CS364A: Algorithmic Game Theory Lecture #14: Robust Price-of-Anarchy Bounds in Smooth Games∗

2. The k inequalities that bound individuals’ equilibrium costs are summed over the players. The left-hand side of the resulting inequality is the cost of the PNE s; the right-hand side is a strange entangled function of s and s∗ (involving terms of the form fef ∗ e ). 3. The hardest step is to relate the entangled term ∑k i=1Ci(si ∗, s−i) generated by the previous step to the only two quantities that we care about, the costs of s and s∗. Specifically, we proved an upper bound of 5 3 cost(s∗) + 1 3 cost(s). Importantly, this step is just algebra, and is agnostic to our choices of s and s∗ as a PNE and an optimal outcome, respectively.