Pure Nash Equilibria in Player-Specific and Weighted Congestion Games

Ackermann, Heiner; Röglin, Heiko; Vöcking, Berthold

doi:10.1007/11944874_6

Cited by 70 publications

(160 citation statements)

References 10 publications

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“…2 Local-effect games. In a local-effect game with player set N = 1 2 n , all players have the same set of available actions, .…”

Section: Preliminariesmentioning

confidence: 99%

“…Assume that A = a 1 a 2 a m . We introduce an arcē 0 between r and the starting node of the second arcē 2 1 , which was one of three arcs replacing the original arc a 1 ∈ A. Furthermore, for 1 ≤ i ≤ m − 1, we create an arcē i connecting the end node ofē 2 i with the start node of e 2 i+1 .…”

Section: Theorem 33 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

“…We introduce an arcē 0 between r and the starting node of the second arcē 2 1 , which was one of three arcs replacing the original arc a 1 ∈ A. Furthermore, for 1 ≤ i ≤ m − 1, we create an arcē i connecting the end node ofē 2 i with the start node of e 2 i+1 . Finally, there is an arcē m from the end node ofē 2 m to the terminal node t. We denote the r-t-path formed by the arcsē k , k = 0 1 m, andē 2 i , i = 1 2 m, by Q 5 .…”

Section: Theorem 33 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

“…Arcsē 1 i andē 3 i , 1 ≤ i ≤ m, are very expensive if the load is greater than or equal to w 3 = m; hence, the only way for p 3 to route her flow in a Nash equilibrium is to use path Q 5 . Because she cannot decrease her cost by switching to a path in the lower part of the network, the cost for using Q 5 must be smaller than K. This implies that at most one of the players p 1 and p 2 uses an arcē 2 i , 1 ≤ i ≤ m (otherwise, the total weight on such an arc would be m + 2). Similarly, neither p 1 nor p 2 can use any arcē k for k ∈ 0 1 m .…”

Section: Theorem 33 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

“…We may assume that B ≥ 12. We will construct a bidirectional local-effect game such that it has a pure-strategy Nash equilibrium if and only if A can be partitioned into m disjoint sets A 1 A 2 A m such that i∈A k w i = B for 1 ≤ k ≤ m. The action set consists of 3m 2 + 2m + 3 actions that are available to 4m + 3 players. For each item i ∈ A, there is a set of m corresponding actions a 1 i a 2 i a m i ; we will make sure that in any Nash equilibrium of the game exactly one player will pick one of these actions, for each item.…”

Section: Theorem 34 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

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On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games

Dunkel

Schulz

2006

Lecture Notes in Computer Science

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Rosenthal's congestion games constitute one of the few known classes of noncooperative games possessing pure-strategy Nash equilibria. In the network version, each player wants to route one unit of flow on a single path from her origin to her destination at minimum cost, and the cost of using an arc depends only on the total number of players using that arc. A natural extension is to allow for players controlling different amounts of flow, which results in so-called weighted congestion games. While examples have been exhibited showing that pure-strategy Nash equilibria need not exist anymore, we prove that it is actually strongly NP-hard to determine whether a given weighted network congestion game has a purestrategy Nash equilibrium. This is true regardless of whether flow is unsplittable or not. In the unsplittable case, the problem remains strongly NP-hard for a fixed number of players. In addition to congestion games, we provide complexity results on the existence and computability of pure-strategy Nash equilibria for the closely related family of bidirectional local-effect games. Therein, the cost of a player taking a particular action depends not only on the number of players choosing the same action, but also on the number of players settling for (locally) related actions.

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“…2 Local-effect games. In a local-effect game with player set N = 1 2 n , all players have the same set of available actions, .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Theorem 33 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

Section: Theorem 33 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

Section: Theorem 33 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

Section: Theorem 34 the Problem Of Deciding Whether A Weighted Netwmentioning

confidence: 99%

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On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games

Dunkel

Schulz

2006

Lecture Notes in Computer Science

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Algorithmic Game Theory and Applications

Nayak

2007

Handbook of Applied Algorithms

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Most of the existing and foreseen complex networks, such as the Internet, are operated and built by thousands of large and small entities (autonomous agents), which collaborate to process and deliver end-to-end flows originating from and terminating at any of them. The distributed nature of the Internet implies a lack of coordination among its users. Instead, each user attempts to obtain maximum performance according to his own parameters and objectives.Methods from game theory and mathematical economics have been proven to be a powerful modeling tool, which can be applied to understand, control, and efficiently design such dynamic, complex networks. Game theory provides a good starting point for computer scientists in their endeavor to understand selfish rational behavior in complex networks with many agents (players). Such scenarios are readily modeled using techniques from game theory, where players with potentially conflicting goals participate in a common setting with well-prescribed interactions.Nash equilibrium [73,74] distinguishes itself as the predominant concept of rationality in noncooperative settings. So, game theory and its various concepts of equilibria provide a rich framework for modeling the behavior of selfish agents in these kinds of distributed or networked environments; they offer mechanisms to achieve efficient and desirable global outcomes in spite of the selfish behavior.Mechanism design, a subfield of game theory, asks how one can design systems so that agents' selfish behavior results to desired systemwide goals. Algorithmic mechanism design additionally considers computational tractability to the set of concerns of mechanism design. Work on algorithmic mechanism design has focused on the complexity of centralized implementations of game-theoretic mechanisms for distributed optimization problems. Moreover, in such huge and heterogeneous networks, each agent does not have access to (and may not process) complete information. 286 ALGORITHMIC GAME THEORY AND APPLICATIONSThe notion of bounded rationality for agents and the design of corresponding incomplete-information distributed algorithms have been successfully utilized to capture the aspect of lack of global knowledge in information networks.In this chapter, we review some of the most thrilling algorithmic problems and solutions, and corresponding advances, achieved on the account of game theory. The areas addressed are the following.Congestion games A central problem arising in the management of large-scale communication networks is that of routing traffic through the network. However, due to the large size of these networks, it is often impossible to employ a centralized traffic management. A natural assumption to make in the absence of central regulation is that network users behave selfishly and aim at optimizing their own individual welfare. One way to address this problem is to model this scenario as a noncooperative multiplayer game and formalize it using congestion game. Congestion games (either unweighted or weighted) offer ...

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Robust clustering for ad hoc cognitive radio network

Fang

Gross

2018

Trans Emerging Tel Tech

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Cluster structure in cognitive radio (CR) networks facilitates cooperative spectrum sensing, routing, and other functionalities. Unlicensed channels, which are temporally available for a group of CR users in one area, consolidate the group into a cluster. More available unlicensed channels in a cluster make the cluster more likely to uphold against the licensed users' influence, making clusters more robust. This paper analyzes the problem of how to form robust clusters in a CR network such that CR systems benefit from collaboration within clusters despite intense primary user activity. We give a formal description of the robust clustering problem, prove it to be NP‐hard, and propose both centralized and distributed solutions. The congestion game model is adopted to analyze the process of cluster formation, which not only contributes to the design of the distributed clustering scheme but also provides a guarantee on the convergence to a Nash equilibrium and the convergence speed. The proposed distributed clustering scheme outperforms state‐of‐the‐art in terms of cluster robustness, convergence speed, and overhead. Extensive simulations are presented supporting the theoretical claims.

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Pure Nash Equilibria in Player-Specific and Weighted Congestion Games

Cited by 70 publications

References 10 publications

On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games

On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games

Algorithmic Game Theory and Applications

Robust clustering for ad hoc cognitive radio network

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