On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games

TL;DR

This paper investigates the existence, properties, and computational complexity of approximate pure Nash equilibria in bandwidth allocation games, providing bounds, hardness results, and efficient algorithms for special cases.

Contribution

It establishes bounds on the threshold for approximate equilibria, proves NP-hardness of the decision problem, and presents polynomial-time convergence results for certain game variants.

Findings

Approximate pure Nash equilibria exist for certain thresholds

Deciding existence of such equilibria is NP-hard

Convergence to near-optimal social welfare states is fast in broader utility games

Abstract

In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $\alpha$ through unilateral strategy changes. There is a threshold $\alpha_\delta$ (where $\delta$ is a parameter that limits the demand of each player on a specific resource) such that $\alpha$-approximate pure Nash equilibria always exist for $\alpha \geq \alpha_\delta$, but not for $\alpha < \alpha_\delta$. We give both upper and lower bounds on this threshold $\alpha_\delta$ and show that the corresponding decision problem is ${\sf NP}$-hard. We also show that the $\alpha$-approximate price of anarchy for BAGs is $\alpha+1$. For a restricted version of the game, where demands of players only differ slightly from each other (e.g. symmetric games), we show that approximate Nash equilibria can be reached (and thus also be computed) in polynomial time using the best-response dynamic. Finally, we show that a broader class of utility-maximization games (which includes BAGs) converges quickly towards states whose social welfare is close to the optimum.