Bottleneck Congestion Games with Logarithmic Price of Anarchy

We study

bottleneck congestion games

where the social cost is determined by the worst congestion on any resource. In the literature, bottleneck games assume player utility costs determined by the worst congested resource in their strategy. However, the Nash equilibria of such games are inefficient since the price of anarchy can be very high and proportional to the number of resources. In order to obtain smaller price of anarchy we introduce

exponential bottleneck games

, where the utility costs of the players are exponential functions of their congestions. In particular, the delay function for any resource

r

is

$\mathcal{M}^{C_r}$

, where

C

r

denotes the number of players that use

r

, and

$\mathcal{M}$

is an integer constant. We find that exponential bottleneck games are very efficient and give the following bound on the price of anarchy:

O

(log|

R

|), where

R

is the set of resources. This price of anarchy is tight, since we demonstrate a game with price of anarchy Ω(log|

R

|). We obtain our tight bounds by using two novel proof techniques:

transformation

, which we use to convert arbitrary games to simpler games, and

expansion

, which we use to bound the price of anarchy in a simpler game.