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].
For the computation of equilibria, Marcotte @cite_30 proposed four numerical algorithms and showed local convergence results. Meunier and Pradeau @cite_10 developed a pivoting-algorithm (similar to Lemkes algorithm) for nonatomic network congestion games with affine player-specific cost functions. Polynomial running time was, however, not shown and seems unlikely to hold. @cite_19 considered nonatomic routing games on parallel links with affine player-specific cost functions. They developed a convex potential function that can be minimized within arbitrary precision in polynomial time. @cite_24 considered general concave games with compact action spaces and investigated algorithms computing an approximate equilibrium. Roughly speaking, they discretized the compact strategy space and use the Lipschitz constants of utility functions to show that only a finite number of representative strategy profiles need to be considered for obtaining an approximate equilibrium (see also @cite_21 for a similar approach). The running time of the algorithm, however, depends on an upper bound of the norm of strategy vectors, thus, implying only a pseudo-polynomial algorithm for our setting.
@cite_30 @cite_21 @cite_24 @cite_19 @cite_10
2030722474 2131967794 2949253038 622918438
We consider the problem of computing a Nash equilibrium in multiple-player games. It is known that there exist games, in which all the equilibria have irrational entries in their probability distributions [19]. This suggests that either we should look for symbolic representations of equilibria or we should focus on computing approximate equilibria. We show that every finite game has an equilibrium such that all the entries in the probability distributions are algebraic numbers and hence can be finitely represented. We also propose an algorithm which computes an approximate equilibrium in the following sense: the strategies output by the algorithm are close with respect to l ∞ -norm to those of an exact Nash equilibrium and also the players have only a negligible incentive to deviate to another strategy. The running time of the algorithm is exponential in the number of strategies and polynomial in the digits of accuracy. We obtain similar results for approximating market equilibria in the neoclassical exchange model under certain assumptions. We consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origin-destination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and propose an extension of Lemke's algorithm able to solve this problem. At the same time, it provides a constructive proof of the existence of an equilibrium in this case. 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. 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. We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s^2 L), where s is the number of strategies and L is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an epsilon We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ--approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5 2 + Oγ for arbitrarily small γ > 0; for latency functions with maximum degree d≥ 2, it is d2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1 γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ-games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications. 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. There has been substantial work developing simple, efficient no-regret algorithms for a wide class of repeated decision-making problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversarially-changing environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a no-regret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games have substantially more structure.In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimal agents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that depends polynomially on the players' regret bounds and the maximum slope of any latency function. We also show that price-of-anarchy results may be applied to these approximate equilibria, and also consider the finite-size (non-infinitesimal) load-balancing model of Azar [2]. We prove the existence of e-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be e-approximated by the payoffs to the players in some such logarithmic support e-Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an e-Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an e-Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payoffs to all players in any m-tuple of mixed strategies can be e-approximated by the payoffs in some m-tuple of constant support strategies.We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an e-equilibrium with small support. It also implies that if the payoff matrices have constant rank we can compute an exact Nash equilibrium in polynomial time. We study the ability of decentralized, local dynamics in non-cooperative games to rapidly reach an approximate Nash equilibrium. For symmetric congestion games in which the edge delays satisfy a "bounded jump" condition, we show that convergence to an e-Nash equilibrium occurs within a number of steps that is polynomial in the number of players and e-1. This appears to be the first such result for a class of games that includes examples for which finding an exact Nash equilibrium is PLS-complete, and in which shortest paths to an exact equilibrium are exponentially long. We show moreover that rapid convergence holds even under only the apparently minimal assumption that no player is excluded from moving for arbitrarily many steps. We also prove that, in a generalized setting where players have different "tolerances" e i that specify their thresholds in the approximate Nash equilibrium, the number of moves made by a player before equilibrium is reached depends only on his associated e i , and not on those of the other players. Finally, we show that polynomial time convergence still holds even when a bounded number of edges are allowed to have arbitrary delay functions. Imitating successful behavior is a natural and frequently applied approach when facing scenarios for which we have little or no experience upon which we can base our decision. In this paper, we consider such behavior in atomic congestion games. We propose to study concurrent imitation dynamics that emerge when each player samples another player and possibly imitates this agents' strategy if the anticipated latency gain is sufficiently large. Our main focus is on convergence properties. Using a potential function argument, we show that these dynamics converge in a monotonic fashion to stable states. In such a state none of the players can improve their latency by imitating others. As our main result, we show rapid convergence to approximate equilibria. At an approximate equilibrium only a small fraction of agents sustains a latency significantly above or below average. In particular, imitation dynamics behave like fully polynomial time approximation schemes (FPTAS). Fixing all other parameters, the convergence time depends only in a logarithmic fashion on the number of agents. Since imitation processes are not innovative they cannot discover unused strategies. Furthermore, strategies may become extinct with non-zero probability. For the case of singleton games, we show that the probability of this event occurring is negligible. Additionally, we prove that the social cost of a stable state reached by our dynamics is not much worse than an optimal state in singleton congestion games with linear latency functions. While we concentrate on the case of symmetric network congestion games, most of our results do not explicitly use the network structure. They continue to hold accordingly for general symmetric and asymmetric congestion games when each player samples within his commodity. In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in @math , @math , and @math biased games, and then use these algorithms to provide strongly polynomial algorithms that find @math , @math , and @math approximations for these norms, respectively. 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 . We consider markets in the classical Arrow-Debreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he she buys) and an initial bundle. At an ldquoequilibriumrdquo set of prices for goods, if each individual buyer separately ex-changes the initial bundle for an optimal bundle at the set prices, the market clears, i.e., all goods are exactly consumed. Classical theorems guarantee the existence of equilibria, but computing them has been the subject of much recent research. In the related area of Multi-Agent Games,much attention has been paid to the complexity as well as algorithms. While most general problems are hard, polynomial time algorithms have been developed for restricted classes of games, when one assumes the number of strategies is constant.For the Market Equilibrium problem, several important special cases of utility functions have been tackled. Here we begin a program for this problem similar to that for multi-agent games, where general utilities are considered. We begin by showing that if the utilities are separable piece-wise linear concave (PLC) functions, and the number of goods(or alternatively the number of buyers) is constant, then we can compute an exact equilibrium in polynomial time.Our technique for the constant number of goods is to de-compose the space of price vectors into cells using certain hyperplanes, so that in each cell, each buyerpsilas threshold marginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the main result - that for general (non-separable) PLC utilities, an exact equilibrium can be found in polynomial time provided the number of goods is constant. The starting point of the algorithm is a ldquocell-decompositionrdquo of the space of price vectors using polynomial surfaces (instead of hyperplanes).We use results from computational algebraic geometry to bound the number of such cells. For solving the problem inside each cell, we introduce and use a novel LP-duality based method. We note that if the number of buyers and agents both can vary, the problem is PPAD hard even for the very special case of PLC utilities - namely Leontief utilities. We study the speed of convergence of decentralized dynamics to approximately optimal solutions in potential games. We consider α-Nash dynamics in which a player makes a move if the improvement in his payoff is more than an α factor of his own payoff. Despite the known polynomial convergence of α-Nash dynamics to approximate Nash equilibria in symmetric congestion games [7], it has been shown that the convergence time to approximate Nash equilibria in asymmetric congestion games is exponential [25]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with linear and polynomial delay functions, the convergence time of α-Nash dynamics to an approximate optimal solution is polynomial in the number of players, with approximation ratio that is arbitrarily close to the price of anarchy of the game. In particular, we show this polynomial convergence under the minimal liveness assumption that each player gets at least one chance to move in every T steps. We also prove that the same polynomial convergence result does not hold for (exact) best-response dynamics, showing the α-Nash dynamics is required. We extend these results for congestion games to other potential games including weighted congestion games with linear delay functions, cut games (also called party affiliation games) and market sharing games. We show that in any n -player, m -action normal-form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three types of equilibria: Nash, correlated, and coarse-correlated. For each one we obtain upper and lower bounds on the number of samples required for the empirical distribution over the sampled action profiles to form an approximate equilibrium with probability close to 1. These bounds imply that using a small number of samples we can test whether or not players are playing according to an approximate equilibrium, even in games where n and m are large. In addition, our results substantially improve previously known upper bounds on the support size of approximate equilibria in games with many players. In particular, for the three types of equilibria we show the existence of approximate equilibrium with support-size polylogarithmic in n and m , whereas the previously best-known upper bounds were polynomial in n ( [Hemon S, de Rougemont M, Santha M (2008) Approximate nash equilibria for multi-player games. Algorithmic Game Theory (Springer), 267–278], Germano and Lugosi [Germano F, Lugosi G (2007) Existence of sparsely supported correlated equilibria. Econom. Theory 32(3):575–578], [Hart S, Mas-Colell A, Babichenko Y (2013) Simple Adaptive Strategies: From Regret-Matching to Uncoupled Dynamics , Vol. 4 (World Scientific Publishing Company)]). A recent body of experimental literature has studied empirical game-theoretical analysis, in which we have partial knowledge of a game, consisting of observations of a subset of the pure-strategy profiles and their associated payoffs to players. The aim is to find an exact or approximate Nash equilibrium of the game, based on these observations. It is usually assumed that the strategy profiles may be chosen in an on-line manner by the algorithm. We study a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games. We give basic results for exact equilibria of bimatrix and graphical games. We then study the query complexity of approximate equilibria in bimatrix games. Finally, we study the query complexity of exact equilibria in symmetric network congestion games. For directed acyclic networks, we can learn the cost functions (and hence compute an equilibrium) while querying just a small fraction of pure-strategy profiles. For the special case of parallel links, we have the stronger result that an equilibrium can be identified while only learning a small fraction of the cost values. Abstract We study the computation of Nash equilibria of anonymous games, via algorithms that use adaptive queries to a game's payoff function. We show that exact equilibria cannot be found via query-efficient algorithms, and exhibit a two-strategy, 3-player anonymous game whose exact equilibria require irrational numbers. We obtain positive results for known sub-classes of anonymous games. Our main result is a new randomized query-efficient algorithm for approximate equilibria of two-strategy anonymous games that improves on the running time of previous algorithms. It is the first to obtain an inverse polynomial approximation in poly-time, and yields an efficient polynomial-time approximation scheme. We define a new class of games, congestion games with load-dependent failures (CGLFs). In a CGLF each player can choose a subset of a set of available resources in order to try and perform his task. We assume that the resources are identical but that players' benefits from successful completion of their tasks may differ. Each resource is associated with a cost of use and a failure probability which are load-dependent. Although CGLFs in general do not have a pure strategy Nash equilibrium, we prove the existence of a pure strategy Nash equilibrium in every CGLF with nondecreasing cost functions. Moreover, we present a polynomial time algorithm for computing such an equilibrium. 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.