We present a deterministic polynomial-time algorithm for computing @math -approximate (pure) Nash equilibria in weighted congestion games with polynomial cost functions of degree at most @math . This is an exponential improvement of the approximation factor with respect to the previously best algorithm. An appealing additional feature of our algorithm is that it uses only best-improvement steps in the actual game, as opposed to earlier approaches that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by [FOCS'11, TEAC 2015], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of @math for the Price of Anarchy (PoA) of @math -approximate equilibria in weighted congestion games, where @math is the Lambert-W function. More specifically, we show that this PoA is exactly equal to @math , where @math is the unique positive solution of the equation @math . Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, while our lower bound is simple enough to apply even to singleton congestion games.
The next significant positive result of this kind was obtained for approximate equilibria in polynomial unweighted congestion games by . They showed how to efficiently compute @math -approximate equilibria in such games. Subsequently, extended this result to the weighted case, achieving an approximation factor of @math . To the best of our knowledge, this has remained state-of-the-art ever since. As far as deterministic solutions are concerned. See below for a discussion of the recent randomized algorithm by . In the present paper, we reduce this factor to @math . The algorithm in @cite_1 first transforms the original game into a so-called defined for the same players and states. Specifically, it runs on the so-called @math -game, which is an exact potential game where players' costs are within a factor of @math from their costs in the original polynomial game. Then it finds and returns a state which is a @math -approximate equilibrium in the new @math -game. In the original game, the approximation factor of this state (which is the approximation guarantee of the algorithm) is worse by the mentioned factor of @math and thus becomes @math .
@cite_1
2108444537 2089015050 1517645949 2051206974
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. 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 study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the well- known multiplicative-weights learning algorithm results in convergence to pure equilibria. Our results show that nat- ural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the load-balancing game introduced by Koutsoupias and Pa- padimitriou, which has super-constant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined game-theoretically, but we show that this property holds whenever a stability criterion from the theory of dy- namical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribu- tion). We further extend our results to show that players can use algorithms with different (sufficiently small) learn- ing rates, i.e. they can trade off convergence speed and long term average regret differently. 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.