Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

AbstractWe 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 ℓe(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 Copt(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.