In this work, we consider an interesting variant

of the well-studied KP model [KP99] for selfish

routing that reflects some influence from the much

older Wardrop [War52]. In the new model, user

traffics are still unsplittable, while social cost

is now the expectation of the sum, over all links,

of a certain polynomial evaluated at the total

latency incurred by all users choosing the link;

we call it polynomial social cost. The polynomials

that we consider have non-negative coefficients.

We are interested in evaluating Nash equilibria in

this model, and we use the Price of Anarchy as our

evaluation measure. We prove the Fully Mixed Nash

Equilibrium Conjecture for identical users and two

links, and establish an approximate version of the

conjecture for arbitrary many links. Moreover, we

give upper bounds on the Price of Anarchy.