I have a question regarding selfish routing games.
For the case where we have affine latency functions I was able to calculate a worst case price of anarchy (PoA) of $4/3$. However, now assume $L_d$ is a class of nondecreasing polynomial latency functions with non-negative coefficients: $\ell(x) = \sum_{i=0}^d c_i x^i$ for $c_i\geq 0, i=0,...,d$. I found the upper bound for the PoA on the internet, but I am not able to proof it myself. The result should be $PoA = [1-d(d+1)^{-(d+1)/d}]^{-1}$.
For the affine latency functions ($\ell(z)=az+b$), I just splitted the latency function into the constant part and the $az$ part and did the following maximization, which stems from the variational inequality: $$w:=max_{z>0, y>0}\frac{(\ell(z)-\ell(y))y}{\ell(z)z}$$ For $\ell(z) = b$ I get $w = 0$ (as wardrop is optimal) and for $\ell(z) = az$, I get that $w=1/4$, which results in the $PoA = 4/3$.
How do I proceed in the polynomial case? Can I also split the latency function and for example consider only the term with degree d?