On the Existence of Pure Nash Equilibria in Weighted Congestion Games

Harks, Tobias; Klimm, Max

doi:10.1007/978-3-642-14165-2_8

Cited by 45 publications

(70 citation statements)

References 35 publications

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“…These Shapley value-based cost shares coincide with proportional shares when all per-user cost functions are affine, but not otherwise (Figure 1(a)). These results explain the previously mysterious fact that the traditional proportional cost shares always yield a potential game if and only if all cost functions are affine [7,11].…”

supporting

confidence: 52%

“…For example, in a network context, players could have different durations of resource usage, different bandwidth requirements, or different contracts with the network provider. Almost all research to date has modeled non-uniform players in congestion-type games through proportional cost sharing [1,2,3,4,5,8,11,12,17,18]. The first assumption in proportional cost sharing is that each player i has a positive weight w i , with larger weights indicating larger resource usage.…”

mentioning

confidence: 99%

“…Even when PNE do exist in such games, they can be much costlier (relative to an optimal solution) than in the unweighted case [2,5]. Atomic selfish routing games with weighted players do not generally have PNE [9,11,23], except when all cost functions are affine [7] and in some other isolated special cases [11].…”

mentioning

confidence: 99%

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Restoring Pure Equilibria to Weighted Congestion Games

Kollias

Roughgarden

2015

ACM Trans. Econ. Comput.

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Congestion games model several interesting applications, including routing and network formation games, and also possess attractive theoretical properties, including the existence of and convergence of natural dynamics to a pure Nash equilibrium. Weighted variants of congestion games that rely on sharing costs proportional to players' weights do not generally have pure-strategy Nash equilibria. We propose a new way of assigning costs to players with weights in congestion games that recovers the important properties of the unweighted model. This method is derived from the Shapley value, and it always induces a game with a (weighted) potential function. For the special cases of weighted network cost-sharing and weighted routing games with Shapley value-based cost shares, we prove tight bounds on the worst-case inefficiency of equilibria. For weighted network cost-sharing games, we precisely calculate the price of stability for any given player weight vector, while for weighted routing games, we precisely calculate the price of anarchy, as a parameter of the set of allowable cost functions.

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confidence: 52%

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confidence: 99%

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Restoring Pure Equilibria to Weighted Congestion Games

Kollias

Roughgarden

2015

ACM Trans. Econ. Comput.

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“…Summing up (19) for all 1 ≤ i ≤ h(σ, t * ) − 1 and the LHS of (20) gives the first inequality (note that the quantity k j=1 r j (σ)[t h(σ,t * )−1 (σ), t * ] > 0 is due to the way we define h(σ, t)).…”

Section: But This Contradicts Lemma 12(i) So We Establishmentioning

confidence: 99%

Collusion in Atomic Splittable Routing Games

Huang

2012

Theory Comput Syst

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Abstract. We investigate how collusion affects the social cost in atomic splittable routing games. Suppose that players form coalitions and each coalition behaves as if it were a single player controlling all the flows of its participants. It may be tempting to conjecture that the social cost would be lower after collusion, since there would be more coordination among the players. We construct examples to show that this conjecture is not true. Even in very simple single-source-singledestination networks, the social cost of the post-collusion equilibrium can be higher than that of the pre-collusion equilibrium. This counter-intuitive phenomenon of collusion prompts us to ask the question: under what conditions would the social cost of the post-collusion equilibrium be bounded by the social cost of the pre-collusion equilibrium? We show that if (i) the network is "well-designed" (satisfying a natural condition), and (ii) the delay functions are affine, then collusion is always beneficial for the social cost in the Nash equilibria. On the other hand, if either of the above conditions is unsatisfied, collusion can worsen the social cost. Our main technique is a novel flow-augmenting algorithm to build Nash equilibria. Our positive result for collusion is obtained by applying this algorithm simultaneously to two different flow value profiles of players and observing the difference in the derivatives of their social costs. Moreover, for a non-trivial subclass of selfish routing games, this algorithm finds the exact Nash equilibrium in polynomial time.⋆ Research supported by an Alexander von Humboldt fellowship.

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“…One would expect that this "order of stability" would carry over to the dynamic setting, as is the case in other extensions of the traditional setting. For example, an NE is not guaranteed for weighted cost-sharing games [9] as well as very restrictive classes of multiset cost-sharing games [5], whereas every linear weighted congestion game [12] and even linear multiset congestion game is guaranteed to have an NE [6].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Resource Allocation Games

Avni

Henzinger

Kupferman

2016

Algorithmic Game Theory

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Abstract. In resource allocation games, selfish players share resources that are needed in order to fulfill their objectives. The cost of using a resource depends on the load on it. In the traditional setting, the players make their choices concurrently and in one-shot. That is, a strategy for a player is a subset of the resources. We introduce and study dynamic resource allocation games. In this setting, the game proceeds in phases. In each phase each player chooses one resource. A scheduler dictates the order in which the players proceed in a phase, possibly scheduling several players to proceed concurrently. The game ends when each player has collected a set of resources that fulfills his objective. The cost for each player then depends on this set as well as on the load on the resources in it -we consider both congestion and cost-sharing games. We argue that the dynamic setting is the suitable setting for many applications in practice. We study the stability of dynamic resource allocation games, where the appropriate notion of stability is that of subgame perfect equilibrium, study the inefficiency incurred due to selfish behavior, and also study problems that are particular to the dynamic setting, like constraints on the order in which resources can be chosen or the problem of finding a scheduler that achieves stability.

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On the Existence of Pure Nash Equilibria in Weighted Congestion Games

Cited by 45 publications

References 35 publications

Restoring Pure Equilibria to Weighted Congestion Games

Restoring Pure Equilibria to Weighted Congestion Games

Collusion in Atomic Splittable Routing Games

Dynamic Resource Allocation Games

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