Swap Equilibria under Link and Vertex Destruction

We initiate the study of the destruction or adversary model (Kliemann 2010) using the swap equilibrium (SE) stability concept (Alon et al., 2010). The destruction model is a network formation game incorporating the robustness of a network under a more or less targeted attack. In addition to bringing in the SE concept, we extend the model from an attack on the edges to an attack on the vertices of the network. We prove structural results and linear upper bounds or super-linear lower bounds on the social cost of SE under different attack scenarios. For the case that the vertex to be destroyed is chosen uniformly at random from the set of max-sep vertices (i.e., where each causes a maximum number of separated player pairs), we show that there is no tree SE with only one max-sep vertex. We conjecture that there is no tree SE at all. On the other hand, we show that for the uniform measure, all SE are trees (unless two-connected). This opens a new research direction asking where the transition from “no cycle” to “at least one cycle” occurs when gradually concentrating the measure on the max-sep vertices

Games in Networks under Robustness, Locality, and Coloring Aspects

Decentralization is a key concept in modern networks, such as the Internet, social networks, or wireless phone or sensor networks. The systematic study of how networks are formed by a multitude of non-cooperative or only mildly cooperative players, is a vivid topic in Discrete Mathe- matics, Computer Science, and Economics. In 2010, a variant addressing robustness aspects was introduced. In this model, known as adversary model or destruction model, one link in the network is destroyed at random af- ter the network has been formed. Players anticipate this disruption and try to build a network that gives them good connectivity even after the destruction. How efficiently can this be done in a decentralized setting? This thesis advances our knowledge regarding the adversary model: we bring in the modern equilibrium concept of swap equilibrium and we extend to the destruction of one vertex. This disruption tends to be more severe compared to the case that just one link is destroyed. We characterize several settings, where for some, the formed networks are provably efficient, while for others, we show they can be inefficient up to a certain degree. Apart from the adversary model, we study a model where each player v tries to maximize the number of players at distance at most k from v, for a fixed parameter k. For example, when the network models friendship, then for k = 2, players would try to maximize the number of friends plus friends of their friends, which is an interesting metric in Sociology. We prove results on the structure and efficiency of such networks. In the final chapter, the network is fixed and each player chooses one of k colors. For example, the network might describe the spatial relations between the players and colors might correspond to radio frequencies, so each player’s aim is to choose a frequency that causes as few interference as possible with the frequencies of her neighbors

Strategic Network Formation with Attack and Immunization

Strategic network formation arises where agents receive benefit from connections to other agents, but also incur costs for forming links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization against attack. An agent's benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework is a stylized model of settings where reachability rather than centrality is the primary concern and vertices vulnerable to attacks may reduce risk via costly measures. In the reachability benefit model without attack or immunization, the set of equilibria is the empty graph and any tree. The introduction of attack and immunization changes the game dramatically; new equilibrium topologies emerge, some more sparse and some more dense than trees. We show that, under a mild assumption on the adversary, every equilibrium network with $n$ agents contains at most $2n-4$ edges for $n\geq 4$. So despite permitting topologies denser than trees, the amount of overbuilding is limited. We also show that attack and immunization don't significantly erode social welfare: every non-trivial equilibrium with respect to several adversaries has welfare at least as that of any equilibrium in the attack-free model. We complement our theory with simulations demonstrating fast convergence of a new bounded rationality dynamic which generalizes linkstable best response but is considerably more powerful in our game. The simulations further elucidate the wide variety of asymmetric equilibria and demonstrate topological consequences of the dynamics e.g. heavy-tailed degree distributions. Finally, we report on a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium.Comment: The short version of this paper appears in the proceedings of WINE-1

On Euler Tours in Streaming Models and some Games on Graphs

In this thesis, we take a look at several graph theoretical problems. We present two streaming algorithms for finding an Euler tour in a graph, prove a tight bound on the capture time in the Bridge-Burning Cops and Robbers Game and solve an open problem for the Extreme Vertex Destruction Model. In Chapter 2, we consider the classical Euler tour problem and take a modern look at this problem in the context of the graph streaming model. Here, RAM is of size O(n polylog(n)), where n is the number of nodes, and the graph is given as a stream of its edges. With this restricted memory space, we give a one-pass algorithm for finding Euler tours in the graph streaming model. In Chapter 3, we regard a lesser-known streaming model, the so-called StrSort model, to tackle a downside of our algorithm mentioned above. The algorithm stores an Euler tour on an output tape in form of a successor function. The order of the edges is given, but the edges are not actually sorted in the order of the Euler tour. Therefore, further processing the tour with another streaming algorithm might become difficult. We give an algorithm for sorting the edges of a graph according to a found Euler tour, that has a preparation step in the graph streaming model and a processing step in the StrSort model. The so-called Bridge-Burning Cops and Robbers Game is the topic of Chapter 4. Here, every time the robber traverses an edge, this edge is deleted afterwards. We study winning strategies of a single cop and make statements on the maximum number of turns of such strategies. In Chapter 5, we study networks formed by selfish agents. When a node is ‘destroyed’, i.e. the adjacent edges are deleted, the network is damaged and some players lose the connection to each other. In the Extreme Vertex Destruction Model, we observe the impact of such a deletion on swap equilibrium graphs and make statements on the maximum amount of damage that can cause here

Positions- und Detektionsspiele

When it comes to the interaction of two ore more parties with individual aims, it’s all about finding an appropriate strategy. In most cases, the individual aim boils down to detection of information about the general situation or about your opponents and improvement of your own position. This goal becomes most clear and specific in the field of recreational games. In games like chess or tic-tac-toe, every player has complete information and the player’s position decides over win and loss. On the contrary, in games like poker every player tries to find out the value of the other players’ hands to play accordingly. This uncertainty of the opponent’s hand is the factor that makes the game interesting. Since all results of this thesis are connected to the field of game theory, it seems important to mention that this research field is not about having fun with different kinds of games, but, in the contrary, it’s about analysis of these games. The crucial difference between casual games and formal combinatorial games is that a combinatorial game is always assumed to be played by two players of infinite computational power. If the considered game is of complete information, the outcome of the game is already determined before it even started. The variety of games that are analyzed in this work ranges from popular recreational games as Mastermind over network-formation games to purely abstract games on graphs or hypergraph

Fairness And Feedback In Learning And Games

In this thesis, we study fairness and feedback effects in game theory and machine learning. In game theory and economics, financial or technological networks are analyzed for feedback effects. These studies analyze how the connectivity benefits or risk of contagious shocks affect the individual agents or the structure of the network formed by these rational agents. Towards this direction, in the first part of this thesis, we study a series of novel network formation games and analyze the structural properties of the equilibrium networks. Feedback effects can also occur in machine learning problems such as reinforcement learning or sequential allocation problems where the decisions of an algorithm over time can change the resources or actions available to the algorithm in the future as well as the environment in which the algorithm is operating. In the second part of this thesis, we study the effect of these feedback loops and ways to prevent them while also ensuring that the algorithm\u27s actions and allocations satisfy natural notions of fairness. In particular we are interested in quantifying the cost of imposing fairness on learning algorithms

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set