The Price of Anarchy for Polynomial Social Cost

In this work, we consider an interesting variant of the well studied KP model for selfish routing on parallel links, which reflects some influence from the much older Wardrop model [29]. In the new model, user traffics are still unsplittable and links are identical. Social Cost is now the expectation of the sum, over all links, of Latency Costs; each Latency Cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link. The resulting Social Cost is called Polynomial Social Cost, or Monomial Social Cost when the latency cost function is a monomial. All considered polynomials are of degree d, where d ≥ 2, and have non-negative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the Monomial Price of Anarchy and the Polynomial Price of Anarchy as our evaluation measures. Through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the Stirling numbers of the second kind and the Bell numbers, we obtain a multitude of results: • For the special case of identical users: – The fully mixed Nash equilibrium, where all probabilities are strictly positive, maximizes Polynomial Social Cost. – The Monomial Price of Anarchy is no more than Bd, the Bell number of order d. This immediately implies that the Polynomial Price of Anarchy is no more than ∑ 1≤t≤dBt. For the special case of two links, the Monomial Price of Anarchy is no more than 2d−2 ( 1 + ( 1 n )d−1) , and this bound is tight for n = 2. • The Monomial Price of Anarchy is exactly (2 d − 1) (d− 1)(2 − 2)d−1 ( d− 1 d )d for pure Nash equilibria. This immediately implies that the Polynomial Price of Anarchy is no more than ∑ 2≤t≤d (2 − 1) (t− 1)(2 − 2)t−1 ( t− 1 t )t .