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Bilò, Davide; Lenzner, Pascal On the Tree Conjecture for the Network Creation Game. Theory of Computing Systems 2019
Selfish Network Creation focuses on modeling real world networks from a gametheoretic point of view. One of the classic models by Fabrikant et al. (2003) is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of \($\alpha$\) per edge to minimize their total distance to all other nodes. The model is wellstudied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all \($\alpha$\) and that for \($\alpha$\) ≥ n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edgeprice \($\alpha$\) and employ it to improve on the best known bound for the latter conjecture. In particular we show that for \($\alpha$\) >thinspace4n −thinspace13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of \($\alpha$\). Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

Bilò, Davide; Friedrich, Tobias; Lenzner, Pascal; Melnichenko, Anna Geometric Network Creation Games. Symposium on Parallelism in Algorithms and Architectures (SPAA) 2019: 323332
Network Creation Games are a wellknown approach for explaining and analyzing the structure, quality and dynamics of realworld networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unitweight edges. In practice, e.g. when constructing a fiberoptic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edgeweighted graphs. We incorporate arbitrary edge weights by generalizing the wellknown model by Fabrikant et al.~[PODC'03] to edgeweighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the stateoftheart for the unitweight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight nonconstant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the nonmetric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the gametheoretic analogue of a variant of the classical Network Design Problem. Thus, lowcost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.

Feldotto, Matthias; Lenzner, Pascal; Molitor, Louise; Skopalik, Alexander From Hotelling to Load Balancing: Approximation and the Principle of Minimum Differentiation. Autonomous Agents and Multiagent Systems (AAMAS) 2019

Chauhan, Ankit; Lenzner, Pascal; Molitor, Louise Schelling Segregation with Strategic Agents. Symposium on Algorithmic Game Theory (SAGT) 2018
Schelling’s segregation model is a landmark model in sociology. It shows the counterintuitive phenomenon that residential segregation between individuals of different groups can emerge even when all involved individuals are tolerant. Although the model is widely studied, no pure gametheoretic version where rational agents strategically choose their location exists. We close this gap by introducing and analyzing generalized gametheoretic models of Schelling segregation, where the agents can also have individual location preferences. For our models we investigate the convergence behavior and the efficiency of their equilibria. In particular, we prove guaranteed convergence to an equilibrium in the version which is closest to Schelling’s original model. Moreover, we provide tight bounds on the Price of Anarchy.

Bilò, Davide; Lenzner, Pascal On the Tree Conjecture for the Network Creation Game. Symposium on the Theoretical Aspects of Computer Science (STACS) 2018: 14:114:15
Selfish Network Creation focuses on modeling real world networks from a gametheoretic point of view. One of the classic models by Fabrikant et al. [PODC'03] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is wellstudied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all \($\alpha$\) and that for \($\alpha \geq n$\) all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edgeprice alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for \($\alpha > 4n13$\) all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

Chauhan, Ankit; Lenzner, Pascal; Melnichenko, Anna; Molitor, Louise Selfish Network Creation with NonUniform Edge Cost. Symposium on Algorithmic Game Theory (SAGT) 2017: 160172
Network creation games investigate complex networks from a gametheoretic point of view. Based on the original model by Fabrikant et al. [PODC’03] many variants have been introduced. However, almost all versions have the drawback that edges are treated uniformly, i.e. every edge has the same cost and that this common parameter heavily influences the outcomes and the analysis of these games. We propose and analyze simple and natural parameterfree network creation games with nonuniform edge cost. Our models are inspired by social networks where the cost of forming a link is proportional to the popularity of the targeted node. Besides results on the complexity of computing a best response and on various properties of the sequential versions, we show that the most general version of our model has con stant Price of Anarchy. To the best of our knowledge, this is the first proof of a constant Price of Anarchy for any network creation game.

Friedrich, Tobias; Ihde, Sven; Keßler, Christoph; Lenzner, Pascal; Neubert, Stefan; Schumann, David Efficient Best Response Computation for Strategic Network Formation under Attack. Symposium on Algorithmic Game Theory (SAGT) 2017: 199211
Inspired by real world examples, e.g. the Internet, researchers have introduced an abundance of strategic games to study natural phenomena in networks. Unfortunately, almost all of these games have the conceptual drawback of being computationally intractable, i.e. computing a best response strategy or checking if an equilibrium is reached is NPhard. Thus, a main challenge in the field is to find tractable realistic network formation models. We address this challenge by investigating a very recently introduced model by Goyal et al. [WINE'16] which focuses on robust networks in the presence of a strong adversary who attacks (and kills) nodes in the network and lets this attack spread viruslike to neighboring nodes and their neighbors. Our main result is to establish that this natural model is one of the few exceptions which are both realistic and computationally tractable. In particular, we answer an open question of Goyal et al. by providing an efficient algorithm for computing a best response strategy, which implies that deciding whether the game has reached a Nash equilibrium can be done efficiently as well. Our algorithm essentially solves the problem of computing a minimal connection to a network which maximizes the reachability while hedging against severe attacks on the network infrastructure and may thus be of independent interest.

Friedrich, Tobias; Ihde, Sven; Keßler, Christoph; Lenzner, Pascal; Neubert, Stefan; Schumann, David Brief Announcement: Efficient Best Response Computation for Strategic Network Formation under Attack. Symposium on Parallelism in Algorithms and Architectures (SPAA) 2017: 321323
Inspired by real world examples, e.g. the Internet, researchers have introduced an abundance of strategic games to study natural phenomena in networks. Unfortunately, almost all of these games have the conceptual drawback of being computationally intractable, i.e. computing a best response strategy or checking if an equilibrium is reached is NPhard. Thus, a main challenge in the field is to find tractable realistic network formation models. We address this challenge by establishing that the recently introduced model by Goyal et al.[WINE'16], which focuses on robust networks in the presence of a strong adversary, is a rare exception which is both realistic and computationally tractable. In particular, we sketch an efficient algorithm for computing a best response strategy, which implies that deciding whether the game has reached a Nash equilibrium can be done efficiently as well. Our algorithm essentially solves the problem of computing a minimal connection to a network which maximizes the reachability while hedging against severe attacks on the network infrastructure.

Albers, Susanne; Lenzner, Pascal On Approximate Nash Equilibria in Network Design. Internet Mathematics 2013: 384405
We study a basic network design game where \(n\) selfinterested agents, each having individual connectivity requirements, wish to build a network by purchasing links from a given set of edges. A fundamental cost sharing mechanism is Shapley cost sharing that splits the cost of an edge in a fair manner among the agents using the edge. In this paper we investigate if an optimal minimumcost network represents an attractive, relatively stable state that agents might want to purchase. We resort to the concept of \(\alpha\)approximate Nash equilibria. We prove that for single source games in undirected graphs, any optimal network represents an \(H(n)\)approximate Nash equilibrium, where \(H(n)\) is the \(n\)th Harmonic number. We show that this bound is tight. We extend the results to cooperative games, where agents may form coalitions, and to weighted games. In both cases we give tight or nearly tight lower and upper bounds on the stability of optimal solutions. Finally we show that in general sourcesink games and in directed graphs, minimumcost networks do not represent good states.

Kawald, Bernd; Lenzner, Pascal On dynamics in selfish network creation. Symposium on Parallelism in Algorithms and Architectures (SPAA) 2013: 8392
We consider the dynamic behavior of several variants of the Network Creation Game, introduced by Fabrikant et al. [PODC'03]. Equilibrium networks in these models have desirable properties like low social cost and small diameter, which makes them attractive for the decentralized creation of overlaynetworks. Unfortunately, due to the nonconstructiveness of the Nash equilibrium, no distributed algorithm for finding such networks is known. We treat these games as sequentialmove games and analyze if (uncoordinated) selfish play eventually converges to an equilibrium. Thus, we shed light on one of the most natural algorithms for this problem: distributed local search, where in each step some agent performs a myopic selfish improving move. We show that fast convergence is guaranteed for all versions of Swap Games, introduced by Alon et al. [SPAA'10], if the initial network is a tree. Furthermore, we prove that this process can be sped up to an almost optimal number of moves by employing a very natural move policy. Unfortunately, these positive results are no longer true if the initial network has cycles and we show the surprising result that even one nontree edge suffices to destroy the convergence guarantee. This answers an open problem from Ehsani et al. [SPAA'11] in the negative. Moreover, we show that on nontree networks no move policy can enforce convergence. We extend our negative results to the wellstudied original version, where agents are allowed to buy and delete edges as well. For this model we prove that there is no convergence guarantee  even if all agents play optimally. Even worse, if played on a noncomplete hostgraph, then there are instances where no sequence of improving moves leads to a stable network. Furthermore, we analyze whether costsharing has positive impact on the convergence behavior. For this we consider a version by Corbo and Parkes [PODC'05] where bilateral consent is needed for the creation of an edge and where edgecosts are shared among the involved agents. We show that employing such a costsharing rule yields even worse dynamic behavior..

Lenzner, Pascal On Dynamics in Basic Network Creation Games. Symposium on Algorithmic Game Theory (SAGT) 2011: 254265
We initiate the study of game dynamics in the Sum Basic Network Creation Game, which was recently introduced by Alon et al.[SPAA'10]. In this game players are associated to vertices in a graph and are allowed to "swap" edges, that is to remove an incident edge and insert a new incident edge. By performing such moves, every player tries to minimize her connection cost, which is the sum of distances to all other vertices. When played on a tree, we prove that this game admits an ordinal potential function, which implies guaranteed convergence to a pure Nash Equilibrium. We show a cubic upper bound on the number of steps needed for any improving response dynamic to converge to a stable tree and propose and analyse a best response dynamic, where the players having the highest cost are allowed to move. For this dynamic we show an almost tight linear upper bound for the convergence speed. Furthermore, we contrast these positive results by showing that, when played on general graphs, this game allows best response cycles. This implies that there cannot exist an ordinal potential function and that fundamentally different techniques are required for analysing this case. For computing a best response we show a similar contrast: On the one hand we give a lineartime algorithm for computing a best response on trees even if players are allowed to swap multiple edges at a time. On the other hand we prove that this task is NPhard even on simple general graphs, if more than one edge can be swapped at a time. The latter addresses a proposal by Alon et al..

Antoniadis, Antonios; Hüffner, Falk; Lenzner, Pascal; Moldenhauer, Carsten; Souza, Alexander Balanced Interval Coloring. Symposium on Theoretical Aspects of Computer Science (STACS) 2011: 531542
We consider the discrepancy problem of coloring n intervals with \(k\) colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time \(O(n \log n + k n \log k)\) for its construction. This is in particular interesting because many known results for discrepancy problems are nonconstructive. This problem naturally models a load balancing scenario, where \(n\) tasks with given start and endtimes have to be distributed among \(k\) servers. Our results imply that this can be done ideally balanced. When generalizing to \(d\)dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any \(d \ge 2\) and any \(k \ge 2\) it is NPcomplete to decide if such a solution exists, which implies also NPhardness of the respective minimization problem. In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.