Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games

We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply sampling techniques which allow us to achieve polynomial running time. The algorithm builds on the algorithmic ideas of [7], however, to the best of our knowledge, this is the first algorithmic result on computation of approximate equilibria using other than proportional shares as player costs in this setting. We present a novel relation that approximates the Shapley value of a player by her proportional share and vice versa. As side results, we upper bound the approximate price of anarchy of such games and significantly improve the best known factor for computing approximate pure Nash equilibria in weighted congestion games of [7].Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-71924-5_1

Social Context and Cost-Sharing in Congestion Games

Congestion games are one of the most prominent classes of games in non- cooperative game theory as they model a large collection of important applications in networks, such as selfish routing in traffic or telecommunications. For this reason, congestion games have been a driving force in recent research and my thesis lies on two major extensions of this class of games. The first extension considers congestion games embedded in a social network where players are not necessarily selfish and might care about others. We call this class social context congestion games and study how the social interactions among players affect it. In particular, we study existence of approximate pure Nash equilibria and our main result is the following. For any given set of cost functions, we provide a threshold value such that: for the class of social context congestion games with cost functions within the given set, sequences of improvement steps of players, are guaranteed to converge to an approximate pure Nash equilibrium if and only if the improvement step factor is larger than this threshold value. The second topic considers weighted congestion games under a fair cost sharing system which depends on the weight of each player, the (weighted) Shapley values. This class considers weighted congestion games where (weighted) Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We study the efficiency of this class of games in terms of the price of anarchy and the price of stability. Regard- ing the price of anarchy, we show general tight bounds, which apply to general equilibrium concepts. For the price of stability, we prove an upper bound for the special case of Shapley values. This bound holds for general sets of cost functions and is tight in special cases of interest, such as bounded degree polynomials. Also for bounded degree polynomials, we show that a slight deviation from the Shapley value has a huge impact on the price of stability. In fact, the price of stability becomes as bad as the price of anarchy. For this model, we also study computation of equilibria. We propose an algorithm to compute approximate pure Nash equilibria which executes a polynomial number of strategy updates. Due to the complex nature of Shapley values, computing a single strategy update is hard, however, applying sampling techniques allow us to achieve polynomial running time. We generalise the previous model allowing each player to control multiple flows. For this generalised model, we study existence and efficiency of equilibria. We exhibit a separation from the original model (each player controls only one flow) by proving that Shapley values are the only cost-sharing method that guarantees pure Nash equilibria existence in the generalised model. Also, we prove that the price of anarchy and price of stability become no larger than in the original model

Strong Nash Equilibria in Games with the Lexicographical Improvement Property

We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium

On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games

In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $\alpha$ through unilateral strategy changes. There is a threshold $\alpha_\delta$ (where $\delta$ is a parameter that limits the demand of each player on a specific resource) such that $\alpha$-approximate pure Nash equilibria always exist for $\alpha \geq \alpha_\delta$, but not for $\alpha < \alpha_\delta$. We give both upper and lower bounds on this threshold $\alpha_\delta$ and show that the corresponding decision problem is ${\sf NP}$-hard. We also show that the $\alpha$-approximate price of anarchy for BAGs is $\alpha+1$. For a restricted version of the game, where demands of players only differ slightly from each other (e.g. symmetric games), we show that approximate Nash equilibria can be reached (and thus also be computed) in polynomial time using the best-response dynamic. Finally, we show that a broader class of utility-maximization games (which includes BAGs) converges quickly towards states whose social welfare is close to the optimum

Weighted Congestion Games With Separable Preferences

Players in a congestion game may differ from one another in their intrinsic preferences (e.g., the benefit they get from using a specific resource), their contribution to congestion, or both. In many cases of interest, intrinsic preferences and the negative effect of congestion are (additively or multiplicatively) separable. This paper considers the implications of separability for the existence of pure-strategy Nash equilibrium and the prospects of spontaneous convergence to equilibrium. It is shown that these properties may or may not be guaranteed, depending on the exact nature of player heterogeneity.congestion games, separable preferences, pure equilibrium, finite improvement property, potential.

Congestion Games with Complementarities

We study a model of selfish resource allocation that seeks to incorporate dependencies among resources as they exist in modern networked environments. Our model is inspired by utility functions with constant elasticity of substitution (CES) which is a well-studied model in economics. We consider congestion games with different aggregation functions. In particular, we study $L_p$ norms and analyze the existence and complexity of (approximate) pure Nash equilibria. Additionally, we give an almost tight characterization based on monotonicity properties to describe the set of aggregation functions that guarantee the existence of pure Nash equilibria.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-57586-5_1