Optimal Price of Anarchy of Polynomial and Super-Polynomial Bottleneck Congestion Games

Abstract

We introduce (super) polynomial bottleneck games, where the utility costs of the players are (super) polynomial functions of the congestion of the resources that they use, and the social cost is determined by the worst congestion of any resource. In particular, the delay function for any resource r is of the form $C_{r}^{{\mathcal M}_r}$, where C _r is the congestion measured as the number of players that use r, and the degree of the delay function is bounded as $1 \leq{\mathcal M}_r \leq \log C_r$. The utility cost of a player is the sum of the individual delays of the resources that it uses. The social cost of the game is the worst bottleneck resource congestion: max _r ∈ R C _r , where R is the set of resources. We show that for super-polynomial bottleneck games with ${\mathcal M}_r = \log C_r$, the price of anarchy is $o(\sqrt{|R|})$, specifically $O(2^{\sqrt{\log |R|}})$. We also consider general polynomial bottleneck games where each resource can have a distinct monomial latency function but the degree is bounded i.e ${\mathcal M}_r = O(1)$ with constants $\alpha \leq{\mathcal M}_r \leq \beta$ and derive the price of anarchy as $\min \left( |R|, \max \left( \frac{2\beta}{C^*}, (2|R|)^{\frac{1}{\alpha +1}} \cdot (\frac{2 \beta}{C^*})^{\frac{\alpha}{\alpha+1}} \cdot (2\beta)^{\frac{\beta - \alpha}{\alpha+1}} \right) \right)$, where C ^* is the bottleneck congestion in the socially optimal state. We then demonstrate matching lower bounds for both games showing that this price of anarchy is tight.