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

Dunkel, Juliane; Schulz, Andreas S.

doi:10.1287/moor.1080.0322

Cited by 42 publications

(17 citation statements)

References 30 publications

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“…Further related are unsplittable congestion games [48] where each commodity chooses a single path of the network. For results on the existence of equilibria, see [2,17,20,21,24,42,48]. Computing a pure Nash equilibrium in a congestion game with unweighted players and playerindependent cost functions reduces to find the local minimum of a potential function and is, thus, contained in the complexity class PLS, the class of all local search problems with polynomially searchable neighborhoods as defined by [33].…”

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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“…Allowing the players to have different loads, gives rise to the class of weighted congestion games [47]; this is a natural and very important generalization of congestion games, with numerous applications in routing and scheduling. Unfortunately though, an immediate dichotomy between weighted and unweighted congestion games occurs: the former may not even have pure Nash equilibria [41,28,30,33]; as a matter of fact, it is a strongly NP-hard problem to even determine if that is the case [23]. Moreover, in such games there does not, in general, exist a potential function [43,34], which is the main tool for proving equilibrium existence in the unweighted case.…”

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confidence: 99%

The Price of Stability of Weighted Congestion Games

Christodoulou¹,

Gairing²,

Giannakopoulos³

et al. 2019

SIAM J. Comput.

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We give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer d we construct rather simple games with cost functions of degree at most d which have a PoS of at least Ω(Φ d ) d+1 , where Φ d ∼ d/ ln d is the unique positive root of equation x d+1 = (x + 1) d . This almost closes the huge gap between Θ(d) and Φ d+1 d . Our bound extends also to network congestion games. We further show that the PoS remains exponential even for singleton games. More generally, we provide a lower bound of Ω((1 + 1/α) d /d) on the PoS of α-approximate Nash equilibria for singleton games. All our lower bounds hold for mixed and correlated equilibria as well.On the positive side, we give a general upper bound on the PoS of α-approximate Nash equilibria, which is sensitive to the range W of the player weights and the approximation parameter α. We do this by explicitly constructing a novel approximate potential function, based on Faulhaber's formula, that generalizes Rosenthal's potential in a continuous, analytic way. From the general theorem, we deduce two interesting corollaries. First, we derive the existence of an approximate pure Nash equilibrium with PoS at most (d + 3)/2; the equilibrium's approximation parameter ranges from Θ(1) to d + 1 in a smooth way with respect to W . Secondly, we show that for unweighted congestion games, the PoS of α-approximate Nash equilibria is at most (d + 1)/α.

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“…Rosenthal gives an example that shows that pure Nash equilibria need not exist for integer-splittable congestion games in general. Dunkel and Schulz [8] strengthened this result showing that the existence of a pure Nash equilibrium in integer-splittable congestion games is NP-complete to decide. Meyers [28] proved that in games with linear cost functions, a pure Nash equilibrium is always guaranteed to exist.…”

Section: Related Workmentioning

confidence: 87%

Sensitivity Analysis for Convex Separable Optimization Over Integral Polymatroids

Harks¹,

Klimm²,

Peis³

2018

SIAM J. Optim.

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We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution there is a new optimal solution to new parameters such that the L 1 -norm of the difference of the two solutions is at most two times the L 1 -norm of the difference of the parameters.We apply these sensitivity results to a class of non-cooperative games with a finite set of players where a strategy of a player is to choose a vector in a player-specific integral polymatroid base polytope defined on a common set of elements. The players' private cost functions are regular, convex-separable and the cost of each element is a non-decreasing function of the own usage of that element and the overall usage of the other players. Under these assumptions, we establish the existence of a pure Nash equilibrium. The existence is proven by an algorithm computing a pure Nash equilibrium that runs in polynomial time whenever the rank of the polymatroid base-polytope is polynomially bounded. Both the existence result and the algorithm generalize and unify previous results appearing in the literature.We finally complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any non-polymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players' strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium.

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

Cited by 42 publications

References 30 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

The Price of Stability of Weighted Congestion Games

Sensitivity Analysis for Convex Separable Optimization Over Integral Polymatroids

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