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Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact [18] or even an approximate [34] pure Nash equilibrium is in general PLScomplete, Caragiannis et al. [9] present a polynomial-time algorithm that computes a (2 + )-approximate pure Nash equilibria for games with linear cost functions and further results for polynomial cost functions. We show that this factor can be improved to (1.61 + ) and further improved results for polynomial cost functions, by a seemingly simple modification to their algorithm by allowing for the cost functions used during the best response dynamics be different from the overall objective function. Interestingly, our modification to the algorithm also extends to efficiently computing improved approximate pure Nash equilibria in games with arbitrary non-decreasing resource cost functions. Additionally, our analysis exhibits an interesting method to optimally compute universal load dependent taxes and using linear programming duality prove tight bounds on the PoA under universal taxation, e.g., 2.012 for linear congestion games and further results for polynomial cost functions. Although our approach yield weaker results than that in Bilò and Vinci [6], we remark that our cost functions are locally computable and in contrast to [6] are independent of the actual instance of the game. Keywords: Congestion games • Approximate pure Nash equilibria • Price of anarchy • Universal taxes V. R. Vijayalakshmi-This work is supported by the German research council (DFG) Research Training Group 2236 UnRAVeL

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We consider a scheduling game in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We characterize four classes of instances in which a pure Nash equilibrium (NE) is guaranteed to exist, and show, by means of an example, that none of these characterizations can be relaxed. We then bound the performance of Nash equilibria for each of these classes with respect to the makespan of the schedule and the sum of completion times. We also analyze the computational complexity of several problems arising in this model. For instance, we prove that it is NP-hard to decide whether a NE exists, and that even for instances with identical machines, for which a NE is guaranteed to exist, it is NP-hard to approximate the best NE within a factor of 2 − 1 m − for all > 0.In addition, we study a generalized model in which players' strategies are subsets of resources, each having its own priority list over the players. We show that in this general model, even unweighted symmetric games may not have a pure NE, and we bound the price of anarchy with respect to the total players' costs.

Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact [16] or even an approximate [33] pure Nash equilibrium is in general Caragiannis et al. [9] present a polynomial-time algorithm that computes a (2 + ǫ)-approximate pure Nash equilibria for games with linear cost functions and further results for polynomial cost functions. We show that this factor can be improved to (1.61+ǫ) and further improved results for polynomial cost functions, by a seemingly simple modification to their algorithm by allowing for the cost functions used during the best response dynamics be different from the overall objective function. Interestingly, our modification to the algorithm also extends to efficiently computing improved approximate pure Nash equilibria in games with arbitrary non-decreasing resource cost functions. Additionally, our analysis exhibits an interesting method to optimally compute universal load dependent taxes and using linear programming duality prove tight bounds on PoA under universal taxation, e.g, 2.012 for linear congestion games and further results for polynomial cost functions. Although our approach yield weaker results than that in Bilò and Vinci [6], we remark that our cost functions are locally computable and in contrast to [6] are independent of the actual instance of the game.