Collusion in Atomic Splittable Routing Games

Huang, Chien‐Chung

doi:10.1007/s00224-012-9421-4

Cited by 14 publications

(10 citation statements)

References 47 publications

(50 reference statements)

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“…Cominetti et al [12] showed that for games with player-independent affine cost functions an equilibrium can be computed efficiently by solving a quadratic program. For special network topologies (including series-parallel graphs), Huang [32] gave a combinatorial algorithm. For the case of parallel links, Harks and Timmermans [25] gave a polynomial algorithm that computes the equilibrium of an atomic splittable congestion game with player-specific affine costs.…”

Section: Introductionmentioning

confidence: 99%

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: 99%

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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“…Using two distinct networks with richer architecture than the four‐node network originally displayed by Braess, our second purpose is to ascertain if, and under what conditions, the BP may be realized in the laboratory. Our third and final purpose, which is the primary concern of this article, is to test whether grouping independent flow units into multiple cohorts that operate as unitary decision makers may reduce cost of travel (Huang ). The novelty of our design is that it allows for the first time in such studies testing hypotheses about individual, rather than only aggregate, route choices.…”

Section: Introductionmentioning

confidence: 99%

Distributed Decisions in Networks: Laboratory Study of Routing Splittable Flow

Rapoport

Gisches

Mak

2014

Production and Operations Management

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We study network games in which users choose routes in computerized networks susceptible to congestion. In the “unsplittable” condition, route choices are completely unregulated, players are symmetric, each player controls a single unit of flow and chooses a single origin–destination (O–D) path. In the “splittable” condition, which is the main focus of this study, route choices are partly regulated, players are asymmetric, each player controls multiple units of flow and chooses multiple O–D paths to distribute her fleet. In each condition, users choose routes in two types of network: a basic network with three parallel routes and an augmented network with five routes sharing joint links. We construct and subsequently test equilibrium solutions for each combination of condition and network type, and then propose a Markov revision protocol to account for the dynamics of play. In both conditions, route choice behavior approaches equilibrium and the Braess Paradox is clearly manifested.

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“…In the setting of single OD (where all the players share the same origin and the same destination), Huang 2013 [14] provides the following example adapted in our language. In the network shown on the left hand side of Figure 2, the social cost at the unique equilibrium with two atomic players of weight 200 and 21 respectively increases when the atomic player of weight 21 deputes her stock to two atomic deputies of weight 20.9 and 0.1 respectively.…”

Section: Discussion and Perspectivesmentioning

confidence: 99%

“…Beyond the coalitions formed by nonatomic players, Cominetti et al 2009 [9], Altman et al 2011 [1], and Huang 2013 [14] consider those formed by atomic players. Their results can be interpreted as the impact of certain kinds of collusion and hence, the "inverse" of it, decentralization, on the social cost.…”

Section: Introductionmentioning

confidence: 99%

Strategic decentralization in binary choice composite congestion games

Wan¹

2016

European Journal of Operational Research

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This paper studies strategic decentralization in binary choice composite network congestion games. A player decentralizes if she lets some autonomous agents to decide respectively how to send different parts of her stock from the origin to the destination. This paper shows that, with convex, strictly increasing and differentiable arc cost functions, an atomic splittable player always has an optimal unilateral decentralization strategy. Besides, unilateral decentralization gives her the same advantage as being the leader in a Stackelberg congestion game. Finally, unilateral decentralization of an atomic player has a negative impact on the social cost and on the costs of the other players at the equilibrium of the congestion game.Although the above results are obtained in the specific setting of binary choice games, the goal of this paper is to introduce the notion of strategic decentralization into composite congestion games, to point out its significance, and to initiate a systematic study of its properties.The paper is organized as follows. Section 2 presents the model, defines decentralization, and shows the special role of single-atomic decentralization strategies. Section 3 proves the existence of an optimal unilateral decentralization strategy, and shows that unilateral decen-tralization gives an atomic player the same advantage as being the leader in a Stackelberg congestion game. Section 4 focuses on the impact of unilateral decentralization on the social cost and the other players' cost. Section 5 concludes. The proofs and auxiliary results are regrouped in Section 6. Related literatureThe "inverse" concept of decentralization -coalition formation or collusion between playershas been extensively studied. Hayrapetyan et al. 2006 [13] first define the price of collusion (PoC) of a parallel network to be the ratio between the worst equilibrium social cost after the nonatomic players form disjoint coalitions and the worst equilibrium social cost without coalitions. Bhaskar et al. 2010 [4] extended this study to series-parallel networks. (A seriesparallel network can be constructed by merging in series or in parallel several graphs of parallel arcs.) This index is closely related to another important notion: the price of anarchy (PoA), which is introduced by Koutsoupias and Papadimitriou 1999 [17] (and [22]) as the ratio between the worst equilibrium social cost and the minimal social cost in nonatomic games. Cominetti et al. 2009 [9] derives the first bounds on the PoA with atomic players. For a specific network structure, one can deduce the PoC by the PoA with atomic players and the PoA with nonatomic players. Further results on the bound of the PoA with atomic players are obtained in Harks 2011 [12], Roughgarden and Schoppmann 2011 [28] and Bhaskar et al. 2010 [4]. Roughgarden and Tardos 2002 [29] and Correa et al. 2008 [10] provide fundamental results on the bound of the PoA with nonatomic players. PoA in nonatomic games with asymmetric costs or elastic demands are studied in [23] and [8], among others.

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Collusion in Atomic Splittable Routing Games

Cited by 14 publications

References 47 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

Distributed Decisions in Networks: Laboratory Study of Routing Splittable Flow

Strategic decentralization in binary choice composite congestion games

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