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

Dunkel, Juliane; Schulz, Andreas S.

doi:10.1007/11944874_7

Cited by 37 publications

(39 citation statements)

References 15 publications

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“…For a general case, Rosenthal (1973b) gave an example of an asymmetric weighted network congestion game that does not have an equilibrium in pure strategies. More recently, Dunkel and Schulz (2008) strengthened this result by showing that the problem of deciding whether a weighted network congestion game with integer-splittable flows admits a PSNE is strongly NP-hard.…”

Section: Related Workmentioning

confidence: 94%

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

Tran-Thanh

Polukarov

Chapman

et al. 2011

Algorithmic Game Theory

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Abstract. We study the existence of pure strategy Nash equilibria (PSNE) in integer-splittable weighted congestion games (ISWCGs), where agents can strategically assign different amounts of demand to different resources, but must distribute this demand in fixed-size parts. Such scenarios arise in a wide range of application domains, including job scheduling and network routing, where agents have to allocate multiple tasks and can assign a number of tasks to a particular selected resource. Specifically, in an ISWCG, an agent has a certain total demand (aka weight) that it needs to satisfy, and can do so by requesting one or more integer units of each resource from an element of a given collection of feasible subsets.1 Each resource is associated with a unit-cost function of its level of congestion; as such, the cost to an agent for using a particular resource is the product of the resource unit-cost and the number of units the agent requests. While general ISWCGs do not admit PSNE (Rosenthal, 1973b), the restricted subclass of these games with linear unit-cost functions has been shown to possess a potential function (Meyers, 2006), and hence, PSNE. However, the linearity of costs may not be necessary for the existence of equilibria in pure strategies. Thus, in this paper we prove that PSNE always exist for a larger class of convex and monotonically increasing unit-costs. On the other hand, our result is accompanied by a limiting asumption on the structure of agents' strategy sets: specifically, each agent is associated with its set of accessible resources, and can distribute its demand across any subset of these resources. Importantly, we show that neither monotonicity nor convexity on its own guarantees this result. Moreover, we give a counterexample with monotone and semi-convex cost functions, thus distinguishing ISWCGs from the class of infinitely-splittable congestion games for which the conditions of monotonicity and semi-convexity have been shown to be sufficient for PSNE existence (Rosen, 1965). Furthermore, we demonstrate that the finite improvement path property (FIP) does not hold for convex increasing ISWCGs. Thus, in contrast to the case with linear costs, a potential function argument cannot be used to prove our result. Instead, we provide a procedure that converges to an equilibrium from an arbitrary initial strategy profile, and in doing so show that ISWCGs with convex increasing unit-cost functions are weakly acyclic.

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Section: Related Workmentioning

confidence: 94%

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

Tran-Thanh

Polukarov

Chapman

et al. 2011

Algorithmic Game Theory

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“…On the positive side, Fotakis et al [16] and Panagopoulou and Spirakis [30] proved the existence of a PNE in games with affine and exponential costs, respectively. Dunkel and Schulz [13] showed that it is strongly NP-hard to decide whether or not a weighted congestion game with nonlinear cost functions possesses a PNE. If the strategy of every player contains a single facility only (singleton games), Fotakis et al [15] showed the existence of PNE for linear cost functions (without a constant).…”

Section: Related Workmentioning

confidence: 99%

On the Existence of Pure Nash Equilibria in Weighted Congestion Games

Harks

Klimm

2010

Automata, Languages and Programming

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We study the existence of pure Nash equilibria in weighted congestion games. Let denote a set of cost functions. We say that is consistent if every weighted congestion game with cost functions in possesses a pure Nash equilibrium. Our main contribution is a complete characterization of consistency of continuous cost functions. We prove that a set of continuous functions is consistent for two-player games if and only if contains only monotonic functions and for all nonconstant functions c 1 c 2 ∈ , there are constants a b ∈ such that c 1 x = a c 2 x + b for all x ∈ ≥0 . For games with at least three players, we prove that is consistent if and only if exactly one of the following cases holds: (a) contains only affine functions; (b) contains only exponential functions such that c x = a c e x + b c for some a c b c ∈ , where a c and b c may depend on c, while must be equal for every c ∈ . The latter characterization is even valid for three-player games. Finally, we derive several characterizations of consistency of cost functions for games with restricted strategy spaces, such as weighted network congestion games or weighted congestion games with singleton strategies.

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“…In contrast to our model, the models above allowed that a player uses a resource with multiple units of demand at the same time. It turns out that allowing for this kind of self-congestion has a severe impact on the existence of pure Nash equilibria [10,24] but for networks of parallel links it is known that pure Nash equilibria are guranteed to exist [16,28].…”

Section: Related Workmentioning

confidence: 99%

Efficiency of Equilibria in Uniform Matroid Congestion Games

Jong

Klimm

Uetz

2016

Algorithmic Game Theory

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Network routing games, and more generally congestion games play a central role in algorithmic game theory, comparable to the role of the traveling salesman problem in combinatorial optimization. It is known that the price of anarchy is independent of the network topology for non-atomic congestion games. In other words, it is independent of the structure of the strategy spaces of the players, and for a ne cost functions it equals 4/3. In this paper, we show that the dependence of the price of anarchy on the network topology is considerably more intricate for atomic congestion games. More specifically, we consider congestion games with a ne cost functions where the strategy spaces of players are symmetric and equal to the set of bases of a k-uniform matroid. In this setting, we show that the price of anarchy is strictly larger than the price of anarchy for singleton strategy spaces where the latter is 4/3. As our main result we show that the price of anarchy can be bounded from above by 28/13 ¥ 2.15. This constitutes a substantial improvement over the price of anarchy bound 5/2, which is known to be tight for arbitrary network routing games with a ne cost functions. ACM Subject Classification IntroductionUnderstanding the impact of selfish behavior on the performance of a system is an important question in algorithmic game theory. One of the cornerstones of the substantial literature on this topic is the famous result of Roughgarden and Tardos [26]. They considered the tra c model of Wardrop [29] in a network with a ne flow-dependent congestion cost functions on the edges. Given a set of commodities, each specified by a source node, a target node, and a flow demand, a Wardrop equilibrium is a multicommodity flow with the property that every commodity uses only paths that minimize the cost. For this setting, Roughgarden and Tardos proved that the total cost of an equilibrium flow is not worse than 4/3 times that of a system optimum. This ratio was coined the price of anarchy by Koutsoupias and Papadimitriou [18] who introduced it as a measure of a system's performance degradation due to selfish behavior. A surprising consequence of the result of Roughgarden and Tardos is that the worst case price of anarchy in congested networks is attained for very simple single-commodity networks already considered a century ago by Pigou [22]. Pigou-style networks consist of only two nodes connected by two parallel links. In fact, Roughgarden [25] proved that for any set of cost functions, the price of anarchy is independent of the network topology as it is always

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

Cited by 37 publications

References 15 publications

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

On the Existence of Pure Nash Equilibria in Weighted Congestion Games

Efficiency of Equilibria in Uniform Matroid Congestion Games

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