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 [23]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with delay functions that satisfy a "bounded jump" condition, 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 as follows.

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 α-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 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 boundedjump condition is restricted to symmetric games only.

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 O(1)-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes (2 + ǫ)-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and 1/ǫ. It also applies to games with polynomial latency functions with constant maximum degree d; there, the approximation guarantee is d O(d) . 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 ρ-approximate equilibria is PLS-complete for any polynomial-time computable ρ.

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This article studies the effects of altruism, a phenomenon widely observed in practice, in the model of atomic congestion games. Altruistic behavior is modeled by a linear trade-off between selfish and social objectives. Our model can be embedded in the framework of congestion games with player-specific latency functions. Stable states are the pure Nash equilibria of these games, and we examine their existence and the convergence of sequential best-response dynamics. In general, pure Nash equilibria are often absent, and existence is NP-hard to decide. Perhaps surprisingly, if all delay functions are affine, the games remain potential games, even when agents are arbitrarily altruistic. The construction underlying this result can be extended to a class of general potential games and social cost functions, and we study a number of prominent examples. These results give important insights into the robustness of multi-agent systems with heterogeneous altruistic incentives. Furthermore, they yield a general technique to prove that stabilization is robust, even with partly altruistic agents, which is of independent interest.In addition to these results for uncoordinated dynamics, we consider a scenario with a central altruistic institution that can set incentives for the agents. We provide constructive and hardness results for finding the minimum number of altruists to stabilize an optimal congestion profile and more general mechanisms to incentivize agents to adopt favorable behavior. These results are closely related to Stackelberg routing and answer open questions raised recently in the literature.