Optimal price of anarchy of polynomial and super-polynomial bottleneck congestion games

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 ofbut the degree is bounded the form C_r^Mr, 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 ≤ M_r ≤ 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_rεR C_r, where R is the set of resources. We show that for super-polynomial bottleneck games with M_r = log C_r, the price of anarchy is o(√|R|), specifically O(2^√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 M_r = O(1) with constants α ≤ M_r ≤ β and derive the price of anarchy as min (|R|, max(2β/C*(2|R|)^1/α+1 ·,(2β/ C*)^α/α+1 · (2β) ^β-α/α+1)), 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.