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

Abstract

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.

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

“…In other cases, all known formulations lead to nonstandard LCPs, which cannot be handled by the "vanilla" version of Lemke's algorithm, and require variants of the algorithm to be devised, e.g., see [Garg et al, 2018;Meunier and Pradeau, 2019]. Finally in some cases, it is not known if the derived LCPs can be solved via any variant of Lemke's algorithm, thus leading to the development of entirely new pivoting algorithms [Klimm and Warode, 2020]. These characteristics of the LCP approach make it somewhat insufficient as a general purpose PPAD-membership technique.…”

“…Indeed, coming up with proofs of existence that also guarantee rationality of solutions has been a topic of interest in the area since the very early days, way before the introduction of the relevant computational complexity classes, e.g., see [Eaves, 1976;Lemke and Howson, 1964;Lemke, 1965;Howson, 1972]. Driven by those classic results, a significant literature in computer science has attempted, and quite often has succeeded in placing the corresponding computational problems in PPAD, for several of the application domains mentioned above, including games [Sørensen, 2012;Hansen and Lund, 2018;Kintali et al, 2013;Klimm and Warode, 2020;Meunier and Pradeau, 2019], markets [Vazirani and Yannakakis, 2011;Garg and Vazirani, 2014;Garg, 2017;Garg et al, 2018], as well as the more recent domain of auto-bidding auctions [Chen et al, 2021a].…”

Efficient Estimation of Equilibria in Large Aggregative Games With Coupling Constraints

“…b) Consumers' constraints and parameters: We consider a set of N = 2000 consumers who have demand constraints of the form: (11) where En is the total energy needed by n, and x n,t , x n,t (physical) bounds on the power allowed at time t. The utility functions have the form un(xn) def = −ωn xn − yn 2 , with yn a preferred charging profile. Simulation parameters are chosen as follows:…”

“…The notion of Nash Equilibrium (NE) [9] has emerged as the central solution concept in game theory. However, the computation of an NE is considered a challenging problem: indeed, recent works have proved the theoretical complexity of the problem (PPAD-completeness [10], [11]). In continuous games, NEs can be characterized as solutions of Variational Inequalities (VI) [12], a characterization we adopt throughout this work.…”

Atomic Splittable Flow Over Time Games

“…Regarding player-specific affine costs, Harks and Timmermans [21] described a polynomial algorithm for parallel-edge networks, and Bhaskar and Lolakapuri [5] presented an exponential algorithm for general convex functions. Very recently, Klimm and Warode showed that the computation with player-specific affine costs is PPAD-complete for general networks [25].…”