text: We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either games and, thus, leads -- via our first algorithm -- to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd [2016].
  @cite_33: Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general PLS -complete. We present a surprisingly simple polynomial-time algorithm that computes @math -approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes @math -approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and @math . It also applies to games with polynomial latency functions with constant maximum degree @math ; there, the approximation guarantee is @math . The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium, the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing @math -approximate equilibria is PLS -complete for any polynomial-time computable @math .
  @cite_7: 
  @cite_22: We study the ability of decentralized, local dynamics in non-cooperative games to rapidly reach an approximate (pure) Nash equilibrium. Our main result states that for symmetric congestion games in which the cost function associated with each resource satisfies a "bounded jump" condition, convergence to an [epsilon]-Nash equilibrium occurs within a number of steps that is polynomial in the number of players and [epsilon]-1. We show moreover that this result holds under a variety of conventions governing the move orders among the players, including the minimal liveness assumption that no player is indefinitely blocked from moving. We also prove that in the generalized setting where players have different "tolerances" [epsilon]i, the number of moves a player makes before equilibrium is reached depends only on his associated [epsilon]i, and not on those of other players. Finally, we show that polynomial time convergence holds even when a constant number of resources have arbitrary cost functions.
  @cite_29: 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 L1-norm of the difference of the two solutions is at most two times the L1 norm of the difference of the parameters. We apply these sensitivity results to a class of non-cooperative polymatroid games and derive the existence of pure Nash equilibria. We 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.
  @cite_32: The complexity of computing pure Nash equilibria in congestion games was recently shown to be PLS-complete. In this paper, we therefore study the complexity of computing approximate equilibria in congestion games. An alpha-approximate equilibrium, for α > 1, is a state of the game in which none of the players can make an α-greedy step, i.e., an unilateral strategy change that decreases the player's cost by a factor of at least α. Our main result shows that finding an α-approximate equilibrium of a given congestion game is sc PLS-complete, for any polynomial-time computable α > 1. Our analysis is based on a gap introducing PLS-reduction from FLIP, i.e., the problem of finding a local optimum of a function encoded by an arbitrary circuit. As this reduction is tight it additionally implies that computing an α-approximate equilibrium reachable from a given initial state by a sequence of α-greedy steps is PSPACE-complete. Our results are in sharp contrast to a recent result showing that every local search problem in PLS admits a fully polynomial time approximation scheme. In addition, we show that there exist congestion games with states such that any sequence of α-greedy steps leading from one of these states to an α-approximate Nash equilibrium has exponential length, even if the delay functions satisfy a bounded-jump condition. This result shows that a recent result about polynomial time convergence for α-greedy steps in congestion games satisfying the bounded-jump condition is restricted to symmetric games only.
  @cite_0: We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time. We show that if the strategy space of each player consists of the bases of a matroid over the set of resources, then the lengths of all best response sequences are polynomially bounded in the number of players and resources. We also prove that this result is tight, that is, the matroid property is a necessary and sufficient condition on the players' strategy spaces for guaranteeing polynomial-time convergence to a Nash equilibrium. In addition, we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games, including first results about the hardness of market sharing games and congestion games for overlay network design. Our approach also yields a short proof for the PLS-completeness of network congestion games. In particular, we show that network congestion games are PLS-complete for directed and undirected networks even in case of linear latency functions.
  @cite_23: We investigate a new class of congestion games, called Totally Unimodular Congestion Games, in which the strategies of each player are expressed as binary vectors lying in a polyhedron defined using a totally unimodular constraint matrix and an integer right-hand side. We study both the symmetric and the asymmetric variants of the game. In the symmetric variant, all players have the same strategy set, i.e. the same constraint matrix and right-hand side. Network congestion games are an example of this class. proved that a pure Nash equilibrium of symmetric network congestion games can be found in strongly polynomial time, while the asymmetric network congestion games are PLS-complete. We give a strongly polynomial-time algorithm to find a pure Nash equilibrium of any symmetric totally unimodular congestion game. We also identify four totally unimodular congestion games, where the players' strategy sets are matchings, vertex covers, edge covers and stable sets of a given bipartite graph. For these games we derive specialized combinatorial algorithms to find a pure Nash equilibrium in the symmetric variant, and show the asymmetric variant is PLS-complete.
  @cite_46: We investigate from the computational viewpoint multi-player games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLS-complete in general. We discuss implications to non-atomic congestion games, and we explore the scope of the potential function method for proving existence of pure Nash equilibria.
  @cite_16: We study competitive resource allocation problems in which players distribute their demands integrally over a set of resources subject to player-specific submodular capacity constraints. Each player has to pay for each unit of demand a cost that is a non-decreasing and convex function of the total allocation of that resource. This general model of resource allocation generalizes both singleton congestion games with integer-splittable demands and matroid congestion games with player-specific costs. As our main result, we show that in such general resource allocation problems a pure Nash equilibrium is guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure Nash equilibrium.
  @cite_25: 
summary: Discrete Congestion Games. As the first seminal work regarding the computational complexity of equilibrium computation in congestions games, @cite_46 showed that the problem of computing a pure Nash equilibrium is PLS-complete for network congestion games. @cite_0 strengthened this result to hold even for network congestion games with linear cost functions. On the other hand, there are polynomial algorithms for symmetric network congestion games (cf. @cite_46 ), for matroid congestion games with player-specific cost functions ( @cite_0 @cite_25 ), for polymatroid congestion games with player-specific cost functions and polynomially bounded demands ( @cite_16 @cite_29 ) and for so-called total unimodular congestion games (see Del @cite_23 ). Further results regarding the computation of approximate equilibria in congestion games can be found in @cite_33 @cite_7 , Chien and Sinclair @cite_22 and Skopalik and V "ocking @cite_32 .