Equilibrium Computation in Atomic Splittable Singleton Congestion Games

Harks, Tobias; Timmermans, Veerle

doi:10.1007/978-3-319-59250-3_36

Cited by 6 publications

(14 citation statements)

References 28 publications

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“…The problem of computing a Nash equilibrium in an atomic splittable congestion game with player-specific affine costs may also be formulated as a linear complementarity problem (LCP). Harks and Timmermans [25] show this explicitly for the singleton case, but it is not hard to convince ourselves that such a formulation is also possible in the general case by using the vertex potentials. However, it is not clear whether the resulting LCP belongs to any of the classes for which it is known that Lemke's algorithm terminates [3,14,18].…”

Section: Introductionmentioning

confidence: 98%

Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians

Klimm¹,

Warode²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We settle the complexity of computing an equilibrium in atomic splittable congestion games with player-specific affine cost functions l e,i (x) = a e,i x + b e,i as we show that the computation is PPAD-complete. To prove that the problem is contained in PPAD, we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique for this method is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix where each entry of the Laplacian is a Laplacian again. Using the properties of this matrix allows to recompute efficiently the Laplacian after the support of the equilibrium changes by matrix pivot operations. These insights give rise to a path following formulation for computing an equilibrium where states correspond to supports that are feasible for some demands and neighboring supports are feasible for increased or decreased flow demands. A closer investigation of the block Laplacian system further allows to orient the states giving rise to unique predecessor and successor states thus putting the problem into PPAD. For the PPAD-hardness, we reduce from computing an approximate equilibrium of a bimatrix win-lose game. As a byproduct of our reduction we further show that computing a multi-class Wardrop equilibrium with class dependent affine cost functions is PPAD-complete as well.On our way, we obtain several new results regarding the multiplicity of equilibria in atomicsplittable games with player-specific cost functions. When the coefficients a e,i are in general position, every game has a finite set of equilibria while without this assumption there may be a continuum of equilibria. When the additive constants b e,i are in general position, games have an odd number of equilibria except for a nullset of demand values.As another byproduct of our PPAD-completeness proof, we obtain an algorithm that computes a continuum of equilibria parametrized by the players' flow demand. For player-specific costs, the continuum may involve several increases and decreases of the demand and yields an algorithm that runs in polynomial space. For games with player-independent costs, only demand increases are necessary yielding an algorithm computing all equilibria as a function of the flow demand that runs in time polynomial in the output.

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Section: Introductionmentioning

confidence: 98%

Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians

Klimm¹,

Warode²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…For studies on the price of anarchy, see Roughgarden and Tardos (2002), Christodoulou and Koutsoupias (2005), and Awerbuch et al (2013). For studies on the computation of a Nash equilibrium, see Caragiannis et al (2011), Chien andSinclair (2011), andHarks andTimmermans (2017). For other studies on congestion games and its special instances, see Feldman and Tennenholtz (2010), Anshelevich et al (2013), and Caskurlu et al (2020a, 2020b, 2021.…”

Section: Related Literaturementioning

confidence: 99%

Location Choice under Spillovers

Ekici¹,

Caskurlu²

2022

berj

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This paper introduces the location choice under spillovers game: A number of firms choose from among a number of alternative locations. A firm's payoff at some location is the sum of two factors: Its location-specific idiosyncratic payoff; and the positive spillover it receives, which is a function of the number of firms choosing the same location. The spillover function is location-specific and monotonically increasing. This game form can be viewed as an extension of the classic "battle of sexes" game. It can also be used to model real-life game-theoretic situations with network effects, such as when app users choose from alternative social media or instant messaging apps. In our main result, we show that the location choice under spillovers game is a potential game, and hence, it always admits a Nash equilibrium in pure strategies. We also show that: A Nash equilibrium outcome need not be Pareto efficient. An outcome that is Pareto efficient need not be a Nash equilibrium. And a Nash equilibrium is not necessarily a strong equilibrium.

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“…Hence, a strategy profile is represented by Since the cost experienced by each MVNO depends on the RRHs they select, the congestion game G has to be weighted and to have player-specific cost functions. Furthermore, a slicing policy is derived by allocating different number of MUs to multiple RRHs in the cluster, implying that G has to be a CG with splittable flows [26]. To summarize, the slicing problem can be modeled as an atomic weighted CG with splittable flows on R parallel links.…”

Section: A Congestion Game Problem Formulationmentioning

confidence: 99%

“…In the previous sections, we have emphasized the importance of deriving theoretical bounds on the efficiency of the NE, and designing effective distributed and convergent algorithms. Due to their interesting properties with respect to the above issues, congestion games [16], and their application to many networking problems such as the network service chaining [15] and load balancing [44], have been investigated in the literature [14,26,28,[45][46][47].…”

Section: Related Workmentioning

confidence: 99%

“…However, those approaches generally require to solve QP problems, and are guaranteed to converge to the unique NE only after an infinite number of iterations. Recently, a quantization approach which can compute the NE in a finite amount of time has been proposed in [26]. However, it suffers from highly polynomial complexity, which makes it hard to implement it in dynamic scenarios.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Low-Complexity Distributed Radio Access Network Slicing: Algorithms and Experimental Results

D’Oro¹,

Restuccia²,

Melodia³

et al. 2018

IEEE/ACM Trans. Networking

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Radio access network (RAN) slicing is an effective methodology to dynamically allocate networking resources in 5G networks. One of the main challenges of RAN slicing is that it is provably an NP-Hard problem. For this reason, we design near-optimal low-complexity distributed RAN slicing algorithms. First, we model the slicing problem as a congestion game, and demonstrate that such game admits a unique Nash equilibrium (NE). Then, we evaluate the Price of Anarchy (PoA) of the NE, i.e., the efficiency of the NE as compared to the social optimum, and demonstrate that the PoA is upper-bounded by 3/2. Next, we propose two fully-distributed algorithms that provably converge to the unique NE without revealing privacy-sensitive parameters from the slice tenants. Moreover, we introduce an adaptive pricing mechanism of the wireless resources to improve the network owner's profit. We evaluate the performance of our algorithms through simulations and an experimental testbed deployed on the Amazon EC2 cloud, both based on a real-world dataset of base stations from the OpenCellID project. Results conclude that our algorithms converge to the NE rapidly and achieve near-optimal performance, while our pricing mechanism effectively improves the profit of the network owner.

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Equilibrium Computation in Atomic Splittable Singleton Congestion Games

Cited by 6 publications

References 28 publications

Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians

Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians

Location Choice under Spillovers

Low-Complexity Distributed Radio Access Network Slicing: Algorithms and Experimental Results

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