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 α through
unilateral strategy changes. There is a threshold αδ​ (where
δ is a parameter that limits the demand of each player on a specific
resource) such that α-approximate pure Nash equilibria always exist for
α≥αδ​, but not for α<αδ​. We give both
upper and lower bounds on this threshold αδ​ and show that the
corresponding decision problem is NP-hard. We also show that the
α-approximate price of anarchy for BAGs is α+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