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Selfish Routing and the Price of Anarchy
 MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
, 2007
"... Selfish routing is a classical mathematical model of how selfinterested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure of this in ..."
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Selfish routing is a classical mathematical model of how selfinterested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure of this inefficiency. We survey recent work that analyzes the price of anarchy of selfish routing. We also describe related results on bounding the worstpossible severity of a phenomenon called Braess’s Paradox, and on three techniques for reducing the price of anarchy of selfish routing. This survey concentrates on the contributions of the author’s PhD thesis, but also discusses several more recent results in the area.
Strong nash equilibria in games with the lexicographical improvement property
 Internet and Network Economics
, 2009
"... Abstract. 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 Le ..."
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Abstract. 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. 1
Characterizing the existence of potential functions in weighted congestion games
 Proc. 2nd Internat. Sympos. Algorithmic Game Theory, volume 5814 of LNCS, pages 97 – 108
, 2009
"... Abstract Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let C be an arbitrary set of locally bounded functions and let G(C) be the ..."
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Abstract Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let C be an arbitrary set of locally bounded functions and let G(C) be the set of weighted congestion games with cost functions in C. We show that every weighted congestion game G ∈ G(C) admits an exact potential if and only if C contains only affine functions. We also give a similar characterization for wpotentials with the difference that here C consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facilitydependent demands and elastic demands, respectively.
Wardrop equilibria and price of stability for bottleneck games with splittable traffic
 In Proceedings of the Second Annual Workshop on Internet and Network Economics (WINE
, 2006
"... Abstract. We look at the scenario of having to route a continuous rate of traffic from a source node to a sink node in a network, where the objective is to maximize throughput. This is of interest, e.g., for providers of streaming content in communication networks. The overall path latency, which wa ..."
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Abstract. We look at the scenario of having to route a continuous rate of traffic from a source node to a sink node in a network, where the objective is to maximize throughput. This is of interest, e.g., for providers of streaming content in communication networks. The overall path latency, which was relevant in other noncooperative network routing games such as the classic Wardrop model, is of lesser concern here. To that end, we define bottleneck games with splittable traffic where the throughput on a path is inversely proportional to the maximum latency of an edge on that very path—the bottleneck latency. Therefore, we define a Wardrop equilibrium as a traffic distribution where this bottleneck latency is at minimum on all used paths. As a measure for the overall system wellbeing—called social cost—we take the weighted sum of the bottleneck latencies of all paths. Our main findings are as follows: First, we prove social cost of Wardrop equilibria on series parallel graphs to be unique. Even more, for any graph whose subgraph induced by all simple startdestination paths is not series parallel, there exist games having equilibria with different social cost. For the price of stability, we give an independence result with regard to the network topology. Finally, our main result is giving a new exact price of stability for Wardrop/bottleneck games on parallel links with M/M/1 latency functions. This result is at the same time the exact price of stability for bottleneck games on general graphs. 1
Competitive Routing over Time
"... Abstract. Congestion games are a fundamental and widely studied model for selfish allocation problems like routing and load balancing. An intrinsic property of these games is that players allocate resources simultaneously and instantly. This is particularly unrealistic for many network routing scena ..."
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Abstract. Congestion games are a fundamental and widely studied model for selfish allocation problems like routing and load balancing. An intrinsic property of these games is that players allocate resources simultaneously and instantly. This is particularly unrealistic for many network routing scenarios, which are one of the prominent application scenarios of congestion games. In many networks, load travels along routes over time and allocation of edges happens sequentially. In this paper we consider two frameworks that enhance network congestion games with a notion of time. We propose temporal network congestion games that use coordination mechanisms — local policies that allow to sequentialize traffic on the edges. In addition, we consider congestion games with timedependent costs, in which travel times are fixed but quality of service of transmission varies with load over time. We study existence and complexity properties of pure Nash equilibria and bestresponse strategies in both frameworks. In some cases our results can be used to characterize convergence for various distributed dynamics. 1
A Stackelberg Strategy for Routing Flow over Time
"... Routing games are used to to understand the impact of individual users ’ decisions on network efficiency. Most prior work on routing games uses a simplified model of network flow where all flow exists simultaneously, and users care about either their maximum delay or their total delay. Both of these ..."
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Cited by 4 (2 self)
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Routing games are used to to understand the impact of individual users ’ decisions on network efficiency. Most prior work on routing games uses a simplified model of network flow where all flow exists simultaneously, and users care about either their maximum delay or their total delay. Both of these measures are surrogates for measuring how long it takes to get all of a user’s traffic through the network. We attempt a more direct study of how competition affects network efficiency by examining routing games in a flow over time model. We give an efficiently computable Stackelberg strategy for this model and show that the competitive equilibrium under this strategy is no worse than a small constant times the optimal, for two natural measures of optimality.
A Competitive Strategy for Routing Flow over Time
"... Network routing games are used to understand the impact of individual users ’ decisions on network efficiency. Prior work on routing games uses a simplified model of network flow where all flow exists simultaneously. In our work, we examine routing games in a flowovertime model. We show that by re ..."
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Network routing games are used to understand the impact of individual users ’ decisions on network efficiency. Prior work on routing games uses a simplified model of network flow where all flow exists simultaneously. In our work, we examine routing games in a flowovertime model. We show that by reducing network capacity judiciously, the network owner can ensure that the equilibrium is no worse than a small constant times the optimal in the original network, for two natural measures of optimality. These are the first upper bounds on the price of anarchy in the flowovertime model for general networks.
selfish users with elastic demands Authors:
, 2006
"... Edge pricing of multicommodity networks for ..."