Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if ? _e (x) is the latency function of an edge e, we replace it by $hat{ell}_{e}(x)$ with $ell_{e}(x) le hat{ell}_{e}(x)$ for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if $hat{C}_{N} (r)$ denotes the cost of the worst Nash flow in the modified network for rate r and C _opt (r) denotes the cost of the optimal flow in the original network for the same rate then $$mathit{ePoA} = max_{r ge 0} frac{hat{C}_N(r)}{C_{mathit{opt}}(r)}. $$ We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.