Alexander Skopalik

Fast convergence to nearly optimal solutions in potential games

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.

Inapproximability of pure nash equilibria

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 players 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.

Efficient Computation of Approximate Pure Nash Equilibria in Congestion Games

Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {\sf PLS}-complete. We present a surprisingly simple polynomial-time algorithm that computes

On the Complexity of Pareto-Optimal Nash and Strong Equilibria

O(1)

On the complexity of pure Nash equilibria in player-specific network congestion games

-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes

Computing pure Nash and strong equilibria in bottleneck congestion games

(2+\epsilon)

Taxing Subnetworks

-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and

Social context in potential games

1/\epsilon

On the complexity of nash dynamics and sink equilibria

. It also applies to games with polynomial latency functions with constant maximum degree

Stability and Convergence in Selfish Scheduling with Altruistic Agents

d