε). T ∗ will then bethe unique optimal solution, and the factor becomes at least

(43 − ε

).

Observe that this lower bound proof applies only to N ≥ 3, because otherwise the angles betweenthe edges of T ∗ amount to less than 120 ◦. Finally, let us conclude the proof with a note on thegeometric structure. Potentially all possible segments in the plane can be used as an alternative,however, for establishing a lower bound the consideration can safely be restricted to the edges ofT ∗ and (s1, tN ). If a payment scheme is a k-approximate Nash in this restricted setting, it is atleast a k-approximate Nash in the setting with more possible deviations, because a player will onlyconsider other segments if they give him a better alternative. 2

Observe that the same argument as in the discussion of Corollary 3.4.1 can be applied. Considerthe payment functions ps purchasing the optimum solution T ∗, which assign payments to player ionly in the subtree T i. For these a lower bound of (2− ε) holds in the geometric environment aswell.

Corollary 4.2.1 For every ε > 0 there is a geometric game, for which any ps is at least a (2− ε)-approximate Nash equilibrium.

Furthermore, if Algorithm 3 for solving the general case is applied directly, it can easily run intoassigning one player three connection sets. Consider for instance the game in Figure 4.12. One caneasily think of bad matching results that make a player pay for 3 edges in total. With the deviationthen a (3− ε)-approximate Nash is achieved. Hence, the unadjusted application of methods for thegeneral case does not yield any worst-case improvement. With adjustments, however, it is likely toget better approximations for the geometric environment. Note that the analysis for Theorem 3.13with 3− 3

N connection sets cannot be translated, which is due to the geometric positioning of theterminals. The position of the terminals in the plane gives then information about the players withpotentially cheap deviations. Actually, by knowing the optimum solution with the positions of theterminals in the plane, the algorithm already ”knows” the whole underlying graph and all possibledeviation lengths! Hence, there is no separation of the cases like in the general case. Furthermorethere is only limited need for bounding techniques that reduce the input size for large graphs.

4.2.2 A Negative Result

For a proof on the cycling of the local improvement strategy, a game from the lower bound exampleis used. This will show that by playing the game iteratively with one player deviating, the networkapproaches the optimal solution, which is guaranteed to have no exact Nash equilibrium.

Theorem 4.5 There is a game, for which any iterative local improvement strategy allowing ineach step exactly one player to deviate will cycle and fail to achieve a Nash equilibrium.

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70 CHAPTER 4. GEOMETRIC CONNECTION GAMES

5

5

1

1

4

4

3

2

3

a)

2

5

5

1

1

4

4

3

2

3

2

3

3

21

1

1

54

21

3

4

4

2

3

1

5

5 12

5

4

b)

Figure 4.13: Network after a) five improvement steps, b) player 1 has removed most parts of d1.

Proof. Consider the game from the proof of Theorem 4.4 with N = 5 players. Any player has thesame cost of his direct connection. Let the player picked in iteration k ∈ N be player ik. Wlogstart with player i1 = 1. He will be picked to establish his direct connection, which is denotedby d1. In the second step, player 1 is satisfied, and another player will be picked to connect histerminals. The cases i2 = 2 or i2 = 5 will result in the same networks. Any of these playerswill use d1 and connect perpendicular to it. In the case i2 = 3 or i2 = 4 they will also use theperpendicular connection. Note, however, that their cost to connect to d1 is more than 2. As longas all connections are perpendicular to d1, player 1 cannot improve his situation (due to Lemma4.1.3). In the remaining 3 steps players 1 and 5 will connect. They will allow players 3 and 4to deviate from the perpendicular connections to connect their terminals directly to the ones ofplayers 1 and 5. This, however, will close a gap and allow player 1 to abandon most parts of d1

through the middle of the cycle. Instead he will connect by purchasing the missing edge on oneside of the cycle. This will then allow either player 3 or 4 to buy the missing edge from the otherside of the cycle and remove a cost of 2. These dynamics lead into a cycle, in which no Nashequilibrium can be achieved.In Figure 4.13 the crucial step is depicted. The numbers at the edges point out which player paysfor a segment. After each player has connected his terminals, and players 3 and 4 have adjustedtheir connection using the segments paid for by player 2 and 5, the network structure is similar toFigure 4.13a. In this example two edges, one between terminals of players 3 and 4 and one betweenterminals of players 4 and 5, are missing. Instead there could also be two different edges missing -the crucial point is that there is always exactly one edge of the cycle from each side of d1 missing.2

4.2.3 Outlook

An important observation in the study of geometric connection games is that for ”line-like” op-timum networks it is unlikely that profitable deviations exist. Suppose the terminals of a playerare located in very distributed locations along an established, nearly straight path. A direct con-nection is unlikely to be much more profitable, thus for good deviations there must be some edgespaid for by other players, with which a player can ”shortcut” his payments. Hence, there mustbe some kind of cyclic structure present in the purchased tree. Otherwise he will not be able to

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4.2. GAMES WITH MORE PLAYERS AND MORE TERMINALS 71

reduce his payments too much. From this point of view the good results in the case with 2 playersand 2 terminals per player seem quite natural. Such a small number of players and terminalsleaves every player more or less ”on his own” with the creation of a network connection. Once thenumber of players increases, however, the geometric case exhibits similar properties as the generalcase. Efficient and profitable deviations might be introduced, as a player now can profit from cheapconnections that might be located in a totally different part of the graph. Consider for instance thedistribution of the terminals in the games of Figure 4.12. These games are ”ill-behaved” becausecyclic dynamics are created. Every player might profit from every edge in the cycle, however onlybecause everybody else provides him with the opportunity. A player might not be able to improverather directly like in the general case - i.e. that at some adjacent terminal there is a cheap edgeof cost 1 crossing the whole cycle. Instead he must rely on the other players to create this situa-tion with the edges they pay for. For such situations the application of an analysis tool like theconnection set combined with geometric observations seems promising.In geometric connection games the structure of T ∗ can provide much more information about theincentives for a player to deviate. For the prices of anarchy notice that the worst-case examplesare located completely outside the convex hull of the instance. But this seems rather unreasonable,given the fact that if the network is newly created, no established link will ever leave the convexhull of the instance. An optimum solution tree will also never leave the convex hull, which canbe derived from the PTAS presented in section 2.2. Hence, with appropriate tools significant im-provement of the quality of Nash equilibria and approximate Nash equilibria (in the convex hull)of geometric connection games over the general case should be achievable. The presented resultscan only provide a starting point for the study of geometric connection games. There is still a lotof work to be done from which a deeper understanding of the underlying dynamics will benefit.

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72 CHAPTER 4. GEOMETRIC CONNECTION GAMES

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Chapter 5

Discussion

The previous observations on connection games have shed some light on how to analyze the moti-vation of selfish agents to pay for cheap networks. In particular, for the general case a number ofdifferent properties of games and algorithms were considered, and results about upper and lowerbounds on approximate Nash equilibria buying the optimal solution were established. In Table5.1 the properties for the general case and the derived factors are summarized. Upper bounds areestablished by algorithms that actually achieve this guarantee, lower bounds are established by thecorresponding theorems in chapter 3. Note that both lower and upper bounds for the case withN players and 2 terminals per player generalize to any number of terminals per player, becausethe algorithm for 3-approximate Nash was derived for this more general scenario. This, however,would not immediately be the case for the bounds suggested in Conjecture 3.1.

Properties of 2 play., 2 term. 2 play., k term. N play., 2 term.the algorithm Upper Lower Upper Lower Upper Lower

Knows only T ∗,no cost-sharing 2 2 2 2 3 3

No cost-sharing 2 2 2 2 3 2

Knows only T ∗ 1.5 1.5 2 2 3 3

Assigns player ito pay only in T i 1.5 1.5 2 1.5 3† 2

No restriction 1.5 1.2 2 1.2 3† 1.5

† Can be lowered to 2 if Conjecture 3.1 holds.

Table 5.1: Upper and Lower bounds for approximate Nash equilibria.

73

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74 CHAPTER 5. DISCUSSION

This reveals that for connection games no significantly improved algorithms exist that rely on anetwork analysis with connection sets only. Hence, more elaborate algorithms have to employinformation about the whole graph G and use additional analytical tools, e.g. the observationspresented in the argument for Conjecture 3.1. Then it is reasonable to hope that the present upperbounds can be effectively reduced.Simple connection games like the ones considered in the previous chapters are well-suited to comeup with some starting ideas about the topology development of networks designed by selfish agents.However, in reality a network like the Internet offers much more features, details and incentivesto consider than just the notion of being connected. Computational aspects like bandwidth con-straints, routing, speed and latency, security, but also physical aspects like distances or installationcosts can influence the decision, whether it is profitable for a party to purchase, install or maintaina certain connection in the topology or not. Therefore it is important to explore the possibilitiesof analyzable and applicable generalizations of the connection game. For the connection gamethe idea of connection sets has exhibited very appealing properties for the design and analysis ofcost allocation algorithms. Hence, it would be desirable to translate this concept to more elaborategames. Unfortunately, in the next paragraphs it will become clear that for some natural extensionsof the connection game the connection set idea cannot be directly applied.

5.1 The 2-Connection Game

Suppose the agents want to make sure that their network is more fault-tolerant. Hence, theywould like to establish two edge-disjoint connections between each pair of terminals. For simplicitysuppose that each of the N players holds only 2 terminals, and that the underlying graph is atleast 2-connected. Here of course the optimum centralized solution is a graph of 2-connectedcomponents. It is easy to observe that in this game the price of anarchy is N . The lower boundinstance can be derived from the connection game. Instead of one there are now 2 connectionsof cost N , which are purchased jointly by the players, and 2 connections of cost 1, which arethe optimum solution. Observe that the basis of the instance containing no Nash equilibrium inFigure 3.1 was the presence of a cycle. Here, however, players have stronger initial requirements(2-connectivity instead of 1-connectivity), and the networks designed will actually be 2-connected.Thus, it is not straightforward how the players can be forced into a cycling situation withoutequilibrium, because they have an incentive to build cycles. In this context notice that the gamein Figure 3.1 has an equilibrium if both players are required to build 2-connected components. Itremains to have an equilibrium if all edges are doubled.Let us examine the applicability of connection sets. Can parts of the optimum solution be usedto lower bound the deviations for a player? The answer to this question is twofold - yes andno. Remember the definition of a connection set in Definition 3.2.1. It made use of the factthat an optimum solution is present. Then parts of the solution network were characterized thatupon removal and (selfish) reconnection resulted in a feasible network. With this construction theoptimality of the network could be used to lower bound all possible deviations. Translating thisidea to the present game a simple adaption of the definition would be as follows.

Definition 5.1.1 A 2-connection set S of player i is a subset of edges of the optimum network,such that for each 2-connected component C in T ∗ \ S either

1. there is a terminal of i in C, or

2. any player that has a terminal in C has all of its terminals in C.

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5.2. THE BOUNDED DISTANCE CONNECTION GAME 75

If now player i removes a 2-connection set and reconnects all his terminals in the different 2-connected components by two edge-disjoint paths, a new feasible solution evolves. Hence, thelower bounding argument can be applied. Notice, however, that the lower bounding example canalso be applied to certain 1-connected components if the reconnection e.g. completes a disconnectedcycle component. But does this cover all 1-connected components? Observe the difficulties in thesimple game depicted in Figure 5.1.

1

12

2

Figure 5.1: A game with no 2-connection sets.

The optimum solution obviously is the outer square. Let each edge in the square have cost 1. It iseasy to notice that no collection of edges is able to obey the properties required in Definition 5.1.1.Hence, one player can have two paths connecting his terminals with a cost of ε � 1 each. In theexample this is player 2. If he removes all payments from the optimum square and buys the ε-costedges, player 1 will still need the whole square to satisfy his connectivity requirement. Note thatactually these edges can be paths and do not need to be parallel in the graph. We only requirethem not to connect to the edges of the square in other points than the terminals of player 2. Analgorithm using only the optimum solution does not know, which player actually has this pair ofcheap edges and how much they cost. So independently of the portion it assigns to player 2 thefactor for the approximate Nash equilibrium can be increased deliberately by simply reducing ε.This is possible, because no bound on these edges can be derived from the solution network.Similar results apply even to a case, in which all players share a terminal. An example for thiscase can be given with a game for 3 players that looks like Figure 5.1. Here one node v contains aterminal of all 3 players and the other three nodes one terminal of one player each. Then the twoedges not connecting to v cannot be used for lower bounding, thus a player paying for them canpossibly achieve an unbounded incentive to deviate.

5.2 The Bounded Distance Connection Game

Suppose the agents are sensitive to the path lengths and would not like to have their terminalsconnected by very long paths. So let there be an additional function δ : E → Q+

0 that defines alength for an edge and a value bi for each player i, which is an upper bound on the distance thata player is willing to accept between any two of his terminals. For simplicity suppose that eachedge length is equal to the respective edge cost, that each player holds only 2 terminals, and thatthe graph allows to establish a connection between the pair of terminals for a player. Note that inthis case all results from the connection game apply for sufficiently high bi.In this case it rather easy to disrupt the connection set idea. All deviations violating the distanceconstraint for some player do not yield a new socially feasible solution and cannot be lower bounded

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76 CHAPTER 5. DISCUSSION

1,3

1,2

2,3

d

Figure 5.2: A game with no connection sets.

by the respective parts of the optimum network. In particular, the following game for 3 playersprovides a simple example.

In the example depicted in Figure 5.2 each player has an upper bound on the distance of b, andeach thick edge in the star has cost and distance b

2 . The thin edge d has cost and distance ε� b.An optimum solution will just include the star, because replacing any edge by d will result in aninfeasible path for player 1 or 2. It is easy to see that no edge from the optimum solution is able tobound the cost of d. Hence, if player 3 is assigned to purchase any portion of the network, then byreducing ε > 0 one can design a game such that d will give player 3 a deliberately high incentiveto deviate. Note that this deviation could also be present for player 2 or 1 instead. Thus, if thealgorithm only knows the optimum solution, it cannot achieve any upper bound on the factor.

5.3 Final remarks and Outlook

As it has become obvious, the idea of lower bounding deviations by some portions of the sociallyoptimum network is not easily applicable to generalized versions of the connection game. This hasbeen presented for 2 games, however, we have tested a number of other generalizations that werenot presented here in detail. Unlike the connection game these games involve additional connec-tivity requirements for a player. It seems that in these environments a game with an optimumnetwork can be constructed, in which a selfish deviation for a player results in the violation of theadditional requirements for other players. Hence, the player is then able to switch to this deviation,however, it will not be considered in the process of constructing an optimum solution. In this wayit cannot be lower bounded in cost by any portion of the optimum network. Hence, an adjustedconcept has to be created or other analytical approaches are needed to deal with the complexityand computational challenges of large networks.In general, the study of network creation games is still a very new research direction. In thisthesis some deeper understanding of the dynamics of general and geometric connection games hasbeen derived. However, a lot of open questions about the incentives of selfish agents in designingnetworks remain. The Internet offers a variety of features that once might be incorporated intonetwork creation models and games. As a first step, it still seems possible to improve the fac-tors for approximate Nash equilibria in the connection game. In addition, the study of geometricconnection games should yield some insight, which might as well be useful in other telecommunica-tions, business or traffic networks. Especially foundational questions about the optimistic price of

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5.3. FINAL REMARKS AND OUTLOOK 77

anarchy or the complexity of finding exact Nash equilibria for more players and terminals shouldbe attacked. Furthermore the extensions of the connection game described in the previous sectionsmight be worth some deeper analysis. Possibly the results can provide interesting observationsabout the behavior of selfish agents with more elaborate connectivity requirements. Perhaps evena mechanism framework for these games could be constructed to model applications in the field ofcost allocation in networks.

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