A Unifying Tool for Bounding the Quality of Non-Cooperative Solutions in Weighted Congestion Games

Abstract

We present a general technique, based on a primal-dual formulation, for analyzing the quality of self-emerging solutions in weighted congestion games. With respect to traditional combinatorial approaches, the primal-dual schema has at least three advantages: first, it provides an analytic tool which can always be used to prove tight upper bounds for all the cases in which we are able to characterize exactly the polyhedron of the solutions under analysis; secondly, in each such a case, the complementary slackness conditions give us a hint on how to construct matching lower bounding instances; thirdly, proofs become simpler and easy to check. For the sake of exposition, we first apply our technique to the problems of bounding the price of anarchy and stability of exact and approximate pure Nash equilibria, as well as the approximation ratio of the strategy profiles achieved after a one-round walk starting from the empty state, in the case of affine latency functions and we show how all the known upper bounds for these measures (and some of their generalizations) can be easily reobtained under a unified approach. Then, we use the technique to attack the more challenging setting of polynomial latency functions. In particular, we obtain the first known upper bounds on the price of stability of pure Nash equilibria and on the approximation ratio of the strategy profiles achieved after a one-round walk starting from the empty state for unweighted players in the cases of quadratic and cubic latency functions.

Appendix: How to Compute Dual Variables: an Example

In this section, we show how to compute the dual variables exploited within the proof of Theorem 1.

In order to find the best possible upper bound on the P o A _𝜖 , we need to find the variables γ and y _i , for each i ∈ [n], such that the dual constraint

$$\sum\limits_{i:e\in s_{i}}\left( y_{i} K_{e}\right)-(1+\epsilon)\sum\limits_{i:e\in s_{i}^{*}}\left( y_{i} (K_{e}+w_{i})\right)+\gamma {O_{e}^{2}} \geq {K_{e}^{2}} $$

is satisfied for each pair of non-negative integers (K _e ,O _e ).

Since in our games no player has a special role which may distinguish her from the others, we can assume, without loss of generality, that all players are indistinguishable, so that we can set y _i = y for each i ∈ [n]. With this simplification, the dual constraint becomes

$$y{K_{e}^{2}}-(1+\epsilon)yO_{e}(K_{e}+ 1)+\gamma {O_{e}^{2}} \geq {K_{e}^{2}}, $$

by which we obtain

$$ \gamma\geq\frac{{K_{e}^{2}}(1-y)+(1+\epsilon)yO_{e}(K_{e}+ 1)}{{O_{e}^{2}}}, $$

(8)

for O _e ≥ 1 and y ≥ 1 for O _e = 0. Moreover, it is also easy to see that, for K _e >> O _e , the dual constraint is satisfied only if y > 1. Thus, from now on, we can assume, without loss of generality, that y > 1, O _e ≥ 1 and focus on the satisfiability of (8).

Let us denote with f(K _e ,O _e ,y) the right-hand side of (8). Our task is to find the value of y minimizing f(K _e ,O _e ,y) for each pair of non-negative integers (K _e ,O _e ) with O _e ≥ 1. We have $\frac {\delta f}{\delta K_{e}}(K_{e},O_{e},y)=\frac {y((1+\epsilon )O_{e}-2K_{e})+ 2K_{e}}{{O_{e}^{2}}}$ which, being a linear function in K _e , is maximized for $K_{e}=\frac {(1+\epsilon )yO_{e}}{2(y-1)}$. By using $K_{e}=\frac {(1+\epsilon )yO_{e}}{2(y-1)}$ in f(K _e ,O _e ,y), we obtain a function which is always decreasing in O _e ; hence, we can claim that the maximum value of f(K _e ,O _e ,y) is attained for $K_{e}=\frac {(1+\epsilon )y}{2(y-1)}$ and O _e = 1.

However, the resulting lower bound on γ might not be tight, as K _e might not be an integer. In order to remain within the realm of the integers, we can assume that the maximum value of f(K _e ,O _e ,y) is attained for O _e = 1 and K _e such that either $K_{e}=\left \lfloor \frac {(1+\epsilon )y}{2(y-1)}\right \rfloor$ or $K_{e}=\left \lceil \frac {(1+\epsilon )y}{2(y-1)}\right \rceil$. By setting $\left \lfloor \frac {(1+\epsilon )y}{2(y-1)}\right \rfloor =z$, this is equivalent to saying that the maximum value of f(K _e ,O _e ,y) is attained for O _e = 1 and K _e such that either K _e = z or K _e = z + 1. By imposing the condition f(z, 1,y) = f(z + 1, 1,y), we obtain $y=\frac {2z + 1}{2z-\epsilon }$ and $\gamma =f(z,1,\frac {2z + 1}{2z-\epsilon })=(1+\epsilon )\frac {z^{2}+ 3z + 1}{2z-\epsilon }$.