Non-Atomic One-Round Walks in Polynomial Congestion Games

Abstract. In this paper we study the approximation ratio of the solutions achieved after an -approximate one-round walk in non-atomic congestion games. Prior to this work, the solution concept of one-round walks had been studied for atomic congestion games with linear latency functions onl

On the Impact of Singleton Strategies in Congestion Games

To what extent does the structure of the players\u27 strategy space influence the efficiency of decentralized solutions in congestion games? In this work, we investigate whether better performance is possible when restricting to load balancing games in which players can only choose among single resources. We consider three different solutions concepts, namely, approximate pure Nash equilibria, approximate one-round walks generated by selfish players aiming at minimizing their personal cost and approximate one-round walks generated by cooperative players aiming at minimizing the marginal increase in the sum of the players\u27 personal costs. The last two concepts can also be interpreted as solutions of simple greedy online algorithms for the related resource selection problem. Under fairly general latency functions on the resources, we show that, for all three types of solutions, better bounds cannot be achieved if players are either weighted or asymmetric. On the positive side, we prove that, under mild assumptions on the latency functions, improvements on the performance of approximate pure Nash equilibria are possible for load balancing games with weighted and symmetric players in the case of identical resources. We also design lower bounds on the performance of one-round walks in load balancing games with unweighted players and identical resources (in this case, solutions generated by selfish and cooperative players coincide)

Nash Social Welfare in Selfish and Online Load Balancing (Short Paper)

In load balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones, in order to execute a certain task. Each resource has a latency function, which depends on its workload, and a client's cost is the completion time of her chosen resource. Two fundamental variants of load balancing problems are selfish load balancing (aka. load balancing games), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and online load balancing, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both problems under the objective of minimizing the Nash Social Welfare, i.e., the geometric mean of the clients' costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal, as it matches the performance of any possible online algorithm

Nash Social Welfare in Selfish and Online Load Balancing

In load balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones, in order to execute a certain task. Each resource has a latency function, which depends on its workload, and a client's cost is the completion time of her chosen resource. Two fundamental variants of load balancing problems are {\em selfish load balancing} (aka. {\em load balancing games}), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and {\em online load balancing}, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both selfish and online load balancing under the objective of minimizing the {\em Nash Social Welfare}, i.e., the geometric mean of the clients' costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal as it matches the performance of any possible online algorithm

Improving Approximate Pure Nash Equilibria in Congestion Games

Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact or even an approximate pure Nash equilibrium is in general PLS-complete, Caragiannis et al. (2011) present a polynomial-time algorithm that computes a ($2 + \epsilon$)-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 $(1.61+\epsilon)$ 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 Bil\`{o} and Vinci (2016), we remark that our cost functions are locally computable and in contrast to Bil\`{o} and Vinci (2016) are independent of the actual instance of the game

The Anarchy-Stability Tradeoff in Congestion Games

This work focuses on the design of incentive mechanisms in congestion games, a commonly studied model for competitive resource sharing. While the majority of the existing literature on this topic focuses on unilaterally optimizing the worst case performance (i.e., price of anarchy), in this manuscript we investigate whether optimizing for the worst case has consequences on the best case performance (i.e., price of stability). Perhaps surprisingly, our results show that there is a fundamental tradeoff between these two measures of performance. Our main result provides a characterization of this tradeoff in terms of upper and lower bounds on the Pareto frontier between the price of anarchy and the price of stability. Interestingly, we demonstrate that the mechanism that optimizes the price of anarchy inherits a matching price of stability, thereby implying that the best equilibrium is not necessarily any better than the worst equilibrium for such a design choice. Our results also establish that, in several well-studied cases, the unincentivized setting does not even lie on the Pareto frontier, and that any incentive with price of stability equal to 1 incurs a much higher price of anarchy.Comment: 27 pages, 1 figure, 1 tabl

Approximation algorithms for distributed and selfish agents

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 157-165).Many real-world systems involve distributed and selfish agents who optimize their own objective function. In these systems, we need to design efficient mechanisms so that system-wide objective is optimized despite agents acting in their own self interest. In this thesis, we develop approximation algorithms and decentralized mechanisms for various combinatorial optimization problems in such systems. First, we investigate the distributed caching and a general set of assignment problems. We develop an almost tight LP-based ... approximation algorithm and a local search ... approximation algorithm for these problems. We also design efficient decentralized mechanisms for these problems and study the convergence of the corresponding games. In the following chapters, we study the speed of convergence to high quality solutions on (random) best-response paths of players. First, we study the average social value on best response paths in basic-utility, market sharing, and cut games. Then, we introduce the sink equilibrium as a new equilibrium concept. We argue that, unlike Nash equilibria, the selfish behavior of players converges to sink equilibria and all strategic games have a sink equilibrium. To illustrate the use of this new concept, we study the social value of sink equilibria in weighted selfish routing (or weighted congestion) games and valid-utility (or submodular-utility) games. In these games, we bound the average social value on random best-response paths for sink equilibria.. Finally, we study cross-monotonic cost sharings and group-strategyproof mechanisms.(cont.) We study the limitations imposed by the cross-monotonicity property on cost-sharing schemes for several combinatorial optimization games including set cover and metric facility location. We develop a novel technique based on the probabilistic method for proving upper bounds on the budget-balance factor of cross-monotonic cost sharing schemes, deriving tight or nearly-tight bounds for these games. At the end, we extend some of these results to group-strategyproof mechanisms.by Vahab S. Mirrokni.Ph.D